Developing mathematics in Third World countries, Volume 33: Proceedings of the international conference held in Khartoum, March 6-9, 1978 (North-Holland Mathematics Studies)

DEVELOPING MATHEMATICS IN THIRD WORLD COUNTRIES

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NORTH-HOLLAND MATHEMATICS STUDIES

33

Developing Mathematics in Third World Countries proceedings of the international conference held in Khartoum, March 6-9, 1978

Editor

M. E. A. EL TOM, FlMA Reader in Mathematics, Khartoum University

1979

NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM

NEW YORK. OXFORD

0 North-Holland Publishing Company, I979

AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 85260 3

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM * NEW YORK. OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILTAVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloglng in Publication Dala

International Conference on Developing V!themtics in Third World Countries, Khartum, 1976. Developing mathematics in Third World countries. (North-Holland mathematics studies ; 33) 1. Yfthematics--Study and teaching--Congresses. L. Mathematical research--Congresses. 3. Underdeveloped areas--Mathematics--Congresses. I. el Tom, M. E. A . , 194111. Title. @ll.A1145 197& 510'.7'101724 76-24359 ISBE 0-444-E5dtO-3

PRINTED IN THE NETHERLANDS

FOREWORD

An International Conference on Developing Mathematics in Third World Countries was held under the auspices of Khartoum University in Khartoum from 6th to 9th March, 1978. It was attended by 117 registrants and 6 associates from 33 countries. The Conference represented the first opportunity ever for third world mathematicians to get together and exchange ideas and

ex-

periences pertaining to the development o f mathematics in their respective countries. A further important feature of the meeting was the strong representation o f the industrialized nations t h u s making the Conference truly international in spirit. This b o o k contains the proceedings of the Conference. It includes 16 invited papers by distinguished mathematicians and educationists most of whom are actively involved in the process o f building mathematics. The main topics dealt with in the book are : school mathematics, university mathematics ins'titutions, mathematics and development, mathematics policy and international cooperation. Also included i s a Final Report embodying a comprehensive set of recommendations and suggestions for action. Althoughthe b o o k is an important historical document, its real value must be judged by its ultimate influence on the future development of mathematics in third world countries. Judging, however, by the calibre of the participants, their enthusiasm for the Conference and the seriousness o f their discussions, one is tempted to feel highly optimistic about its eventual impact.

M.E.A. El Tom

V

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A C K N OW LE D G E M E N T S

Much o f

t h e s u c c e s s of

and c o n s t r u c t i v e

t h e C o n f e r e n c e was due t o t h e e n t h u s i a s m

many d i f f e r e n t c o u n t r i e s ; ted;

t h e p a r t i c i p a n t s , who came f r o m s o

t h i n k i n g of

t o t h e high s t a n d a r d of p a p e r s presen-

t o t h e exchange of views

outside lecture halls;

that

took p l a c e both w i t h i n and

t o t h e Chairmen and S e c r e t a r i e s o f Working

G r o u p s who g a v e s o much o f

t h e i r time;

and t o t h e e f f o r t s of

t h o s e who p r e s i d e d a t p l e n a r y s e s s i o n s . I w i s h , Organizing Committee,

on b e h a l f

of

t o express our deep g r a t i t u d e t o a l l

the

those

involved, Special indebtedness

i s acknowledged t o t h o s e E d i t o r s o f i n t e r n a -

t i o n a l p e r i o d i c a l s who p 1 a c e d ) a n a n n o u n c e m e n t o f

the Conference

in their journals. Several individuals,

i n s t i t u t i o n s and p u b l i s h e r s c o n t r i b u t e d t o

t h e Book E x h i b i t i o n a n d many o f

them have g e n e r o u s l y d o n a t e d t h e i r

To a l l of

e x h i b i t s t o t h e U n i v e r s i t y of Khartoum.

them go o u r

grateful thanks. The e f f o r t s o f

t h o s e c o l l e a g u e s who m a n n e d t h e r e g i s t e r a t i o n d e s k s

a n d h e l p e d mount

t h e Book E x h i b i t i o n a r e g r e a t l y a p p r e c i a t e d .

A s p e c i a l word of

thanks goes t o M r .

B.M.

B a b i k e r and h i s c o l l e a g u e s

i n t h e r e c e p t i o n committee. The O r g a n i z i n g Committee i s p a r t i c u l a r l y a p p r e c i a t i v e

of

the finan-

c i a l s u p p o r t g i v e n by t h e A r a b o r g a n i z a t i o n f o r E d u c a t i o n ,

Culture

a n d S c i e n c e ; t h e E u r o p e a n R e s e a r c h O f f i c e ; UNESCO a n d t h e U n i o n o f Sudanese I n s u r a n c e and R e i n s u r a n c e Companies. Last but not least, in the university,

I wish,

on behalf

of

all

the mathematicians

t o p l a c e on r e c o r d o u r d e e p g r a t i t u d e t o t h e

U n i v e r s i t y o f Khartoum f o r t h e most g e n e r o u s and w h o l e h e a r t e d support i t gave t o mathematics, Without t h i s support t h e Conference would n o t have b e e n p o s s i b l e .

M.E.A.

Chairman of

v ii

E l Tom

t h e O r g a n i z i n g Committee

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TABLE OF CONTENTS

Foreword

V

vii

Acknowledgements

ix

Table of Contents

PART ONE 1.

THE CONFERENCE : ITS BACKGROUND AND WORK Pre-Conference considerations

3

2. Planning for the Conference

5

3. Problems along the way 4 . Publications exhibition

7

a

5 . The Conference in action

9

6 . The work of the Conference

PART TWO

9 21

7 . Conclusions

THE INVITED TALKS

Strategies

&

priorities in mathematical education and

research in developing countries 25

A.A. ASHOUR Adequate mathematics for third world countries : Consideranda and strategies

33

U. D'AMBROSIO

The development of school mathematics : Some general principles 47

B. THWAITES Past, present and future educational technologies

53

P. SUPPES Research and higher education in mathematics : the Philippine experience

67

B.F. NEBRES ix

TABLE OF CONTENTS

X

Aspects of the recent development of functional analysis in Brazil L. NACHBIN

81

Programmed teaching of probability and statistics J. GREN

89

Mathematical models of schistosomiasis

I.

NASELL

I l l

Developing mathematics R.

THOM

127

Computers, mathematics and applications LIONS

J-L.

135

Mathematics research in third world countries : Pitfalls and opportunities S.

SHAHSHAHANI

143

Organising mathematical research in developing countries M.S.

NARASIMHAN

I5 1

La situation actuelle et les potentialites mathematiques de l'hfrique H. HOGBE-NLEND

157

Cooperacion internacional : Una experiencia y algunas reflexiones E. ROFMAN

165

Development of mathematics in Southeast Asia : The experience of the Southeast Asian Mathematical Society * LEE PENG YEE

* This topic w a s introduced by Dr. Tan Wang Seng, President-elect of SEAMS, due to the inability of the author to attend the meeting.

169

TABLE OF CONTENTS

PART THREE

xi

THE FINAL REPORT

I.

School mathematics

11.

University mathematics institutions

185

111.

Mathematics and development

187

IV.

Mathematics policy

I8 I

& international

cooperation

190

APPENDICES

Appendix 1

: L i s t of p a r t i c i p a n t s

195

A p p e n d i x 2 : T i t l e s of c o n t r i b u t e d p a p e r s

203

Appendix 3 : The Conference Committees, Working Groups and Chairmen

205

Appendix 4

207

: Allocution d e Cloture

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PART O N E

T H E C O N F E R E N C E : I T S B A C K G R O U N D A N D WORK

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1. Ere-Conference

considerations t h e view t h a t i n d i g e n o u s , as op-

Third world countries are of posed t o imported, t h e b u i l d i n g of

s c i e n c e i s necessary t o development.

s c i e n c e , u n d e r c o n d i t i o n s of

However,

underdevelopment,

i s a p r o c e s s t h a t r u n s i n t o a number of s e r i o u s d i f f i c u l t i e s . These d i f f i c u l t i e s arise e s s e n t i a l l y because of

the attempt t o

perform a t r ansplantation operation without t h e b e n e f i t of a well-worked

out theory

t o guide one's

actions. Needless t o say

t h a t no s u c h o p e r a t i o n would be s u c c e s s f u l u n l e s s t h e c o n d i t i o n s were r i g h t . What a r e t h e s p e c i f i c p r o b l e m s t h a t c o n f r o n t t h i r d w o r l d m a t h e m a t i c i a n s when a t t e m p t i n g t o p r o m o t e t h e i r d i s c i p l i n e w i t h i n t h e i r r e s p e c t i v e s o c i e t i e s ? Sweeping g e n e r a l i z a t i o n s a r e obvious l y not permissible here.

For,

the problems r e l a t i n g t o t h e

promotion of mathematics i n B r a z i l a r e very d i f f e r e n t from those i n Yemen a n d t h o s e i n I n d i a b e a r l i t t l e r e s e m b l a n c e t o t h o s e i n Malawi,

t o g i v e o n l y two e x a m p l e s . H o w e v e r ,

from t h e Sudanese e x p e r i e n c e ,

i f one a b s t r a c t s

then,for several

t h i r d world

c o u n t r i e s , t h e s e problems would a p p e a r t o t e l e s c o p e i n t o a problem of

leadership

: a s m a l l group of young m a t h e m a t i c i a n s

a r e e x p e c t e d t o i n i t i a t e , i n s p i r e a n d b u i l d when t h e y h a v e n e i t h e r t h e e x p e r i e n c e n o r t h e knowledge t h a t seem t o be n e c e s sary for efficient leadership. T y p i c a l l y , upon s u c c e s s f u l c o m p l e t i o n of a d o c t o r a t e program i n mathematics r e t u r n s home

i n a Western country,

( o f t e n one a t a time)

the f i r s t generation

t o j o i n a u n i v e r s i t y mathe-

m a t i c s u n i t and f i n d themselves f a c e d w i t h t h e f o l l o w i n g concret e problems.

(i)

A t the university level

they are expected t o develop

curricula;

that the library i s adequately

t o see t o it

s t o c k e d ; t o h e l p r e c r u i t new m e m b e r s o f represent t h e i r mathematics un i t

staff;

and t o

in various university

bodies. (ii)

Soon a f t e r a r r i v a l , problems of

t h e i r advice on,

teaching,

curricula,

and a s s i s t a n c e i n ,

examinations and t e x t s

a r e k e e n l y s o u g h t by o t h e r i n s t i t u t i o n s o f h i g h e r e d u c a -

3

4

THE CONFERENCE : ITS BACKGROUND AND WORK

tion and the Ministry of Education.

(iii) They are expected to help create a national society of mathematicians and/or scientists and in case such a society already exists then they will be expected to play an active and prominent role in running i t . They themselves expect to carry o n doing research in their respective fields of specialization. Their working environment i s likely to be most unstimulating : few good students studying for a first degree in mathematics; n o graduate program; no two members of staff working in the same area of mathematics and limited possiblities of establishing regular contacts with the outside community of mathematicians. The well known and muchtalkedabout problem of friction and jealousy is likely to arise and to lead, eventually, to the kind of fragmentation which will undoubtedly have a detrimental effect on the development of the subject. In short, it is demanded of these mathematicians to

mathema-

tics as well as provide for i t when they have only been trained to perform the former job. Considerations such as the above together with our determination to

face u p t o the challenge posed by the reality of underdevelop-

ment and our firm belief that a trial and error approach, under third world conditions, can lead to disastrous results, led us to the idea of holding a conference to look into the problems of building mathematics in these countries. A legitimate question might arise here. Given the nature of the problem t o be investigated and the fact that the status of mathematics in third world countries is characterized by uneven development, would i t not be more productive if the meeting were to be limited to a more homogeneous grouping, say, along geographical or cultural boundaries ? I would like to advance, i n answer to this question, the following argument. Mathematics, and science i n general, interact in an important way with the socio-economic reality of the country where it is practiced. An adequate understanding of this mode o f interaction seems to be a pre-requisite for the successful planning of

5

THE CONFERENCE : ITS BACKGROUND AND WORK

p r o d u c t i v e m a t h e m a t i c a l a c t i v i t y . However, world c o u n t r i e s

t h e socio-economic

f o r most t h i r d

r e a l i t y is fundamentally

t h e same a n d e v i d e n t l y c u t s a c r o s s b o t h g e o g r a p h i c a l a n d cultural boundaries.

On t h e e d u c a t i o n a l l e v e l , u n d e r l y i n g s o c i o - e c o n o m i c

structures

i n t h i r d world c o u n t r i e s play a fundamental r o l e i n t h e generaa number o f m a j o r problems which a r e e s s e n t i a l l y s i m i -

t i o n of

l a r f o r most of

them.

In t h i s c o n n e c t i o n i t i s i m p o r t a n t t o add

t h a t the educational systems of

the majority of

t h i r d world

countries are s t i l l influenced t o a g r e a t e r o r lesser degree by t h e c o r r e s p o n d i n g s y s t e m s of

their ex-colonialists.

influence is evidently independent of

This

geographical location

and c u l t u r a l h e r i t a g e . Finally, while the relationship that all hear t o the industrialized nations and technology, dency,

t h i r d world countries

i n most f i e l d s of

and d e f i n i t e l y i n mathematics,

science

i s one of depen-

they a s p i r e t o achieve independence i n these f i e l d s i n

the shortest possible

time.

W i t h t h e s e i d e a s i n m i n d , we a p p r o a c h e d t h e a u t h o r i t i e s

i n the

U n i v e r s i t y o f Khartoum t o a s k f o r t h e i r s u p p o r t and t h e i r r e s p o n s e was h i g h l y e n c o u r a g i n g .

2.Planning

f o r the Conference

Having p r e s e n t e d our i d e a t o t h e a p p r o p r i a t e b o d i e s of

the

u n i v e r s i t y who w e r e q u i c k i n r e s p o n d i n g p o s i t i v e l y t o i t , we were t h e n f a c e d w i t h t h e problem of p r e p a r i n g f o r t h e meeting. Although a few o f

t h e u n i v e r s i t y mathematicians have p a r t i c i p a -

ted i n international s c i e n t i f i c conferences of ture,

s p e c i a l i z e d na-

none of u s had p r e v i o u s l y been i n v o l v e d i n t h e o r g a n i z a -

t i o n of such meetings.

How t h e n s h o u l d we move a b o u t o n a l a n d

w i t h w h i c h w e were h a r d l y f a m i l i a r ? We s t a r t e d b y p r e p a r i n g a d o c u m e n t o u t l i n i n g t h e o b j e c t i v e s o f t h e m e e t i n g a n d t h e t h e m e s i t w a s e x p e c t e d t o t a c k l e . We e n v i s a ged t h e Conference e s s e n t i a l l y as a p l a t f o r m f o r t h e exchange and i n t e r a c t i o n of

i d e a s and experiences r e l a t i n g t o t h e develop-

ment of m a t h e m a t i c s i n t h i r d w o r l d c o u n t r i e s . S i n c e t h e c e was t h e f i r s t o f

Conferen-

i t s k i n d , e v e r t o b e h e l d , we d e c i d e d t h a t i t

should look i n t o a l l of

t h e major problems p e r t a i n i n g t o i t s

THE CONFERENCE : ITS BACKGROUND AND WORK

6

main theme. In this way the scene would be better set for any subsequent conferences that might be planned for dealing with any proper subset of these problems. In all six major problem areas were identified : School mathematics; The role of university mathematics institutions; Research and graduate education; Mathematics and development; Mathematics policy and International cooperation (more details about these are given in Section 6 below).

Suffice it to note here that participants were expected

to, among other things, clarify the question o f objectives, identify major problems in each area and look at some case studies. We approached the rather thorny problem of invited speakers with two basic considerations in mind. First, that the three major continents of the area (Africa, Asia and Latin America) should be adequately represented. Secondly, since mathematics is an international activity with its present centre in the industrialized countries, the meeting must be truly international and include representatives from this part of the world as well. A s a result of contacts with a number of distinguished

scien-

tists as well as national, regional and international organizations, a pool of candidates was formed out of which 21 speakers were finally selected. What kind of format should we adopt for the scientific program ? The nature of the problems to be investigated demanded that considerable time be devoted to discussions. The scope of the meeting and financial considerations made i t necessary to limit the duration of the Conference to four days. We further envisaged participants to work for six hours per day, excluding breaks and ceremonial functions. It was thus rather attractive to divide the resulting 2 4 working hours as follows : give each of the 21 invited speakers a maximum of 40 minutes followed by a minimum of 20 minutes of discussion s o that the total comes to one hour and devote ih? remaining three hours for the working out of recommendations and suggestions for action. Having decided upon objectives, themes and an organizational format; having received prompt and highly encouraging responses from most invited speakers and firmly believing that the Conference belonged to all third world countries and that i t ought to

THE CONFERENCE : ITS BACKGROUND AND WORK

7

be o f c o n c e r n t o t h e i n t e r n a t i o n a l community o f m a t h e m a t i c i a n s a t large,

t h e organizing committee decided t o send t h e f i r s t

announcement t o , institutions,

among o t h e r s ,

a l l t h i r d world mathematics

m i n i s t r i e s of e d u c a t i o n and h i g h e r e d u c a t i o n

a s w e l l a s t o r e l e v a n t n a t i o n a l and i n t e r n a t i o n a l o r g a n i z a t i o n s i n a l l i n d u s t r i a l i z e d c o u n t r i e s . Our v e n t u r e was t h u s f i n a l l y l a u n c e d on J u n e 4 ,

1977.

3 . Problems a l o n g t h e way. The r e s p o n s e t o o u r f i r s t a n n o u n c e m e n t , America,

exceeded our expectations.

s p e c i a l l y from L a t i n

This response,

however,

brought with it three not e n t i r e l y unforseen problems. To s t a r t w i t h many a p p l i c a n t s a s k e d t h e O r g a n i z i n g C o m m i t t e e t o p r o v i d e them w i t h t r a v e l o g r a n t s

a n d i t was l a t e r o b v i o u s t o

u s t h a t many t h i r d w o r l d c o u n t r i e s w o u l d n o t b e r e p r e s e n t e d a t

t h e meeting i f

they could n o t r e c e i v e a i d from t h e Committee.

Given t h e u n i v e r s i t y ' s

financial situation,

t h e m o s t t h e Com-

m i t t e e c o u l d d o was t o a p p r o a c h r e g i o n a l a n d i n t e r n a t i o n a l o r g a n i z a t i o n s and ask t h e u n i v e r s i t y t o a g r e e t o o u r u s i n g w h a t e v e r a i d we e v e n t u a l l y r e c e i v e d f o r h e l p i n g p a r t i c i p a n t s most i n need of

such h e l p . A t

t h e e n d w e were o n l y p a r t i a l l y

s u c c e s s f u l i n s o l v i n g t h i s problem. S e c o n d l y , a l a r g e number o f a p p l i c a n t s w i s h e d t o c o n t r i b u t e papers t o the meeting and,

indeed,

many o f

them i n d i c a t e d t h a t

t h e i r o b t a i n i n g the necessary funds from t h e i r r e s p e c t i v e

insti-

t u t i o n s was c o n d i t i o n a l on t h e i r p r e s e n t i n g a p a p e r a t t h e Conference.

A s t h e p r e s s u r e mounted, we f e l t t h a t we could n o t

b u t y i e l d t o i t . A s a r e s u l t i t was n e c e s s a r y t o c o m p l e t e l y r e v i s e o u r f i r s t programme and a d o p t t h e plenary sessions' For,

f o r m a t . T h i s was a h a r d d e c i s i o n t o t a k e .

i t m e a n t t h a t e x c e p t f o r two t a l k s

themes of

'working groups-

the conference,

t o u c h i n g upon a l l s i x

i n v i t e d t a l k s would be p r e s e n t e d i n

working groups s e s s i o n s thus robbing t h e p a r t i c i p a n t s of undoubted value of

the

s e e i n g and h e a r i n g d i s t i n g u i s h e d speakers.

But t h i s c o u l d p a r t l y be compensated f o r by d u p l i c a t i n g i n v i t e d t a l k s and d i s t r i b u t i n g them t o a l l p a r t i c i p a n t s p r i o r t o t h e Conference.

THE CONFERENCE : ITS BACKGROUND AND WORK

8

As

for working groups, constraints of space, time,expected

number of participants and overlap of certain themes led u s to the decision of having four working groups each dealing with one of the following themes :

- School mathematics

-

University mathematics institutions Mathematics and development Mathematics policy and International cooperation.

The revision o f the programme and the selection of working groups' Chairmen and secretaries put a final seal o n o u r scientific programme. Finally, we were hard pressed by several colleagues from Africa and Latin America to adopt besides English, French and Spanish as official languages of the Conference. However, in view of the lack of adequate facilities in Khartoum for simultaneous translation and our feeling that English is the most widely used language in mathematics, made it imperative for u s to abide by our earlier dec sion concerning the use of English as the official language of the conference while allowing the u s e of French only when and

f necessary.

4 . Publications exhibition

Somewhere along the way we came to realize how little w e k n e w about the nature of work that goes on in many third world mathematics institutions. For instance, I doubt very much i f many African and Latin American mathematicians were aware, prior to the Conference, o f the work of the South East Asian Mathematical Society (SEAMS).

It thus occured to us to use the Confe-

rence as a means of introducing third world mathematicians and mathematics institutions to each other through their work. Realizing, however, that there are many non-third world institutions some of whose work might be of importance to the Conference,we

decided to g o beyond the initial idea of a li-

mited display with a limited purpose and mount an exhibition o f publications relevant to the theme o f the Conference. Several

individuals, institutions as well as international publishers were then contacted and asked to contribute to the exhibi.tion.

THE CONFERENCE : ITS BACKGROUND AND WORK

9

5. The C o n f e r e n c e i n a c t i o n T h e C o n f e r e n c e was o f f i c i a l l y o p e n e d a t 8 h . 3 0 with a brief of

on M a r c h 6 ,

1978,

s p e e c h o f welcome by t h e Deputy V i c e - C h a n c e l l o r

t h e U n i v e r s i t y o f Khartoum.

A l l i n v i t e d and c o n t r i b u t e d

t a l k s were p r e s e n t e d d u r i n g t h e f i r s t day.

The s e c o n d day o f

t h e C o n f e r e n c e was e n t i r e l y d e v o t e d t o W o r k i n g G r o u p s d i s c u s s i o n s . Preliminary reports of

t h e f o u r Working Groups were

d i s c u s s e d i n ple.nary s e s s i o n on t h e t h i r d day. s e s s i o n of

The morning

t h e f o u r t h a n d f i n a l day w a s d e v o t e d t o Working

Groups d i s c u s s i o n s and p r e p a r a t i o n o f

A unified

final reports.

f i n a l r e p o r t prepared by t h e Chairmen of

t h e Working Groups

was d i s c u s s e d i n t h e f i n a l p l e n a r y m e e t i n g o f

the Conference.

The c l o s i n g s p e e c h w a s g i v e n by P r o f e s s o r Hogbe-Nlend capacity as President of

t h e , A f r i c a n Mathematical Union.

T h e s o c i a l p r o g r a m m e s t a r t e d on M a r c h 5 , g i v e n by t h e V i c e - C h a n c e l l o r

1978, with a r e c e p t i o n

o f Khartoum U n i v e r s i t y and l a s t e d

t h r o u g h March 1 0 , 1978. I t f u r t h e r i n c l u d e d a one-hour s e e i n g t o u r and a r e c e p t i o n by t h e Dean, and A r c h i t e c t u r e ,

sight-

Faculty of Engineering

on M a r c h 6 ; a f o l k d a n c e show on M a r c h 7 ;

a b o a t t r i p a n d l u n c h g i v e n by t h e Dean,

on March 8 ;

in his

Faculty of Science,

a p i c n i c o r g a n i z e d by t h e C o n f e r e n c e O r g a n i z i n g

C o m m i t t e e a n d a r e c e p t i o n b y t h e M i n i s t e r o f E d u c a t i o n on March 9 ;

a n d a t o u r o f Omdurman o n F r i d a y 10 M a r c h .

An e x t r a

programme was a l s o o r g a n i z e d f o r a c c o m p a n y i n g p e r s o n s . T h e C o n f e r e n c e w a s a t t e n d e d b y 1 2 3 m e m b e r s o f whom 6 w e r e a c companying p e r s o n s .

6 . T h e work o f

In a l l 3 3 c o u n t r i e s w e r e r e p r e s e n t e d .

the Conference

In P a r t T h r e e o f t h i s book i s p r e s e n t e d a r e p o r t o n t h e m a i n f i n d i n g s of

the Conference h i g h l i g h t i n g major p o i n t s of agreement

between p a r t i c i p a n t s .

However,

t h i s r e p o r t does not i n d i c a t e ,

f o r o b v i o u s r e a s o n s , how t h e v a r i o u s G r o u p s a p p r o a c h e d t h e i r themes nor does it i n c l u d e r e f e r e n c e s t o p o i n t s o f disagreements no^

t o t o p i c s w h i c h c o u l d not b e d i s c u s s e d f o r l a c k o f

The ,purpose of

time.

t h i s s e c t i o n i s t o make up f o r t h e s e e n t i r e l y

n a t u r a l ommissions from t h e F i n a l R e p o r t by f o c u s i n g a t t e n t i o n

THE CONFERENCE : ITS BACKGROUND AND WORK

10

on t h e d i s c u s s i o n s of t h e f o u r Working Groups. 6.1.

School mathematics t h i s i s t h e a r e a where t h i r d w o r l d n a t i o n s encounLer

Perhaps,

problems which v a r y l i t t l e from one c o u n t r y t o a n o t h e r ; and the greater is the similarity

t h e lower t h e educational s t a g e ,

b e t w e e n t h e s e t s o f p r o b l e m s e n c o u n t e r e d . The s c a r c i t y o f d e d i c a t e d and competent t e a c h e r , w i t h t h e u s e of

the mother-tongue

and classroom c o n d i t i o n s ,

the

the difficulties associated a s a medium o f

instruction

are t h r e e important examples of

p r o b l e m s t h a t s e e m t o b e common t o t h e m a j o r i t y o f t h e s e c o u n tries,

I n c o n t r a s t t o i n d u s t r i a l i z e d n a t i o n s , primary education s i x years of

(first

s c h o o l ) i s t e r m i n a l f o r t h e m a j o r i t y of p u p i l s

i n most d e v e l o p i n g c o u n t r i e s .

T h i s f a c t w a s s i n g l e d o u t by

members o f

t h e Working Group a t t h e o u t s e t of

tions and,

consequently,

their delibera-

they devoted t h e g r e a t e r p a r t of

their

t i m e f o r c o n s i d e r a t i o n o f p r i m a r y m a t h e m a t i c s e d u c a t i o n . The F i n a l Report i n c l u d e s a number of p e r t i n e n t o b s e r v a t i o n s and recommendations on o b j e c t i v e s , c u r r i c u l a , e v a l u a t i o n , a n d t h e m a t h e m a t i c s t e a c h e r ( s e e p.183).

language

An i s s u e t h a t h a s b e e n

the subject of vigorous debate f o r the l a s t f i f e t e e n years o r

s o a n d w h i c h was t h e r e f o r e b o u n d t o b e b r o u g h t up d u r i n g t h e Group's meetings, maths'.

i s t h a t of

the so-called

'modern'

or

'new

I n f a c t t h e i s s u e w a s o f c o n s i d e r a h l e i n t e r e s t t o many

p a r t i c i p a n t s and, moreover,

representatives

from s e v e r a l coun-

t r i e s , where a 'modern m a t h s ' h a s j u s t b e e n o r i s a b o u t t o be i n t r o d u c e d i n s e c o n d a r y s c h o o l s were v e r y e a g e r t o l e a r n a b o u t t h e e x p e r i e n c e of o t h e r c o u n t r i e s . countries,

R e p r e s e n t a t i v e s f r o m two

Costa Rica and t h e P h i l i p p i n e s ,

h a v i n g i n t r o d u c e d a n American v e r s i o n o f a s 1 9 6 5 , used t h e word of

'disastrous'

t h e i r two c o u n t r i e s .

However,

the l a t t e r country

'new m a t h s '

the Group's a t t i t u d e t o the

i s s u e may b e summed up i n t h r e e o b s e r v a t i o n s .

F i r s t , many d i f -

f e r i n g i n t e r p r e t a t i o n s a r e a s s o c i a t e d w i t h t h e terms and 'modern'

as e a r l y

t o describe the experience

'traditional'

m a t h e m a t i c s a n d v i e w i n g t h e two a s m u t u a l l y

e x c l u s i v e c o n c e p t s would n o t b e h e l p f u l t o a d i s c u s s i o n of curricula

( s e e p.187).

Secondly,

i t i s certainly of

greater

i m p o r t a n c e now t o l o o k m o r e d e e p l y a t p r i n c i p l e s o f g o o d m a t h e -

THE CONFERENCE : ITS BACKGROUND AND WORK

11

m a t i c s t e a c h i n g drawn from t h e t e a c h e r ' s e x p e r i e n c e . F i n a l l y , t h e p r o v i s i o n i n a d e q u a t e numbers o f more c r u c i a l

than the d e t a i l s of

competent t e a c h e r s i s

curricula.

I n s e v e r a l A s i a n and L a t i n American c o u n t r i e s and i n most African countries, be l a r g e

--

q u e s t i o n of

t h e number o f

often over a hundred. t h e medium o f

languages spoken tends I n such countries,

to

the

i n s t r u c t i o n obviously poses d i f f i c u l t

p o l i t i c a l a n d e d u c a t i o n a l p r o b l e m s . From a n e d u c a t i o n a l p o i n t of view,

however,

t h e b a s i c problem is t o a r r i v e a t a s a t i s f a c -

tory understanding of of

t h e r e l a t i o n s h i p between t h e l e a r n i n g

a s u b j e c t ( i n our c a s e mathematics) and t h e language through

which i t i s l e a r n t .

For, even i f one a c c e p t s t h e hypothesis

t h a t t h e l e a r n i n g of mathematical

concepts is greatly f a c i l i t a -

t e d when t h e y a r e t a u g h t i n ' t h e p u p i l ' s m o t h e r - t o n g u e , c o m e s a s t a g e when i t b e c o m e s p r a c t i c a l l y

there

impossible for a

country with l i m i t e d r e s o u r c e s t o have every p u p i l taught i n h i s o r h e r mother-tongue.

The Working Group c o u l d do no more

here than urge each country concerned t o engage i n r e s e a r c h

on t h i s f u n d a m e n t a l p r o b l e m . The m o s t i m p o r t a n t e l e m e n t i n an e d u c a t i o n a l s y s t e m i s e v i d e n t l y A s long a s t h e t e a c h e r remains unmotivated and

the teacher.

inadequately prepared, a l l attempts a t ameliorating other fact o r s i n t h e s y s t e m a r e doomed t o f a i l u r e . S e v e r a l d e l e g a t e s s i n g l e d o u t low s a l a r i e s a n d l i m i t e d o p p o r t u n i t i e s f o r promot i o n as t h e main f a c t o r s c o n t r i b u t i n g t o t h e p r o b l e m s o f teaching profession.

However,

the

s i g n i f i c a n t improvements i n

s a l a r i e s i n a c o n t i n u o u s l y e x p a n d i n g e d u c a t i o n a l s y s t e m , may w e l l be beyond o n e member o f ration of

t h e c a p a b l i t i e s o f many c o u n t r i e s .

Noting t h i s ,

t h e Group saw a s o l u t i o n t o t h e p r o b l e m i n r e s t o -

the teacher's

self-esteem.

But is i t p o s s i b l e t o a c h i e -

v e t h i s when m a t e r i a l c o n d i t i o n s a r e r a t h e r p o o r ? I f n o t , we t o c o n c l u d e t h a t we a r e iri t h e p r e s e n c e o f o n e o f

are

the vicious

c i r c l e s t h a t a r e c i t e d by some s c h o l a r s t o c h a r a c t e r i z e u n d e r development ? The G r o u p ' s

appreciation of the magnitude and seriousness of

t h e t e a c h e r p r o b l e m i s i n d i c a t e d by c h e i r r e c o m m e n d a t i o n t o UNESCO t h a t a y e a r

s h o u l d be ' d e s i g n a t e d as

'World T e a c h e r s ' Y e a r '

THE CONFERENCE : ITS BACKGROUND AND WORK

12

during which the public at large would be made aware o f the problem. Notwithstanding these difficulties, the quality of mathematics teaching may be greatly improved through the implementation of certain measures which are entirely feasible within existing conditions. One such measure emphasized by the Group was the training o f servicing teachers.

Indeed, the urgency

of in-service training was s o strongly felt by one member o f the Group that he proposed the closing down of all schools for one year during which teachers would be available for training. A less extreme proposal was that one day o f the school's working week should be free for retraining

so

teachers are constantly exposed

new ideas and new techniques.

to

as to make sure that

One topic which the Working Group did not have sufficient time to look into was the role of educational technology (in the restricted sense of technological products) in the countries of the area. Almost every village and every group of nomads has its radio and, in some cases, television sets. Yet, radios and TV sets are hardly used for purposes pertaining to mathematical education in most third world countries. While these technologies appear to have a great potential for mathematical education in developing countries, i t also seems that conditions of underdevelopment impose important limitations on their effectiveness. Therefore, it is important to get adequately informed about both their potential and limitations. One paper in this volume gives a n extensive survey o f innovative educational technologies and points out a number of philosophical and social implications arising from their use.

-6.2. -

University mathematics institutions.

In industrialized countries, the development of mathematics on a national scale is a joint (not necessarily planned) effort undertaken by different institutions : ministries o f education, governmental agencies, universities, mathematics societies, research centres, etc. However, most third world countries will

THE CONFERENCE : I T S BACKGROUND AND WORK

have to depend, for reasons peculiar to them

*,

13

o n university

mathematics institutions for initiating and leading the process of building mathematics. Are these institutions, where they exist, well-equipped to shoulder their responsibilities ? Where they do not yet exist o r are in the making (in the case of planned and new un-iversities, respectively), one would like to get informed about efficient ways of establishing them. Here, one needs to look a t , among other things, forms o f organization, curricula, degree structures, graduate education, faculty specializations and research work. Regarding forms of organization, several models suggest themselves : a department of 'pure' mathematics ('applied' mathematics being attached to some other relevant units in the university, for instance, physics, engineering and economics);

several

departments of mathematics one each attached to a particular faculty; one single unit under whose roof are grouped a number of mathematical sciences. The choice of a model seems to be decisively influenced by a combination of three factors : the colonial heritage, the vitality of mathematical activity in industrialized countries and the dependence o n these countries for the training of highly qualified mathematicians. Indeed, in many third world countries, forms of organization of university mathematics institutions are replicas of corresponding forms in Western countries. After considering various models, the Group felt that the form of 'School of Mathematical Sciences' including, besides 'pure' and 'applied' mathematics, disciplines such as computer science, statistics, operations research and mathematical education, is more appropriate for universities in third world countries. By allowing for flexiblity and a pooling of resources, this form of organization would encourage, at least in theory, interdisciplinary and team work as well as make possible a more efficient use of resources. Several papers presented to the Group dealt with certain aspects of university mathematics in Brazil, India, Nigeria, Philippines and Turkey. The case studies on the last three countries strongly *For instance, all Ph.D. holders in mathematics in the Philippines and in Sudan are employed by the universities.

14

THE CONFERENCE : ITS BACKGROUND AND WORK

suggest t h a t t h e development of u n i v e r s i t y mathematics i n m o s t t h i r d w o r l d c o u n t r i e s i s h a m p e r e d by t w o r e l a t e d f a c t o r s . F i r s t , n o t enough good s t u d e n t s c h o o s e t o r e a d f o r a f i r s t degree i n mathematics". of

Secondly, while undergraduate

courses

s t u d y compare f a v o u r a b l y w i t h t h o s e o f f e r e d i n t h e i n d u s -

trialized countries,

t h e l e v e l and q u a l i t y of g r a d u a t e educa-

t i o n a re rather poor.

Lack of

j o b o p p o r t u n i t i e s and low

s a l a r i e s w e r e g i v e n as two of

t h e f a c t o r s t h a t prompt

good

s t u d e n t s t o o p t f o r d i s c i p l i n e s o t h e r t h a n m a t h e m a t i c s . But s u r e l y t h e q u a l i t y o f mathematics education i n pre-university s t a g e s i s r e l e v a n t t o t h i s phenomenon. Moreover,

the experien-

c e of I n d i a , where t h e d e v e l o p m e n t o f m a t h e m a t i c s i s n o t h a m p e r e d by a l a c k o f

talent,

indicates that the intellectual

t r a d i t i o n of a country i s a f a c t o r which must be taken i n t o account i f one i s t o provide a s a t i s f a c t o r y e x p l a n a t i o n of t h i s phenomenon**.At t y of

any r a t e ,

t a l e n t i n mathematics

it

is evident that the scarci-

i s a problem t h a t m e r i t s s e r i o u s

s t u d y f o r i t s i m p l i c a t i o n s on t h e f u t u r e o f m a t h e m a t i c s i n the countries concerned are far-reaching. The b r o a d c o n t e n t s o f a n u n d e r g r a d u a t e p r o g r a m o f

study were

r a t h e r vigorously debated i n t h e Group's meetings.

Initially,

t h e debate c e n t r e d around t h e o l d and f a m i l i a r d i v i s i o n of mathematics i n t o 'pure'

and ' a p p l i e d ' .

The u s u a l compromise

of a b a l a n c e d program triumphed a t t h e e n d . However, p o i n t e d o u t t h a t i n designing such a program, mathematics rieeds of

t h e Group

university

i n s t i t u t i o n s must always t a k e i n t o account t h e

the s o c i e t i e s within which they function. For instan-

c e , w h i l e t h e r e i s an a c u t e s h o r t a g e of mathematics

teachers

i n N i g e r i a n s c h o o l s , t h e market seems t o b e a t s a t u r a t i o n point i n India. Furthermore, any program a d e q u a t e modelling.

Presumably,

room m u s t b e p r o v i d e d f o r m a t h e m a t i c a l i n t h o s e i n s t i t u t i o n s where t h e r e i s

a n o t i c e a b l e s h o r t a g e of

--

*

t h e Group emphasized t h a t i n

s t a f f q u a l i f i e d both i n mathematics

In t h e c a s e o f T u r k e y , i t was p o i n t e d o u t t h a t g o o d s t u d e n t s o p t f o r e n g i n e e r i n g . T h i s i s a l s o t r u e of Sudanese s t u d e n t s ,

**

O r i s t h e c a s e of I n d i a simply t h a t of large population ?

sampling from a

THE CONFERENCE : ITS BACKGROUND AND WORK

and a significant area of application, i t would be necessary to look into the possiblity of designing integrated courses of study. Scarcity of talent in mathematics, inadequate facilities, academic isolation, heavy teaching loads, low level of graduat e education, presence o f extra-academic and immediately pres-

sing problems - - all of these and other factors combine i n varying degrees to make worthwhile research in mathematics in third world countries a n extremely difficult task. The Group noted that whatever research is done in these countries is usually the result o f sustained labour under difficult conditions. It is evident that significant efforts o n the part of outstanding individuals and indigenous organizations as well as aid from external sources are all necessary if research activity, in those countries where i t i s nonexistent or dormant, is to achieve reasonably quickly the kind of vigour and vitality that will make i t bear fruit.

6.3.

Mathematics and development.

Whether i t is conceded o r not that mathematics is necessary to development, the important role which mathematics plays in almost all spheres of human activity remains one o f the outstanding features of our civilization. Members of the recently created African Mathematical Union (AMU) and the Union of Arab Physicists and Mathematicians (UAPM) have laid great emphasis, in their respective statutes, on this role. But w h y , i t may be asked, is this role of mathematics s o evidently important in the case o f industrialized societies while practically indiscernable in the case of other societies ? Is i t because mathematics becomes significantly applicable only beyond a certain threshold of economic development or are the reasons to be sought rather in the nature o f the training of mathematicians, on the one hand, and the weakness of relevant institutional linkages o n the other ? That there is no dearth in third world countries of developmental problems amenable to meaningful mathematical analysis may easily be proved by reference to the relevant mathematical literature. Indeed, the

15

THE CONFERENCE : I T S BACKGROUND AM, WORK

16

paper by I. Nzsell i n this volume (see p.111)

is an excellent

instance of such problems. The Group noted with great concern that the developmental problems of underdeveloped countries often attract a proportionately far greater number o f mathematicians in the industrialized countries than i n the countries that face the problems. This rather curious phenomenon

*

was analysed by the Group

and a number of conclusion which seem to provide a satisfactory explanation were given (see p.189).

Incidentally, this

phenomenon does not seem to b e peculiar to mathematicians

**.

But the desire, when it exists, o f mathematicians to grapple with developmental problems has a number of important implications. To start with, mathematicians who are interested in development must first acquire adequate knowledge of the major developmental problems of their countries. This would often entail the cooperation o f non-mathematicians. H o w to enlist their cooperation is by n o means a simple problem.

The Group

identified consulting as an effective tool towards this end. Secondly, the usual system (of promotion) by which academics are rewarded for their activities needs to be revised to allow for proper recognition of work which might not be publishable. Third, the training of mathematicians, both at the undergraduate and graduate levels, must be s o designed as to provide the necessary motivation and orientation. Finally, university mathematics institutions must take an active interest in service courses. F o r it is quite evident that the less adequate the mathematical training of non-mathematicians, the weaker will the impact of mathematics on society be.

*

Is it not curious that a tropical disease su,ch a s b i l h a r z i a (Schistomiasis) should interest a Swedish mathematician and not a single one in those countries where i t represents an obstacle to development not to speak o f the human suffering caused by i t ?

**

F o r a number of examples s e e the article by C. Nader i n :

C. Nader and A.B. Zahlan ( E d s . ) ,

Science and Technology in

Developing Countries, Cambridge University Press (1969).

THE CONFERENCE : ITS BACKGROUND AND WORK

17

In emphasizing the importance for mathematics of its role in national development, the Group was concerned lest this emphasis should be misinterpreted as providing the only valid reason for doing mathematics. That members of the Group were mindful of the non-utilitarian aspects of mathematics is evidenced by the first of the three objectives on p.188.

__ 6.4.

Mathematics policy and International cooperation.

The combination of two important themes and the presentation of a proportionately larger number of papers in this Working Group helped to make i t the largest in the Conference. Mathematics policy is essentially concerned with the creation of conditions favourable to productive and creative mathematical activity. While in some' countries the weight of tradition and certain value systems combine to obviate the need for a clearly defined policy, the enunciation of guidelines for the development of mathematics in those countries with no tradition in the subject seems to be of prime importance. However, for the formulation o f any meaningful policy, one must have available adequate information about basic inputs : manpower, education, research, finance, job opportunities and facilities. The collection of data and the formulation of polic y can obviously only be satisfactorily carried out by the community of mathematicians of the country concerned. Moreover, for any worthwhile activity to take place, i t seems that the existence o f a "critical mass" curs in several papers

--

-- a notion that oc-

is an absolute necessity.

Once the requisite information and a "critical mass" of mathematicians are available, a number of conflicting policy options arise. Are we committed, first and foremost, to mathematics or to the community within which mathematicians function

*

? While the pratogonists of the first option would

typi-

cally claim that tying down mathematics to concrete problems can only lead to its enfeeblement, those siding with the second

*

This is reminiscent of the 'Art f o r life and Art for Art's sake' conflict.

THE CONFERENCE : ITS BACKGROUND AND WORK

18

option would assert that the development of mathematics, and science in general, in isolation of the socio-cultural milieu within which i t is placed is doomed to failure. The above conflict is clearly related to the 'pure' v 'applied' conflict. It was suggested in the Group's meetings, however, that the advent of computers has so radically changed the structure of the mathematical sciences and the relations between their constituent parts o n the one hand, and between the mathematical and other sciences on the other,

so

much s o that this

classical dichotomy is n o longer meaningful. Where should the starting point for a strategy be located : at the lower end o f the educational ladder or at the level of research and higher education ? One view w a s that the starting point should be the formulation of a strategy for research related to national goals. According to this view, once such a strategy is worked out, i t would be relatively easy to elaborate the content of university and pre-university education. Another view was that it would be more natural to take care first of the base of the pyramid rather than its apex. The 'quantity'

2,

'quality' conflict leads to, among other things,

the well-known concept of "centres of excellence". Should these centres excel in a broad range of fields o r simply in one particular area ? An important parameter that must be taken into account here is obviously the size of the educational system in the country concerned. Finally, should mathematics be built around individuals or specific topics ? Here, i t is instructive to reflect upon the influence of great mathematicians like Abel and Caratheodory, to name only two, o n Norwegian and Greek mathematics, respectively.

Training abroad is an important ingredient in the policies of many developing countries. Is it more effective, from the point of view of achieving basic policy objectives, to send students

abroad at an early or advanced stage of their studies ? Besides the financial costs, sending students abroad at an early stage

THE CONFERENCE : ITS BACKGROUND AND WORK

seems t o e n c o u r a g e t h e ' b r a i n d r a i n ' d o e s r e t u r n home,

and,

i n case the student

he o r she i s l i k e l y t o have s p e c i a l i z e d

i n a n a r e a i n c o m p a t i b l e w i t h p o l i c y o b j e c t i v e s . The g e n e r a l view of

t h e Group was t h a t t r a i n i n g programmes a b r o a d s h o u l d

b e d e e m p h a s i z e d and l i m i t e d , as f a r a s i s p r a c t i c a b l e , mature s t u d e n t s with well-defined

to

i n t e r e s t s and f o r r e l a t i v e -

ly short periods. The i m p o r t a n c e o f b o t h i n t e r d i s c i p l i n a r y a n d t e a m work w e r e e m p h a s i z e d in t h e G r o u p ' s d i s c u s s i o n s . t h a t the building of and,

therefore,

The Group n o t e d , however,

teams i s a d i f f i c u l t and c o s t l y p r o c e s s it is of

once e s t a b l i s h e d ,

t h e utmost importan-

c e t o e n s u r e t h e i r s t a b i l i t y . Being a f u n c t i o n of p o l i t i c a l , economic,

s o c i a l and academic v a r i a b l e s ,

t h e s t a b i l i t y of

teams must be s e e n a s t h e j o i n t r e s p o n s i b l i t y of t h e i n s t i t u t i o n and t h e l e a d e r s of work.

the

government,

t h e u n i t s where t h e teams

T h i s makes s t a b i l i t y t h e more d i f f i c u l t t o r e a l i z e i n

practice, When w o r k i n g o u t p o l i c i e s

for scientific research,

it i s not

uncommon f o r p o l i c y m a k e r s i n t h i r d w o r l d c o u n t r i e s fo r recipes i n the experiences of Canada and J a p a n .

To what e x t e n t

l e a r n from t h e e x p e r i e n c e s of comers ? The f a c t t h a t ,

countries

like Australia,

can t h i r d world c o u n t r i e s

these so-called

except

t o look

for Japan,

s u c c e s s f u l new-

t h e s e new-comers

have always had s t r o n g c u l t u r a l a f f i n i t i e s w i t h Europe o b v i o u s l y imposes a l i m i t a t i o n on t h e usefu1nes.s of most t h i r d w o r l d c o u n t i r e s . however,

A more f u n d a m e n t a l l i m i t a t i o n ,

would seem t o be t h a t none of

'underdeveloped'

t h e i r models t o

t h e s e c o u n t r i e s was e v e r

i n t h e s e n s e i n which t h i s t e r m i s u s u a l l y

understood. An i - n s t a n c e o f

an i n d i r e c t ,

a n d p e r h a p s m o r e i n s t r u c t i v e , way

of u t i l i z i n g t h e e x p e r i e n c e s of

industrialized countries

s u g g e s t e d b y t h e f o l l o w i n g . A t some p o i n t cussions, of

i t was a s k e d w h e t h e r t h e r e h a d e v e r b e e n a n i n s t a n c e

a s u c c e s s f u l school of mathematics

70 % o f

is

i n the Group's dis-

i n a c o u n t r y where o v e r

t h e p o p u l a t i o n w e r e i l l i t e r a t e ? "We a r e n o t t r y i n g t o

form s c h o o l s " was one r e p l y .

A

second, but i n d i r e c t ,

reply

p o i n t e d o u t t h a t i l l i t e r a c y was r a t h e r h i g h i n s e v e r a l W e s t

THE CONFERENCE : ITS BACKGROUND AND WORK

20

European countries at the turn of the 18th century. The question, however, is worthy of generalization in the following way : What are the socio-cultural prerequisites, if a n y , for mathematics ?

.

The drafting of policies obviously needs to be supplemented by methods for the evaluation of results. Here one may usefully learn from the methodologies used in those industrialized countries where a reasonable degree of planning exists. The discussion of the experiences of others naturally leads to that of cooperation. The importance o f

'international coope-

ration' stems from three fundamental factors : All mathematicians are ultimately working toward the same g o a l , namely, the development of the subject - - though the means obviously differ; the international charachter of the mathematical activity and, finally, the necessity, for developing nations, of accelerating the pace of mathematics development in their countries. There are various levels at which international cooperation operates. First, there are the informal ties between individual mathematicians of different countries. These usually take the form of exchange of preprints, technical reports, doctoral theses, etc.; and although such links may not be formalized, their importance c a n not be overemphasized. Second, cooperation may take place between the institutions and organizations of different countries. These may assume one of various forms ranging from informal ties between two departments of mathematics to formal agreements between universities stipulating the exchange of students and academic staff. A third mode of cooperation is that between national organizations on the one hand, and international ones on the other. Naturally, all three modes of cooperation should reinforce each other in the endeavour to achieve specific goals of a mathematics policy. The role of indigenous groups is crucial here. The choice o f individual visitors, of institutions and organizations with which formal links are to be established and the decision as to the type of aid required must all be taken by them if the

THE CONFERENCE : ITS BACKGROUND AND WORK

e f f e c t i v e n e s s of i n t e r n a t i o n a l c o o p e r a t i o n is to h e m a x i m i z e d T h e Group's e m p h a s i s o n the r o l e o f

l o c a l m a t h e m a t i c i a n s is

e v i d e n t i n the f i r s t two r e c o m m e n d a t i o n s o n p . 1 9 0 . I n t e r n a t i o n a l c o o p e r a t i o n , l i k e team w o r k , a l s o n e e d s to be s t a b i l i z e d and for this to be p o s s i b l e a n e c e s s a r y s t e p w o u l d s e e m t o h e its i n s t i t u l i z a t i o n . T h e e x i s t e n c e o f r e g i o n a l b o d i e s s u c h a s S E A M S , A M U and U A P M s h o u l d be of g r e a t h e l p i n the c o n s t r u c t i o n o f a s t a b l e n e t w o r k f o r the c h a n e l l i n g o f c o o p e r a t i v e e f f o r t s w i t h i n the t h i r d w o r l d . T h r o u g h the e x c h a n ge o f p u b l i c a t i o n s a n d i n f o r m a t i o n , both f o r m a l a n d i n f o r m a l r e g u l a r r o n t a c t s b e t w e e n e x e c u t i v e s a n d the s p o n s o r s h i p o f joint meetings, regional organizations can obviously do an i n v a l u a b l e s e r v i c e t o the d e v e l o p m e n t of m a t h e m a t i c s in t h e i r respective regions. T w o r e c o m m e n d a t i o n s of t h e G r o u p s t a n d

out

a s b e i n g of f u n d a -

mental importance for the further strengthening and stabilizat i o n of i n t e r n a t i o n a l c o o p e r a t i v e e f f o r t s . T h e s e c o n c e r n its c a l l f o r the c r e a t i o n w i t h i n I M U ( I n t e r n a t i o n a l M a t h e m a t i c s Union) o f an " I n t e r n a t i o n a l C o m m i s s i o n o n M a t h e m a t i c s a n d D e v e l o p m e n t " and t h e i r s u p p o r t f o r the e s t a b l i s h m e n t by U N E S C O of a n " I n t e r n a t i o n a l C e n t r e o f P u r e a n d A p p l i e d M a t h e m a t i c s " . I f the Group's r e c o m m e n d a t i o n s i n t h i s r e s p e c t a r e f u l f i l l e d then international cooperation would have undeniably taken g r e a t s t e p s toward the g o a l o f p r o m o t i n g m a t h e m a t i c a l a c t i v i t y i n third w o r l d c o u n t r i e s a n d e v e n t u a l l y toward that o f p r o m o t i n g m a t h e m a t i c s itself.

7.

Conclusions F o u r d a y s is too b r i e f a time f o r a l l the i m p o r t a n t q u e s t i o n s to be a s k e d and a n a l y s e d . N o n e t h e l e s s , m a j o r p r o b l e m s o f m a t h e matics development i n third world countries were identified. T h e s e r a n g e f r o m the p r o b l e m o f t h e p o o r q u a l i t y of the s c h o o l m a t h e m a t i c s t e a c h e r to t h a t o f the s c a r c i t y o f talent. W h i l e m a n y o f the p r o b l e m s a r e f a i r l y w e l l u n d e r s t o o d , s o m e a r e i n

need o f f u r t h e r c l a r i f i c a t i o n (for i n s t a n c e , t h e p r o b l e m o f the m e d i u m o f i n s t r u c t i o n at all levels). It w a s n o t t h e p u r p o s e of the C o n f e r e n c e , n o r c o u l d i t p o s s i b l y

21

THE CONFERENCE : ITS BACKGROUND AND WORK

22

have b e e n , t o p r o d u c e a magical f o r m u l a that w o u l d t a k e e a c h d e v e l o p i n g c o u n t r y a l o n g a r o y a l road (does o n e e x i s t ? ) leading t o t h e k i n g d o m of m a t h e m a t i c s . T h e C o n f e r e n c e a i m e d a t p r o v i d i n g a p l a t f o r m f o r the e x c h a n g e of i d e a s and e x p e r i e n c e s p e r t a i n i n g t o its theme. T h e r e i s surely g r e a t v a l u e i n h e a r i n g a t first h a n d a b o u t the s u c c e s s e s and f a i l u r e s of i n d i v i dual and c o l l e c t i v e e f f o r t s for p r o m o t i n g m a t h e m a t i c s in countries whose socio-economic structures share a number o f fundamental characteristics. T h e s i m i l a r i t y and d i f f i c u l t y of the p r o b l e m s f a c e d by many

d e v e l o p i n g c o u n t r i e s , o n t h e o n e h a n d , and the n a t u r e o f m a t h e m a t i c s a n d m a t h e m a t i c a l a c t i v i t y , o n the o t h e r , m a k e the s u b j e c t of c o o p e r a t i o n of great i m p o r t a n c e a n d u r g e n c y . T h e C o n f e r e n c e laid g r e a t e m p h a s i s o n r e g i o n a

and international coope-

ration. I t s r e c o m m e n d a t i o n s to I M U and U N E S C O c o n s t i t u t e important s t e p s toward the s t r e n g t h e n i n g a n d s t a b i l i z a t i o n of international cooperation. If o n e w e r e to s i n g l e o u t only o n e o f

he Conference's conclu-

s i o n s as the m o s t i m p o r t a n t , t h e n that w o u l d u n d o u b t e d l y b e i t s s y s t e m a t i c e m p h a s i s o n t h e r o l e of i n d i g e n o u s g r o u p s i n b u i l d i n g m a t h e m a t i c s . W h e t h e r a t the l e v e l o f i d e n t i f y i n g problems o r p r e s c r i b i n g s o l u t i o n s , there w a s u n a n i m o u s a g r e e m e n t that local m a t h e m a t i c i a n s , w h e r e a c r i t i c a l m a s s e x i s t s , m u s t a l w a y s a s s u m e full r e s p o n s i b l i t y . I t may o f t e n be n e c e s s a r y f o r them

to

s e e k t h e a s s i s t a n c e of f o r e i g n e x p e r t s o r e x t e r n a l

b o d i e s , but this type o f a s s i s t a n c e s h o u l d n e v e r be a l l o w e d to be m o r e than w h a t it is : a n e c e s s a r y aid f o r the a c h i e v e ment of clearly defined goals. T h e e n c o u n t e r s a t c o f f e e b r e a k s and d u r i n g s o c i a l f u n c t i o n s , the F i n a l R e p o r t a n d the d e t e r m i n a t i o n o f a l 1 , p a r t i c i p a n t s to s e e to i t that t h e i r r e c o m m e n d a t i o n s a r e n o t only w i d e l y diss e m i n a t e d b u t also a c t e d upon

*,

all d e m o n s t r a t e t h a t i n t e r n a -

tional m e e t i n g s s u c h as the K h a r t o u m o n e a r e very m u c h w o r t h h a v i n g i n the future. June, 1978

*

M.E.A.

El T o m

I n the c l o s i n g s e s s i o n o f the C o n f e r e n c e , a F o l l o w - u p C o m m i t t e e was elected,

P A R T TWO

THE I N V I T E D TALKS

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Deoeloping Mathematics i n Third WorZd Countries, M.E.A. EZ Tom ( e d . ) 0 North-Holland Publishing Company, 1979

S'I'ItAM G I ES & PHIOHIT I ES I N IlATllE?4A'PI C A L EDUCATION AND HESEARCII I N UEVELOE'ING C O U N T I U E S

BY A. A. ASIIOUR C A I I W U N I V E H S I T Y AN U B E I I W T AItAB U N I V E K S I T Y

L a d i e s and Gentlemen,

P a m i n d e b t e d t o t h e O r g a n i s i n g Committee o f t h i s Confer e n c e f o r i n v i t i n g me t o give t h i s t a l k . Allow me f i r s t t o s a y t h a t most of my t a l k w i l l n o t b e l i m i t e d t o mathematical e d u c a t i o n . It i s i n f a c t v a l i d for scient i f i c e d u c a t i o n and r e s e a r c h i n g e n s r a l . U h i v e r s i t y e d u c a t i o n i s a key f a c t o r i n t h e development o f any c o u n t r y and s p e c i a l l y o f d e v e l o p i n g ones. However, i t must be emphasised t h a t t h i s does n o t mean t h a t t h e r e s h o u l d n o t b e d i f f e r e n c e s between t h e system of e d u c a t i o n i n d e v e l o p i n g and developed c o u n t r i e s . I f we f o l l o w t h e same methods, c u r r i c u l a , s t r a t e g i e s and so f o r t h as i n t h e developed world, i t w i l l t a k e u s a l o n g time t o a c h i e v e a n y t h i n g and c e r t a i n l y w e s h a l l n o t be a b l e t o b r i d g e t h e s c i e n t i f i c and t e c h n o l o g i c a l gap which i s widen i n g e v e r y day a t a n i n c r e a s i n g rate. To have a new s t r a t e g y f o r u n i v e r s i t y e d u c a t i o n one needs t o have a wider s t r a t e g y f o r t h e development o f t h e c o u n t r y . It i s e s s e n t i a l t h a t i n every developing country a s c i e n t i f i c p o l i c y be drawn o u t , T h i s of c o u r s e w i l l d i f f e r from one c o u n t r y t o a n o t h e r . However, i n any country such development w i l l b e c l o s e l y connected and dependent on s c i e n t i f i c r e s e a r c h and t e c h n o l o g i c a l advancement. That i s why I p r o p o s e , when d e a l i n g w i t h c u r r i c u l a and u n d e r g r a d u a t e t e a c h i n g t o s t a r t f i r s t w i t h r e s e a r c h . Once a s t r a t r g y i s formed f o r s c i e n t i f i c r e s e a r c h and i t s r e l a t i o n t o development, i t w i l l be r e l a t i v e l y e a s y t o move downwards t o u n d e r g r a d u a t e and p r e - u n i v e r s i t y levels.

Here w e s t a r t w i t h f e w b u t e s s e n t i a l p o i n t s : ( i ) If t h e r e is a hope t o b r i d g e t h e gap between t h e developed indeed i f t h e r e and t h e underdeveloped r e g i o n s o f t h e world

-

25

26

A . A . ASHOUR

i s a hope t o keep the p o p u l a t i o n of' c e r t a i n r e g i o n s s u r v i v i n g , t h e n this hope l i e s i n S c i e n c e & Technology.

(ii)That made from ning they

S c i e n c e and Technology cannot j u s t be imported as a r e a d y comodety, Ln f a c t t h i s what we are b e i n g d r i v e n t o have i n d u s t r i a l c o n c e r n s i n the advanced world, t h u s remaialways dependent on them. What we g e t i n t h i s way i s w h a t are w i l l i n g t o g i v e u s , n o t what w e r e a l l y need,

S c i e n c e and Technology c a n n o t be bought o r s o l d , they must be developed l o c a l l y through a d i f f i c u l t , t e d i o u s and l o n g

process, H B r e we cannot a v o i d wondering a b o u t t h e 13r.N. Agencies a c t i v i t y i n t h e f i e l d " t r a n s f e r of technology." P am a f r a i d u n l e s s c o n d i t i o n s of t h e r e c i p e n t c o u n t r y a r e such t h a t t h e y can k e e p and develop the technology t r a n s f e r e d i t i s a waste.

( i i i ) F o r c o u n t r i e s s e e k i n g p o l i t i c a l independence and s o v e r e i g n t y , -it i s n e c e s s a r y t o a c h i e v e " s c i e n t i f i c independence". -Here independence does n o t imply i s o l a t i o n . Thus s c i e n t i f i c research i s n o t o n l y a means o f a c c e l e r a t i n g economic development b u t a n e c e s s a r y c r i t e r i o n € o r p o l i t i c a l independence. I f we a c c e p t these c r i t e r i a , w e have t o face a v e r y importhow c a n a c o u n t r y b u i l d i t s own S c i e n c e a n t q u e s t i o n , namely and Technology i f i t i s going t o be independent i n t h e s e n s e developed above? The answer t o t h i s q u e s t i o n i n my o p i n i o n i s t h a t a c o u n t r y c a n n o t s t a r t t o do this e x c e p t a f t e r r e a c h i n g o r a c q u i r i n g a c e r t a i n minimum of momentum i n s c i e n t i f i c research and t e c h n o l o g i c a l e x p e r i e n c e , T h i s i s t h e f i r s t s t a g e . -In r e a c h i n g t h i s c r i t i c a l l e v e l i t does n o t m a t t e r what research i s done and whether i t d i r e c t l y r e f l e c t s on t h e needs of t h e c o u n t r y of c o u r s e i t would be b e t t e r i f i t does but t h e i m p o r t a n t t a s k i s t o form a h a r d n u c l e u s of s c i e n t i f i c researchers t r a i n e d i n t h e methods and sys terns of research, aware of i t s needs and r e q u i r e m e n t s ( l i b r a r y , documentation, p e r s o n n e l etc. ,) Another v e r y i m p o r t a n t a s p e c t needed i n t h i s n u c l e u s is the a c c e p t a n c e t h a t research i s b e i n g done i n d i f f e r e n t and much more d i f f i c u l t c o n d i t i o n s t h a n i n t h e developed world where a l a r g e m a j o r i t y of t h e members may have o b t a i n e d

-

-

-

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MATHEMATICAL EDUCATION AND RESEARCH IN DEVELOPING COUNTRIES

27

t h e i r own t r a i n i n g . If a s c i e n t i s t i n a d e v e l o p i n g c o u n t r y assumes t h a t r e s e a r c h c a n n o t be done e x c e p t w i t h t h e f a c i l i ties available i n the highly industralised countries then he w i l l n e v e r be u s e f u l i n h i s own, The c h a l l e n g e f o r u s i s t o pursue research f i r s t class r e s e a r c h i n the d i f f i c u l t c o n d i t i o n s w i t h i n which w e l i v e . 1 might add h e r e t h a t p o l i t i c a l c o n s c i o u s n e s s and d e d i c a t i o n t o ones own c o u n t r y i s n e c e s s a r y . Fn t h e mean time, l e a d e r s and a u t h o r i t i e s i n t h e d e v e l o p i n g c o u n t r i e s must r e a l i s e t h a t any q u a l i f i e d s c i e n t i s t is invaluable a t t h i s stage.

-

-

-

-

Once t h e mentioned minimum of momentum i s a c h i e v e d it w i l l be p o s s i b l e f o r l o c a l s c i e n t i s t s t o t a k e p a r t i n forming t h e s c i e n t i f i c p o l i c y of t h e c o u n t r y , i n c o n n e c t i o n w i t h i t s development. It w i l l a l s o be p o s s i b l e f o r t h e s c i e n t i f i c commu n i t y t o o r i e n t i t s e l f toward t h e l o c a l problems t o be s o l v e d , t o "breed" o t h e r s c i e n t i s t s w i t h t h e p r o p e r t r a i n i n g which i s b i a s e d toward t h e s p e c i a l problems of t h e c o u n t r y i n short t o grow up t o t h e r i g h t s i z e and i n t h e r i g h t d i r e c t i o n .

-

Scientific level What a b o u t t h e l e v e l o f s c i e n t i f i c r e s e a r c h i n d e v e l o p i n g c o u n t r i e s ? I f we are g o i n g t o be choosy and c o n c e n t r a t e on o u r problems does t h i s mean t h a t w e s h a l l be a t t a c k i n g problems nf a lower l e v e l ? - l!n o t h e r words w i l l t h e p i o n e e r i n g r e s e a r c h and c h a l l e n g i n g problems be l i m i t e d t o workers i n t h e i n d u s t r i a l i s e d c o u n t r i es ?

-

The answer t o t h i s q u e s t i o n s h o u l d d e f e n i t e l y be no. The c h o i c e i s not i n t h e level of t h e s c i e n t i f i c research but r a t h e r i n its a p p l i c a t i o n . I n t h e d e v e l o p i n g world, due t o l i m i t e d r e s o u r c e s s c i e n t i f i c r e s e a r c h may n o t o r perhaps s h o u l d n o t , embrace a l l p o s s i b l e f i e l d s , b u t s h o u l d be of t h e h i g h e s t p o s s i b l e l e v e l .

-

R e l a t i o n w i t h t h e world s c i e n t i f i c cominunity As mentioned b e f o r e t h e l o c a l development o f s c i e n t i f i c resear c h does n o t mean s c i e n t i f i c i s o l a t i o n . On t h e c o n t r a r y i t i s nec e s s a r y f o r t h e s c i e n t i f i c community i n t h e d e v e l o p i n g c o u n t r i e s t o have access t o world s c i e n t i f i c l i t e r a t u r e t o exchange i d e a s

-

28

A . A . ASHOUR

w i t h c o l l e a g u e s i n t h e i n d u s t r a l i e s e d world by a t t e n d i n g confe-

r e n c e s and by h o l d i n g such c o n f e r e n c e s i n the d e v e l o p i n g c o u n t r i e s . It i s a l s o n e c e s s a r y t o a t t r a c t h i g h l y q u a l i f i e d s c i e n t i s t s from anywhere i n t h e world t o work w i t h t h e l o c a l s c i e n t i s t s . For t h i s t o succeed and n o t become j u s t a h o l i d a y f o r t h e f o r e i g n s c i e n t i s t , a s i t is v e r y o f t e n , we have t o o f f e r c h a l l e n g i n g s c i e n t i f i c problems t o t h o s e s c i e n t i s t s . A p o i n t which d e s e r v e s mentioning h e r e i s t h a t up t i l l now

most d e v e l o p i n g c o u n t r i e s have l i m i t e d t h e i r s c i e n t i f i c r e l a t i o n s w i t h h i g h l y developed c o u n t r i e s . T h i s g i v e s r i s e t o a s o r t o f Ildiscontinuity" i n t h e process of s c i e n t i f i c research t h a t r e s u l t s i n t h e two a s p e c t s of t h e " b r a i n d r a i n " ( a ) S c i e n t i s t s from t h e d e v e l o p i n g c o u n t r i e s r e s i d i n g i n t h e hil?;hly developed c o u n t r i e s , and ( b ) S c i e n t i s t s i n t h e d e v e l o p i n g c o u n t r i e s working on problems s e r v i n g t h e i n d u s t r y of t h e developed c o u n t r i e s where t h e y had t h e i r h i g h e r e d u c a t i o n . Although we d e f i n e t e l y would l i k e t o keep i n t e r a c t i o n between t h e s c i e n t i f i c communities i n b o t h t h e developed and d e v e l o p i n g worlds- i t i s u s e f u l t o have r e l a t i o n s between t h e s c i e n t i f i c communities i n the d e v e l o p i n g c o u n t r i e s themselves. t h e wording "developing c o u n t r i e s " does n o t r e a l l y Incidentally mean one l e v e l of development. S c i e n t i s t s i n a c e r t a i n l e v e l of development s h o u l d i n t e r a c t w i t h t h e i r c o l l e a g u e s i n t h e n e x t l e v e l . They w i l l b e n e f i t from e x p e r i e n c e i n c o n d i t i o n s similar t o t h e i r s and w i l l have no f r u s t r a t i o n due t o t h e non-existnnce of a n immense d i f f e r e n c e i n f a c i l i t i e s .

-

R e f l e c t i o n on mathematics I t h i n k i t i s h i g h time f o r me t o s a y how what h a s been s a i d r e f l e c t s on o u r p a r t i c u l a r f i e l d : Mathematics ( i ) The f e w q u a l i f i e d mathematicians i n a d e v e l o p i n g c o u n t r y are v e r y p r e c i o u s . They s h o u l d b e encouraged f i n a n c i a l l y and t h e i r i m p o r t a n t r o l e r e c o g n i s e d by t h e i r government t o guard them from t h e d a n g e r s of immigration.

( i i ) T h e main r o l e of mathematicians i n d e v e l o p i n g c o u n t r i e s s h o u l d be i n t h e a p p l i c a t i o n s . There are immense f i e l d s a t o u r d i s p o s a l where App- Mathematicians can e x c e l themselves. L e t me mention o n l y a f e w :

MATHEMATICAL EDUCATION AND RESEARCH IN DEVELOPING COUNTRIES

29

( i )A g r i c u l t u r e

Food i s t h e main problem i n d e v e l o p i n g c o u n t r i e s and t h e i n c r e a s e and improvment of h a r v e s t s and animal s t o c k s i s o f prime importance. The new f i e l d s i n Molecular B i o l o g y - G e n e t i c Engineer i n g need t h e o r e t i c a l f o r m u l a t i o n and modelling. The o p t i m s i a t i o n of t h e u s e o f l a n d and water r e s o u r c e s ( i n c l u d i n g water r e s e r v o i r s ) t h e d e s e r t i f i c a t i o n , i t s p r e d i c t i o n and c o n t r o l ; d e s a l i n a t i o n pose i n t e r e s t i n g and i n t r i g u i n g mathematical problems. ( i i ) N a t u r a l d i s a s t e r s s u c h as E a r t h q u a k e s , Monsoons and H u r r i c a n e s

When s u c h a d i s a s t e r f a l l s on a developed c o u n t r y , t h e damage c a n be o v e r come, b u t when t h e c o u n t r y i s p o o r , t h e damage i s o f t e n beyond r e p a i r . T h i s i s b e c a u s e t h e magnitude of t h e damage does n o t depend on whether t h e a r e a i s r i c h o r poor. When a monsoon s t a r t s a f t e r t h e s e e d s have been p l a n t e d , a y e a r s h a r v e s t i s spoiled when a n e a r t h q u a k e h i t s a poor c o u n t r y , i t s i m p l y d e s t r o y s a l l i t s housing a l l i n d u s t r y e t c . . Hence we i n t h e d e v e l o p i n g c o u n t r i e s have s t r o n g e r r e a s o n s t o s t u d y t h e p r e d i c t i o n s of such disasters.

-

Weather p r e d i c t i o n , Climate c o n t r o l Earthquake p r e d i c t i o n , P h y s i c s of t h e d e s e r t , are a l l f i e l d s whose development c a n n o t be c o n t i n u e d w i t h o u t mathematical m o d e l l i n g and t h e u s e of v e r y sophi s t i c a t e d mathematics: Theory o f i n f o r m a t i o n - A r t i f i c i a l i n t e l l i g e n c e - P a t t e r n recognition-Optimsation-Catastrophe theory-System a n a l y s i s and of c o u r s e P r o p a b i l i t y tlieory-Numerical a n a l y s i s and approximation theory , (iii) Enerpy The a q u s i t i o n o f new s o u r c e s o f energy and the development of e x i s t i n g s o u r c e s i s o b v i o u s e l y a n e c e s s i t y f o r d e v e l o p i n g count r i e s . Thus t h e s t u d y of s o l a r energy-thermal energy-energy from t h e t i d e s and waves a g a i n pose mathematical problems of lst grade.

These are o n l y examples. The matlieinatical problems i n v o l ved a r e of such l e v e l a s t o a t t r a c t matliernnticjans from t h e devel o p e d a s w e l l as the developin: c o u n t r i e s t o work on them b u t may be f o r d i f f e r e n t aims. I t i s o u r r e s p o n s i b i l i t y t o see t h a t t h e

30

A . A . ASHOUR

a p p l i c a t i o n s a r e t o t h e b e n e f i t of our development and n o t t h e opposit e e A word about computers i s n e c e s s a r y h e r e . W e s h a l l no doubt need computers t o work o u t t h e r e s e a r c h problems mentioned, Howe v e r , we must u t i l i s e computers n o t t o s a v e manpower as i s o f t e n done i n developed c o u n t r i e s where i t i s scarce, b u t t o s o l v e problems which o t h e r w i s e cannot be s o l v e d . T h i s would j u s t i f y t h e c o s t involved.

Curricula A t t h e moment, moat u n i e r s i t i e s i n t h e d e v e l o p i n g world f o l l o w e x a c t l y t h e same c u r r i c u l a of u n i v e r s i t i e s i n c e r t a i n devel o p e d c o u n t r i e s , mainly B r i t a i n and France, simply b e c a u s e t h e F a c u l t y h a s been e d u c a t e d a t l e a s t p a r t l y i n t h e s e c o u n t r i e s .

We r e c o g n i s e v e r y w e l l t h a t mathematics i s an e n t i t y and t h a t almost e v e r y branch of i t i s dependent on t h e o t h e r s . However, i t i s n o t j u s t i f i e d t h a t i n c e r t a i n c o u n t r i e s s c h o o l math. t e a c h e r s are imported w h i l e t h e f e w members of u n i v e r s i t y s t a f f are busy d o i n g r e s e a r c h i n some high brow mathematics and t h e u n i v e r s i t y c o u r s e s do n o t i n c l u d e any down t o E a r t h mathematics. While we do n o t a d v o c a t e t h e o m i t t i n g of any p a r t i c u l a r c o u r s e , what one s h o u l d i n s i s t on i s t h a t c o u r s e s i n s t a t i s t i c s probability-numerical analysis-approxfmation theory-information t h e o r y and o p t i m i s a t i o n s h o u l d have p r i o r i t y . illso w e s h o u l d have c o u r s e s s u i t a b l e f o r producing t h e kind of mathematicians needed f o r c e r t a i n r e s p o n s i b i l i t i e s i n t h e c o u n t r y , s u c h as t e a c h e r s s u r v e y o r s - s t a t i s t i c i a n s - a c t u a r i a n s and s o on. It i s a f i r s t p r i o r i t y t h a t t h e d e v e l o p i n g c o u n t r i e s s h o u l d n o t spend t h e i r r e s o u r c e s f o r h i r i n g f o r e i g n e r s t o do s t a n d a r d and s i m p l e j o b s ,

If we a r e g o i n g t o g i v e p r i o r i t y t o t h e a p p l i c n t i o n s of mathematics, t h e mathematician s h o u l d have enough knowledge o f t h e f i e l d s i n which he i s a p p l y i n g h i s mathematics. Hence t h e r e s h o u l d be more c o u r s e s i n biology-mechanics- economics etc. which are t a u g h t t o mathematical g r a d u a t e s . S i m i l a r l y more mathematicians s h o u l d be t a u g h t t o e n g i n e e r s - b i o l o g i s t s - economists-geophysic i s t s e t c . t o produce t h e o r e t i c a l people i n t h e s e f i e l d s as i t is

MATHEMATICAL EDUCATION AND RESEARCH I N DEVELOPING COUNTRIES

s u c c e s s f u l l y done w i t h p h y s i c i s t s .

It may be a good t h i n g t o have a g e n e r a l degree i n mathe m a t i c s and one or more of t h e s e d i s c i p l i n e s . A g r a d u a t e who i s e q u a l l y q u a l i f i e d i n b o t h matliernatics and t h c f i e l d of a p p l i c a t i o n might be t h e k i n d of p e r s o n we r e a l l y nccd. I did not give d e t a i l e d courses here but only general guiding ideas. R e g i o n a l and I n t e r n a t i o n a l O r g a n i z a t i o n s s u c h as t h e A f r i c a n Mathematical Union, t h e I n t e r n a t i o n a l hiathematical Union and i t s Committee on E d u c a t i o n IChlI c a n o f f e r immense h e l p f o r us i n drawing up t h e d e t a i l e d c o u r s e s f o r each p a r t i c u l a r r e g i o n and c o u n t r y i n c o n s u l t a t i o n w i t h t h e l o c a l people. L a d i e s and Gentlemen thank you f o r y o u r l i s t i n i n g .

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Developing Mathematics i n Third World Countries, M.E.A. E l Tom led.) 0 North-Holland Publishing Company, 1979

A D E Q U A T E M A T H E M A T I C S FOR T H I R D WORLD COUNTRIES C O N S I D E P A N D A A N D STRATEGIES. U b i r a t a n D ' Ambrosio I n s t i t u t o de M a t e m a t i c a ,

Es t a t i s t i c a e C i g n c i a

ComputaClao d a U n i v e r s i d a d e E s t a d u a l de C a m p i n a s Sao P a u l o

-

de

.

Brazil.

Introduction: This paper i s mainly concerned w i t h the r e l a t i o n s h i p o f mathematics and t h e s o c i o - c u l t u r a l

context i n which i t i s placed.

Although tempt

e d t o e l a b o r a t e on some o f t h e c o n s i d e r a n d a i n i t i a t e d i n

ID'Ambrosio

1 9 7 6 a ) , we w i l l b r i n g i n t o t h e d i s c u s s i o n some m o r e s p e c i f i c s t r a t e g i e s t o b u i l d up m a t h e m a t i c s i n c l o s e r e l a t i o n s h i p t o t h e r e a l w o r l d i n a l l levels o f education.

Although most o f the a v a i l a b l e m a t e r i a l

a l r e a d y d e v e l o p e d b y t h e " I n s t i t u t o de M a t e m a t i c a ,

Estatistica

C i e n c i a de Computaczo" o f t h e S t a t e U n i v e r s i t y o f C a m p i n a s , i s f o r t h e p r i m a r y and secondary l e v e l s ,

e

i n Brazil

the teacher t r a i n i n q s t r a

-

tegy and t h e modular concept have been developed w i t h t h e aim o f p r g p a r i n g u n i v e r s i t y t e a c h e r s and o f b u i l d i n g up models f o r r e s e a r c h training,

respectively.

We o p e n w i t h some b a s i c q u e s t i o n s w h i c h f a c e e v e r y o n e w o r k i n p c l o s i n g t h e gap b e t w e e n t h e h a v e s r o o t o f o u r concern.

and have-nots

which i s i n

i n the

We l o o k f o r m o d e l s w h i c h w i l l a l l o w a c c e l e r a t -

i n g the process o f b r i n g i n g mathematical development t o a l e v e l which

w i l l be t r a n s f o r m e d i n a companion o f o t h e r s c i e n c e s and i n t h e developmental e f f o r t s ,

techniques

a n d n o t as a n a c a d e m i c e x e r c i s e

"per

se", which i s , i n o u r s i t u a t i o n , o f l e s s p r i o r i t y , o f l e s s urgency We a r e t h e n f a c e d w i t h some q u e s t i o n s : 1.

Is i t possible t o i d e n t i f y directions o f mathematical

research

r e l a t e d t o n a t i o n a l p r i o r i t i e s and g o a l s ? 2.

I f so,

i s i t possible t o conduct the t r a i n i n q o f mathematicians

i n the i d e n t i f i e d directions?

3.

I s t h e r e t h e d a n g e r t h a t t h e s p e e d w i t h w h i c h new p r o b l e m s

and

new n e e d s a p p e a r w i l l n o t a l l o w f o r t h e a d e q u a t e p r e p a r a t i o n o r adaptation o f mathematicians,

trained according with i d e n t i f i e d

p r i o r i t i e s and i m m e d i a t e needs, and needs?

33

t o c o p e w i t h t h e s e nww p r o b l e m s

.

34

U. D'AMRROSIO

W i t h r e s p e c t t o 3 . , i t h a s b e e n l o n a r e c o q n i z e d t h a t d e v e l o p m e n t gen e r a t e s needs n e v e r h i t h e r t o f e l t , and t h i s i s p a r t i c u l a r l y t r u e i n s c i e n t i f i c and t e c h n o l o g i c a l development (see ID'Ambrosio 1971 I ) . On t h e o t h e r h a n d , t h e l e v e l o f s o p h i s t i c a t i o n n e e d e d f o r t e c h n o l o -

g i c a l advancement, i n p a r t i c u l a r w i t h r e s p e c t t o mathematics

,

may

be of a d i f f e r e n t nature o f the mathematics which are most a t t r a c t i v e t o young mathematicians from developing c o u n t r i e s . Undoubtfully, t o be r e c o g n i z e d and a c c e p t e d i n t h e i n n e r c i r c l e o f t h e m a t h e m a t i c a l community o f t h e d e v e l o p e d c o u n t r i e s i s more a p p e a l i n g and o f a more i m m e d i a t e r e w a r d t h a n f a c i n g t h e a p p a r e n t l y t r i v i a l a n d

old

-

f a s h i o n e d mathematical problems p r e s e n t e d b y t h e s u f f e r i n q r e a l ?ty o f underdevelopment. countries,

Although t h i s i s n o t l e s s t r u e i n developed

i n o u r case t h i s b r i n g s t h e m a t t e r a sense o f

urgency,

s i n c e we h a v e t o d e c i d e a b o u t t h e a l l o c a t i o n o f b a d l y n e e d e d human and f i n a n c i a l resources t o a g r o w i n g s c i e n t i f i c and

technological

establishment, We w i l l l e a v e t h e t h r e e m a j o r q u e s t i o n s l i s t e d a b o v e

to further

d i s c u s s i o n i n a l a t e o p p o r t u n i t y and w i l l p r e s e n t a few s t r a t e g i e s w h i c h may c r e a t e a b a c k g r o u n d a n d p i l o t g r o u n d f o r a d e e p e r i n s i g h t i n t o t h e s e same q u e s t i o n s .

I n d e a l i n g w i t h s t r a t e g i e s , we h a v e t o i n s i s t on g e n e r a t i n g

inter-

d i s c i p l i n a r y c o n t e x t s i n which mathematics can be b r o u g h t i n t o p l a y . The p e c u l i a r i t y o f o u r s i t u a t i o n c a l l s f o r h i g h l e v e l o f c r e a t i v i t y i n t h e a p p l i c a t i o n o f e x i s t i n g m a t h e m a t i c s t o an unknown p r o b l e m a t i c s . T h e p o i n t i s much o f a p h i l o s o p h i c a l n a t u r e , a n d i t was a m p l y d i s c u s sed i n (D'Ambrosio 19771 S c i e n t i f i c c r e a t i v i t y i n f u l l r e l a t i on

.

ship w i t h the c u l t u r a l environment i n which the c r e a t i v i t y

takes

p l a c e and f i n d s i t s m e a n i n q f u l n e s s , can h a r d l y b e a c h i e v e d i n a f r a g m e n t e d d i s c i p l i n a r y c o n t e x t . The c u l t u r a l p l a c e m e n t o f an e d u c a t i o n a l system and o f s c i e n t i f i c s t r u c t u r e s i s , p r o b a b l y , t h e most r e l e v a n t f a c t i n t h e modern development o f e d u c a t i o n , m a i n l y i n u n d e v e l o p e d countries.

An e x t e n s i v e a n a l y s i s o f t h e n a t u r e a n d p u r p o s e s

science education i n developing countries

,

of

i t s o b j e c t i v e s and g o a l s ,

and t h e need t o a c c e l e r a t e t h e p r o c e s s o f p r e p a r i n g s c i e n c e s p e c i a l i s t s and b r i n g i n g s c i e n t i f i c l i t e r a c y t o a l a r q e o o r t i o n o f t h e population i s given i n

I D ' A m b r o s i o 19761

and

I D ' A m b r o s i o 19751

.

ADEQUATE MATHEMATICS FOR THIRD WORLD COUNTRIES

35

M a t h e m a t i c s and n a t u r e : S i n c e s c i e n c e p r e s e n t s i t s e l f as a g l o b a l v i e w o f n a t u r a l p h e n o m e n a , w i t h s t r o n g i n p u t f r o m t h e s o c i o c u l t u r a l momentum, i n a v e r y u n s o p h i s t i c a t e d way,

we m i g h t c o n s i d e r

the mental process which

generates

s c i e n t i f i c a t t i t u d e and s c i e n t i f i c u n d e r s t a n d i n g a c c o r d i n g

to

the

Figure 1.

FORMULATED

Figure 1

I n t h i s p r o c e s s , m a t h e m a t i c s as a l a n g u a g e i s t h e m a i n i s s u e s h o u l d be g i v e n a s p e c i a l p l a c e , i n c u r r i c u l a , e m p h a s i s on c o n t e n t s .

In fact,

a shift

i n s t e a d of

an

from content towards

and over more

36

U. D'AMBROSIO

e m p h a s i s on p r o c e s s a n d m e t h o d , seems t o be a n i d e n t i f i e d t r e n d i n s c i e n c e e d u c a t i o n t o d a y . I n t h i s c o n t e x t t h e view p o i n t o f Rene Thom i n p l a c i n g m a t h e m a t i c s a s a f i n e r l a n g u a g e t h a n n a t u r a l l a n g u a g e t o d e s c r i b e e x t e r n a l phenomena i s q u i t e a d e q u a t e t o a c o n t e n t - f r e e s c i e n c e e d u c a t i o n (Thorn 1 9 7 5 1 . M i t h o u t any d o u b t , t o d e f i n e " a p r i o r i " a p r o g r a m o f s t u d i e s i n m a t h e m a t i c s bounds t h e s t u d e n t s a n d t h e t e a c h e r , as w e l l a s t h e e d u c a t i o n a l e x p e r i e n c e , t o an u n f a v o u r a b l e c o n d i t i o n f o r t r u l y s c i e n t i f i c a t t i t u d e . The process of s c i e n t i f i c d i s c o v e r y , meets a very s e r i o u s handicap w i t h i n a p r o g r a m a t i c c o n t e x t . The f i g u r e a b o v e r e f l e c t s an open s i t u a t i o n , w h e r e a c c u m u l a t e d k n o w l e d g e i s b r o u g h t i n t o the e d u c a t i o n a l c o n t e x t i n t h e m e a s u r e i t i s n e e d e d . T h i s i s a f o r m of " i n f o r m a t i o n r e t r i e v a l " , which c o u l d be t r a c e d back t o t h e i n v e n t i o n o f p r i n t i n g a n d more r e c e n t l y t o t h e f u l l use of e l e c t r o n i c s . Although t h e p r a c t i c a l u s e of s u c h form o f r e t r i e v a l o f s t o r e d knowledge i s , as y e t , somewhat d i f f i c u l t t o a c h i e v e , i t seems t o me t h a t i n t h e f i n d i n g o f c h e a p e r and e a s y - t o - m a n i p u l a t e f o r m s of r e t r i e v a l of s t o r e d knowledge r e s i d e s t h e f u t u r e of e d u c a t i 0 n . A t o t a l c h a n g e of e m p h a s i s from c o n t e n t s , o r from t h e t r a d i t i o n a l informative t e a c h i n g , t o an a c t i v e p a r t i c i p a t i o n of s t u d e n t s i n a d i s c o v e r y pro c e s s , w i t h t e a c h e r s s e r v i n g as g u i d e s a n d r e s o u r c e s , a n d f o r m a l t e a c h i n g b e i n g r e d u c e d t o m o t i v a t i o n a l a s p e c t s and m e t h o d o l o g y o f a c c e s s t o i n f o r m a t i o n , seems t o b e t h e key e l e m e n t f o r a r e l e v a n t , dynamic and u p d a t e d e d u c a t i o n . M a i n l y f o r d e v e l o p i n g c o u n t r i e s , I s e e n o hope o f b r i n g i n g a d v a n c e d s c i e n c e a n d t e c h n o l o q y t o u s e f u l a n d r e a d y a c c e s s u n l e s s we s h i f t t h e p a t t e r n o f e d u c a t i o n t o an r e t r i e v a l b a s e d scheme. In t h i s c o n t e x t , h o w does mathematics f i t i n t o t h e s c h o o l s y s t e m , and how i s i t r e l a t e d t o o t h e r s c h o o l s u b j e c t s ? I t seems t o me, t h e q u e s t i o n would b e somewhat t r i v i a l i z e d i n a new e d u c a t i o n a l s t r u c t u r e , b a s e d on a model i n which s t o r e d knowledqe i s r e t r i e v e d i n some form. The r o l e o f c o m p u t e r s i n t h i s i s a l r e a d y c l e a r . We mention e s p e c i a l l y t h e e f f o r t s of t h e p r o j e c t C H E L S E A SCIENCE EDUCATION PROJECT, conducted b y the Center f o r Science Education, C h e l s e a C o l l e g e . O t h e r f o r m s o f r e t r i e v a l , r a n q i n g f r o m home comput a r i zed t e l e v i s i o n t o a l p h a - n u m e r i c p o c k e t c a l c u l a t o r s , a n d t o personal computers, w i l l g i v e t h e concept of r e t r i e v a l o f s t o r e d knowledge i t s p r o p e r p l a c e i n t h e e d u c a t i o n o f t h e f u t u r e , b r i n g i n g m a t h e m a t i c s t o a n a t u r a l r e l a t i o n s h i p w i t h o t h e r knowledge s y s t e m s represented in school.

37

ADEQUAI'E MATHEMATICS FOR T H I R D WORLD C O U N T R I E S

I t seems t h a t t h e t r a d i t i o n a l p a t t e r n o f m a t h e m a t i c a l e d u c a t i o n , b e i t the so-called

c l a s s i c a l o r t r a d i t i o n a l approach,

mathematics approach,

c a r r i e s an o v e r - e m p h a s i s

be i t t h e modern

on t h e v a l u e o f m a t h g

-

m a t i c s b y i t s e l f , w i t h a c l e a r t e n d e n c y t o w a r d s o v e r s t a t i n g a some w h a t r o m a n t i c i m p o r t a n c e o f m a t h e m a t i c s as t h e b u i l d e r o f c l e a r thinking,

as t h e r i o o r o u s s c i e n c e p a r e x c e l l e n c e ,

cinaly possible p r a c t i c a l value.

-

a n d an u n c o n v i n

On t h e o t h e r h a n d ,

m a t h e m a t i c s as a

process, so w e l l e x e m p l i f i e d by the mathevatization processes which a r e g o i n g on w i t h s c i e n c e ,

and mathematical m o d e l l i n a , which i s p l a y

i n g an u n p r e c e d e n t r o l e i n t h e a n a l y s i s o f n a t u r e , Growing c a p a b i l i t y o f f a s t e r c a l c u l a t i o n , i n t o t h e c u r r i c u l u m a t a much s l o w e r

m a i n l y due t o t h e

has been f i n d i n g i t s

Dace.

r e a l a n d u n d e n i a b l e v a l u e o f m a t b e v a t i c s as a d i s c i p l i n e , i n f u l l integration with a l l other subjects. r a t h e r e n c o u r a g i n g examples o f 'mathematics sciences,

Mr.

as i n t h e G h a n i a n

Oheae-Asare,

t h e i r own s a k e b u t , explain,

areas,

i n as much as t h e y a i v e m e a n i n a t o ,

i n 1973-74,

by "mea-

not

and

for help

t h e general p a t t e r n n o t i c e d i s matheAn e f f o r t b y t h e Y i n i s t r y o f E d u c a t i o n

lead t o several curricular units related

o u r r e a l i t y I D ' A m b r o s i o 19751 c o n d u c t e d by UNESCO,

the

as r e p o r t e d

volumes a r e o i v e n p r o m i n e n c e ,

matics unrelated t o science. of B r a z i l ,

obviously

being integrated i n

school system where,

s c i e n t i f i c concepts",

the

A l t h o u s h we f i n d

f r o m Ghana A s s o c i a t i o n o f S c i e n c e T e a c h e r s ,

surements o f l e n g t h s ,

way

T h i s we s t e e m i s

.

to

Ye m u s t v e n t i o n t h e e f f o r t s

through the Montevideo Penional O f f i c e o f

Science and Technology

f o r L a t i n America and t h e C a r i b b e ,

t o aenerate

c u r r i c u l a r m a t e r i a l t o b r i n a mathematics t o a c l o s e r r e l a t i o n s h i p t o applications.

Indeed,

a s e r i e s o f modules f o r a p p l i c a t i o n s o f mathe-

matics are r e s u l t i n g from several projects.

I+'e m e n t i o n i n p a r t i c u -

l a r t h e e f f o r t s o f t h e p r o j e c t UYAP ( U n d e r a r a d u a t e M a t h e m a t i c s

and

i t s A p p l i c a t i o n s P r o j e c t ) , developed by the Education Development Center,

Newton, Massachusetts,

which represents,

t o o u r knowledge,

t h e most comprehensive e f f o r t f o r t h e development o f u n i t s o f mathematics w i t h a r e a l i t y flavour.

With the special goal o f developing

c o u n t r i e s , some o f s u c h u n i t s h a v e b e e n d e v e l o p e d b y UNESCO's Montevideo O f f i c e , limited.

b u t t h e e x t e n t and number o f s u c h u n i t s a r e s t i l l

U. D'AMBROSIO

38

Model B u i l d i n g .

Spec f i c E x a m p l e s .

One o f t h e m o s t d i f f c u l t a s p e c t s o f b r i d g i n g t h e g a p b e t w e e n r e a l i t y and mathematics r e s i d e s i n t h e c o n c e p t s o f model b u i l d i n g .

This re-

q u i r e s some s p e c i f i c s t r a t e g i e s , w h i c h a r e b a s e d on t h e m e n t a l i z a t i o n p r o c e s s d e s c r i b e d i n F i o u r e 1. The i d e a o f d e v e l o p i n g s t r a t e g i e s f o r c u r r i c u l u m d e v e l o p m e n t i n rnathg m a t i c s , w i t h a v i e w t o w a r d s r e a l i t y , h a s b e e n m e t b y some

projects

d e v e l o p e d i n L a t i n A m e r i c a , w h i c h we d i s c u s s b e l o w . I t has b e e n l o n g r e c o g n i z e d t h a t m a t h e m a t i c a l e d u c a t i o n h a s n o t b e e n

s u f f i c i e n t l y dynamic i n d e v e l o p i n g c o u n t r i e s , changes needed f o r development and, m a i n l y ,

t o meet t h e r a p i d

t o r e s p o n d t o new n e e d s

and p r i o r i t i e s w h i c h development i t s e l f g e n e r a t e s . To a l a r g e e x t e n t , d e v e l o p m e n t g e n e r a t e s r e q u i r e m e n t s f o r more a d v a n c e d a n d

updated

mathematics i n the school l e v e l , b r i n s i n g mathematical l i t e r a c y t o an u n p r e c e d e n t p o s i t i o n i n u n d e v e l o p e d c o u n t r i e s . The s e l f a c c e l e r a t i n g process o f development p r e s e n t s two f a c e s . W h i l e

it

p r e s e n t s t h e w e l l known b e n e f i t s o f m a k i n g l i f e more d i g n i f y i n g a n d enhancing q u a l i t y o f l i f e , i t creates i n t h e e d u c a t i o n a l systems a g r o w i n g n e e d f o r more u p d a t e d and r e l e v a n t t r a i n i n g .

I n particular,

development b r i n g s a h i g h need f o r s c i e n t i f i c and t e c h n o l o a i c a l a w a r e n e s s a n d c o m p e t e n c y , w h i c h h a s a p r o f o u n d i n f l u e n c e on m a t h e mati cal education. L o o k i n g f o r a b e t t e r s u i t e d m a t h e m a t i c s f o r t h e r a p i d chanaes

i n

t h e s o c i o - e c o n o m i c p a t t e r n o f t h e c o u n t r y , we a r e d e v e l o p i n g

since

1973, a s e r i e s o f c u r r i c u l a r development p r o j e c t s i n t h e I n s t i t u t e o f M a t h e m a t i c s , S t a t i s t i c s and Computer S c i e n c e o f t h e S t a t e U n i v e r s i t y o f Campinas. secondary schools, t o 20 h o u r s ,

These p r o j e c t s , b o t h f o r elernentary

and

h a v e g e n e r a t e d u n i t s o f i n s t r u c t i o n , e a c h o f 15

touching c r i t i c a l t o p i cs o f t h e t r a d i ti onal c u r r i c u l u m ,

and which have been " i n s e r t e d " i n t h e e x i s t i n g o f f i c i a l c u r r i c u l a . The u n d e r l y i n g p h i l o s o p h y o f t h e p r o j e c t s , seminar

I D ' A m b r o s i o 19741

,

advanced i n a

national

aims a t b r i n q i n a m a t h e m a t i c s t o

close relationship w i t h the real world.

a

Basically, i t i s saught t o

d e r i v e f r o m t h e s u r r o u n d i n g r e a l i t y , enouqh m o t i v a t i o n f o r a mathematization process,

v e r y much i n t h e s c h e m e p r e s e n t e d i n F i g u r e 1.

We c o n s i d e r a n i n f o r m a l c a p a b i l i t y o f r e c o q n i z i n g m a t h e m a t i c s

i n

r e a l s i t u a t i o n s as a f i r s t s t e p f o r a t t a c k i n g a m a t h e m a t i c a l p r o b l e m . This i n t u i t i v e p r e - m o d e l l i n g approach,

on t h e e l e m e n t a r y l e v e l , i s

a b a s i c tone o f our c u r r i c u l a r development.

Wuch o f t h i s p h i l o s o p h y

39

ADEQUATE MATHEMATICS FOR THIRD WORLD COUNTRIES

i s d e s c r i b e d i n o u r address t o t h e 3 r d I n t e r n a t i o n a l Congress o f M a t h e m a t i c a l E d u c a t i o n , K a r l s r u h e , Germany, A u g u s t 1976 I D ' A m b r o s i o 1976al

.

We w i l l b r i e f l y d e s c r i b e t h e a d o p t e d s t r a t e g i e s

for

the

project. The s t r a t e g y t o d e v e l o p t h e u n i t s f o l l o w s a D a t t e r n as d e s c r i b e i n Figure 2

. I

1 1

I t i s important t o mention t h a t these projects, althouoh d i r e c t e d t o pre-university

schools,

are located i n a U n i v e r s i t y I n s t i t u t e ,

40

U. D'AMBROSIO

whose p r i m a r y aims a r e t e a c h i n g , b o t h a t t h e u n d e r g r a d u a t e and g r a d u a t e l e v e l s , a n d r e s e a r c h . O u r e x p e r i e n c e shows t h a t t h e r e s u l t s of m i x i n g c u r r i c u l a r d e v e l o p m e n t team w i t h r e s e a r c h and u n i v e r s i t y p r o f e s s o r s i s a most h e a l t h y o n e . The c r o s s f e r t i l i z a t i o n o f i d e a s which r e s u l t from t h i s i s most r e w a r d i n a , b o t h f o r t h e new e l e m e n t a r y c u r r i c u l a and f o r t h e u n i v e r s i t y a c t i v i t i e s i n r e s e a r c h and t e a c h i n g . A f t e r a few weeks o f m u t u a l " m i s t r u s t " , b o t h g r o u p s s t a r t t o r e s p e c t e a c h o t h e r and t h e i n t e r e s t o f r e s e a r c h mathemat i c i a n s i n p r o b ems of t e a c h i n g shows a g r o w i n g p a t t e r n . The p r e s e n c e of c h i d r e n , d u r i n g t h e e x p e r i m e n t a l p h a s e o f t h e p r o j e c t i n t h e u n i v e r s i y i s a l s o most r e w a r d i n g . We w i l l b r i e f l y d e s c r i b e some o f t h e u n i t s a l r e a d y d e v e l o p e d : 1 . E x p e r i m e n t a l Geometry : S p e c i a l e m p h a s i s i s g i v e n t o a n Archimedean a p p r o a c h t o v o l u m e s , w i t h o b s e r v a t i o n s of p h y s i c a l n a t u r e , s u c h as d e n s i t y , w e i g h t , e t c . , The o b j e c t i v e of r e c o g n i z i n g w h a t i s a mathematical p r o p e r t y i n a n e x p e r mental s i t u a t i o n as c o n t r a s t i n g w i t h a p h y s i c a l o r a c h e m i c a p r o p e r t y , i s a m a j o r one. O t h e r s i t u a t i o n s a r e b r o u g h t i n t o t h e e xpe r i men t . Mathematical p r o p e r t i e s l i k e E u l e r ' s r e l a t i o n a r e f e l by t h e students i n dealing with polyhedra. 2 . Functions : An u n d e r s t a n d i n g of t h e q u a n t i f i c a t i o n o f the re l a t i o n s h i p c a u s e - e f f e c t i s t h e main o b j e c t i v e . S e v e r a l e x p e r i mental s i t u a t i o n s , mainly d e a l i n g w i t h p l a n t - g r o w i n g , a r e sugg e s t e d . The s t e p of t r a n s l a t i n g i n t o m a t h e m a t i c a l l a n g u a g e n a t u r a l phenomena, a n d t h e r e c o g n i t i o n o f t h e e x i s t e n c e and c h o i c e of p a r a m e t e r s f o r t h i s m a t h e m a t i z a t i o n i s a m o s t important o b j e c t i v e of t h i s u n i t . 3 . E q u a t i o n s a n d I n e q u a t i o n s : The a i m i s t o b r i n g i n t o p l a y l i n e a-r i t y t h r o u g h s y s t e m s o f e q u a t i o n s and b u i l d i n g u p f o r a v e r y elementary i n t r o d u c t i o n t o t h e techniques of l i n e a r programing. 4 . I n i t i a t i o n t o M a t h e m a t i c s : The o b j e c t i v e i s t o l e t c h i l d r e n f e e l t h e growth o f m a t h e m a t i c a l i d e a s . N u m b e r s , o r d e r r e l a t i o n s , t o p o l o g i c a l n o t i o n s , r e s u l t from e x p e r i m e n t a l s i t u a t i o n s . T h i s u n i t i s i n d e v e l o p m e n t , a n d s h a l l be f i n i s h e d by t h e e n d of 1 9 7 8 . 5 . P r i m e r b o o k f o r t h e Amazon r e g i o n : ( f o r m a t h e m a t i c a l l i t e r a c y ) : T h i s p r o j e c t i s i n t h e i n i t i a l p l a n n i n g s t e p , and i t i s aimed a t b r i n g i n g e c o l o g i c a l and e n v i r o n m e n t a l r e l e v a n c e t o t h e f i rs t s t e p s of m a t h e m a t i c a l s t u d i e s , S t r o n g e n v i r o n m e n t a l b a c k g r o u n d a s s o c i a t e d w i t h l i n g u i s t i c s c o n c e r n s , s h a l l g e n e r a t e more m a t h e -

-

ADEQUATE MATHEMATICS FOR THIRD WORLD COUNTRIES

41

matical c r e a t i v i t y , which w i l l probably influence future applications o f mathematics.

The t o n e i s t o g e n e r a t e more s e n s e o f r e l e v a n c e f o r

mathematical concepts. The u n i t s t h e m s e l v e s h a v e l e s s v a l u e b e c a u s e o f t h e c o n t e n t s , much m o r e as c e n t r a l

and r e a l i t y i n t h e e l e m e n t a r y l e v e l .

T h e f i v e u n i t s w h i c h we

d e s c r i b e d above summarize w h a t w o u l d be t h e b a s i c c o n t e n t s mathematical syllabus for primary school, world, i n

,

I S p o t o r n o 19761 1976 1

t h e UMAP m o d u l e s ,

1 D' A m b r o s i o

and

A special place i s reserved,

and r e a l i t y ,

of a

o r i e n t e d towards t h e r e a l

associated t o a modelling concept,

1 D' Ambrosi o

but

themes w h i c h c l o s e t h e gap b e t w e e n m a t h e m a t i c s

as f o r e x a m p l e d e s c r i b e d [ L i g h t h i l l 19771

19771

and

.

i n b r i d g i n g t h e gap b e t w e e n m a t h e m a t i c s

t o f u l l use o f h a n d - h e l d c a l c u l a t o r s .

Undoubtedly, f o r

the f i r s t time s i n c e r e c e n t developments o f science,

mathematical

e d u c a t i o n can be changed from a p u r e l y d e d u c t i v e and d e s c r i p t i v e p a t t e r n t o q u a n t i f i e d reasoninq, method.

i n t h e f u l l essence o f s c i e n t i f i c

I n d e e d , m a t h e m a t i c a l model l i n g c o u l d n o t b e

previously

a c h i e v e d a t an e l e m e n t a r y l e v e l e x c e p t i n a t o k e n way, l a c k o f c o m p u t a t i o n a l power. to heuristics calculators,

I P o l y a 19621

,

because o f

We s e e t h e r e v i v a l o f G . P o l y a ' s approach a s s o c i a t e d w i t h f u l l use o f h a n d - h e l d

as a m o s t p r o m i s i n g t r e n d i n b r i d g i n g t h e g a p b e t w e e n

mathematical e d u c a t i o n and r e a l w o r l d

I D'Ambrosio

.

1977al

I n the u n i v e r s i t y l e v e l , t h e approach t o c a l c u l u s combined n u m e r i c a l m e t h o d s , w i t h some o v e r t u r e f o r c o m p u t i n g ,

with

gives i n t r o

-

ductory courses the p o s s i b i l i t y of b r i n g i n g i t s f u l l p o t e n t i a l t o practical applications. and c o n t i n u i n g ,

G i v i n g up d e t a i l e d d e f i n i t i o n s

of

l i m i t

and a d o p t i n g t h e approach o f v i e w i n g d i f f e r e n t a t i o n

as a l i n e a r i z a t i o n t e c h n i q u e , we a r e a b l e t o b r i n g a f a s t e r deeper understanding o f the concept o f f i n e r n a t u r a l phenomena t h r o u g h Taylor expansions.

,

f o r example,

an

approximations

of

elementary approach t o

We a d o p t e d t h i s a p p r o a c h i n

Calculus which brings,

and

a version

i n a somewhat s h o r t e n e d v e r s i o n ,

to

the concept

o f a p p r o x i m a t i o n t o a more i m m e d i a t e use I D ' A m b r o s i o 1 9 7 5 a l

.

An e f f o r t t o make m a t h e m a t i c a l t h e o r i e s a n d c o n c e p t s o f a s o m e w h a t advanced l e v e l a v a i l a b l e

f o r f a s t e r a p p l i c a t i o n seems t o b e a c h a l -

lenge which mathematicians from d e v e l o p i n g c o u n t r i e s have t o face.

A c o l l e c t i o n o f m a t h e m a t i c a l methods w h i c h a r e b o t h advanced

and

s e l f - c o n t a i n e d s h o u l d b e make a v a i l a b l e t o y o u n g m a t h e m a t i c i a n s i n developing countries,

s o t h a t t h e y m i g h t b e made a w a r e o f t h e o r i e s

w h i c h s e r v e as r e s o u r c e f o r f u t u r e u s e I D ' A m b r o s i o 1 9 7 7 b l

.

42

U. D'AMBROSIO

o f t h e a c t i v i t i e s b a s e d on p r e v i o u s l y d i s t r i b u t e d q u e s t i o n a i r e s . This discussion generates r e v i s i o n , n i n g f o r t h e n e x t 3-month p e r i o d .

o r confirmation,

of the plan-

E q u a l l y , e v e r y 2 months t h e r e i s

a formal evaluation o f students through i n d i v i d u a l presentation o f progress

,

p r o j e c t s and i n d i v i d u a l l y r e q i s t e r e d o b s e r v a t i o n b y t h e

instructors.

The r e s u l t s o f t h e a c t i v i t i e s a r e amply d i s c u s s e d w i t h

each s t u d e n t , w i t h a p p r o p r i a t e measures t o o b t a i n b e t t e r a c h i e v e m e n t as an e s s e n t i a l r e s u l t o f t h e e v a l u a t i o n .

The f i n a l e v a l u a t i o n i s

an o v e r a l l a n a l y s i s o f t h e p a r t i a l e v a l u a t i o n s . The c o u r s e p r e s e n t s many n o v e l f e a t u r e s a n d i n c o r o o r a t e many p r e v i o u s e x p e r i e n c e s on h i g h e r e d u c a t i o n .

We m e n t i o n s p e c i a l l y t h e

pos t - g r a d u a t e t r a i n i n a p r o g r a m k n o w n as P r o j e c t "CPS-Bamako" , f o r the t r a i n i n g o f s p e c i a l i s t s i n

r e s e a r c h and t e a c h i n g o f s c i e n c e i n

u n i v e r s i t i e s a t the doctoral level,

This p r o j e c t sponsored by

U N E S C O i n t h e R e p u b l i c o f M a l i , b e g a n i n 1 9 7 1 a n d i s b a s i c a l l y an

in-service,

student oriented,

program.

By i t s c o s t e f f e c t i v e n e s s

and o v e r a l l r e s u l t s i n i m p r o v i n g h i g h e r e d u c a t i o n and s c i e n t i f i c research i n the country,

t h e r e s u l t s may b e c o n s i d e r e d h i g h l y

s ati s f a c t o r y .

W i t h emphasis i n s c i e n c e e d u c a t i o n ,

t h e p r o g r a m d e s c r i b e d above has

b e e n i n o p e r a t i o n s i n c e 1 9 7 5 , a n d t h e r e s u l t s a c h i e v e d u p t o now are highly encouraging. F i n a l i z i n g , w h i l e a competent and w e l l t r a i n e d team o f m a t h e m a t i c i a n s i s an i n d i s p e n s a b l e c o m p o n e n t t o b u i l d u p m a t h e m a t i c a l r e s e a r c h , t h e s e a r c h o f a d e q u a t e s t r a t e g i e s t o b u i l d up t h e

desirable

a n d n e e d e d l e v e l o f c o m p e t e n c e i n a f a s t e r a n d m o r e p r o d u c t i v e way i s a c h a l l e n g e w h i c h we, i n d e v e l o p i n g c o u n t r i e s ,

have t o f a c e w i t h

urgency. References : I D ' A m b r o s i o 1971 I

-

D'Ambrosio,

and F o r e i g n Aid, Studies,

S.U.N.Y.,

I D ' A m b r o s i o 19741

-

U b i r a t a n : U n i v e r s i t y , Development

C o n f e r e n c e on t h e S e m i n a r on L a t i n A m e r i c a n a t B u f f a l o , S p r i n o 1971 (mirneo)l2 paaes.

D ' A m b r o s i o , U b i r a t a n : S o b r e a I n t e g r a C Z o do

e n s i n o de c i e n c i a s e m a t e m s t i c a IOn t h e i n t e g r a t i o n o f t h e t e a c h i n q o f s c i e n c e and mathematics] CiGncia e C u l t u r a , 2 6 ( 1 1 ) , n o v e m b e r 1974, p.1003-1010,

I D'Ambrosio

1975

I -

D ' Ambrosio,

U b i r a t a n : P r o j e t o s I n t e q r a d o s de

m a t e m a t i c a e c i E n c i a s e uma o p q a o p a r a a formaCIao de m e s t r e s p a r a o e n s i n o de c i s n c i a s I I n t e g r a t e d p r o j e c t s on m a t h e m a t i c s

43

ADEQUATE MATHEMATICS FOR THIRD WORLD COUNTRIES

The a p p r o a c h t o t h e s e t h r e e i s i n t e g r a t e d . b ) S e n s i b i l i z i n g s u b j e c t s : Generate problems and m o t i v a t e t h e study o f specialized topics i n sciences,

mathematics

,

education

and t e a c h i n g , w h i c h a r e n o r m a l l y s t u d i e d i n t h e t r a d i ti o n a l d i s c i p l i n e s o f u n i v e r s i t y c u r r i c u l a . The m e t h o d o f t e a c h i n g t h e s e n s i b i l i z i n g s u b j e c t s i s i n f o r m a l , m a i n l y r e l y i n g on s e m i n a r s , workshops, panel d i s c u s s i o n s and l e c t u r e s b y s p e c i a l l y i n s p i r i n g and i n q u i s i t i v e s p e a k e r s .

T h e b a c k g r o u n d o f t h e s t u d e n t s i s an

important source o f motivation f o r the s e n s i b i l i z i n g subjects. The f o l l o w i n g d i s c i p l i n e s a r e i n c l u d e d i n t h i s c a t e g o r y : ( i )D i s c u s s i o n o f m o d e l l i n g t e c h n i q u e s

( i i ) Science and technology p r o j e c t s ( i i i ) N a t i o n a l goals and p r i o r i t i e s ( i v ) Cultural expansion. c ) S u p p o r t i n g s u b j e c t s : a r e u n i t s o f m a t h e m a t i c s a n d o t h e r sciences g e n e r a t e d b y t h e s e n s i b i l i z i n p d i s c i p l i n e s . The u n i t s a r e d e s i g n e d a c c o r d i n g t o t h e b a c k g r o u n d a n d i n t e r e s t o f t h e students h e n c e a v a i l a b l e t o i n d i v i d u a l s or g r o u p s .

D u r a t i o n and c o n t e n t

are p e r s o n a l i z e d t o s u i t these i n d i v i d u a l s o r groups and

the

format o f t h e courses range from t r a d i ti onal exoosi t o r y l e c t u r e s t o personalized instruction,

i n c l u d i n g workshops,

seminars,

r e a d i n g courses and programed i n s t r u c t i o n w i t h use of computer a i d e d i n s t r u c t i o n and a u d i o - v i s u a l

methods.

Methodology o f

access

t o i n f o r m a t i o n , s u c h as i n f o r m a t i o n r e t r i e v a l a n d a u d i o - v i s u a l collections,

a r e emphasized.

The s c i e n c e a n d t e c h n o l o g y p r o j e c t s i n t h e f i r s t phase,

( ( i i ) o f b ) w h i c h may b e b e g u n

g r a d u a l l y i n c r e a s e i n t h e time and a t t e n t i o n

d e v o t e d t o t h e m as t h e p r o g r a m e v o l v e s .

I t i s e x p e c t e d t h a t an

o v e r a l l 300 h o u r s w i l l b e d e v o t e d t o t h e p r o j e c t s ( o r p r o j e c t ) b y each i n d i v i d u a l s t u d e n t , project,

i n c l u s i v e o f research methodology.

This

w h i c h w i l l g e n e r a t e a m o n o g r a p h , may b e t h e i n i t i a l s t e p

towards a more e l a b o r a t e r e s e a r c h p r o j e c t l e a d i n g t o a M a s t e r ' s thesis

.

A d y n a m i c a l a n d c o n s t r u c t i v e e v a l u a t i o n i s an i n t e q r a l p a r t o f t h e course.

Away f r o m t r a d i t i o n a l e v a l u a t i v e t e c h n i q u e s ,

evaluation i n t o the planning o f the course.

we i n c o r p o r a t e

There i s a t o t a l of

f i v e course and s t u d e n t e v a l u a t i o n e x e r c i s e s , w i t h t h e f o l l o w i n g s t r u c t u r e and method. P e r i o d i c a l l y , i d e a l l y e v e r y t h r e e months, f u l l day i s d e v o t e d t o an open d i s c u s s i o n ,

a

following or followed

up b y s e m i n a r s a n d p a n e l d i s c u s s i o n s on t h e s t r u c t u r e a n d p l a n n i n g

44

U. D'AMBROSIO

and t e c h n o l o g i c a l e s t a b l i s h m e n t i n t h e i r c o u n t r i e s o r s t a t e s o f o r i g i n , a n d w i l l b e a b l e t o s e r v e as s t i m u l a t i n g e l e m e n t s f o r t h e e s t a b l i s h m e n t o f r e g i o n a l l e a d e r s h i p . The c a n d i d a t e s m u s t h a v e a b a c h e l o r ' s degree o r e q u i v a l e n t i n m a t h e m a t i c s and, i d e a l l y , funtions which w i l l f a c i l i t a t e leadership: f a c u l t y member i n a h i g h l e v e l i n s t i t u t i o n , i n s t i t u t i o n , etc..

f o r example,

have

t o be

a

t o work i n a research

A11 s t u d e n t s w o r k w i t h f u l l t i m e d e d i c a t i o n t o

the course. C o n s i d e r i n g the f a c t t h a t the s t u d e n t s have degree i n mathematics, possibly with specialization,

t h e c o u r s e i s s t r u c t u r e d i n s u c h a way

as t o s t i m u l a t e i n t e g r a t i o n a n d d e v e l o p m e n t o f i n q u i s i t i v e a n d c r e a t i v e a t t i t u d e s , w i t h i n d i v i d u a l i z e d c u r r i c u l a , i n s u c h a way

that

a l l o w s f o r t h e m e e t i n g o f i n d i v i d u a l c h a r a c t e r i s t i c s and i n t e r e s t s o f each s t u d e n t , o f f e r i n g them t h e p o s s i b i l i t y o f a t t a c k i n g s p e c i f i c problems generated i n t h e i r previous experiences.

The s t u d e n t s w o r k

on a l t e r n a t i v e s o l u t i o n s f o r p r o b l e m s D o s e d d u r i n g t h e c o u r s e , i n s u c h a way t h a t w i l l e n a b l e t h e m t o o p t i n f u n c t i o n o f s i q n i f i c a n t v a r i a b l e s i n the contents toward i d e n t i f y i n g major oroblems a f f e c t i n g t h e i r r e g i o n s a n d a n a l y s i n g t o w h a t e x t e n t m a t h e m a t i c s may c o n t r i b u t e t o s o l v e them. control

,

Examples o f such p r o b l e m s a r e f o o d production,epidemic

environmental preservation,

d e v e l o p m e n t o f n a t u r a l resources,

etc., T o a t t a i n t h e s e ob j e c t i v e s t h e c o u r s e h a s a l e s s c o n v e n t i o n a l structure.

I t c o n s i s t s o f t h r e e i n t e r o e n e t r a t i n q phases,

and d u r i n g

each o f t h e t h r e e phases t h e r e a r e i n t e a r a t e d a c t i v i t i e s , w i t h a p p r o p r i a t e c o n t e n t and m e t h o d o l o g i c a l approach. a l l a c t i v i t i e s a d d up t o 1 , 5 0 0 h o u r s o f w o r k ,

I t i s planned t h a t

d u r i n g 10 m o n t h s .

One o f t h e m a i n f e a t u r e s o f t h e c o u r s e i s t h e c r e a t i o n o f a m u l t i c u l t u r a l and i n t e r d i s c i p l i n a r y l e a r n i n g environment, w i t h emphasis

on c o o p e r a t i v e e f f o r t a n d j o i n t i n t e l l e c t u a l v e n t u r e s t h r o u g h a dynamical and u n s t r u c t u r e d programminp. The a c t i v i t i e s o f t h e c o u r s e g e n e r a t e r e s e a r c h p r o j e c t s w h i c h may culminate i n a Master's thesis, P a r a l l e l l y , there are courses, s t r u c t u r e d i n a more c l a s s i c a l s e n s e , subjects;

b) s e n s i b i l i z i n g subjects;

a) Instrumental subjects:

o f three types:a)

instrumental

c) supporting subjects.

s e r v e as a means o r l a n o u a q e f o r t h e s u b -

sequent s t u d i e s , and i n c l u d e t h e f o l l o w i n p : ( i ) Computer s c i e n c e ( i i ) Scientific English ( i i i ) Mathematical methods.

45

ADEQUATE MATHEMATICS FOR THIRD WORLD COUNTRIES

Summing u p ,

i t seems t h a t w o r k a b l e k n o w l e d a e o f a f e w c h a p t e r s

of

more i m m e d i a t e a p p l i c a b i l i t y s h o u l d t a k e p r i o r i t y o v e r o t h e r t h e o r i e s o f l e s s immediate i n t e r e s t , which n e v e r t h e l e s s s h o u l d be given a p u r e l y i n f o r m a t i v e , somewhat e n c y l o p a e d i c - l i k e

form o f presentation.

R i g o r and exhaustiveness m i a h t g i v e p l a c e , i n t h i s f o r m o f presentation,

t o m e t h o d o l o g i c a l and i n f o r m a t i v e

motivation, derived from r e a l i t y , o f s t o r e d knowledge.

aspects, which a l l i e d t o

t r i a a e r s the process o f r e t r i e v a l

The c h a l l e n g e r e s i d e s o n s p e c i a l e f f o r t o f

mathematicians from developing countries t o aenerate accessible t e a c h i n g m a t e r i a 1 o f a d v a n c e d t o p i c s on m a t h e m a t i c s . Regarding t r a i n i n g o f researchers a t the graduate l e v e l ,

a pilot

p r o j e c t g o i n g on a t t h e S t a t e U n i v e r s i t y o f C a m p i n a s s i n c e 1 9 7 5 , under j o i n t sponsorship o f the B r a z i l i a n P i n i s t r y o f Education the O r g a n i z a t i o n o f American S t a t e s (see

I D’Ambrosio

enphasis f r o m academic t r a i n i n g a t t h e p o s t - a r a d u a t e

and

19751 )

,

level

t o an

changes

a c t i o n program. The p r o g r a m ,

i n n o v a t i v e b o t h i n i t s o b j e c t i v e s and m e t h o d o l o g y ,

as i t s p r i n c i p a l a i m t h e t r a i n i n g o f l e a d e r s i n m a t h e m a t i c a l

has

-

re

s e a r c h a n d e d u c a t i o n , w h o w i 1 1 h o p e f u l l y i n f l u e n c e t h e i r own e d u c a t i o n a l and s c i e n t i f i c systems i n a c r e a t i v e d i r e c t i o n , p o i n t o f view o f c o n t e n t s and methodology,

both from the

as w e l l as t r y i n g t o r e -

l a t e t e a c h i n q and r e s e a r c h t o n a t i o n a l d e v e l o p m e n t a l p r i o r i t i e s and goals. The p r o g r a m r e c e i v e s e a c h y e a r a m i x e d s t u d e n t c o m p o n e n t , w i t h

20

p a r t i c i p a n t s f r o m B r a z i l a n d 12 f r o m o t h e r L a t i n A m e r i c a n c o u n t r i e s . I t i s p a r t o f a 2-3 y e a r proqram l e a d i n g t o a M a s t e r ’ s degree, w i t h

t h e 1 0 m o n t h t r a i n i n g p e r i o d d e s c r i b e d b e l o w as t h e b a s i s f o r a f o l low-up which i s b a s i c a l l y r e s e a r c h and t h e w r i t i n g o f a M a s t e r ‘ s thesis

,

t o b e d e v e l o p e d i n e a c h s t u d e n t ’ s home e n v i r o n m e n t .

The p r o g r a m h o p e s t o d i r e c t human r e s o u r c e s a l r e a d y a c t i v e i n some c a p a c i t y i n L a t i n America c o u n t r i e s ,

i n t o an i n n o v a t i v e m a s s i v e

e f f o r t , demis ti f i e d and c r e a t i v e enouqh t o have s c h o o l s and r e s e a r c h i n s t i t u t e s d i r e c t l y envolved i n crossing the b a r r i e r o f under development.

-

T h e p a r t i c i p a n t s w i l l b e i n v o l v e d i n d e v e l o p i n g l e a d e-r

s h i p i n t h e analysis, adaptation and e l a b o r a t i o n o f c u r r i c u l a , i n p r o m o t i n g courses and campaigns t o improve t h e g e n e r a l l e v e l o f m a t h e m a t i c s and t o p r o d u c e m a t e r i a l s and l a b o r a t o r i e s f o r t h e t e a c h i n g o f s c i e n c e and mathematics. I t i s h o p e d t h a t p r o f e s s i o n a l s T i n i s h i n a t h e c o u r s e wi.11 b e e n q a g e d

i n s p e c i f i c research and i n t h e q l o b a l a n a l y s i s o f the s c i e n t i f i c

46

U. D'AMBROSIO

and s c i e n c e a n d an o p t i o n f o r t h e p r e p a r a t i o n o f M a s t e r s i n t h e T e a c h i n g o f S c i e n c e l , Enselianza i n t e q r a d a de l a s c i e n c i a s e_ n _ America i n a- -- 2 , U N E S C O , O f i c i n a R e g i o n a l de C i e n c i a y _ _ L~a t T e c n o l o g i a p a r a America L a t i n a y e l C a r i b b e , M o n t e v i d e o , 1 9 7 5 , p.lOC-112. ID'Ambrosio 1 9 7 5 a l - D ' A m b r o s i o , U b i r a t a n : C a ' l c u l o onm i n t r o d u c l a o 'i A n S l i s e I C a l c u l u s w i t h I n t r o d u c t i o n t o P n a l y s i s I Companhia Edi t o r a N a c i o n a l , S a o P a u l o , 1 9 7 5 . ID'Ambrosio 19761 - D ' A m b r o s i o , U b i r a t a n : O p t i o n s p o u r I ' e n s e i q n e ment e t l a r e c h e r c h e m a t h g m a t i q u e en vue d u d z v e l o p p e m e n t , l e r C o n g r e s P a n - A f r i c a i n des M a t h g m a t i c i e n s , Union Y a t h e matique A f r i c a i n e , Rabat, 1976. ID'Ambrosio 1 9 7 6 a l - D ' A m b r o s i o , U b i r a t a n : O v e r a l l 9 o a l S and o b j e c t i v e s f o r mathematical e d u c a t i o n , ChaDter IX i n Trends i n Mathematical Education I V Y U Y E S C O , P a r i s ( t o --a p p e a r ) : A v a i l a b l e i n an e x t e n d e d v e r s i o n f r o m : I n s t i t u t o de M a t e m z t i c a , E s t a t i s t i c a e C i s n c i a de Computacao,UNICAMP Campi n a s , B r a z i 1 . ID'Ambrosio 19771 - D ' A m b r o s i o , U b i r a t a n : E n s i n o de C i e n c i a s e D e s e n v o l v i m e n t o lThe T e a c h i n g o f S c i e n c e a n d Development1 . C i G n c i a e C u l t u r a , 29 ( 2 ) , f e v e r e i r o 1 9 7 7 , p . 1 4 3 - 1 5 0 . ID'Ambrosio 1 9 7 7 a l - D ' A m b r o s i o , U b i r a t a n : I s s u e s a r i s i n g i n t h e use o f hand-held c a l c u l a t o r s . Panel o r e s e n t a t i o n t o t h e 3rd I n t e r n a t i o n a l C o n g r e s s on M a t h e m a t i c a l E d u c a t i o n , K a r l s r u h e 1976 ( m i m e o ) , l o p . ID'Ambrosio 1 9 7 7 b / D ' A m b r o s i o , U b i r a t a n : E & d o s de T o p o l o g i a : I nt r o d G o e A p l i S Ge s IMethods o f T o p o l o q y : I n t r o d u c t i o n a n d Applications1 , l i v r o s Tecnicos e C i e n t i f i c o s Editora S . A . , Rio de J a n e i r o , 1977. [ L i g h t h i l l 19771 - L i q h t h i l l , James e t a l : Newer Uses o f P a t h e --m a t i c s , P e n g u i n B o o k s , L o n d o n , 1977. I P o l y a 1962 I - P o l y a , G e o r g e : Y a t h e m a t i c a l D i s c o v e r y , 2 v o l u m e s , John Wiley & S o n s L t d . Yew York,1962(vo1.1),1964(vol.Z), I S p o t o r n o 19761 - S p o t o r n o , Bruno e V i l l a n i V i n i c i o : M o n d o R e a l e e M o d e l l i M a t e m a t i c i I R e a l World and M a t h e m a t i c a l Y o d e l s ] --_.----_ La Nuova I t a l i a , F i r e n z e , 1 9 7 6 . I Thom 19751 Thom RenE: Les m a t h g m a t i q u e s e t l ' i n t e l l i g i b l e IMathematics a n d t h e i n t e l i q i b l e l , D i a l e c t i c a , v o l . ? O , n . l , 1975, p.71-80.

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Developing Mathematics in Third World Countries, M.E.A. E l Tom ( e d . ) 0 North-Holland publishing Company, 1979

THE DEVELOPMENT OF SCHOOL MATHEMATICS Some General Principles by Eryan Thwaites?'

Introduction This very brief communication is intended to establish certain fundamental principles for the Conference's discussion of mathematics at the school level. The brevity is only partly in deference to the wishes of the organisers; it stems also from my belief that there are very few principles in education which are universally o r unquestionably valid - and those that there are, are simple and do not require any great elaboration. This being the first, or introductory, paper to be given to the Working Group on school mathematics, I shall try also to set out each guiding principle under its own heading, so that discussion can proceed in an orderly tshion.

What is Mathematics? It is necessary (I suggest) to attempt an answer to this question at the outset, because without doubt there will be many conflicting opinions expressed during the next four days as to what particular topic does, or does not, come within mathematics. I will offer a definition by way of a diagram designed to reflect the fact that, at least in the foreseeable future, the Third-World Countries are bound to view their educational systems against the background of their developmental problems and aspirations. I thus start with Real Life, that is to say, the totality of life of a country in which new problems of all kinds are continually arising.

itprincipalof Westfield College, University of London; formerly Professor of Mathematics, Southampton; co-founder, formerly Director and now Chairman of The School Mathematics Project.

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THE DEVELOPMENT OF SCHOOL MATHEMATICS

49

The principal features of this diagram are: "Pure" mathematics means intrinsic mathematical knowledge which exists more or less independently of anything else; Applied mathematics is not put in a box: for it is an attitude of mind (with associated skills) which enables mathematical knowledge to advance and explore and regularise mathematical models. Similarly, Science is not knowledge as such but another attitude and skill which synthesise mathematical models and actual experiments. Technology, again, is more an attitude and skill rather than a definable corpus of knowledge, and it is technology which, with money and managerial skill and labour, ultimately produces the action needed as a result of the original practical problem. By Mathematics, then, is meant all that within the broken line - that is to say, (i) all intrinsic mathematical knowledge, (ii) a knowledge or capability of at least some types of model building, and (iii) the capacity or art of thinking in the manner of applied mathematics. What is School Mathematics? From any fundamental point of view, there is no difference between Mathematics as just defined and School Mathematics. If there is a difference, it arises from the obvious distinctions of age and ability of school pupils, and character of teachers. National Differences There is however a highly important difference between School Mathematics as interpreted in highly developed countries and in the Third-World, at least at the lower, or primary, stages. In the former countries, virtually all children attend primary school and then proceed to secondary school; in the latter, this is very far from the case. This contrast itself means that the detailed interpretation of School Mathematics may vary between the Developed and the ThirdWorld. Indeed, it needs saying at this early stage of the Conference that, after some twenty years or so of attempts to help developing countries by exporting the ideas and methods of the developed countries, it is now becoming realised that such attempts need substantial rethinking and that the differences between countries are often greater than can be bridged by common ideas and methods.

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B. THWAITES

Differences in Time Another dominating influence on our subject of school mathematics is the fact that attitudes and needs are constantly changing: what is good and appropriate at one time may not merely appear to become, but actually be, no longer good or appropriate in 10, 25 o r 50 years time. Even in our own personal lifetimes, many of us have listened to, o r participated in, discussions on specific mathematical topics or methods which have moved backwards, forwards and sideways, and which will continue to do so. In short, the development of school mathematics does not constitute an academic discipline as we normally know it: its objective content is relatively slight: it depends far more on local conditions, on current sociological trends, and on attitudes of parents as well as pupils and teachers. If only for this reason, is my list of fundamental principles so slight. Syllabus, Content, MethodoloRy, Curriculum. as well as other words.

I make a plea for as accurate a use as possible of certain very common terms: misunderstandings often arise from ambiguous meanings being attached to such familiar terms as these. I make no attempt at definitive explanations which would satisfy teachers as well as educationalists, administrators and psychologists; but perhaps the following might be a simple guide: a syllabus is a brief list of topics (whether for an examination or otherwise); by content is meant a longer and descriptive account of a syllabus; methodology is the description of the interaction between the teacher and his pupils; the curriculum is the totality of the teaching and learning activity. Mathematics: Traditional, New and Modern The terms just roughly defined should enable us to compare and analyse, if we wished, the many curricula which are in existence in the world at present. If there were such analysis, it would show (I suggest) a more-or-less continuous spectrum of practice in which it would be found pointless to distinguish between what is traditional, o r new o r modern. Apart from the non-trivial observation that what is new or modern today will be old and out-of-date tomorrow, it remains that the discontinuity in content which occurred in so many countries in the 1960's and early 1970's was historically unusual and its effect already dissipated; there is a growing awareness that the development of school mathematics B a continuous process and that at any one time responsible teachers and

THE DEVELOPMENT OF SCHOOL MATHEMATICS

Administrators may reasonably be expected to do what they feel is best for both the short and the long term. I suggest, therefore, that we eschew, now and for ever, the words of the last heading. Time-Scales A further reason for allowing the exceptional 1960's to slip to the backs of our minds is that the characteristic time-scales of education are very long - of the order of a generation, or 25 years: an educational decision taken now (for example, to write a new series of secondary text books) is likely to be having an effect a quarter of a century later (i.e. at least some of these books will still be in use). Change should be made deliberately and on the basis of as great a consensus as possible. False Dichotomies May I suggest, next, that in this Conference we avoid what may be called false dichotomies? Skill and understanding; education and training; elitism and egalitarianism; abstract and concrete; structure and problem-solving - these and many others are not examples of opposites but only reflect the necessity for balance in a curriculum. Just to exemplify from the first of the pairs just mentioned - skill and understanding: the one is meaningless without the other: neither is easily definable or measureable; yet we are surely all convinced, without further argument, that everyone, from the start of the primary stage onwards, should possess skill and understanding in mathematics. Compulsory Education Finally, let us contemplate for a moment the implication of compulsory education (which, even though it is by no means fully implemented in many countries, is nevertheless their goal in the longer term). Whether such a compulsory stage lasts for 5 or 15 years, it represents not only a severe restriction on the natural liberty of the young citizen but an increasingly heavy financial burden on the State.

For both these reasons, it seems to me essential that education

should be capable of justification through an assessment of the output of the educational process in terms of the capabilities of the individual pupil. We cannot avoid, in this Conference, a careful discussion of how the ultimate output of Education can be quantified.

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PAST, PRESENT AND FUTURE EDUCATIONAL TECHNOLOGIES P a t r i c k SUPPES I n s t i t u t e f o r Mathematical S t u d i e s i n t h e S o c i a l Sciences S t a n f o r d U n i v e r s i t y . S t a n f o r d , C a l i f o r n i a , USA

What I w o u l d l i k e t o d o t o d a y i s t o t r y t o p u t t h e c u r r e n t e f f o r t s i n e d u c a t i o n a l technology i n h i s t o r i c a l p e r s p e c t i v e . I n f a c t , I want t o t a k e a v e r y l o n g h i s t o r i c a l v i e w i n m a k i n g t h e s e r e m a r k s a n d t h e n show how t h e y are r e l a t e d t o t h e p r e s e n t .

_ P a_ st

Educational Technologies

I c a n i d e n t i f y f o u r major t e c h n o l o g i c a l i n n o v a t i o n s of t h e p a s t t h a t a r e comparable t o t h e c u r r e n t computer r e v o l u t i o n .

W r i t t e n Records ~The f i r s t i s t h e i n t r o d u c t i o n o f w r i t t e n r e c o r d s f o r t e a c h i n g p u r p o s e s i n a n c i e n t t i m e s . We do n o t know e x a c t l y when t h e u s e of w r i t t e n r e c o r d s f o r i n s t r u c t i o n a l p u r p o s e s b e g a n b u t w e d o h a v e , a s e a r l y as P l a t o ’ s D i a l o g u e s , w r i t t e n i n t h e f i f t h c e n t u r y B . C . , s o p h i s t i c a t e d o b j e c t i o n s t o t h e u s e of written records. Today n o o n e would d o u b t t h e v a l u e o f w r i t t e n m a t e r i a l i n e d u c a t i o n , b u t t h e r e were v e r y s t r o n g a n d c o g e n t o b j e c t i o n s t o t h i s v e r y e a r l i e s t i n n o v a t i o n a w r i t t e n r e c o r d is v e r y i m p e r s o n a l ; i n education. The o b j e c t i o n s were t h e s e : i t is very uniform; i t does not adapt t o t h e i n d i v i d u a l s t u d e n t ; i t does not I n o t h e r words, S o c r a t e s and t h e a n c i e n t e s t a b l i s h rapport with t h e student. S o p h i s t s , t h e t u t o r s of s t u d e n t s i n a n c i e n t A t h e n s , o b j e c t e d t o i n t r o d u c i n g w r i t t e n r e c o r d s and d e s t r o y i n g t h e k i n d of p e r s o n a l r e l a t i o n between s t u d e n t a n d t u t o r t h a t was a p a r t o f t h e i r main r e a s o n f o r b e i n g .

I t h a s become a f a m i l i a r s t o r y i n o u r own t i m e t h a t a t e c h n o l o g i c a l i n n o v a t i o n has s i d e e f f e c t s t h a t a r e not always uniformly b e n e f i c i a l . It is important t o r e c o g n i z e t h a t t h i s i s n o t a new a s p e c t o f i n n o v a t i o n b u t h a s b e e n w i t h 11s from t h e beginning:

Books The s e c o n d i n n o v a t i o n of g r e a t h i s t o r i c a l i m p o r t a n c e i n e d u c a t i o n was t h e I n t h e western world we i d e n t i f y t h e move f r o m w r i t t e n r e c o r d s t o b o o k s . b e g i n n i n g d a t e of t h i s i n n o v a t i o n w i t h t h e p r i n t i n g o f t h e G u t e n b e r g B i b l e I t i s a l s o i m p o r t a n t t o r e c o g n i z e t h a t t h e r e was e x t e n s i v e u s e o f i n 1452. b l o c k p r i n t i n g i n Korea a n d C h i n a t h r e e or f o u r hundred y e a r s e a r l i e r . N e a r l y h a l f a m i l l e n n i u m l a t e r i t i s d i f f i c u l t t o h a v e a v i v i d s e n s e of how i m p o r t a n t I n t h e a n c i e n t w o r l d of t h e t h e i n n o v a t i o n of p r i n t i n g t u r n e d o u t t o b e . M e d i t e r r a n e a n t h e r e were o n l y a few m a j o r l i b r a r i e s , a n d by a few I mean a number t h a t you c o u l d c o u n t o n t h e f i n g e r s o f o n e h a n d . One of t h e f a m o u s a s p e c t s of A l e x a n d r i a , f o r e x a m p l e , was t h e w e a l t h a n d m a g n i t u d e of i t s l i b r a r i e s , a n d t h e A l e x a n d r i a of 100 B.C. h a d v e r y few c o m p e t i t o r s . The I t was i m p o s s i b l e t o h a v e l a r g e n u m b e r s of c o p i e s r e a s o n f o r t h i s is o b v i o u s :

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of m a n u s c r i p t s r e p r o d u c e d when a l l of t h i s ha d t o b e done t e d i o u s l y by ha nd. The i n t r o d u c t i o n of p r i n t i n g i n t h e 1 5 t h c e n t u r y p r o d u c e d a r a d i c a l innovation--indeed a r e v o l u t i o n - - i n t h e d i s t r i b u t i o n of i n t e l l e c t u a l a n d e d u c a t i o n a l materials. By t h e m i d d l e of t h e 1 6 t h c e n t u r y , 100 y e a r s l a t e r , t h r o u g h o u t E u r o pe n o t o n l y i n s t i t u t i o n s b u t w e a l t h y f a m i l i e s as w e l l had l i b r a r i e s of s e r i o u s p r o p o r t i o n s . t h e r e were d e f i n i t e t e c h n o l o g i c a l s i d e e f f e c t s t h a t were not uniformly b e n e f i c i a l . T hose of you who know t h e a r t a nd t h e b e a u t y of t h e m e d i e v a l m a n u s c r i p t s t h a t p r e c e d e d t h e i n t r o d u c t i o n of p r i n t i n g c a n a p p r e c i a t e t h a t i t was r e g a r d e d by some as a d e g r a d a t i o n of t h e s t a t e of r e p r o d u c t i o n t o move t o mass p r i n t i n g f r o m t h e b e a u t i f u l hand-produc e d m a n u s c r i p t s of t h e e a r l i e r period.

O n t h e o t h e r hand,

I t is a l s o i m p o r t a n t t o h a v e a s e n s e of how s l o w t h e i m p a c t of a t e c h n o l o g i c a l i n n o v a t i o n c a n somet i mes b e . I t w a s n o t u n t i l t h e e n d of t h e 1 8 t h c e n t u r y I n t h e case of t h a t books were u s e d e x t e n s i v e l y f o r t e a c h i n g i n s c h o o l s . a r i t h m e t i c , f o r e x a m p l e , most t e a c h e r s c o n t i n u e d t o u s e o r a l me thods t h r o u g h o u t t h e 1 9 t h c e n t u r y and i t was n o t u n t i l a l m o s t t h e b e g i n n i n g of t h i s c e n t u r y t h a t a p p r o p r i a t e e l e m e n t a r y t e x t b o o k s i n m a t h e m a t i c s were a v a i l a b l e . I t is c e r t a i n l y my hope t h a t i t w i l l n o t t a k e u s 500 y e a r s t o d i s t r i b u t e c o m p u t e r s i n t o s c h o o l s , a f i g u r e c o m p a r a b l e t o what i t t o o k t o d i s t r i b u t e b o o k s i n t o s c h o o l s f o r t h e t e a c h i n g of m a t h e m a t i c s . F o r t u n a t e l y t h e s c a l e of d i s s e m i n a t i o n i n t h e modern w o r l d is of a n e n t i r e l y d i f f e r e n t o r d e r f r o m w ha t i t was i n the past. P e r h a p s my f a v o r i t e exampl e of t h i s i s t h e estimate t h a t i t t o o k o v e r f i v e y e a r s f o r t h e news of J u l i u s C a e s a r ’ s a s s a s s i n a t i o n t o r e a c h t h e f u r t h e s t c o r n e r s of t h e Roman Empire. Today s u c h a n a s s a s s i n a t i o n would b e known t h r o u g h o u t t h e w o r l d i n a matter of m i n u t e s . With r e g a r d t o t h e p a c e a t whi ch books h a v e be e n i n t r o d u c e d i n t o e d u c a t i o n , i t would b e a m i s t a k e t o t h i n k t h a t t h e r e was s o m e t h i n g p e c u l i a r a b o u t t h e u s e of methods of r e c i t a t i o n i n t h e e l e m e n t a r y s c h o o l u n t i l l a t e i n t h e 1 9 t h c e n t u r y ; s t o r i e s of a comparable s o r t a l s o h o l d a t t h e u n i v e r s i t y l e v e l . According t o a t l e a s t o n e a c c o u n t , t h e l a s t p r o f e s s o r a t t h e U n i v e r s i t y of Ca mbridge i n E n g l a n d who i n s i s t e d on f o l l o w i n g t h e r e c i t a t i v e t r a d i t i o n t h a t d a t e s b a c k t o t h e M i d d l e Ages was C . D . B road. He i n s i s t e d on d i c t a t i n g a n d t h e n r e p e a t i n g e a c h s e n t e n c e s o t h a t s t u d e n t s would h a v e a d e q u a t e t i m e t o w r i t e e a c h s e n t e n c e e x a c t l y as d i c t a t e d . I c a n n o t i m a g i n e c o n t e m p o r a r y u n i v e r s i t y s t u d e n t s ‘ t o l e r a t i n g s u c h met hods.

Mass S c h o o l i n g The t h i r d i n n o v a t i o n , and a g a i n one t h a t w e now a c c e p t a s a c o m p l e t e a n d n a t u r a l p a r t o f o u r s o c i e t y , i s mass s c h o o l i n g . We h a v e a t e n d e n c y i n t a l k a b o u t o u r s o c i e t y t o p u t s c h o o l s a n d f a m i l i e s i n t o t h e same c a t e g o r y of m a j o r institutions. But i t i s e x t r e m e l y i m p o r t a n t t o r e c o g n i z e t h e g r e a t ps yc hol o g i c a l d i f f e r e n c e b e t w e e n t h e s t a t u s of t h e f a m i l y a n d t h e s t a t u s of s c h o o l s . F a m i l i e s are r e a l l y d e e p i n t o o u r b l o o d a n d o u r c u l t u r e . The e v i d e n c e of f a m i l i e s ’ b e i n g i n one f o r m o r a n o t h e r t h e most i m p o r t a n t c u l t u r a l u n i t g o e s b a c k t h o u s a n d s of y e a r s . S c h o o l s a r e n o t a t a l l c o m p a r a b l e ; t h e y a r e , we m i g h t s a y , v e r y much Johnny-come-l at el y to our culture. A hundred y e a r s ago i n 1870, f o r e x ampl e, o n l y two p e r c e n t of young p e o p l e g r a d u a t e d f r o m h i g h school i n t h e United S t a t e s . A hundred y e a r s b e f o r e t h a t o n l y a v e r y small p e r c e n t a g e e v e n f i n i s h e d t h i r d or f o u r t h g r a d e . I c a n n o t g i v e you a n e x a c t percentage because our recordkeeping, t h a t is, our s o c i a l statistics, are not much more t h a n a h u n d r e d y e a r s o l d and we h a v e no s e r i o u s i d e a of how many s t u d e n t s were a c t u a l l y i n s c h o o l two h u n d r e d y e a r s a g o , e x c e p t t h a t w e do know t h a t t h e number was q u i t e s m a l l .

PAST, PRESENT AND FUTURE ED~JCATIONAL TECHNOLOGIES

Even a s s h o r t a p e r i o d a s f i f t y y e a r s a g o , i n most of t h e w o r l d l e s s t h a n one p e r c e n t of t h e p o p u l a t i o n completed se c o n d a r y s c h o o l . During t h e r e c e n t upheavals connected w i t h t h e " c u l t u r a l r e v o l u t i o n " i n China t h e el ement ary s c h o o l s , n o t t o s p e a k of c o l l e g e s and s e c o n d a r y s c h o o l s , were c l o s e d f o r several years. I n o u r s o c i e t y a s we now t h i n k of i t , i t i s u n b e l i e v a b l e t o From a c o n t e m p l a t e c l o s i n g t h e e l e m e n t a r y s c h o o l s f o r s u c h a p e r i o d of t i m e . C h i n e s e h i s t o r i c a l p e r s p e c t i v e , h o w e v e r , i t was n o t s u c h an i m p o r t a n t m a t t e r , f o r C h i n e s e c u l t u r e e x t e n d s b a c k c o n t i n u o u s l y s e v e r a l t h o u s a n d y e a r s and t h e r e i s i n t h a t c u l t u r a l t r a d i t i o n no s a l i e n t p l a c e f o r s c h o o l i n g .

I n many d e v e l o p i n g c o u n t r i e s o f t h e w o r l d t o d a y t h e b e s t t h a t c a n b e hoped i s t h a t t h e m a j o r i t y of t h e young p e o p l e w i l l b e g i v e n f o u r g r a d e s of e l e m e n t a r y s c h o o l . U n t i l t h e p o p u l a t i o n g r o w t h i s b r o u g h t i n c h e c k , i t w i l l t a k e a l l of t h e r e s o u r c e s a v a i l a b l e t o a c h i e v e t h i s much. The p o s i t i o n of America a s a w o r l d l e a d e r i n e d u c a t i o n i s s o m e t i m e s n o t a d e q u a t e l y r e c o g n i z e d by my f e l l o w A m e r i c a n s , b e c a u s e we a c c e p t a s s o much a p a r t o f o u r c u l t u r e t h e c o n c e p t o f a l l young p e o p l e c o m p l e t i n g s e c o n d a r y s c h o o l and a h i g h p e r c e n t a g e g o i n R on t o college. I n f a c t , o u r l e a d e r s h i p i n c r e a t i n g a s o c i e t y w i t h mass e d u c a t i o n i s p e r h a p s one of t h e most i m p o r t a n t a s p e c t s of American i n f l u e n c e i n t h e w o r l d . A s r e c e n t l y a s t h e l a t t e r p a r t of t h e 1 9 t h c e n t u r y t h e d i s t i n g u i s h e d B r i t i s h p h i l o s o p h e r , J o h n S t u a r t M i l l , d e s p a i r e d of d e m o c r a c y ' s e v e r r e a l l y w o r k i n g anywhere i n t h e w o r l d f o r o n e r e a s o n - - i t was s i m p l y n o t p o s s i b l e t o e d u c a t e t h e m a j o r i t y of t h e p o p u l a t i o n . In h i s v i e w i t was n o t p o s s i b l e t o h a v e a s i g n i f i c a n t p e r c e n t a g e of t h e p o p u l a & i o n a b l e t o r e a d a n d t o h e i n f o r m e d a b o u t p o l i t i c a l e v e n t s . A s i n t h e c a s e of many s u c h p r e d i c t i o n s , h e was v e r y much in error. The r e v o l u t i o n i n mass s c h o o l i n g i s one of t h e most s t r i k i n g phenomena of t h e 2 0 t h c e n t u r y . I would c e r t a i n l y p l a c e i t on my own l i s t of t h e f i v e most i m p o r t a n t modern i n n o v a t i o n s i n w o r l d c u l t u r e .

Testing The f o u r t h e d u c a t i o n a l i n n o v a t i o n i s t e s t i n g , w h i c h i s i n many ways o l d e r t h a n t h e c o n c e p t o f mass s c h o o l i n g . The g r e a t t r a d i t i o n of t e s t i n g was f i r s t e s t a b l i s h e d i n C h i n a ; t e s t i n g t h e r e b e g a n i n t h e f i f t h c e n t u r y A.D. and became f i r m l y e n t r e n c h e d by t h e 1 2 t h c e n t u r y A . D . There i s a continuous h i s t o r y from t h e 1 2 t h c e n t u r y t o t h e e n d of t h e 1 9 t h c e n t u r y i n t h e u s e of t e s t s f o r t h e s e l e c t i o n of m a n d a r i n s - - m a n d a r i n s c o n s t i t u t e d t h e c i v i l s e r v a n t s who r a n t h e i m p e r i a l government of C h i n a , T h e s e c i v i l s e r v i c e p o s i t i o n s h e l d by m a n d a r i n s were r e g a r d e d a s t h e e l i t e s o c i a l p o s i t i o n s i n t h e s o c i e t y . The i m p o r t a n c e of t h e s e t e s t s i n C h i n e s e s o c i e t y i s w e l l a t t e s t e d t o by t h e l i t e r a t u r e of various periods. I f one e x a m i n e s , f o r e x a m p l e , t h e l i t e r a t u r e of t h e 1 5 t h o r 1 6 t h c e n t u r y , t h e n one i s i m p r e s s e d by t h e c o n c e r n e x p r e s s e d f o r p e r f o r m a n c e on t e s t s . A v a r i e t y of l i t e r a r y t a l e s f o c u s e d o n t h e q u e s t i o n of w h e t h e r s o n s would s u c c e s s f u l l y c o m p l e t e t h e t e s t s a n d what t h i s would mean f o r t h e f a m i l y . (As you m i g h t e x p e c t , i n t h o s e d a y s women had no p l a c e i n t h e manap,ement of t h e s o c i e t y a n d no p l a c e a s a p p l i c a n t s f o r c i v i l s e r v i c e p o s i t i o n s . ) The p r o c e d u r e s of s e l e c t i o n w e r e a s r i g o r o u s a s t h o s e found i n a c o n t e m p o r a r y m e d i c a l s c h o o l o r a g r a d u a t e s c h o o l of b u s i n e s s i n t h e U n i t e d S t a t e s , and i n many p e r i o d s less t h a n t w o p e r c e n t o f t h o s e t h a t began t h e t e s t s , which w e r e a r r a n g e d i n a c o m p l i c a t e d h i e r a r c h y , s u c c e s s f u l l y c o m p l e t e d t h e s e q u e n c e and w e r e p u t o n t h e list of e l i g i b l e m a n d a r i n s . A l t h o u g h t e s t i n g h a s a h i s t o r y t h a t g o e s b a c k h u n d r e d s of y e a r s , i n many ways i t i s p r o p e r t o r e g a r d t e s t i n g a s a 2 0 t h c e n t u r y i n n o v a t i o n b e c a u s e i t was o n l y i n t h i s c e n t u r y t h a t t h e s c i e n t i f i c a n d t e c h n i c a l s t u d y of t e s t s b e g a n . I t i s o n l y i n t h i s c e n t u r y t h a t t h e r e h a s been a s e r i o u s e f f o r t t o underst and a n d t o d e f i n e what c o n s t i t u t e s a good t e s t f o r a g i v e n a p t i t u d e , a g i v e n a c h i e v e m e n t , o r a g i v e n s k i l l . M o r e o v e r , t h i s i n t e n s i v e s t u d y of t e s t i n g f r o m

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P. SUPPES

a t e c h n i c a l s t a n d p o i n t was p r i m a r i l y a f o c u s of American r e s e a r c h by s u c h e d u c a t i o n a l p s y c h o l o g i s t s as Edward L . T h o r n d i k e . The t r a d i t i o n t h a t T h o r n d i k e b e g a n h a s become a m a j o r o n e i n o u r s o c i e t y and i t i s a s o u r c e of c o n t i n u a l c o n t r o v e r s y in terms of i s s u e s of f a i r n e s s , o b j e c t i v i t y , e t c . C e r t a i n l y t h e c u r r e n t d i s c u s s i o n s of t h e r e a s o n s f o r t h e d e c l i n e s i n t h e v e r b a l a n d m a t h e m a t i c a l s c o r e s on t h e S c h o l a s t i c A p t i t u d e T e s t s t a k e n f o r c o l l e g e e n t r a n c e p r o v i d e a n e x c e l l e n t e x a m p l e of t h e k i n d of d e t a i l e d s c r u t i n y w e g i v e t o o u r t e s t s t h a t i s c o m p l e t e l y u n c h a r a c t e r i s t i c of any t r a d i t i o n of t e s t i n g w h e t h e r i n C h i n a , Europe o r t h e U n i t e d S t a t e s p r i o r t o t h i s c e n t u r y .

The f o u r i n n o v a t i o n s t h a t I h a v e d i s c u s s e d - - w r i t t e n r e c o r d s , b o o k s , s c h o o l s , and tests--are t h e v e r y f a b r i c of o u r e d u c a t i o n a l s y s t e m t o d a y . It is almost u n t h i n k a b l e t o c o n t e m p l a t e a modern e d u c a t i o n a l s y s t e m w i t h o u t e a c h of t h e s e innovations playing an important p a r t . Of t h e s e f o u r t e c h n o l o g i e s , none h a d b e e n i n any way a d e q u a t e l y f o r e c a s t or o u t l i n e d a t t h e t i m e i t was i n t r o d u c e d . Of c o u r s e , a few i n d i v i d u a l s f o r e s a w t h e c o n s e q u e n c e s a n d had s o m e t h i n g t o s a y a b o u t t h o s e c o n s e q u e n c e s , b u t c e r t a i n l y t h e d e t a i l s of t h e u s e of any of t h e s e f o u r t e c h n o l o g i e s h a d n o t I am c e r t a i n t h a t t h e same t h i n g w i l l b e t r u e of been a d e q u a t e l y f o r e s e e n . t e c h n o l o g i e s now d e v e l o p i n g f o r u s e i n t h e f u t u r e , and so I do n o t want t o a p p e a r c o n f i d e n t t h a t what I s a y is a c o r r e c t s c e n a r i o f o r t h e f u t u r e . But I want t o s a y s o m e t h i n g a b o u t e a c h of t h e f o u r . F i r s t , I h a v e m e n t i o n e d , a n d I want t o e m p h a s i z e a g a i n , t h e v e r y r e c e n t and h i s t o r i c a l l y v e r y t r a n s i e n t c h a r a c t e r of s c h o o l s . I t i s a phenomenon i n a g e n e r a l s e n s e of t h e l a s t h u n d r e d y e a r s i n t h e most d e v e l o p e d p a r t s of t h e w o r l d , a n d a phenomenon of t h e l a s t t h i r t y y e a r s ( t h a t i s , s i n c e World War 1 1 ) i n t h e u n d e r d e v e l o p e d In p a r t s of t h e w o r l d . Now, a n i m p o r t a n t q u e s t i o n f o r t h e f u t u r e i s t h i s : f i f t y y e a r s o r one h u n d r e d y e a r s , w i l l w e a b o l i s h s c h o o l s ? Will w e d e l i v e r i n t o t h e home, o r i n t o s m a l l n e i g h b o r h o o d u n i t s by t e c h n o l o g i c a l means a l l c u r r i c u l u m a n d i n s t r u c t i o n ? F u r t h e r , w i l l t h e w i s h of t h e i n d i v i d u a l , t h e f a m i l y , t h e p a r e n t s , o r t h e neighborhood group be such t h a t they w i l l n o t want t h e i r c h i l d r e n i n s c h o o l , b u t r a t h e r want them a t home o r i n t h e n e i g h b o r h o o d ? I t h i n k t h e a n s w e r s t o t h e s e q u e s t i o n s are n o t e a s y t o predict o r t o foresee. The same k i n d of f o r e c a s t may b e made f o r books. The i m p o r t a n c e of books t h a t w e h a v e f e l t f o r s e v e r a l h u n d r e d y e a r s , s i n c e t h e b e g i n n i n g of t h e R e n a i s s a n c e , a n d t h a t h a s b e e n a s s o c i a t e d w i t h t h e d e v e l o p m e n t and e d u c a t i o n of a n i n f o r m e d c i t i z e n r y , may o n c e a g a i n f a d e away. I t h i n k t h a t a l l of u s , a t l e a s t t h o s e Some r e c e n t of my a g e , h a v e seen t h i s a l r e a d y i n t h e c a s e of young s t u d e n t s . s t u d i e s have i n d i c a t e d t h a t t h e c u l t u r a l r e f e r e n c e p o i n t s f o r t h e younger g e n e r a t i o n a r e no l o n g e r t o b e f o u n d i n b o o k s , o r i n c u r r e n t n o v e l s , b u t on t e l e v i s i o n a n d i n movies. I n t h e c a s e of t e s t s , I a l s o p r e d i c t t h a t t h i s c l a s s i c a l t e c h n o l o g y w i l l decrease i n importance. I b e l i e v e t h a t tests w i l l d e c r e a s e i n importance b e c a u s e we w i l l h a v e t h e t e c h n o l o g i c a l means t o keep a much more s a t i s f a c t o r y a n d much more d e t a i l e d r e c o r d of t h e l e a r n i n g of i n d i v i d u a l s t u d e n t s . Thus o u r i n f e r e n c e s a b o u t t h e p e r f o r m a n c e of s t u d e n t s a n d t h e i r c a p a b i l i t i e s f o r t a k i n g n e x t s t e p s w i l l depend upon a much more s u b s t a n t i a l r e c o r d , a much b e t t e r b a s i s of i n f e r e n c e t h a n w e h a v e i n c u r r e n t tests. F i n a l l y , what a b o u t t h e w r i t t e n r e c o r d ? The w r i t t e n r e c o r d w i l l u n d o u b t e d l y c o n t i n u e t o h a v e i m p o r t a n c e , b u t I t h i n k t h a t when i t comes t o t e a c h i n g , t h e o b j e c t i o n s f o u n d i n P l a t o ’ s D i a l o g u e s t o t h e c o l d and n e u t r a l w r i t t e n word as o p p o s e d t o t h e w a r m and f r i e n d l y v o i c e of t h e t e a c h e r w i l l o n c e a g a i n b e h e a r d and f e l t a s s e r i o u s o b j e c t i o n s . What I am s a y i n g i s t h a t , i n s t a r t i n g t o think about t h e future, we can forecast obsolescence o r semiobsolescence f o r a l l of t h e g r e a t t e c h n o l o g i e s of t h e past--and t h a t i s p r o p e r a n d a p p r o p r i a t e .

PAST, PRESENT AND FUTURE EDUCATIONAL TECHNOLOGIES

Present Educational Technologies T h e r e a r e t h r e e s a l i e n t new t e c h n o l o g i e s i n e d u c a t i o n : and c o m p u t e r s .

radio, television,

The u s e of r a d i o f o r i n s t r u c t i o n g o e s b a c k more t h a n t w e n t y y e a r s , b u t i t i s not e x t e n s i v e l y used i n t h e United S t a t e s a t t h e p r e s e n t time. I am c u r r e n t l y a s s o c i a t e d with a l a r g e p r o j e c t i n Nicaragua using r a d i o f o r teaching e l e m e n t a r y - s c h o o l m a t h e m a t i c s , and I t h i n k t h a t i n s t r u c t i o n a l r a d i o h a s a n important r o l e t o play i n developing countries. It i s f a i r t o s a y t h a t t h e most i m p o r t a n t w o r l d w i d e r e v o l u t i o n i n c o m m u n i c a t i o n i s n o t t h e t e l e p h o n e a n d n o t t e l e v i s i o n b u t r a d i o . Most d e v e l o p i n g c o u n t r i e s a r e s a t u r a t e d w i t h r a d i o s t a t i o n s , and t h e s p r e a d o f t r a n s i s t o r r a d i o s e t s , e s p e c i a l l y s m a l l p o r t a b l e o n e s , i s a w o r l d w i d e phenomenon.

Television Twenty y e a r s a g o t h e r e were o p t i m i s t i c f o r e c a s t s a b o u t t h e e x t e n s i v e u s e of t e l e v i s i o n f o r e d u c a t i o n a l p u r p o s e s . F o r a v a r i e t y of r e a s o n s t h i s e x t e n s i v e e d u c a t i o n a l u s e of t e l e v i s i o n h a s n o t d e v e l o p e d a s r a p i d l y a s f o r e c a s t . The g e n e r a l i m p a c t o f t e l e v i s i o n i s r e c o g n i z e d by e v e r y o n e and i s o n e of t h e most i m p o r t a n t and most s a l i e n t c u l t u r a l a s p e c t s of o u r s o c i e t y . R e c e n t l y t h e e d u c a t i o n a l u s e of t e l e v i s i o n i n t h e U n i t e d S t a t e s , e s p e c i a l l y a t t h e c o m m u n i t y - c o l l e g e l e v e l , h a s begun t o show a marked i n c r e a s e . I would p r e d i c t t h a t e s p e c i a l l y f o r f i n a n c i a l reasons t h i s i n c r e a s e w i l l continue; it i s s i m p l y g o i n g t o b e v e r y much more e c o n o m i c a l t o b r o a d c a s t c o u r s e s t o o f f - c a m p u s s t u d e n t s t h a n t o i n v e s t i n a d d i t i o n a l c a p i t a l p l a n t f o r h o u s i n g s t u d e n t s . The c o n c e r n f o r e n e r g y c o n s e r v a t i o n , t h e p r o b l e m s o f t r a n s p o r t a t i o n i n many u r b a n a r e a s , t h e i n c r e a s i n g f o c u s on making t h e home a l e a r n i n g c e n t e r - - a l l of t h e s e f a c t o r s s h o u l d c o n t r i b u t e t o t h e more w i d e s p r e a d u s e of e d u c a t i o n a l telev i s i o n , e s p e c i a l l y a t t h e a d u l t l e v e l . The i m p a c t a t t h e o t h e r end of t h e s c a l e of s u c h p r o g r a m s as Sesame S t r e e t h a s a l s o b e e n w i d e l y r e c o g n i z e d , a n d p r o g r a m s of t h i s c h a r a c t e r w i l l c o n t i n u e t o p l a y a n i m p o r t a n t r o l e . A l t h o u g h I r e c o g n i z e t h e importance t h a t t e l e v i s i o n h a s i n o u r s o c i e t y and i t s i n c r e a s i n g l y p r o m i n e n t r o l e i n e d u c a t i o n , I s h a l l n o t s a y more a b o u t i t i n t h e p r e s e n t t a l k b u t move on t o c o m p u t e r s ,

Computers I b e g i n w i t h a q u i c k s u r v e y of o u r work a t S t a n f o r d . Work i n c o m p u t e r a s s i s t e d i n s t r u c t i o n (CAI) a t t h e I n s t i t u t e f o r M a t h e m a t i c a l S t u d i e s i n t h e S o c i a l S c i e n c e s , S t a n f o r d U n i v e r s i t y , b e g a n i n 1963. D u r i n g t h e f i r s t d e c a d e , most o f t h e work was c o n c e r n e d w i t h t h e d e v e l o p m e n t o f c o m p u t e r - b a s e d c u r r i c u l u m s f o r e l e m e n t a r y s c h o o l s , e s p e c i a l l y i n m a t h e m a t i c s and r e a d i n g .

Elementary-school Mathematics

1 d e s c r i b e b r i e f l y h e r e t h e c u r r e n t v e r s i o n of elementary-school mathematics a t t h e d r i l l - a n d - p r a c t i c e l e v e l o f f e r e d by Computer C u r r i c u l u m C o r p o r a t i o n (CCC). T h i s c u r r i c u l u m h a s a c l o s e r e s e m b l a n c e t o t h e c u r r i c u l u m t h a t was d e v e l o p e d a t t h e I n s t i t u t e i n t h e 1960s. I t i s s i m i l a r i n g e n e r a 1 , s t r u c t u r e b u t d i f f e r s i n many s p e c i f i c d e t a i l s . d e s c r i p t i o n of i t w i l l p r o v i d e a good example of t h e u s e of c o m p u t e r s f o r i n s t r u c t i o n a t t h e e l e m e n t a r y - s c h o o l l e v e l on a c u r r e n t b a s i s i n t h e U n i t e d S t a t e s .

57

58

P. SUPPES

The c u r r i c u l u m i s o r g a n i z e d i n t o s t r a n d s . The e x e r c i s e s , d e s c r i b e d i n more d e t a i l below, a r e g e n e r a t e d randomly f r o m i t e m - t y p e d e s c r i p t i o n s i n t h e CCC c u t r i c u l u m , w h e r e a s i n t h e I n s t i t u t e ' s c u r r i c u l u m t h e y were f i x e d e x e r c i s e s o f a f i x e d number. T h i s means t h a t t h e m o t i o n of a s t u d e n t t h r o u g h t h e CCC c u r r i c u l u m i s p r o b a b l y a c l o s e r a p p r o x i m a t i o n t o h i s a c t u a l l e v e l of a c h i e v e ment t h a n i s t h e case w i t h t h e I n s t i t u t e ' s c u r r i c u l u m . The CCC c u r r i c u l u m d i f f e r s i n a number of o t h e r r e s p e c t s t h a t w i l l n o t b e d e s c r i b e d h e r e .

A s t r a n d r e p r e s e n t s one c o n t e n t a r e a w i t h i n a c u r r i c u l u m . For example, a d i v i s i o n s t r a n d , a d e c i m a l s t r a n d , and a n e q u a t i o n s t r a n d a r e i n c l u d e d i n t h e M a t h e m a t i c s S t r a n d s c u r r i c u l u m . Each s t r a n d i s a s t r i n g of r e l a t e d items whose d i f f i c u l t y p r o g r e s s e s Erom e a s y t o h a r d . A c o m p u t e r program k e e p s r e c o r d s of t h e s t u d e n t ' s p o s i t i o n a n d p e r f o r m a n c e s e p a r a t e l y € o r e v e r y s t r a n d By comparing a s t u d e n t ' s r e c o r d o f p e r f o r m a n c e o n t h e material i n o n e s t r a n d w i t h a p r e s e t performance c r i t e r i o n , t h e program d e t e r m i n e s i f t h e s t u d e n t n e e d s more p r a c t i c e a t t h e same l e v e l of d i f f i c u l t y w i t h i n t h e s t r a n d , s h o u l d move b a c k t o a n easier l e v e l f o r r e m e d i a l work, or h a s m a s t e r e d t h e c u r r e n t c o n c e p t a n d c a n move a h e a d t o a more d i f f i c u l t l e v e l . Then t h e p r o g r a m automatically adjusts the student's position to the correct level within t h e s t r a n d . The p r o c e s s of e v a l u a t i o n a n d a d j u s t m e n t a p p l i e s t o a l l s t r a n d s a n d is c o n t i n u o u s throughout e a c h s t u d e n t ' s i n t e r a c t i o n w i t h a c u r r i c u l u m . E v e n l y s p a c e d g r a d a t i o n s in t h e d i f f i c u l t y l e v e l of t h e m a t e r i a l a l l o w p o s i t i o n s w i t h i n a s t r a n d t o b e matched t o s c h o o l g r a d e p l a c e m e n t s by t e n t h s of a y e a r . G r a d e p l a c e m e n t i n a s p e c i f i c s u b j e c t a r e a c a n t h e n b e d e t e r m i n e d by e x a m i n i n g a s t u d e n t ' s p o s i t i o n i n t h e s t r a n d r e p r e s e n t i n g t h a t a r e a . S i n c e p e r f o r m a n c e i n e a c h s t r a n d is r e c o r d e d a n d e v a l u a t e d s e p a r a t e l y , t h e s t u d e n t may h a v e a d i f f e r e n t g r a d e p l a c e m e n t i n e v e r y s t r a n d o f a c u r r i c u l u m . T e a c h e r s ' r e p o r t s , a v a i l a b l e a s p a r t of e a c h c u r r i c u l u m , r e c o r d p r o g r e s s by showing t h e s t u d e n t ' s g r a d e p l a c e m e n t i n e a c h s t r a n d a t t h e time of t h e report. I n a c u r r i c u l u m b a s e d on t h e s t r a n d s i n s t r u c t i o n a l s t r a t e g y , a n o r m a l l e s s o n c o n s i s t s of a m i x t u r e of e x e r c i s e s f r o m d i f f e r e n t s t r a n d s . Each t i m e a n item f r o m a p a r t i c u l a r c u r r i c u l u m i s t o b e p r e s e n t e d . a c o m p u t e r program randomly s e l e c t s t h e s t r a n d f r o m which i t w i l l draw t h e e x e r c i s e . Random s e l e c t i o n of s t r a n d s e n s u r e s t h a t t h e s t u d e n t w i l l r e c e i v e a m i x t u r e of d i f f e r e n t t y p e s o f items i n s t e a d of a s e r i e s of s i m i l a r i t e m s . Each c u r r i c u l u m a l s o p r o v i d e s f o r r a p i d g r o s s a d j u s t m e n t of p o s i t i o n i n a l l t h e s t r a n d s a s t h e s t u d e n t is f i r s t b e g i n n i n g work i n t h e c o u r s e . S t u d e n t s who p e r f o r m v e r y w e l l a t t h e i r e n t e r i n g g r a d e l e v e l s a r e moved up i n h a l f - y e a r s t e p s u n t i l t h e y r e a c h a more c h a l l e n g i n g l e v e l . S t u d e n t s who p e r f o r m p o o r l y are moved down i n h a l f - y e a r s t e p s , T h i s a d j u s t m e n t o f o v e r a l l g r a d e l e v e l e n s u r e s t h a t s t u d e n t s are c o r r e c t l y p l a c e d i n t h e c u r r i c u l u m a n d i s i n e f f e c t only during a student's f i r s t ten sessions. M a t h e m a t i c s S t r a n d s , G r a d e s 1-6, c o n t a i n s 1 4 s t r a n d s , o r c o n t e n t areas. T a b l e 1 l i s t s t h e s t r a n d s i n t h e m a t h e m a t i c s c u r r i c u l u m . The c u r r i c u l u m b e g i n s a t t h e f i r s t - g r a d e l e v e l a n d e x t e n d s t h r o u g h g r a d e l e v e l 7.9. The seventh-grade m a t e r i a l does n o t c o n s t i t u t e a complete curriculum f o r t h a t g r a d e y e a r b u t is i n t e n d e d a s e n r i c h m e n t f o r s t u d e n t s who c o m p l e t e t h e s i x t h - g r a d e material. (A s e p a r a t e CCC c u r r i c u l u m is d e v o t e d t o s e v e n t h and eighth-grade mathematics.)

PAST, PRESENT AND FUTURE EDUCATIONAL TECHNOLOGIES

59

Table 1 The S t r a n d s i n M a t h e m a t i c s S t r a n d s , G r a d e s 1-6

Strand

Name

Abbreviation

--____ 1 2

3 4

5 6 7 8

9 10 11

12 13 14

Number C o n c e p t s Horizontal Addition Horizontal Subtraction V e r t i c a l Addition Vertical S u b t r a c t i o n Equations Measurement Horizontal Multiplication Laws of A r i t h m e t i c Ver t i c a 1 Mu 1t i p l i c a t i o n Division Fractions D e c i ma 1s N e g a t i v e Numbers

NC HA HS VA

vs EQ

MS HM LW

VM DV FR DC NG

Each s t r a n d is o r g a n i z e d i n t o e q u i v a l e n c e c l a s s e s , o r s e t s of e x e r c i s e s of s i m i l a r number p r o p e r t i e s a n d s t r u c t u r e , D u r i n g e a c h C A I s e s s i o n i n mathematics, s t u d e n t s r e c e i v e e x e r c i s e s from a l l t h e s t r a n d s t h a t c o n t a i n e q u i v a l e n c e classes a p p r o p r i a t e t o t h e i r g r a d e l e v e l s . For example, a s t u d e n t a t g r a d e l e v e l 2.0 w i l l r e c e i v e e x e r c i s e s f r o m s e v e n s t r a n d s : N C , HA, HS, VA, VS, E q , a n d MS. S t u d e n t s d o n o t r e c e i v e a n e q u a l number of e x e r c i s e s f r o m a l l s t r a n d s . The p ro gr a m a d j u s t s t h e p r o p o r t i o n of e x e r c i s e s f r o m e a c h s t r a n d t o ma tc h t h e p r o p o r t i o n of e x e r c i s e s c o v e r i n g t h a t c o n c e p t i n a n a v e r a g e t e x t b o o k . The c u r r i c u l u m material i n M a t h e m a t i c s S t r a n d s , G r a d e s 1-6, i s n o t s t o r e d i n t h e c o m p u t e r ’ s memory b u t t a k e s t h e f o r m of a l g o r i t h m s t h a t u s e random-number t e c h n i q u e s t o g e n e r a t e e x e r c i s e s . When a p a r t i c u l a r e q u i v a l e n c e c l a s s i s s e l e c t e d , a program g e n e r a t e s t h e n u m e r i c a l v a l u e s u s e d i n t h e e x e r c i s e , p r o d u c e s t h e r e q u i r e d f o r m a t i n f o r m a t i o n f o r t h e p r e s e n t a t i o n of t h e e x e r c i s e , and c a l c u l a t e s t h e c o r r e c t r e s p o n s e f o r comparison w i t h s t u d e n t i n p u t . As a r e s u l t , t h e a r r a n g e m e n t of t h e l e s s o n a n d t h e a c t u a l e x e r c i s e s p r e s e n t e d d i f f e r b e t w e e n s t u d e n t s a t t h e same l e v e l a n d b e t w e e n l e s s o n s f o r a s t u d e n t who r e m a i n s a t a c o n s t a n t g r a d e p l a c e m e n t f o r s e v e r a l l e s s o n s . E x t e n s i v e e v a l u a t i o n of t h i s c u r r i c u l u m h a s be e n r e p o r t e d i n t h e l i t e r a t u r e . A good r e c e n t r e f e r e n c e i s Macken a n d S u p p e s (11.

Un iv e r si ty - -

Computer-assisted

Instruction

B e g i n n i n g i n t h e l a t e 1960s, work a t t h e I n s t i t u t e b e g a n t o f o c u s more on t h e p o s s i b i l i t i e s of CAI f o r u n i v e r s i t y - l e v e l c o u r s e s . The f i r s t major e f f o r t was u n d e r t h e d i r e c t i o n of P r o f e s s o r J o s e p h Van Campen o f t h e D e p a r t m e n t o f S l a v i c L a n g u a g e s a t S t a n f o r d , a n d was c o n c e r n e d w i t h t h e d e v e l o p m e n t of t w o y e a r s o f e l e m e n t a r y R u s s i a n a t t h e u n i v e r s i t y l e v e l . The v e r y p o s i t i v e e v a l u a t i o n r e s u l t s of t h e f i r s t - y e a r R u s s i a n c o u r s e a r e r e p o r t e d i n S u p p e s a n d M o r n i n g s t a r 121. The work i n S l a y i c l a n g u a g e s h a s now s h i f t e d f r o m t h e b e g i n n i n g c o u r s e s t o i n t e r m e d i a t e c o u r s e s t h a t h a v e small e n r o l l m e n t . Computer-assisted i n s t r u c t i o n i s o f f e r e d i n such r e l a t i v e l y e s o t e r i c t o p i c s as Ol d C h u r c h S l a v o n i c a n d t h e l i n g u i s t i c h i s t o r y of t h e R u s s i a n l a n g u a g e .

60

P. SUPPES

One o f t h e e a r l i e s t c u r r i c u l u m e f f o r t s f o r t h e e l e m e n t a r y s c h o o l was t h e development of a c o u r s e i n l o g i c f o r g i f t e d elementary-school s t u d e n t s . In t h e l a t e 1 9 6 0 s , e f f o r t s t u r n e d t o t h e d e v e l o p m e n t of a c o r r e s p o n d i n g c o u r s e i n l o g i c f o r u n i v e r s i t y s t u d e n t s . Over t h e p a s t few y e a r s t h e c o n t i n u e d d e v e l o p m e n t a n d improvement of t h i s c o u r s e h a s b e e n o n e of t h e c e n t r a l e f f o r t s i n C A I a t t h e I n s t i t u t e . R e c e n t l y , t h e e f f o r t s i n l o g i c h a v e moved t o t h e i n t e r m e d i a t e l e v e l a s a p r i m a r y f o c u s of r e s e a r c h a nd d e v e l o p m e n t . A greater e f f o r t i s now b e i n g p u t i n t o a x i o m a t i c set t h e o r y , a n d f u t u r e e f f o r t s a t a s i m i l a r l e v e l a r e a n t i c i p a t e d . C u r r i c u l u m o u t l i n e s o f t h e s e two c o u r s e s a r e g i v e n i n T a b l e s 2 and 3 .

Table 2 C u r r i c u l u m f o r I n t r o d u c t i o n t o Logic ( C A I V e r s i o n )

Chapter

Topic

Lesson

'P ass ' S e q u e n c e

I

A.

B. C. D.

I1

B. C.

D.

IV

Semantics Syntax Inference rules, derivations Validity, counterexamples 410-420

INTEGER ARITHMETIC A.

111

401-409

PROPOSITIONAL L O G I C

Equality Axioms, d e f i n i t i o n s , a n d t h e o r e m s Commut at i ve g r o u p s Noncommutative g r o u p s

FINDING AXIOMS

42 1

PREDICATE LOGIC A.

B. C. D.

E.

422-429

Symbolization R u l e s of i n f e r e n c e , d e r i v a t i o n s Interpretations Consistency, independence L o g i c of i d e n t i t y , s o r t s

Letter-grade

sequences

__ VI

BOOLEAN ALGEBRA

VII

QUALITATIVE PROBABILITY

VI 11

SOCIAL DECISION THEORY

601-603

8001

501-506

PAST, PRESENT AND FUTURE EDUCATIONAL TECHNOLOGIES

Table 3 C u r r i c u l u m f o r A x i o m a t i c S e t T h e o r y (CAI V e r s i o n )

Chapter

I I1

Topic

INTRODUCTION AND HISTORICAL SURVEY

ELEMENTARY SET THEORY A. B. C.

D. 111

RELATIONS AND FUNCTIONS A. B. C.

IV V

VI

Ordering r e l a t i o n s Well-ordering Functions

EQUIPOLLENCE, FINITE AND INFINITE SETS CARDINAL NUMBER THEORY O R D I N A L NUMBEK THEORY

A. B. C. VII

The a l g e b r a of s e t s Abstraction Power sets, C a r t e s i a n p r o d u c t s , g e n e r a l i z e d u n i o n s a n d intersections The axiom of r e g u l a r i t y

Ordinals T r a n s f i n i t e induction D e f i n i t i o n by r e c u r s i o n

THE AXIOM OF C H O I C E AND ITS EQUIVALENTS

--_ C o u r s e s i n music a n d computer programming h a v e been d e v e l o p e d f o r u s e a t t h e I n d i v i d u a l i z e d C A I i n music h a s been a p p l i e d t o u n i v e r s i t y l e v e l as w e l l . t h e o r e t i c a l a n d i n s t r u c t i o n a l i n v e s t i g a t i o n s i n a number of d i f f e r e n t c o u r s e s . P r e v i o u s C A I work i n p r o g r a m m i n g h a s i n c l u d e d c o u r s e s i n t h e BASIC, A I D , SIMPER, a n d LOGO l a n g u a g e s ; c u r r e n t p r o j e c t s d e a l w i t h a n e n t i r e l y new a p p r o a c h t o t e a c h i n g BASIC a n d t h e i n t e g r a t i o n o f a C A I c o u r s e i n LISP i n t o t h e u n i v e r s i t y curriculum.

Productivity i n CAI The r e a s o n f o r t h e e m p h a s i s on s m a l l - e n r o l l m e n t c o u r s e s i s t o a v e r y c o n s i d e r a b l e e x t e n t a matter o f p r o d u c t i v i t y 131. The i n i t i a l i m p e t u s f o r t h e course i n axiomatic set theory was t h e loss of a f a c u l t y p o s i t i o n i n t h i s area d u e t o b u d g e t c u t s a t S t a n f o r d . I n t h e f a c e of d e c l i n i n g o r f i x e d b u d g e t s , i t h a s become a p p a r e n t t h a t f a c u l t y s i z e s w i l l p r o b a b l y d e c r e a s e during t h e remainder of t h i s century. Some A m e r i c a n s t a t e u n i v e r s i t i e s r e q u i r e t h a t a s p e c i f i c l e v e l of e n r o l l m e n t b e m a i n t a i n e d f o r a l l c o u r s e s o f f e r e d , and t h e r e is considerable p r e s s u r e a g a i n s t s p e c i a l i z e d courses with low e n r o l l m e n t s . S t i l l , s u c h c o u r s e s r e p r e s e n t a n i m p o r t a n t f u n c t i o n of t h e u n i v e r s i t y i n t r a n s m i t t i n g i n t e l l e c t u a l knowledge and s k i l l s from o n e generation t o t h e next.

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One of o u r main a i m s in C A I r e s e a r c h a n d d e v e l o p m e n t a t t h e I n s t i t u t e is t o show how t h e s e s p e c i a l i z e d c o u r s e s c a n b e m a i n t a i n e d a t r e a s o n a b l e c o s t i n t h e f u t u r e by a p p r o p r i a t e u s e of computer t e c h n o l o g y . Our aim i s t o i n c r e a s e t h e t e a c h i n g l o a d of f a c u l t i e s i n terms of c o u r s e s o f f e r e d , and o n e of o u r s u b s i d i a r y aims, c o n s e q u e n t l y , i s t o i m p r o v e t h e a b i l i t y of C A I t o p r o v i d e e f f e c t i v e i n s t r u c t i o n w i t h l i t t l e o r no i n t e r v e n t i o n r e q u i r e d of t h e i n s t r u c t o r . A few t e a c h i n g a s s i s t a n t s , a v a i l a b l e a t s c h e d u l e d h o u r s t h r o u g h t h e d a y a n d e v e n i n g , work in c o n j u n c t i o n w i t h e a c h c o u r s e ; t h e i n s t r u c t o r h i m s e l f is f r e e t o s u p e r v i s e f u r t h e r d e v e l o p m e n t of C A I c o u r s e s o r t o c o n s u l t on a n i n d i v i d u a l b a s i s w i t h s t u d e n t s a n d t e a c h i n g a s s i s t a n t s .

Complex I n s t r u c t i o n a l S y s t e m s I have e m p h a s i z e d t h u s f a r t h e i m p o r t a n c e o f p r o d u c t i v i t y c o n s i d e r a t i o n s i n a t t i t u d e t o w a r d C A I a t t h e c o l l e g e l e v e l a n d i n o u r a p p r o a c h t o making i t a v i a b l e a n d permanent p a r t of c o l l e g e - l e v e l i n s t r u c t i o n . T h e r e is, however, a n i m p o r t a n t c o r o l l a r y t o t h i s a p p r o a c h t h a t n e e d s t o b e e m p h a s i z e d a n d t h a t g e n e r a t e s a number of f u n d a m e n t a l a n d i n t e l l e c t u a l l y c h a l l e n g i n g p r o b l e m s of c o m p u t e r s c i e n c e .

OUK

I t seems l i k e l y t h a t e l e m e n t a r y c o u r s e s o f s e v e r a l k i n d s c a n b e g i v e n w i t h o u t r a d i c a l l y a d v a n c i n g t h e p r e s e n t t e c h n i q u e s of C A I and w i t h o u t c a l l i n g upon d e e p e r methods of program c o n s t r u c t i o n f o r i n t e r a c t i o n between s t u d e n t and computer.

However, as w e push toward i n c r e a s i n g l y t u t o r i a l t o p i c s a n d t o p i c s a t a n i n t e r m e d i a t e l e v e l of d i f f i c u l t y ( e x e m p l i f i e d , f o r i n s t a n c e , by c o u r s e s t h a t a r e p r i m a r i l y m a t h e m a t i c a l l y b a s e d i n t h e i r f o u n d a t i o n s s u c h as t h e c o u r s e i n a x i o m a t i c set t h e o r y ) , t h e need f o r work a t t h e f r o n t i e r s of c o m p u t e r s c i e n c e becomes e v i d e n t . The c o u r s e s a t t h i s l e v e l w i l l b e v i a b l e and t h e r e f o r e p r o d u c t i v e o n l y i f t h e t e c h n i q u e s of i n s t r u c t i o n a r e r i c h enough t o p r o v i d e a c h a l l e n g e t o t h e s t u d e n t s a n d a r e a l i s t i c r a n g e of o p t i o n s c o r r e s p o n d i n g t o what o n e m i g h t e x p e c t i n s u c h a c o u r s e t a u g h t by t r a d i t i o n a l methods. I n t h e case of m a t h e m a t i c a l l y b a s e d c o u r s e s , t h e p r i m a r y need is c l e a r . T e c h n i q u e s of p r o o f t h a t a p p r o a c h i n f o r m a l methods of m a t h e m a t i c a l a r g u m e n t a r e a b s o l u t e l y e s s e n t i a l i n order not t o involve t h e s t u d e n t i n an i n o r d i n a t e amount of t e d i o u s d e t a i l . Moving f r o m f o r m a l l y e x p l i c i t p r o o f s t o i n f o r m a l o n e s is a c e n t r a l i n t e l l e c t u a l p r o b l e m o f a r t i f i c i a l i n t e l l i g e n c e a n d c o m p u t e r science: t h e c o n s t r u c t i o n of a model of i n f o r m a l m a t h e m a t i c a l r e a s o n i n g . Our c u r r e n t e f f o r t s h e r e h a v e borrowed f r o m a r t i f i c i a l i n t e l l i g e n c e t e c h n i q u e s of It t h e o r e m p r o v i n g a n d a x i o m a t i c r e p r e s e n t a t i o n of m a t h e m a t i c a l r e a s o n i n g . h a s a l s o been n e c e s s a r y t o add t o t h e s e t e c h n i q u e s , a n d o u r c u r r e n t p r o o f c h e c k e r f o r s e t t h e o r y is b e l i e v e d t o be t h e most s o p h i s t i c a t e d s u c h program i n t h e w o r l d b e i n g u s e d by s t u d e n t s w i t h o u t any t r a i n i n g o r e x p e r i e n c e i n programming ( a n d n o t r e q u i r i n g a n y ) . C l o s e l y a s s o c i a t e d w i t h t h e p r o b l e m of i n f o r m a l p r o o f methods i s t h e p r o b l e m of t h e l a n g u a g e i n which t h e s t u d e n t g i v e s a p r o o f , a n d t h e ways i n which t h e program c a n e x p l a i n a n d e l u c i d a t e a p r o o f t o t h e s t u d e n t . A p r o o f c h e c k e r f o r t h e s e a d v a n c e d c o u r s e s must b e a b l e t o a c c e p t p r o o f s i n a n i n f o r m a l s t y l e t h a t I f e e l t h a t by c o n c e n t r a t i n g o n t h e approaches a fragment of English. r e s t r i c t e d domain of m a t h e m a t i c a l p r o o f s i t w i l l be f e a s i b l e t o h a v e a r e l a t i v e l y r i c h a n d a d e q u a t e u s e of n a t u r a l l a n g u a g e i n t h i s s e t t i n g , a n d t h a t w i t h r e a s o n a b l e time a n d e f f o r t w e s h a l l b e a b l e t o d u p l i c a t e t h e k i n d s of u s e of n a t u r a l l a n g u a g e c h a r a c t e r i s t i c of t h e w r i t i n g of m a t h e m a t i c a l p r o o f s i n t h e best textbooks.

PAST, PRESENT AND FUTURE

EDUCCATIONAL

TECHNOLOGIES

A p r o b l e m o f g r e a t e r d i f f i c u l t y is t h e d e v e l o p m e n t of t e c h n i q u e s f o r c o n d u c t i n g a d i a l o g u e b e t w e e n s t u d e n t a n d p r o g r a m a b o u t a g i v e n p r o o f on w h i c h t h e s t u d e n t i s engaged. We want t h e p r o g r a m t o b e a b l e t o make h e l p f u l a n d u s e f u l comments c o n t i n g e n t upon t h e s t u d e n t ' s work. We want t h e s e comments t o b e r e l e v a n t , p e r t i n e n t , and n a t u r a l i n tone. We do n o t b e l i e v e t h a t t h e s e d i a l o g u e p r o b l e m s w i l l b e s o l v e d i n any r e a s o n a b l e c o m p l e t e n e s s i n t h e n e a r f u t u r e . T h e r e a r e a s y e t v e r y few e x a m p l e s of n a t u r a l d i a l o g u e s b e t w e e n s t u d e n t a n d c o m p u t e r p ro gr a m. Two s y s t e m s t h a t show g r e a t p o t e n t i a l . a r e SCHOLAR 141 a n d S O P H I E 151. W h il e b o t h o f t h e s e p r o g r a m s a r e s i g n i f i c a n t a r t i f i c i a l i n t e l l i g e n c e p r o g r a m s , t h e y h a v e n o t y e t b e e n u s e d on a r e g u l a r b a s i s by s t u d e n t s i n a s t a n d a r d e d u c a t i o n a l s e t t i n g . An i m p o r t a n t f e a t u r e of o u r work a t t h e I n s t i t u t e i s t h a t a l l o f o u r e f f o r t s a r e d i r e c t e d t o w a r d r e a l c o u r s e s a n d a n s w e r known pedagogical needs.

A n o t h e r i n t e r e s t i n g u s e of c o m p u t e r - s c i e n c e t e c h n i q u e s o c c u r s i n t h e B A S I C I n s t r u c t i o n a l P r o gram ( R I P ) . Here t h e f o c u s of o u r e f f o r t h a s b e e n t o s e l e c t i n s t r u c t i o n a l m a t e r i a l and o r d e r i t s p r e s e n t a t i o n f o r t h e i n d i v i d u a l s t u d e n t . T h i s i s d o n e by u s i n g n e t w o r k s r e p r e s e n t i n g s k i l l s a nd t a s k s , w h i c h a r e compared t o p r o f i l e s f o r e a c h i n d i v i d u a l s t u d e n t . The e m p h a s i s on t h e a p p l i c a t i o n of t h e s e c o m p u t e r - s c i e n c e t e c h n i q u e s t o u n i v e r s i t y i n s t r u c t i o n h a s n e c e s s i t a t e d a d i f f e r e n t progra mming e n v i r o n m e n t t h a n most o t h e r C A I p r o j e c t s . For example, w e have chosen t o u s e t h e h i g h l y i n t e r a c t i v e TENEX t i m e s h a r i n g s y s t e m o n o u r PUP-10 c o m p u t e r , a n d h a v e i m p l e m e n t e d a n d e x t e n d e d s e v e r a l programmi ng l a n g u a g e s o r i g i n a l l y d e s i g n e d f o r a r t i f i c i a l intelligence research. These l a n g u a g e s c o n t a i n d a t a and c o n t r o l s t r u c t u r e s t h a t a r e f a r more a d v a n c e d t h a n t h e s t r u c t u r e s f o u n d i n t r a d i t i o n a l C A I l a n g u a g e s s u c h a s TUTOR, PILOT, INST, a n d COURSELJRITER.

I n t e l l e c t u a l Problems of t h e F u t u r e Co mp u t e r s T h a t T a l k L e t m e b r e a k t h i s d i s c u s s i o n of i n t e l l e c t u a l p r o b l e m s i n t o f o u r p a r t s t h a t w i l l t a k e u s b a c k t h r o u g h some of t h e e a r l i e r t e c h n o l o g i e s . The f i r s t p r o b l e m i s s i m p l y t h a t o f t a l k i n g ( o r a l s p e e c h ) . What d o e s i t t a k e t o g e t a c o m p u t e r t o t a l k ? The f a c t i s t h a t t h e t e c h n i c a l i s s u e s a r e a l r e a d y p r e t t y w e l l i n h an d . P e r h a p s t h e r e a d e r saw on t e l e v i s i o n "The F o r b i n P r o j e c t " - - a movie a b o u t two l a r g e c o m p u t e r s i n t h e S o v i e t Union a n d t h e U n i t e d S t a t e s g e t t i n g together t o dominate t h e world. To t h e c o g n o s c e n t i who h a v e s e e n t h a t m o v i e , l e t m e make a c a s u a l remark a b o u t t a l k i n g . A t e c h n i c a l c r i t i c i s m oE t h e movie i s t h a t t h e two v e r y l a r g e a n d s o p h i s t i c a t e d c o m p u t e r s were c o n d u c t i n g o n l y one c o n v e r s a t i o n a t a t i m e . Already i n o u r computer system a t S t a n f o r d w e h a v e e i g h t e e n c h a n n e l s of i n d e p e n d e n t s i m u l t a n e o u s t a l k a n d t h e c o m p u t e r t a l k s i n d e p e n d e n t l y a n d d i f f e r e n t l y t o e i g h t e e n s t u d e n t s a t t h e same t i m e . So you see, w e h a v e t h e c a p a c i t y f o r t h e c o m p u t e r t o t a l k . IJhat w e n e e d , h o w e v e r , i s b e t t e r i n f o r m a t i o n a b o u t what i s t o b e s a i d . F o r e x a m p l e , when I s e r v e a s a t u t o r , t e a c h i n g o n e of y o u , o r when o n e of you is t e a c h i n g m e , i n t u i t i v e l y a n d n a t u r a l l y we follow cues and say t h i n g s t o each o t h e r without having a n e x p l i c i t t h e o r y of how w e s a y what w e s a y . IJe s p e a k as p a r t of o u r h u m a n n e s s , i n s t i n c t i v e l y , on t h e b a s i s o f o u r p a s t e x p e r i e n c e . But t o s a t i s f a c t o r i l y t a l k w i t h a c o m p u t e r w e n e e d o n e x p l i c i t t h e o r y of t a l k i n g .

Co mp u te r s T h a t L i s t e n The r e p l a c e m e n t o f t h e w r i t t e n r e c o r d , t h e k i n d of r e c o r d t h a t was o b j e c t e d t o i n P l a t o ' s Phaedrus, c a n b e a v a i l a b l e t o u s i n t h e t a l k i n p , computer. The o t h e r s i d e of t h a t c o i n which S o c r a t e s a l s o emphasized, o r s h o u l d have

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emphasized, c o nc e r n s l i s t e n i n g . I t i s a much more d i f f i c u l t t e c h n i c a l problem. The p r o b l e m of d e s i g n i n g a c o m p u t e r t h a t c a n l i s t e n t o a s t u d e n t t a l k is much more d i f f i c u l t t h a n h a v i n g a s t u d e n t l i s t e n t o t h e c o m p u t e r t a l k . However, t h e p r o b l e m s a r e s o l v a b l e . For e x a m p l e , we j u s t c o m p l e t e d o u r f i r s t e x p e r i m e n t s w i t h t e n - y e a r - o l d s t a l k i n g t o t h e computer i n o r d e r t o g i v e a n s w e r s t o e l e m e n t a r y m a t h e m a t i c a l p r o b l e m s . Our c u r r e n t e x e r c i s e , t o g i v e you a s e n s e of w h e r e we a r e i n a c l a s s i c a l c u r r i c u l u m , i s s p e l l i n g o v e r t h e t e l e p h o n e . We c a l l t h e s t u d e n t a t home and g i v e him f i v e o r t e n m i n u t e s of s p e l l i n g l e s s o n s . D u r i n g t h i s p e r i o d h e s p e l l s o r a l l y and t h e c o m p u t e r t r i e s t o r e c o g n i z e what h e h a s s p e l l e d . T h i s r e c o g n i t i o n p r o b l e m h a s p r o v e d t o be d i f f i c u l t b u t n o t im p o ssib le o p e r a t i o n a l l y . The Use o f ---

Knowledge

The t h i r d p a r t of my d i s c u s s i o n c o n c e r n s t h e p r o b l e m of u n d e r s t a n d i n a t h e knowledge b a s e . To h a v e a n e f f e c t i v e c o m p u t e r - b a s e d s y s t e m o f i n s t r u c t i o n , w e must t r a n s c e n d m i n d l e s s t a l k i n g and l i s t e n i n g and l e a r n t o u n d e r s t a n d and use a l a r g e knowldge b a s e . For example, i f we w e r e s i m p l y t o r e q u i r e i n f o r m a t i o n r e t r i e v a l f r o m a knowledge b a s e , i t would b e r e l a t i v e l y s i m p l e i n t h e n e a r f u t u r e t o p u t t h e e n t i r e American L i b r a r y o f C o n g r e s s in e v e r y e l e m e n t a r y s c h o o l . The c a p a c i t y t o s t o r e i n f o r m a t i o n i s i n c r e a s i n g s o r a p i d l y t h a t we w i l l b e a b l e t o s t o r e much more i n f o r m a t i o n t h a n c o u l d e v e r p o s s i b l y be used.

A d i f f e r e n t and more d i f f i c u l t q u e s t i o n i s how t o g e t t h e s i z a b l e knowledge b a s e t o i n t e r a c t w i t h t h e s t u d e n t . A s we come t o u n d e r s t a n d how t o h a n d l e s u c h a knowledge b a s e , t h e s c h o o l c o m p u t e r of t h e f u t u r e s h o u l d b e a b l e t o a n s w e r any wayward q u e s t i o n t h a t t h e s t u d e n t might l i k e t o a s k . M o r e o v e r , as w e a l l know, o n c e a s t u d e n t uses s u c h a c a p a b i l i t y , h e w i l l h a v e a s t r o n a t e n d e n c y t o p u r s u e s t i l l f u r t h e r q u e s t i o n s t h a t a r e more d i f f i c u l t a n d more i d i o s y n c r a t i c . I t w i l l , I t h i n k , be w o n d e r f u l t o s e e how c h i l d r e n i n t e r a c t w i t h s u c h a s y s t e m ; i n a l l l i k e l i h o o d , we w i l l s e e c h i l d r e n g i v e t o l e a r n i n g t h e h i g h d e g r e e of c o n c e n t r a t i o n and t h e s u s t a i n e d s p a n of a t t e n t i o n t h e y now g i v e t o commercial t e l e v i s i o n . T h e r e is one r e l a t e d p o i n t I want t o e m p h a s i z e . From t h e v e r y b e g i n n i n g of s c h o o l , s t u d e n t s l e a r n q u i c k l y t h e "law of t h e l a n d " a n d know t h e y s h o u l d n o t a s k q u e s t i o n s t h e t e a c h e r c a n n o t a n s w e r . T h i s t a s k of d i a g n o s i n g t h e l i m i t a t i o n s o f t e a c h e r s b e g i n s e a r l y and c o n t i n u e s t h r o u g h c o l l e g e a n d g r a d u a t e s c h o o l . So, o n c e we h a v e t h e c a p a c i t y f o r a n s w e r i n g out-of-the-way q u e s t i o n s , i t w i l l be m a r v e l o u s t o s e e how s t u d e n t s w i l l t a k e a d v a n t a g e of t h e o p p o r t u n i t y and t e s t t h e i r own c a p a c i t i e s w i t h a r e l e n t l e s s n e s s t h e y d a r e n o t e n g a g e in now. Need f o r --

T h e o r i e s of L e a r n i n g and I n s t r u c t i o n

The f o u r t h p r o b l e m , and i n many ways t h e l e a s t - d e v e l o p e d f e a t u r e of t h i s t e c h n o l o g y , is t h e t h e o r y of l e a r n i n g a n d i n s t r u c t i o n . Ne c a n make t h e c o m p u t e r t a l k , l i s t e n , a n d a d e q u a t e l y h a n d l e a l a r g e knowledge d a t a b a s e , b u t w e s t i l l n e e d t o d e v e l o p a n e x p l i c i t t h e o r y of l e a r n i n g and i n s t r u c t i o n . In t e a c h i n g a s t u d e n t , young o r o l d , a g i v e n s u b j e c t matter o r a g i v e n s k i l l , a computer-based l e a r n i n g s y s t e m c a n k e e p a r e c o r d of e v e r y t h i n g t h e s t u d e n t d o e s . I t c a n know c o g n i t i v e l y a n enormous amount of i n f o r m a t i o n a b o u t t h e s t u d e n t . The p r o b l e m i s how t o u s e t h i s i n f o r m a t i o n w i s e l y , s k i l l f u l l y , and e f f i c i e n t l y t o t e a c h t h e s t u d e n t . T h i s i s s o m e t h i n g t h a t t h e v e r y b e s t human t u t o r d o e s w e l l , e v e n t h o u g h h e d o e s n o t u n d e r s t a n d a t a l l how h e d o e s i t , j u s t a s h e d o e s n o t u n d e r s t a n d how h e t a l k s . None of u s u n d e r s t a n d s how we t a l k and none of u s u n d e r s t a n d s how we i n t u i t i v e l y i n t e r a c t w i t h someone we

PAST, PRESENT AND FUTURE EDUCATIONAL TECHNOLOGIES

a r e t e a c h i n g on a one-to-one b a s i s . S t i l l . e v e n t h o u g h o u r p a s t and p r e s e n t t h e o r i e s o f i n s t r u c t i o n h a v e n o t c u t v e r y d e e p , i t d o e s n o t mean t h a t we h a v e n o t made some p r o g r e s s . F i r s t , we a t l e a s t r e c o g n i z e t h a t t h e r e i s a s c i e n t i f i c p r o b l e m ; t h a t a l o n e i s p r o g r e s s . One h u n d r e d f i f t y y e a r s a g o t h e r e was no e x p l i c i t r e c o g n i t i o n t h a t t h e r e was e v e n a p r o b l e m . T h e r e i s n o t s t a t e d i n t h e e d u c a t i o n l i t e r a t u r e of 150 y e a r s a g o a n y v i e w t h a t i t is i m p o r t a n t t o u n d e r s t a n d i n d e t a i l t h e p r o c e s s o f l e a r n i n g on t h e p a r t of t h e s t u d e n t . Only i n t h e 2 0 t h c e n t u r y do we f i n d a n y s y s t e m a t i c d a t a o r a n y s y s t e m a t i c t h e o r e t i c a l i d e a s a b o u t t h e d a t a . What p r e c e d e s t h i s p e r i o d i s romance and f a n t a s y u n s u b s t a n t i a t e d by any s o p h i s t i c a t e d r e l a t i o n t o e v i d e n c e . So at l e a s t we c a n s a y t h a t w e h a v e begun t h e t a s k .

C o n c l u d i n g Note The f i n a l i s s u e I w i s h t o d i s c u s s i s t h e p l a c e of i n d i v i d u a l i t y a n d human f r e e d o m i n a modern t e c h n o l o g i c a l s o c i e t y . The c r u d e s t f o r m o f o p p o s i t i o n t o w i d e s p r e a d u s e of t e c h n o l o g y i n e d u c a t i o n and i n o t h e r p a r t s of s o c i e t y i s t o c l a i m t h a t w e f a c e t h e real d a n g e r of men becoming s l a v e s t o m a c h i n e s . T h i s a r g u m e n t i s o r d i n a r i l y made i n a r o m a n t i c a n d n a i v e f a s h i o n by t h o s e who seem t h e m s e l v e s t o h a v e l i t t l e u n d e r s t a n d i n g o f s c i e n c e o r t e c h n o l o g y a n d how i t i s u s e d i n o u r s o c i e t y . The b l a t a n t n a i v e t e of some of t h e s e o b j e c t i o n s i s w e l l i l l u s t r a t e d by t h e s t o r y of t h e man who was o b j e c t i n g t o a l l f o r m s of technology i n o u r s o c i e t y and th e n i n t e r r u p t e d h i s d i a t r i b e t o s a y t h a t he had t o r u s h o f f t o t e l e p h o n e h i s d e n t i s t a b o u t a n a p p o i n t m e n t . No s c i e n t i f i c a l l y i n f o r m e d p e r s o n s e r i o u s l y b e l i e v e s t h a t o u r s o c i e t y c o u l d s u r v i v e i n a n y t h i n g l i k e i t s p r e s e n t f o r m w i t h o u t t h e w i d e s p r e a d u s e of t e c h n o l o g y . I t i s o u r p r o b l e m t o u n d e r s t a n d how t o u s e t h e t e c h n o l o g y a n d t o b e n e f i t w i s e l y from t h a t u se . I n d e e d , t h e c l a i m about s l a v e r y i s j u s t t h e o p p o s i t e of t h e t r u e s i t u a t i o n . I t i s o n l y i n t h i s c e n t u r y t h a t w i d e s p r e a d u s e of s l a v e r y h a s b e e n a b o l i s h e d , and i t may b e c l a i m e d by h i s t o r i a n s o f t h e d i s t a n t f u t u r e t h a t mankind c o u l d n o t do w i t h o u t s l a v e r y , b e c a u s e j u s t a s human s l a v e s are b e i n g a b o l i s h e d , w i t h i n a s h o r t t i m e s p a n t h e y w i l l b e r e p l a c e d by machine s l a v e s whose u s e w i l l n o t v i o l a t e o u r e t h i c a l p r i n c i p l e s and moral s e n s i b i l i t i e s . One c a n i n d e e d i m a g i n e a h i s t o r i c a l t e x t of 2500 o r 3000 A.D. a s s e r t i n g t h a t f o r a s h o r t p e r i o d i n t h e l a t t e r p a r t o f t h e 2 0 t h c e n t u r y t h e r e was l i t t l e s l a v e r y p r e s e n t on e a r t h . But t h e n i t was d i s c o v e r e d t h a t m a c h i n e s c o u l d b e made t h a t c o u l d do a l l t h e work of human s l a v e s , and so i n t h e 2 1 s t c e n t u r y t h e l u x u r y of s l a v e s , and t h e p e r s o n a l s e r v i c e t h e y a f f o r d e d , was b r o u g h t n o t t o t h e p r i v i l e g e d f e w , a s had h i s t o r i c a l l y b e e n t h e c a s e b e f o r e t h e 2 0 t h c e n t u r y , b u t t o a l l p e o p l e on e a r t h . I n o u r j u d g m e n t , t h e t h r e a t t o human i n d i v i d u a i t y a n d f r e e d o m d o e s n o t come f r o m t e c h n o l o g y , b u t f r o m a n o t h e r s o u r c e t h a t was w e l l d e s c r i b e d by J o h n S t u a r t M i l l i n h i s famous e s s a y & L i b e r t y , He said,

...

t h e g r e a t e s t d i f f i c u l t y t o be encount ered does n o t l i e i n t h e a p p r e c i a t i o n of means t o w a r d a n a c k n o w l e d g e d e n d , b u t i n t h e i n d i f f e r e n c e of p e r s o n s i n g e n e r a l t o t h e e n d i t s e l f . I f i t were f e l t t h a t t h e f r e e d e v e l o p m e n t o f i n d i v i d u a l i t y is o n e of t h e l e a d i n g e s s e n t i a l s of w e l l - b e i n g ; t h a t is n o t o n l y a c o - o r d i n a t e e l e m e n t w i t h all t h a t is d e s i g n a t e d by t h e terms c i v i l i z a t i o n , i n s t r u c t i o n , e d u c a t i o n , c u l t u r e , b u t is i t s e l f a n e c e s s a r y p a r t a n d c o n d i t i o n of a l l t h o s e t h i n g s ; t h e r e would b e no d a n g e r t h a t l i b e r t y s h o u l d b e u n d e r v a l u e d : and t h e a d j u s t m e n t o f t h e ' b o u n d a r i e s b e t w e e n i t and s o c i a l c o n t r o l would p r e s e n t no e x t r a o r d i n a r y difficulty

.

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J u s t a s books f r e e d s e r i o u s s t u d e n t s f r o m t h e t y r a n n y of o v e r l y s i m p l e methods o f o r a l r e c i t a t i o n , so c o m p u t e r s c a n f r e e s t u d e n t s f r o m t h e d r u d g e r y of d o i n g e x a c t l y similar t a s k s u n a d j u s t e d and u n t a i l o r e d t o t h e i r i n d i v i d u a l n e e d s . A s i n t h e case of o t h e r p a r t s of o u r s o c i e t y , o u r new a n d wondrous t e c h n o l o g y is t h e r e f o r b e n e f i c i a l u s e . I t is o u r p r o b l e m t o l e a r n how t o u s e i t w e l l . When a c h i l d o f s i x b e g i n s t o l e a r n i n s c h o o l u n d e r t h e d i r e c t i o n of a t e a c h e r , h e h a r d l y h a s a c o n c e p t of a f r e e i n t e l l i g e n c e a b l e t o r e a c h o b j e c t i v e knowledge He d e p e n d s h e a v i l y upon e v e r y word a n d g e s t u r e of t h e t e a c h e r of t h e w o r l d . t o g u i d e h i s own r e a c t i o n s a n d r e s p o n s e s . T h i s i n t e l l e c t u a l weaning of c h i l d r e n is a c o m p l i c a t e d p r o c e s s t h a t we do n o t y e t manage o r u n d e r s t a n d v e r y w e l l . T h e r e a r e t o o many a d u l t s among u s who a r e n o t a b l e t o e x p r e s s t h e i r own f e e l i n g s o r t o r e a c h t h e i r own j u d g m e n t s , We would claim t h a t t h e w i s e u s e of technology and s c i e n c e , p a r t i c u l a r l y i n e d u c a t i o n , p r e s e n t s a major o p p o r t u n i t y a n d c h a l l e n g e . We do n o t want t o claim t h a t w e know v e r y much y e t a b o u t how t o r e a l i z e t h e f u l l p o t e n t i a l of human b e i n g s ; b u t w e do n o t d o u b t t h a t o u r modern i n s t r u m e n t s c a n b e u s e d t o r e d u c e t h e p e r s o n a l t y r a n n y of o n e i n d i v i d u a l o v e r a n o t h e r , wherever t h a t t y r a n n y depends upon ignorance.

References

111 E. Macken a n d P. Suppes: E v a l u a t i o n s t u d i e s of CCC e l e m e n t a r y - s c h o o l c u r r i c u l u m s , CCC E d u c a t i o n a l S t u d i e s 1 ( 1 9 7 6 ) 1-37. 121 P. Suppes a n d M. M o r n i n g s t a r : C o m p u t e r - a s s i s t e d i n s ' t r u c t i o n , S c i e n c e 166 ( 1 9 6 9 ) 343-350. 131 P. Suppes: I m p a c t of c o m p u t e r s o n c u r r i c u l u m i n t h e s c h o o l s a n d u n i v e r s i t i e s , i n 0. Lecarme a n d R. L e w i s ( E d s . ) , _Computers i n e d u c a t i o n , p a r t I: E. Amsterdam: N o r t h - H o l l a a n d , 1975. 141 J. R. C a r b o n e l l a n d A. M. C o l l i n s : N a t u r a l s e m a n t i c s in a r t i f i c i a l i n t e l l i g e n c e , P r o c e e d i n g s of t h e T h i r d I n t e r n a t i o n a l Joint C o n f e r e n c e on A r t i f i c i a l I n t e l l i g e n c e , S t a n f o r d , C a l i f . , August 1973. 151 J . S. Brown, R. R. B u r t o n , and A. B e l l : An i n t e l l i g e n t C A I s y s t e m t h a t R o l t Reranek r e a s o n s a n d u n d e r s t a n d s (BBN R e p o r t 2 7 9 0 ) . Cambridge, Mass.: a n d Newman, 1974.

Developing Mathematics i n Third World Countries, M.E.A. E l Tom l e d . ) 0 North-Ho 1land Publishing Company, I9 79

RESEARCH AND HIGHER EDUCATION IN MATHEMATICS: THE PHILIPPIhF EXPERIENCE* Bienvenido F. Nebres

ABS'IRACT

The paper divides into three parts. The first part analyzes the present status of higher mathematics and research in the Philippines, concluding that while undergraduate mathematics training (for the B.S. in mathematics) has reached acceptable levels, graduate training is inadequate and research is barely existent, because professional mathematicians are few and the few that are around are engaged in many non-mathematical tasks. The second part traces the development of mathematics in the Ateneo de Manila: through the initial successes of the undergraduate program, and the difficulties and limitations of the graduate programs. A clear problem area that emerges is the lack -of fit between mathematics and Philippine society: unclear career opportunities for mathematicians, relatively low pay for professors and researchers, insufficient funds for the development of mathematics and science [even at minimal levels). The third and final part outlines plans and programs for the future: recruitment, fostering, training of mathematical talent from primary school on; improvement and standardization of the B.S. mathematics in the colleges by concentrating on three core courses as well as a systematization of the applied mathematics area (statistics, actuarial science, computer science, operations research) in the undergraduate curriculum; stabilization and expansion of the present M.S. and Ph.D. programs to meet the need for adequately trained college mathematics teachers and for a school of higher mathematics and mathematical research.

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Preliminary Remarks, The following effort at a description of the state of mathematical science in the Philippines today is in the nature of a personal and historical reflection on the work of individual mathematicians and mathematical groups. This may seem like a rather unscientific method but I have felt encouraged by the fact that in conversations with various mathematicians from more advanced countries, they tend to describe their own growth and development in the same fashion. This underscores, perhaps, the importance of tradition in mathematics and the growth of the discipline around individuals and schools - - a characteristic of the discipline that continues to this day. During one of the off days at a Sunnner School in Logic at Monash University, Australia, in 1974, I had occasion to talk with Prof. A. Mostowski on the development of mathematics in Poland. He dated its present development to the period when a few young Polish mathematicians in particular, Janiszewski, returned from studies in France and built a small school of disciples, starting research in specialized sections of topology and analysis (where they had a chance of competing with the outside world). From this school has grown Fundamenta Mathematicae and Fundamenta, of course, continues to reflect this initial tradition. Prof. B.H. Neumann of Canberra recalled similar beginnings in Australian mathematics and the strength of Australian mathematics in group theory, bears this stamp. More recently Prof. Y. Kawada of Japan was telling me of the beginnings of modern mathematics in Japan 11977 was the centennial year of the Mathematical Society of Japan) with its roots in the first few Japanese mathematicians to learn Western mathematics and develop schools of mathematics in Japan. Thus the examples here are efforts of individuals and groups with which we are most familiar - - the Ateneo de Manila and the Mathematical Society of the Philippines, the beginnings of serious mathematics at the Ateneo de Manila with the work of Profs. Wallace Campbell and Federico Sioson, the birth of the Mathematical Society of the Philippines in meetings of concerned mathematicians in a Manila club called the Philippine Colunbian. The intention is to try to understand where we are, to learn from the initial successes and present difficulties, to analyze failures and see what paths we are to explore for the future. The paper divides into three parts: Where are we today? - - the current status of research and graduate education in mathematics; How did we get here? - - past efforts,

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their successes and failures; Where do we go from here? - - the future of research and higher mathematics education in the Philippines. I.

Where are we today? The current status of research and higher education in mathematics in the Philippines.

How would we describe the state of mathematical development in the Philippines today? This question could be answered in two ways: we could compare the state of mathematics in the Philippines with that in other countries or we could discuss its internal role in the country, its interaction with Philippine Society. First in comparison with other countries. On the undergraduate level (college mathematics majors), the quality and training of students compare favorably. The Ateneo de Manila and the University of the Philippines graduate mathematics majors, who do quite well in graduate studies abroad [mostly in the United States). The only problem here is the relative fewness of the students, but this is more because of future job opportunities thgn interest or talent. -9

8

When we come to graduate education or research, however, the situation changes drastically. Graduate mathematics in the Philippines is in great part geared towards teacher training - - for college as well as primary and secondary school teachers. This is, of course, a necessary and important task. But it does not create much new mathematics nor does it encourage much research. To make the comparison more precise, we have found from the several mathematicians from Singapore who have spoken in our Sununer Institutes that the level of higher mathematical activity in our country does not even measure up to that in Singapore (with its population of 2 millions). Tney do serious mathematical research, they publish a research journal (Nanta Mathernatica), while we have not yet reached the point where mathematical research can be institutionalized. This is not to say that we have not made any progress in mathematics. The state of undergraduate mathematics education [for mathematics majors) has improved; on a limited level so has teacher education. But the total number of active mathematics Ph.D.'s is woefully inadequate. Many of those who have studies abroad have elected, for professional reasons, not to return. Those who are in the Philippines are busy at duties that are more innnediately pressing than the continuing study of mathematics, It would be quite correct to say that there is hardly a single person in the country today who is devoting all his efforts at mathematical

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study. How about the internal role of mathematics? What information do we have about the interaction between mathematics and Philippine society? What we know are data that point to lack of fit (between mathematics and Philippine Society).

Item.

Much of the vigor of mathematics in Singapore owes to the support of the Lee Kong Chian Institute at Nanyang University. It is difficult to believe that a country less than half the size of Metro Manila can invest in a venture, which the Philippines at the moment finds it cannot support. Or supports only after much persuasion - - if the amount of time and effort which Filipino mathematicians have to expend seeking necessary funds for their work could be spent on teaching and research, mathematics would be much further along the way. Item. There is the more general problem of the whole of Philippine Society's attitudes towards teachers and education. Salaries in education are so pitifully low (one-third to one-fifth what one can get in business and certain sectors of government). The consequences of such an undervaluation of education are obvious. Item, Few bright young people go into mathematics. Talent in the field is scarce. It is true that mathematics can survive in an economically hostile environment (witness but this requires the force of genius - - and this is not very much in evidence at the present.

w),

Item. The few trained mathematicians find themselves pursuing different avocations upon return to the country (for diverse and quite pressing reasons). The conclusion is that the mathematical (and generally the intellectual) tradition is weak or barely alive. Thus one finds that environment is at best not particularly encouraging for mathematics. Despite these general difficulties, there are some genuine breakthroughs, though rather fragile and unstable. On the level of research and higher mathematics, there is now a Ph.D. program which is to train ten mathematicians in the next five years. This is being undertaken by a consortium of three universities

RESEARCH AND HIGHER EDUCATION IN MATHEMATICS

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(University of the Philippines, Ateneo de Manila University, De La Salle University) and is being supported by the National Economic and Development Authority (NEDA) and the National Science Development Board (NSDB). On the Master's level, there are programs which are producing more teachers for college mathematics, mainly at the Ateneo de Manila and at the University of the Philippines, with support from government and private agencies. There is growing cooperation between different institutions and agencies to tackle the problem of mathematics education at the primary and secondary levels. Manila will host a Southeast Asian Conference on Mathematical Education, May 29 - June 3, 1978. This conference should serve as a focal point for efforts to improve the quality of mathematics education at all levels. These accomplishments are signs on the horizon than the fit between mathematics and Philippine society may improve over the coming years. But this remains a promise rather than a reality. For while these achievements are real indeed, they are fragile, resting more on the vision and efforts of individuals, than on the institutionalized goals of the larger society. 11. How did we get here? Past Successes and Failures We describe here the growth of mathematics at the Ateneo de Manila and the development of the Mathematical Society of the Philippines, because these are the efforts we are most familiar with. We feel, however, that they are representative of many other parallel efforts throughout the country (particularly at the University of the Philippines). The experience, we hope, can serve as a basis for the future analysis. THE A m 0 DE MANILA M4THDNTICS PROGRAM

The history of the program consisted of two phases: First phase. The beginnings of serious mathematics at the Ateneo de Manila can be quite precisely dated - - it came with the efforts (in the late fifties and early sixties) of two men: Professors Wallace Campbell and Federico Sioson. They formed a remarkable, complementary team, with Prof. Sioson as the inspiring teacher and mathematician, initiating us into the beauties and mysteries of higher and lower mathematics and Prof. Campbell as the organizer, logistics man, recruiter and dynamic mover. During this period of foundation, Prof. Campbell set up the mathematics programs in the Ateneo Grade School and High School (introducing the Addison-Wesley series). I personally participated in this effort by helping introduce the first new mathematics textbooks in the Ateneo High School

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in 1963-65. At the same time, the mathematics major in the college was built up to 60 units, a dynamic faculty was recruited and perhaps most improving, the program began to draw some of the best talent in the college. This period saw quite a few Ateneans go on for higher degrees in mathematics. As one reflects on the dynamic of this period, one can discern various components of the process of development of mathematics through the college level: - - introduction of a new curriculum and more rigorous textbooks in the grade school and high school (these were introduced in many grade schools and high schools through Summer Seminars)

-- together with intensive teacher training seminars (mostly during summers) to prepare teachers for the new mathematics - - personal monitoring of the process by Prof. Campbell and other staff of the College Mathemtics Department

- - introduction of a strong mathematics major in the college

- - together with recruitment, training, encouragement of highly trained faculty - - and recruitment of talented majors (Prof. Campbell used to teach the senior honors class in the Ateneo High School mainly to try to recruit the best students for mathematics)

Within a few years, the task of the development of mathematics education through the college level was complete.

-Phase Two:

The next task was to initiate and develop a graduate program - - towards the M.S. and Ph.D. This was necessary if we were to achieve stability a d not to be completely dependent on graduate training abroad for further sources of teachers and researchers. Dr. Sioson had initiated in the late sixties a highpowered graduate program leading to the M.S. and the Ph.D. Unfortunately, this was at the period when he was discovered to have terminal cancer and he never had the chance to develop the program. It is difficult to judge how far it might have gone, had he been around to direct and inspire it.

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When I returned in 1970, the graduate program was just getting off the ground, with few students for the M.S. and only one f o r the Ph.D. It was also the period of intense student activism prior to the declaration of martial law and the academic atmosphere was not very conducive to intellectual excursions into homological algebra o r topological groups. In any case I worked with the M.S. students to help them finish their degrees and started off a few others on their graduate work. More importantly, we felt the need to expand the numbers of students working for the M.S., as well as to initiate more intensive mathematical activity. These culminated in two institutional means: the establishment of a center for the training of college mathematics teachers at the Ateneo de Manila (supported by the Fund for Assistance to Private Education) and the beginning of the Mathematical Society of the Philippines. The expanded Graduate Program (for the training of college and mathematics teachers) ran into immediate difficulties because of the backgrounds of the applicants for the M.S. in Mathematics (who did not come from the Ateneo de Manila o r the University of the Philippines). It became clear that because of inadequate undergraduate preparation, it was not possible to give the usual M.S. program to the greater number of the students. Thus, while the graduate program has, indeed, expanded, it has not yet become the training ground for future research mathematicians - - at least not in the numbers needed for a research school to begin in the country. There is an important sequel to the initial success of the undergraduate mathematics program. In the latter part of the sixties, a new undergraduate program fcalled management engineering), with heavy concentration in operations-researchtype mathematics and behavioural science and geared for students going into management was introduced at the Ateneo de Manila. The program soon began to attract many of the students who previously were going into B.S. Mathematics. T h i s development is important because it tried to come to terns with the problem of career opportunities for students with a strong mathematics background. For it was soon found that relatively few of the students with a B.S. mathematics went on to become professional mathematicians. The others entered some kind of management job, often going on for further training in management. This may make the professional mathematician flinch, but we have to respect the fact that awareness in Philippine society is largely agricultural and commercial. Neither the industrial awareness with its higher technological requirements nor a straightforward intellectual awareness with its pursuit of creative thought is present to a sufficient

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In f a c t , it i s the problematic of appropriate career i n s e r t i o n t h a t has

extent.

made the development of a graduate program so d i f f i c u l t . How would we evaluate the above experience? We begin with the general evaluation t h a t the first phase met generally with success, but t h e second phase (graduate program and higher l e v e l research) met with many obstacles and though it is making some progress now, i t has not reached a s t a b l e s t a t u s . How do we account f o r these events? Why d i d mathematics develop with r e l a t i v e ease through the college l e v e l , but f a l t e r so badly on higher l e v e l s ? The f i r s t answer has t o be t h a t on t h e undergraduate l e v e l t h e student has t h e luxury of not having t o worry too much about f u t u r e career - - he can "indulge" i n serious mathematics and still have t h e option of going on t o engineering o r business. I n f a c t , t h i s has been i n s t i t u tionalized i n the Management Engineering Program a t t h e Ateneo de Manila, a program which one might legitimately describe as an offspring of the mathematics program, though a r a t h e r more vigorous offspring a t t h e moment. A t t h e graduate l e v e l , however, the student has t o take mathematics s e r i o u s l y as a c a r e e r . Unfortunately (the lack of f i t t h a t we talked of above), it is not too c l e a r w h a t mathematics means as a career i n the Philippines today. The lower s a l a r y s c a l e s i n u n i v e r s i t i e s , heavy loads which make research d i f f i c u l t - - these m i l i t a t e against the c l a s s i c career of a professional mathematician i n the u n i v e r s i t i e s . I t i s d i f f i c u l t , therefore, t o develop a r a t i o n a l e f o r greater numbers i n higher mathematics (everybody says we need them, but there r e a l l y a r e not many places f o r them -- places where they can earn a livelihood commensurate with t h e i r s t a t u s and be a b l e t o pursue mathematics a c t i v e l y ) . 111. Where do we go from here?

Plans and Prospects f o r the Future

The Mathematical Society of t h e Philippines was founded i n March 1973.

I t deve-

loped from meetings and discussions of concerned mathematicians throughout the year 1972. Fortunately, t h i s period of foundation coincided with t h e f i r s t b i annual meeting of the Southeast Asian Mathematical Society held i n Singapore i n July of 1972. Thus, from t h e very beginning t h e Mathematical Society of the Philippines and the Southeast Asian Mathematical Society worked together very closely. This has been of immense help i n the development of Philippine mathema-

tics. The MSP launched inunediately an ambitious series of seminars culminating i n the Southeast Asian Summer I n s t i t u t e i n Graph Theory held i n Manila i n May of 1975. Many of t h e l e c t u r e s preliminary t o the Summer I n s t i t u t e and during the Summer I n s t i t u t e itself came from Nanyang University, Singapore, with the assistance of the contacts we had made through the Southeast Asian Mathematical Society

.

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A National Congress was held in May 1973, a series of seminars in various univer-

sities through 1973-74, lecture series in Graph Theory in the Sumner of 1974, more seminars in 1974-75 and finally the S m e r Institute in 1975 with Prof. Claude Berge of the University of Paris, as Director. The goal of all these activities was to gather enough momentum so it would be possible to start at least a nucleus of intensive mathematical activity. Both the graduate program and the seminars had the goal of generating enough mathematical activity to create a stable, self-regenerating system. For at the moment, most of the inputs into the mathematical system are from the outside: either in the form of Filipinos returning from studies abroad or visiting mathematicians. While such inputs will continue to be needed and appreciated, there is urgent need for more stability and creativity on the local scene. Otherwise, there is danger that instability could make even the present achievements lose momentum and disappear. This search for a stable local group has led to the newest phase in these efforts to build up higher mathematics in the Philippines. The initiation of a Ph.D. program within the University of the Philippines-Ateneo de Manila-De La Salle University consortium in June 1977. The program expects to train ten Ph.D.'s in the next five years and is being supported by the Philippine Government. There are presently six students in the program drawn from the junior faculty of the three universities in the consortium, An important facet of this program is its linkage with the expected regional coordi ting center for mathematics in Southeast Asia to be established in Nanyang University, Singapore. We expect the students in the program to be doing their dissertation research with the assistance of faculty and facilities in the center. More specifically, Prof. Chew K i m Lin of Nanyang will be visiting Manila April-May this year to initiate a research seminar among the six students with a view to generating possible research topics in O.R. Hopefully, the students could pursue the necessary reading and preliminary work afterwards. They may then pursue their research in its final stages at Nanyang University. This was the pattern we followed in the case of the first Ph.D. in mathematics at the Ateneo de Manila: initial work in graph theory through contact with visiting professors from Nanyang and completing the work over a three month stay in Singapore. We are hopeful that this Ph.D. program will become the nucleus for the stable research and teaching group in mathematics which has been eluding us for the past many years.

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If one looks at the development of mathematics in other countries, one finds an interplay between breakthroughs of genius against environmental odds and the flowering of talent due to supportive measures. One may cite the heroic work of Abel in Norway as an example of the first case. He was often on the verge of starvation and died at the early age of 27. But his work and example planted the seed of a Norwegian mathematical tradition, For the second case, there was the systematic effort of Japan to send mathematicians and scientists for training abroad and the development of structures for training and research at home. In any case, it is clear that both forces are needed: the force of genius o r talent and structures that will train and foster mathematical talent. Without the first, structures w i l l degenerate into a bureaucracy. Without the second, mathematical tradition will not stabilize. The Philippine has to aim for an interplay of both forces: search for and development of mathematical talent and an effort to create the proper working conditions for trained mathematicians, There have been attempts at both [Science High Schools, scholarships for mathematics students in college, etc.) - - but the ultimate goal of establishing a mathematics school and tradition has not been clearly defined. Thus much effort and money has been dissipated without permanent achievement. In the concrete, the tasks are: (1) The identification and training of mathematically talented students from the

earliest years and through grade school and high school. B.S. mathematics courses in colleges, so there is substantial uniformity of training [with minimum standards applicable to all). (3) Stabilization and expansion of M.S. programs for college mathematics teachers and the Ph.D. program to create a core of mathematicians to do graduate teaching and research. (4) Eventual establishment of a Mathematics Institute or its equivalent to be the home of a genuine school of mathematics in the Philippines.

(2) The standardization of

The First Southeast Asian Mathematics Education Conference is to be held in Manila May 29-June 3, 1978. One of its major goals is towards the improvement of mathematical education in the primary and secondary grades. One hopeful outcome would be the development of centers in the various regions in the country, where talented students could get a superior education in mathematics and science in the elementary and secondary levels. Previous recommendations for the establishment of

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such centers have sometimes come under criticism of education specialists and officials as smacking of elitism. But an underdeveloped country which does not consider its talented children [particulary in the sciences) as a major national resource to be developed and fostered is condemned to continue underdeveloping. One of the greatest constraints to development in the country is the scarcity of trained talent at the top (except possibly in management) - - so that the few capable people are forced to spread their resources much too thin to achieve optimal results. This effort at establishing centers of mathematical excellence at the lower levels will produce a larger pool of students, who can then go on to further scientific training.

On the college level, there are two tasks. First, in terms of the preparation of future mathematicians there is need .to standardize the B.S. Mathematics in the different schools. It is proposed that we develop three hardcore courses: Linear algebra, advanced calculus, and number theory. Advanced calculus as a background for engineering, statistics, systems analysis. M b e r theory [taken in the broader combinatorial/algorithmic sense) for a sense of algorithms. Linear algebra, aside from its many applications, would provide a sense of abstract mathematics. The graduates of such programs should have the minimum prerequisites to go on for further training as college mathematics teachers or professional mathematicians. The mistake in the past was to try to do too much, i.e., to aim for a common curriculum - - and end up achieving too little - - since the contents and levels of complexity of all the courses in curriculum could not be prescribed. By concentrating on only three major courses, detailed specifications and standards can be endforced. The second task is to clarify the contents and directions of what we might call applied mathematics offerings (computer science, actuarial science, statistics, operations research). The mathematically talented students are increasingly turning to these applied areas if only because of the clearer career opportunities. Much work, however, has to be done to optimize the use of our resources (teaching manpower, computer time, etc.) for these training programs. Finally, on the graduate level, the task is for the M.S. and Ph.D. programs to stabilize and expand. For the M.S. program to grow sufficiently so as to able to meet the need for qualified college mathematics teachers in the country. Such expansion requires both additional funds and manpower. For the Ph.D. program to finally succeed in producing a core of mathematicians in research and higher level teaching. A necessary component of the latter effort is research publication - through journals here or abroad and through departmental monographs.

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Much, therefore, remains to be done. The only real basis of hope for the future is that genuine (though rather sloni and painful) progress has been made in the past. We would like to end by situating our above analyses and remarks in the larger national context. During the SEAMS meeting in Bandung, Indonesia in July 1976, I had occasion to talk with Prof. Jean Dhombres about our work. After listening patiently to the many non-mathematical tasks that have to be done to develop mathematics in the country, he remarked, "Yes, it is difficult to build a nation". We realize that mathematics is not the world. It certainly is not our developing nations. It is but a part (perhaps a modest part) of our national realities. And yet the building up of mathematics is part of the building up of our nations -- and we hope we can play OUT role (modes o r otherwise) in the task of building up our people to the stature and dignity that is our birthright. References 1. Sir James Lighthill, FRS, FIMA, "The Interaction between Mathematics and Society," Bulletin of the Institute of Mathematics and its Applications, vol. 12, no. 10, October 1976. (Reprinted in SEAMS Bulletin, vol. 1, no. 2 November 1977). 2. Lee Peng Yee, Development &Mathematics & Southeast Asia: The Fxperience -of the Southeast Asian Mathematical Society, International Conference on Developing Mathematics in Third World Countries, Khartoum 1978. 3. B.F. Nebres, "Mathematics and Mathematicians in the Philippines," Philippine Studies 21 (1973), pp. 409-23. 4. , v'Mathematics in the Philippines: Beginnings and Growth of the Mathematical Society of the Philippines, I 1 Southeast Asian Bulletin of Mathematics, vol. 1 , no. 1, May 1977, pp. 3-8.

Ateneo de Manila University

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APPENDIX: Some Notes on Mathematical Tradition The following are some reflections on the task of building a mathematical tradition in countries where it is weak or non-existent. In particular, we want to address ourselves to the young mathematics Ph.D. returning to this homeland after graduate studies abroad. His usual temptation is to build something in the image and likeness of what he has seen abroad only to find that this is not quite possible. So he throws up his hands in despair and joins the brain drain. There is a passage from E.F. Schumacher, which is apropos of this problem: Let us imagine a visit to a modern industrial establishment, say a great refinery. As we walk around in its vastness, through all its fantastic complexity, we might well wonder how it was possible for the human mind to conceive such a thing. What an immensity of knowledge, ingenuity and experience is here incarnated in equipment! How is it possible? The answer is that it did not spring ready-made out of m y person's mind - - it came by a process of evolution. It started quite simply, then this was added and that was modified, and so the whole thing became more and more complex. But even what we actually see in this refinery is only, as we might say, the tip of an iceberg. What we cannot see on our visit is far greater than what we can see: the immensity and complexity of the arrangements that allow crude oil to flow into the refinery and ensure that a multitude of consignments of refined products, properly prepared, packed and labelled, reaches innumerable consumers through a most elaborate distribution system. All this we cannot see. Nor can we see the intellectual achievements behind the planning, the organizing, the financing and marketing. Least of all can we see the great educational background which is the precondition of all, extending from primary schools to universities and specialized research establishments, and without which nothing of what we actually see would be there. A s I said, the visitor sees only the tip of the icebery: there is ten times as much somewhere else, which he cannot see, and without the "ten," the "one is worthless. And if the "ten" is not supplied by the country or society in which the refinery has been erected, either the refinery simply does not work or it is, in fact, a foreign body depending for most of its life on some other society. Now, all this is easily forgotten, because the modern tendency is to see and become conscious of only the visible and to forget the invisible things that are making the visible possible and keep it going."

*E.F. Schumacher, Small is Beautiful, Harper and Row, 1975,.pp. 164-65

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A fundamental problem to the task of building a mathematical tradition is to be aware of the rest of the iceberg, of the infrastructure that is a condition of possibility of the great mathematical centers in the world. A school of mathematical research requires at least as much societal infrastructure as Schumacher's factory. Unless we are aware of this need and try to identify and build such infrastructure, our mathematical house will be built on shifting sand. It is important, therefore, that we try to identify what is needed to build mathematics in our countries. That we also identify what is possible at a given stage of our history. If progress seems slow and inch by inch, this need not be a cause for discouragement. So long as the structures we are bulding are taking firm root. A corollary to the above remarks is that mathematics in Manila o r Khartoum need not look exactly like mathematics in Paris or London. At least not at this present stage of our histories. There is need to consider what might be appropriate mathematics in a given developing country. All too often the tendency is to transplant the mathemai$ics we learned abroad, without critical analysis as to the viability of the transplant in our radically different social and intellectual milieu. We have to give more thought to the type of mathematics that can take root in our countries and institutions. More attention to the task of identifying and building requisite infrastrmcture and "appropriate" mathematics, which has a chance of taking root in our given social and intellectual environment, will hopeful1 assist each one of us in the task of building in our countries a mathematical tradition that will endure.

Developing Mathematics i n Third World Countries, M . E . A . E l Tom (ed.) 0 North-Holland Publishing Company, 1979

ASPECTS OF THE RECENT DEVELOPMENT OF FUNCTIONAL ANALYSIS IN BRAZIL

Leopoldo Nachbfn Universidade Federal do Rio de Janeiror Brazil and University of Rochesterr USA

I would like to start out by expressing my sincere thanks to the University of Khartoum, and to the organizing committee of this inter national conference on developing mathematics in third world m u n tries, in particular to Professor H. E. A. El Tornr for inviting me to Sudan, and offering me this opportunity of giving a talk. Although I have had many opportunities of lecturing at universities in the h e r & canr Asian and European continentsr this is my first chance of visiting a university in Africa. As you may know, I am a native of a country, namely Brazil, whose population was originally to a large extent a mixture of white colonizers coming from Portugal and a negro influx from Africa. We do have in Brazil a strong African influence in our musicalr religious and popular culture. As a result, although I am of European descent

- my father was from Poland and my mother was

from Austria

- and

this

is my first stay in an African country, I somehow have the impression of being at home while I am here in Sudan. As a matter of factr we in Brazil as a whole, and our mathematical camunity there in particular# have a keen interest in aaking closer and stronger our ties with Africa.

The title of my lecture may be misleading# as it might give some of you the wrong impression that I am going today to describe theo-

rems proved by the functional analysis school in Brazil. This, however, would not be suitable for this conference. Regarding my query to Profeesor El Tomr about the type of lecture I was supposed to de81

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liver. he kindly told me that perhaps it is imgortant to note in this respect that our audience ia predominantly formed by mathematicians. and that we expect the conference to produce some concrete proposals. Accordingly. I am going to describe some aspects of my personal experience in attempting to develop mathematics in Brazil ( a country of the so-called third world according to a terminology that I do not like, I am sorry for that). Hy description starts in 1945. vhen Andre W d l went to Brazil

for a stay of three years. He waa followed in 1946 by Jean Dieudonnh. vho visited Brazil for two years. It was under their strong influence in Braril that I acquired my Bourbaki-like training, which by the way was extremely suitable to my natural inclination as a youngster. Well, in 1947 we had a three month visit by Marshall Stone (who, by the way, is with us today attending this conference). I remember very well the inspiring course on rings of continuous functions that he then offered. Stone had Just finished h i s famous paper "A generalized Weierstrass approximation theorem" published in volume 21 (1948)

of Mathematics Magazine. Among other things. he presented in that course his extension of the classical Weierstrass approximation theorem. I mean what is well-known as the Weierstrass-Stone theorem. I was very impressed by Stone*s course and it had a lasting influence

on my career. as I now will describe. Bit by bit, through an interesting psychological process that I do not have the t h e to describe here in full detail. I got interest-

ed in Serge Bernstein*e approximation problem. which is a nore g m e r -

a1 way of looking at the Weierstrass approximation theor= through the use of wcsightls.

I then succeeded in extending the Bernstein problem and its solp tion along the same spirit that Stone had used in connection with the Weierstrass theorem. The point now was the use of modules of continu-

RECENT DEVELOPMENT OF

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ous functions, instead of just algebras of such functions, and the use of general weight functions and weighted topologies. instead of just characteristic functions and compact-open topologies. My first note on this research was published in volume 47 (1961) of the Proceedings of the National Academy of Sciences of the USA. At the 1962 International Congress of Mathematicians at Stockholm, I gave an invited lecture which appeared in the Proceedings of the Congressr on this and related subjects. It was followed by an article in volume 01 (1965) of the Annals of Mathematics. and by my monograph "Elements of approximation theory" published in the USA in 1967. in Paul Halmose series Van Nostrand Mathematical studies. There is in Brazil a flourishing research school devoted to approximation theory including some of my former doctoral students. namely Silvio Machado. Joao Bosco Prolla and Guido Zapatan and their doctoral students. As some of the highlights of our recent activities

in this direction, may I quota Prolla's monograph "Approximation of vector valued functions" published in Netherlands in 1977, and an International Symposium on Approximation Theory held in 1977 in Brazil, the Proceedings of vhich has Prolla as its Editor and will be publish ed in Netherlands in 1979. Let me turn now to a different aspect of our research activities in functional analyeis in Brazil. While I vas visiting the University

of Chicago at the invitation of Marshall Stone. for two years during 1948-1950. I was told by Andrd Weil that 1 should learn distribution theory as it was going to become very important, As a matter of fact. weil handed to me copies of Laurent Schwartz' famous papers on distri butions that appeared in the Annales de l'fnstftut

Fourier, in vol-

ume 21 (1945) and volume 23 (1947-1948). Later onr Laurent Schvartt vieited Brazil for three months in 1952. and offered ua a remarkable course on diutribution themrp. It uas under his influence. and his subsequent work

on integration in tQ

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pological spaces that are not necessarily locally compact. that I became strongly interested in extending distribution theory from finite to infinite dimensions. My set of notes "Lectures on the theory of distributions" appeared in the USA in 1963. I must admit that so far

I have failed in such an attempt. There has been work along this line by Paul Kree and Bernard Lascar, besides others, but we are still far

from a final theory a8 needed, say by mathematical physics. Hovever, the classical Paley-Hiener-Schwartt theorem in finite dimensions concerning the Fourier transforms of distributions with bounded supports led me from distributions in infinite dimensions to the study of holomorphy in infinite dimensions. In thia respect, I would like to quote my book 'Topology on spaces of holomorphic mappings" published in West Germany in 1969, and my euxvey lecture at the 1971 Winter meeting o f the American Hathematical Society on nRecent developments in infinite dimensional holomorphyn. There is in Brazil a flourishing research school devoted to infi nite dimensional holomorphy including some of my former doctoral students, namely Jorge Albert0 Barroso. Hirio Matos and Jorge Mujica, as well as other doctoral students of theirs and mine. As some of the highlights of our recent activities in this direction, may I quote Barroso's manograph of an "Introduction to holomorphy between normed spaces" in Spanish plblished in Spain in 1976s an International sympq sium on Infinite Dimensional Holomorphy held in Brazil in 1975, the Proceedings of which has Xatoa as its Editor. published in Netherlands in 1977; and an International Seminar on Holomorphy held in Brazil in 1977, the Proceedings of which has Barroso as its Editor, to be published in Netherlands in 1979. L e t me turn now to another aspect having to do with teaching and

not with research proper. In 1948. we had in Brazil a visit of thrmonths by Warren Ambrose. He offered us a nice course on harmonic analysis and group representation. It was under its influence and also

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of Andrd Weil*s book on the integration on topological groups that,

in 1960, I published in Brazil an elementary text on "The Haar integral" in Portuguese. Copies of my text were sent to Marshall Stone and Paul Halmos. As a result, my book was translated into English and published in the USA in 1965 in Stone.8 The University Series in High er Mathematice. This elementary book of mine wile successful among mathematics students as a first reading on this subject. To my big surprise hoveverr it also became successful among mathematical statis ticians who use integration on homogeneous spaces. What is the reason

for this success? It is very simple. I just wrote an elearentary text in Portuguese for the Brazilian students, full of motivation, examples and historical notes. Had I written a more sophisticated book in English for international studentsr I most likely would have had much less success. This is an example of a humble enterprise meant only for a developing country, but actually useful in developed countries. So farr I have described some of my experimces in Brazil under

the direct influence of leading mathematicians visiting the country. Let me turn now to an aspect of my activities that I did on my ownr let us say in a sort of intellectual isolation. In 1947, before I left Brazil for the first time in 1948, I got interested in the study of topological ordered spaces as an extension of general topology, which is the case when inequality becomes equality. I published three notes siunmarizing my vork in volume 226 (1948) of the Comptes Rendus Be l*Acad&ie

des Sciences de Paris. In 1950r at the very end of my

tvo year stay at the University of Chicago during 1948-1950r I prepared a thesis on that subject to present, the same year, to the Federal University of RiO de Janeiro (then known as University of Brazil) as a candidate for a professorship there. I got appointed only in 1972..., twenty two years later, after a series of events which are typical of a developing country. This monograph on *'Topology and order" M S printed in Portuguese in the USA. Later on, this volume

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was translated into English and published in the USA in 1965 in Paul

Halnos* series Van Nostrand Mathematical studies. Since thenr my work on topological ordered spaces became better h o r n r and subsequently was used by workers in differential equations. probability theory, pp tential theory and mathematical statistics. What have X then tried to prove in this lecture, in terms of prQ spective proposals resulting from this conference? Well. that it is possible to further teaching and research in mathematics in a develog ing country. and still win some international recognition, provided certain conditions are met. Firstly, it ia absolutely essential to have frequent and close personal contact with distinguished mathematicians from the major international centers, to get information and inspiration from them cop cerning the main stream of mathematics and applicatians. and also to acquire high standards resulting from sound tradition. Secondly. it is also quite fundamental to have local ingenuity, in order to be able to concentrate on w e l l selected areas of reseach, and develop them in a m y . so to speak. independent of the major international centers. It is a balance between these two apparently opposite tendencies that leads to the establishment of a university school of mathematics in developing countries. Last, but not least, one needs a lot of good luck:..,

REFERENCES 1. J. A. BARROSO.

*~Introducci6na la holomorfia entre espacios nof-

madose@, Universidad de Santiago de Compostela, Spain (1976). 2 . J. A. BARROSO (Editor),

"Adnnces in holomorphynr Notas de Hate-

mbtica. to appear. North-Holland. Netherlands ( 1979).

3. N. C. MANS (Editor).

"Infinite dimensional holomorphy and appli

cations". Notas de Matel&tiCa.

W l , 54. North-Holland. Netherlands

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a7

(1977).

4. L. NACHBIN,

"Elements of approximation theory**,Van Nostrand

Mathematical Studies, vol. 14, Van Nostrand, USA (1967). Reprinted by Robert Krieger, USA (1976).

5. L. NACHBIN,

"Lectures on the theory of distributions", Universi-

ty of Roche~ter,USA (1963). Reprinted in Textos de Matembtica, ~ 1 1.5 , Univeraidade do Recife+ Brasil (1964). 6. L. NACHBINr

Vopology on spaces of holomrphic mappings'** Erge-

bnisse der Hathematik und ihre Grenzgebiete, vol. 47, SpringerVerlag, west Germany (1969).

7. L, NACHBIN,

"Recent developments in infinite dimensional holouq

phy", Bulletin of the American Mathematical Societyr vol. 79, pp. 625-640, USA (1973). 8. L. NACHBXN,

"Integral de Haar", Textos de Matemzitica, vol. 7, U-

niversidade do Recifer Brasil (1960). Translated as "The Haar intg gral". The University Series in Higher Mathematics, vol. 15, Van Nostrand, USA (1965). Reprinted by Robert Krieger, USA (1976). 9. L. NACHBIN,

nTopologia e ordem*@,University of Chicago Press,

USA (1950). Translated as T q ~ ~ l o gand y order", Van Nostrand Hathematical Studies, vol. 4, Van Nostrand, USA (1965). Reprinted by Robert Krieger, USA (1976). 1O.J. B. PROUA,

"Approximation of vector valued functionsn, Notae

de Mateardtica, vol. 61, North-Holland, Netherlands (1977). 1l.J. B. PRO=

(Editor),

"Approximation t h e m y and functional anal

ysis", Notas de Hatemdtica, to appear, North-Holland, Netherlands (1979).

This Page Intentionally Left Blank

Developing Mathematics in Third World C o m t r i e s , M.E.A. E l Tom (ed.) @ North-Holland Publishing Company, 1979

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS Jerzy GREN Central School of Planning and Statistics Warsaw POLAND

1 . Introduction

It is well known that a quick application of the scientific and technical progress to the practice depends to a great extent on the highly qualified researchers and managers employed in modern institutes and enterprises. Working for the above organizations they should often use t

modern mathematical and statistical methods, so they have to be acquainted, among others, with the two basic sections of applied mathematics, i.e. the probability theory and mathematical statistics. The modern mathematical and statistical methods are playing really important role in the economics and engineer ing.

i)

We may give here three main examples:

econometric models -explaining the structure of processes in the economy or enabling their prediction,

ii)

optimal decision models for decisions under uncertainty,

iii)

methods of statistical quality control.

The process of a fast and wide education of the scientific cadre (both in th’e form of regular or extramural and 89

90

J. GREN

postgraduate s t u d i e s ) i n t h e f i e l d of a p p l i e d mathematics, and e s p e c i a l l y i n p r o b a b i l i t y c a l c u l u s and s t a t i s t i c s , req u i r e s t h e a p p l i c a t i o n of m o r e and m o r e e f f i c i e n t d i d a c t i c methods.

The programmed t e a c h i n g i s a v e r y e f f i c i e n t method

f o r t e a c h i n g mathematics.

A s p e c i a l b l o c k method o f program-

med t e a c h i n g w a s d e v e l o p e d i n P o l a n d .

The a i m of t h i s p a p e r

i s t o p r e s e n t a c o n c e p t o f a programmed t e x t b o o k o n mathe-

matical s t a t i s t i c s which w a s b a s e d o n t h e above method. The a u t h o r h a s w r i t t e n t h i s manual f o r s t u d e n t s a t t h e C e n t r a l S c h o o l of P l a n n i n g and S t a t i s t i c s i n Warsaw.

2 . Block Method o f P r o g r a r n m e d T e a c h i n q

Every d i d a c t i c p r o c e s s s h o u l d r e a l i z e t h e two b a s i c goals : A) Teaching t h e i n f o r m a t i o n , i . e . ,

t h e elements of

a g i v e n domain o f knowledge. B ) T r a i n i n g i n a b i l i t i e s and h a b i t s o f p r a c t i c a l

activities. Those aims

-

-

a s many d i d a c t i c e x p e r i m e n t s h a v e shown

c o u l d b e r e a l i z e d w i t h t h e h e l p o f v a r i o u s methods o f

t e a c h i n g and c o n t r o l l i n g t h e p r o g r e s s made by s t u d e n t s . However, i n c o m p a r i s o n w i t h t r a d i t i o n a l methods, t h e soc a l l e d programmed t e a c h i n g methods ( d e v e l o p e d s i n c e f i f t i e s ) e n a b l e t h e more g e n e r a l and f a s t e r r e a l i z a t i o n o f b o t h these goals. A s opposed t o t h e t r a d i t i o n a l d i d a c t i c methods,

programmed t e a c h i n g methods a r e c h a r a c t e r i z e d by t h e following principles:

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

1.

T e a c h i n g ma te ria l i s d i v i d e d i n t o some b a s i c d o s e s o f information.

2.

The s t u d e n t g o e s t o t h e n e x t amount o f i n f o r m a t i o n o n l y a f t e r d i g e s t i n g t h e information contained i n t h e former one.

3.

I n o r d e r t o check whether t h e s t u d e n t i s accustomed t o a g i v e n amount of i n f o r m a t i o n , p r o p e r t e s t s a r e i n t r o d u c e d a n d t h e a n s w e r s compared.

4.

The r a p i d i t y of l e a r n i n g i s a d e q u a t e t o t h e i n d i v i d u a l a b i l i t i e s of a student.

5.

The d i d a c t i c t e x t ( i n t h e form o f a p r o g r a m ) i s i n t r o d u c e d i n programmed t e x t b o o k s o r w i t h t h e h e l p o f s p e c i a l d i d a c t i c machines. T h e r e a r e many d i d a c t i c programming m e t h o d s , among

o t h e r s a r e t h e main l i n e a r d i d a c t i c p r o g r a m s ( i n t r o d u c e d by B. S k i n n e r , H a r v a r d U n i v e r s i t y ) , b r a n c h e d p r o g r a m s (N.A.

Crowder, C h i c a g o U n i v e r s i t y ) and mixed d i d a c t i c

programs

( K . Austwick,

Sheffield University).

A l l t h e s e methods a r e b a s e d o n t h e f i v e p r i n c i p l e s

given above.

I n e a c h of t h o s e methods t h e whole t e a c h i n g

m a t e r i a l i s d i v i d e d i n t o l a r g e r o r s m a l l e r amounts of i n f o r m a t i o n , a f t e r which d i f f e r e n t ,

s p e c i f i c f o r a given

method, t e s t s a r e g i v e n ( e . g . f i l l i n g o f g a p s , s e l e c t i o n of a proper answer, e t c . ) . According t o t h e o b t a i n e d r e s u l t s t h e s t u d e n t i s d i r e c t e d t o t h e n e x t p o r t i o n o f m a t e r i a l or t o t h e a d d i t i o n a l e x p l a n a t i o n s a n d r e p e t i t i o n s , d e p e n d i n g o n his i n d i v i d u a l abilities.

I n t h i s way t h e c h a r a c t e r i s t i c , f o r a g i v e n

method, p r o c e s s a r i s e s w h i c h i s c a l l e d a d i d a c t i c p r o g r a m ,

91

92

J. GREN

and which enables more individual activities of a student in comparison with traditional methods of learning. Many years didactic experiences have proved that programmed teaching methods are particularly efficient in teaching foreign languages or sciences.

Especially in mathematics

where the student meets a large number of definitions and theorems which should be remembered. The elements of a didactic program, independently of the applied methods, are the amounts of basic information, proper tests and instructions concerning the sequence of students passing through particular amounts of information and tests.

In a didactic program the tests are playing

the most essential role since those are the tools for discovering individual errors in student's considerations, and for displaying his gaps in digesting basic information and abilities of its practical use. In the early seventies, at Warsaw University, a group of experts in dydactics, headed by Professor Czeslaw Kupisiewicz, has developed a Polish type of dydactic programming.

A starting point for this method was the assumption

'that a dydactic program should include tests in the form of small scientific problems which require from the student an intense intellectual effort and connecting various elements of the knowledge acquired so far.

Such, similar

to the "case studies", tests, being rather scientific problems, are developing well a scientific activity of a student during the teaching process, and are especially useful at the higher school levels.

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

The Polish method of didactic programming/programmed teaching is called the block method, because it contains, for each part of teaching material, six blocksmutually connec ted : I

-

information block

-

containing some amount of basic

information (notions, definitions, theorems) , TI

-

information testsblock

-

checking the extent of

digesting the information f r o m block I by the student, KI

-

correction-information block

-

containing additional

instructions and explanation in order to correct wrong decisions of a student, P

-

problem block

-

containing the formulation of the

so-called problem task to be solved individually by the student, TP

-

problem test block

-

checking the correctness of

a problem task solution, KP

-

correction-problem block - containing additional explanation in case of wrong/false solution of a problem task. Of course, when the student has gone through the

test block, and, if necessary, through the correction blocks, he may go to the next amount of basic material. Among the six blocks mentioned, four blocks: I, TI, P, TP are of a basic character, whereas KI and KP are auxiliary blocks, i.e. they are used by the student who had some difficulties with information test or with the solution of problem task.

93

J. GREN

94

A g e n e r a l flow-diagram of a b l o c k method of program-

med t e a c h i n g i s as f o l l o w s ( k

-

t h e number of a s e q u e n t i a l

amount of i n f o r m a t i o n ) :

Test results

Basic

L F

Problem 7

KP-k

1

I

I

Solution

+

Additional information

As w e may see i n t h i s d i a g r a m , t h e s h o r t e s t p a t h f o r a b l e s t u d e n t s i s i n t h i s method:

(I-k)---&TI-k)

-4P-k)

(TP-k)---c(I-k+l). I t i s o b v i o u s t h a t programmed t e a c h i n g b a s e d o n t h i s

b l o c k method r e q u i r e s a s p e c i a l , programmed manual.

Applied

m a t h e m a t i c s , and p a r t i c u l a r l y t h e p r o b a b i l i t y t h e o r y and mathematical s t a t i s t i c s a r e p a r t i c u l a r l y f i t t i n g t o t h i s method of p r o g r a m e d t e a c h i n g .

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

According to this assumption the author has tried recently to apply this method for programed teaching of mathematical statistics. The first task was to write a proper, programed manual.

3. General concept of the programed textbook on mathematical statistics Basing on the block method of didactic programing the author has written recently a programmed textbook on mathematical statistics for University students in Poland, especially for the students at the Warsaw Central School of Planning and Statistics where he is lecturing, and where an experimental group was organized for testing and verification of this textbook.

The text book is now in print in the

State Scientific Publishers (PWN) in Warsaw.

It consists

of about 500 pages and covers such essential topics in probability theory and mathematical statistics which are necessary in economic and industrial practice.

Those are

presented in the form of a didactic block program according to the mentioned above Polish block method of programed teaching. The list of 40 basic information blocks (I-k for k = 1,.,.,40),

appearing in the program, is as follows:

1-1

Random variables and their probability distributions

1-2

Random vectors and their probability distributions

1-3

Expectation and moments of random variable

1-4

Correlation matrix and regression functions

1-5

Some important discrete distributions

95

J. GREN

96

1-6

Some important continuous distributions

1-7

Statistical populations and random samples

1-8

Empirical frequency distribution from a sample

1-9

Sample statistics and their distributions

1-10 Methods for determination of sampling distributions 1-11 Sampling distributions of the sample mean 1-12 Sampling distributions of the sample variance 1-13 Sampling distributions of the sample proportion 1-14 Sampling distributions of sample moments and order

statistics 1-15 Sampling distributions of correlation and regression

coefficients 1-16 Statistical point estimation of parameters, unbiased

estimator 1-17 Efficiency of estimators 1-18 Consistency of estimators 1-19 Sufficiency of estimators 1-20 Moment method of estimation 1-21 Maximum likelihood estimators 1-22 Least Squares estimators 1-23 Bayes and minimax estimators 1-24 Interval estimation of parameters 1-25 Confidence intervals for the population mean 1-26 Confidence intervals for the population variance 1-27 Confidence interval for the population proportion

(or probability) 1-28 Confidence intervals for correlation and regression

coefficients

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

1-29 Nonpar ametr ic estimation 1-30 Estimation in finite populations 1-31 General concept of testing statistical hypotheses 1-32 Most powerful parametric tests 1-33 Likelihood ratio tests 1-34 Significance tests of means and proportions 1-35 Siginificance tests of variances 1-36 Analysis of variance 1-37 Significance tests of correlation and regression coefficients 1-38 Chi-square and Kolmogorow tests of goodness of fit I- 39 The Chi-square test of independence 1-40 Nonparametric tests for two samples According to the adopted in this textbook the block method of programmed teaching each of the above basic information blocks I-k is completed with blocks: TI-k, KI-k, P-k, TP-k and KP-k which contain either the proper tests or additional, correcting and explaining information, while the problem block P-k contains more complicated problem task connected with a real economic and engineering pratice.

In order to explain a detailed internal structure of the dydactic block program adopted in the discussed textbook we introduce a selected part concerning the outline of the block 1-18 (Consistency of estimators) as well as the detailed contents of the blocks TI-18, KI-18, P-18, TP-18

and KP-18.

97

J. GREN

98

4. Excerpt from the Author's Programmed Textbook on Mathematical Statistics -

1-18

1

Consistency of estimators (The outline of the contents of 6 pages in the original version of this block in the textbook)

1. Definition: An estimator Z is said to be a consistent

estimator of parameter

8

if, for all

lim P { ( Z , - 8 n-tm 2 . Explanation of this definition the equality

(<E)

-

E >

0 holds

= 1

a consistent estimator

produces the sequence estimates which approaches population parameter as the sample size

n

increases

3 . Utilization of notation of probability limit (convergency

in probability to 8

):

plim Zn = 8 n-tm

4.

Equivalence of the definition of consistency with the law of large numbers for statistic

Zn

5. The Bernoulli law of large numbers and consistency of relative frequency-m as the estimator of probability p n 6. The Kchinchin law of large numbers and consistency of

a sample moment

A,=

2.

xr

as the estimator of

i=l r-th moment of population mr

7. The theorem on consistency of the adequate order statistic as estimator quantile in population

X

P

of the order

p

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

8.

Slutsky theorem: If

plim Xn = c, and g(x) is continuous

nor rational, then

9.

Theorem: If E(Zn) =

0

lim D'(Zn) n+m of 0 .

= Of

(unbiased) or then Zn

lim E(Zn) = 8 , and n-tm is a consistent estimator

10. Examples of consistency: sample median as an estimator

of population mean

m

in normal population N ( m f o 2 ) 3

and sample variance variance o

TI-18

2

SL

as an estimator of population

.

In this block we shall check if the reader understands the notion of consistency of an estimator and if he is able to utilize adequate theorems given in the block 1-18 in order to check if any estimator is consistent or not.

The test consists

of three parts with adequate auxiliary frames. Please find and write into the frame A the

A 1-[.

[

[

number made from the digits corresponding to the sequential numbers of sentences (from the below set) which are not true:

1. The consistent estimator assures for large sample a small

probability for a large estimation error. 2 . The consistent estimator is such an estimator which is

stochastically convergent to zero.

99

100

J. GREN

3 . A sufficient condition for consistency of estimator is

the convergence to zero of its variance a

n-. n in 4 . Order statistic 2;") which is of the order 10 10 n-elements simple sample is a consistent estimator of the first centile of population. 5.

Standard deviation

S

from a simple sample is a consistent

estimator of the parameter

in the normally distributed

a

population. B. 1 Statistical population has probability dis:

tribution with the probability density function as follows:

Parameter 6 is known. As the estimator of the estimated parameter

s= n n-l XI

N

we take

where 2 is arithmetic mean from the n-

elements simple sample taken from this population. Answer if 2 is a consistent estimator of parameter Please write into the frame B the number 1 answer is: yes, or the number 2

-

-

N

?

if your

if the answer is: not

(for nonconsistent estimator). Statistical population has a geometrical distribution defined by the probability function: p(x) = ~ ( 1 - p ) ~for

x

= 0~1,2~...

101

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

A s a n e s t i m a t o r o f t h e unknown p a r a m e t e r

zn

-

- , l+Z

where

x is

w e propose

p

t h e a r i t h m e t i c mean f r o m n - e l e m e n t s

s i m p l e s a m p l e which w a s t a k e n from t h i s p o p u l a t i o n .

Check

i f t h i s estimator i s c o n s i s t e n t f o r parameter p? P l e a s e w r i t e i n t o t h e f r a m e C t h e number 5

-

i f your

r e s u l t i s t h a t e s t i m a t o r Z n i s c o n s i s t . e n t o r t h e number 1 0

-

i f t h e i n v e s t i g a t e d estimator i s n o n c o n s i s t e n t .

Now p l e a s e sum u p t h e t h r e e numbers f r o m f r a m e s A , 3 a n d C i n t h e t e s t TI-18,

a n d t h e o b t a i n e d sum s h o u l d be p u t i n t o

t h e f i n a l frame:

Now p l e a s e c h e c k i f t h e number you

obtained i s 240?

I f y e s , t h e whole t e s t TI-18 i s d o n e c o r r e c t l y and you may

go t o t h e b l o c k P-18.

I f t h e number you h a v e f o u n d i f d i f f e r e n t f r o m 2 4 0 , p l e a s e go t o t h e b l o c k KI-18.

1 KI-18

T e s t TI-18 h a s shown t h a t you d o n o t u n d e r s t a n d

as y e t t h e n o t i o n o f e s t i m a t o r c o n s i s t e n c y and

you a r e n o t a b l e t o u s e p r o p e r t h e o r e m s c o n c e r n i n g t h i s notion.

W e s h a l l t r y now t o f i n d and improve y o u r errors

made i n t h e t e s t TI-18.

F i r s t of a l l you s h o u l d c h e c k w h e t h e r

t h e f r a m e s A , B a n d C ( t e s t TI-18) 2 3 4 , 1 and 5 , r e s p e c t i v e l y .

i n c l u d e t h e numbers:

O t h e r numbers, w r i t t e n b y y o u ,

s h a l l i n d i c a t e i n which p a r t o f t h e t e s t TI-18 you h a v e made an e r r o r .

-------------- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ --------- ---------

102

J. GREN

The frame A of the test TI-18 should include the number 234 since in the given set of sentences only the first and

the last one are true, while the second, third and fourth are not true. Sentence 1 is true since the limit appearing in the esti~mator consistency may be also written in the form: lim P{IZn - 0 [ > ~ =1 0 for E>O which means that n-tm the larger sample, the smaller is the probability of a large (exceeding the number

E)

estimation error of the parameter

8, with the help of a consistent estimator Zn.

Sentence 2 is false since a consistent estimator is the

-_.__

estimator stochastically convergent to the estimated parameter 8 , but not to zero. The difference IZn

-

81

is

stochastically convergent to zero for a consistent estimator of parameter 8. Sentence 3 is not true since the convergence to zero of the estimator variance is not enough for that, that the estimator is consistent.

Moreover, the estimator should be at least

asymptotically unbiased. : " ) Sentence 4 is false since order statistic 2

10

as the

first decile from a sample, is a consistent estimator of the first decile, but not the population centile. Sentence 5 is true since as we have stated in the block 1-12, the standard deviation from a simple sample derived

from normal population N(m,02), has for the large samples 2

an asymptotically normal distribution N ( o ,U~ ; i ) .

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

103

This is then the estimator unbiased asymptotically with a variance concurrent to zero with n-,

and therefore it is

the consistent estimator of parameter u of the population.

............................................................ The frame B (test TI-18) should include the number 1 since for given in exercise B the distribution of population with a known parameter 5,estimator of the form ?=

52

is a consistent estimator of parameter a . In order to show this it is enough to notice that this is the estimator asymptotically unbiased with a variance tending to zero, as n-. It is easy to notice that the given function of population distribution density corresponds to the distribution of a gamma type with parameters b =

8 and p

Hence the mean population amounts to

= 8. = a,

m = E(X) =

and variance in population amounts to a 2 = D 2 (X)

=

E2

=

22 B .

2

and D2 (X) = n , for the investigated estimator we obtain an expected value n n E ( X ) = n-l E ( X ) = n-l a , and variance

Since for each population E(Z)= m

D 2 (X)

2 =

D 2 (X) =

(n-1) Estimator

2

2

n

(n-1)

(n-11

a

2

2 B '

x is then the biased estimator of parameter a , -

n n- 1 a

-

1 a = n- 1 a+O when i.e. this is an asymptotically unbiased estimator

but its bias is: bn n-,

n ~

of parameter a.

= E(X)

a

=

J . GREN

104

Since simultaneously

D

2

( i ) +0

as

,

n+m

then by

virtue of a proper theorem we obtain that estimator

?

is

a consistent estimator of parameter a.

The frame C (test TI-18) should include the number 5, since given in exercise C estimator

Zn -

-

is a con-

l+Z

sistent estimator of parameter p of the population in the geometric distribution.

In order to show that, we should

use the Slutsky theorem on a stochastic convergence of the sequence

g(Xn)

for

plim Xn = c and rational function n+m g, since as it is easy to notice, estimator Zn is a rational function of the mean from sample

about which we

know that it is a consistent estimator of the mean value m of population.

For appearing in exercise

C a geometric

distribution of population there occurs, as we know,

-

g = l-p i.e. the arithmetic mean X from P P , a simple sample taken at random from this population holds

m = E(X)

=

the relation

plim n+m theorem we obtain

plim n-

Zn

=

p lim n-

?

=

P

1 -1+Z

.

Hence by virtue of Slutsky

1

-

lxplim 2 n+m

1

1

+

- 1

1

-

P.

P

Since plim Zn = p then the estimator Zn is a consistent n+m estimator of parameter p, and therefore the frame C (test TI-18) should include number 5 .

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

105

Now you s h o u l d solve t h e f o l l o w i n g a d d i t i o n a l p r o b l e m : A general population has the exponential d i s t r i b u t i o n

f ( x ) = a e-ax

with d e n s i t y f u n c t i o n o f t h e form: x>O.

P a r a m e t e r a i s unknown.

As i t s

t h e r e c i p r o c a l o f a r i t h m e t i c mean

?

for

estimator w e t a k e

from n-elements

s i m p l e sample t a k e n a t random f r o m t h i s p o p u l a t i o n .

w r i t e i n t o t h e f r a m e g i v e n b e l o w t h e symbol

2

Please

i f the

g i v e n estimator i s a c o n s i s t e n t e s t i m a t o r of p a r a m e t e r

o r p l e a s e w r i t e t h e symbol

N,

a,

i f t h i s e s t i m a t o r i s non-

consistent.

Now p l e a s e c h e c k i f t h e symbol i s

Z

s i n c e s u c h s h o u l d be

y o u r d e c i s i o n i f you h a v e a p p l i e d t h e p r o p e r t h e o r e m t o determine

1

plim n+m

X

.

I f you h a v e w r i t t e n t h e symbol

Z

i n t o t h e f r a m o f addi-

t i o n a l e x e r c i s e p l e a s e g o t o t h e b l o c k P-18.

I f you w e r e n o t a b l e t o e s t a b l i s h i f t h i s e s t i m a t o r i s cons i s t e n t o r n o n c o n s i s t e n t e s t i m a t o r o f p a r a n e t e r a, o r you h a v e p u t a n

N

-

e s t i m a t o r i s nonconsistent, p l e a s e once

more s t u d y t h e b l o c k 1-18 and t r y t o s o l v e p r o p e r l y t h i s additional exercise.

7

Now you h a v e t h e f o l l o w i n g problem t o s o l v e .

106

J. GREN

In some electronic equipment plant the radio lamps are produced.

The receiver is investigating first of all the

lifetime of produced volume, basing on a large sample of lamps taken at random.

It is knownthat the lifetime of

a radio lamp is a random variable with the distribution defined by density function of the form: f(x) = k 2 xe-kx for x>O, where parameter

k

is unknown.

In order to estimate the parameter

k

of a radio lamp

lifetime distribution, the producer is proposing the application of one of the two estimators of parameter determined from a simple sample of

n

k,

lamps:

To which of these two estimators of parameter

k

should

agree the receiver of lamps? Please write into the frame below the letter

if, in your

opinion, the receiver should agree to the estimator or write the letter

B

,:Z

if you think that the receiver

should choose the estimator

:2

of the parameter

k.

Please go now to the block TP-18.

TP-18

In order t o solve properly t h e problem P-18

107

P R O G U D TEACHING OF PROBABILITY AND STATISTICS

the sample is large), and then in case when both estimators are consistent, the selection of a more efficient one is possible.

Is that your way of thinking?

If not, please try

again to solve the problem P-18.

_-___-_-------- _---------------............................ If you have properly based the solution of problem P-18 on investigating at first the consistency, and eventually after, the efficiency of proposed estimators of parameter k, you should come to the conclusion that the receiver of lamps B should accept the estimator Zn.

you have put a

B

If such is your answer and

into the frame, please go to the next

information block 1-19.

............................................................ If you are not able to choose which of the two proposed estimators of parameter

k

of the lamp lifetime distribu-

tion should be chosen by the receiver or your choice is wrong (you have put an

A

into the frame), please go now

to the block KP-18.

I---KP- 18

A proper solution of the problem P-18 should be

started with the investigation of consistency

of the two proposed estimators of parameter

k

of the

radio lamp lifetime distribution. The property of consistency is deciding here, taking into account a large sample.

The

estimator which would be nonconsistent is not worthy of recommendation.

Only if it has turned out that both estimators

3 are consistent you should choose a more efficient Zt and Zn

one.

J. GREN

108

The investigation of consistency of the estimators proposed by the producer should be based on the Slutsky theorem (in block 1-18) since both estimators

and

2 ;

are rational

2:

functions of the arithmetic mean 2 from sample:

It should be noticed that a given, in the problem P-18, form of the function of a lamp lifetime distribution density, is the function of distribution density of a gamma type with parameters

b = k, p

= 2,

amounts to

m = E(X)

=

the expected value of which

.

=

From the Kchinchin law of

large numbers there results a consistency of the mean 2 as an estimator of parameter

m

of the general population,

i.e. for the population given in problem P-18 occurs:

-

plim X = m = n+Applying for

p lim n-t-

zA n

=

=

.

2

-

k

the theorem

estimator :2

p lim n- 1 n-tm nX

=

n- 1

.

lim n-tm

n- 1 p lim n n-

.

1 1

-

X

:=I. X

plim n-t-

, we have:

1 ~

plim 2 n-tm

This result means the nonconsistency of estimator :2 parameter 2;

k.

for

Whereas similar calculations for estimator

give the result plim n

: , 7

=

k, i.e. the estimator

2;

PROGRAMMED TEACHING OF PROBABILITY AND STATISTICS

is a consistent estimator of parameter

k,

I09

and only this

estimator for a large sample should be accepted by the receiver of lamps already produced.

This is the reason why

the final frame in problem P-18 should include a letter B. Now please repeat your calculations in order to prove the consistency of estimator Z Bn which should be chosen as a result for problem P-18. Then you may go to the next information block 1-19 (Sufficiency of estimators).

5. Conclusion

a) In teaching mathematics and applied mathematics (e.g. statistics) the programmed teaching methods should be applied more and more widely since they enable differentiated, efficient studies, especially for the extramural students or for those who are making up their arrears. b) The Polish block method of programmed teaching, due to the problem block P - K ,

is strongly developing the acti-

vity of students in solving the tasks similar to those which could be met in practice. c) Programmed teaching method requires writing and verification of a special programmed textbook. d) Similarly like in the case of a textbook on mathematical statistics discussed here it is possible to write programmed manuals for teaching other mathematical topics, e.g. econometrics, mathematical analysis, geometry, etc.

J. GREN

110

e) Programmed teaching may be applied not only at the university level, but also at lower levels of teaching.

References 1. Gren J.

- "Mathematical Statistics - Programmed Text-

book" (-tobe published in Polish by State Scientific Publishers (PWN), Warsaw). 2. Lumsdaine A., Glaser R.

med Learning.

-

"Teaching Machines and Program-

A Source Book", Washington 1960, National

Education Association of the US. 3. Kupisiewicz Cz.

-

"Methods of Didactic Programming"

(in Polish), Warsaw 1974, PWN.

Developing Mathematics in Third World C o w t r i e s , M.E.A. E l Tom (ed.) Q North-HoZland Publishing Company, 1979

MATHEMATICAL MODELS O F SCHISTOSOMIASIS

I n g e m a r NASELL Department o f Mathematics, Royal I n s t i t u t e o f Technology, S-100

4 4 S t o c k h o l m 7 0 , Sweden

s UMMARY A r e v i e w i s g i v e n o f m a t h e m a t i c a l models f o r t h e t r a n s m i s s i o n o f

s c h i s t o s o m i a s i s . The models a r e u s e d t o s t u d y the’ i n f l u e n c e o f biol o g i c a l and e n v i r o n m e n t a l f a c t o r s o n e n d e m i c i n f e c t i o n l e v e l s , c o n t r o l e f f i c i e n c i e s , and e r a d i c a t i o n e f f o r t s . Q u a l i t a t i v e f e a t u r e s a n d t h r e s h o l d e f f e c t s a r e e m p h a s i z e d . The m o d e l s r e v i e w e d a c c o u n t f o r e x t e r n a l i n f e c t i o n , l a t e n c y a n d d i f f e r e n t i a l m o r t a l i t y among t h e s n a i l s , d i f f e r e n t modes o f m a t i n g , c o n c o m i t a n t i m m u n i t y i n t h e h o s t p o p u l a t i o n s , and a g g r e g a t i o n a s c a u s e d e . g .

by nonhomogeneous e x p o s u r e o f

human b e i n g s t o i n f e c t i o n .

1. I n t r o d u c t i o n Schistosomiasis is a tropical p a r a s i t i c disease t h a t is estimated to a f f e c t more t h a n 2 0 0 m i l l i o n p e o p l e . I t i s , t o g e t h e r w i t h o t h e r t r o pical diseases,

r e g a r d e d a s a major i m p e d i m e n t t o t h e a l l e v i a t i o n o f

poverty i n developing c o u n t r i e s o f t h e t h i r d world. S c h i s t o s o m i a s i s p o s e s a s e r i o u s world h e a l t h problem. I t i s one o f t h e s i x d i s e a s e s i n c l u d e d i n t h e S p e c i a l Programme f o r R e s e a r c h and T r a i n i n g i n T r o p i c a l D i s e a s e s r e c e n t l y l a u n c h e d j o i n t l y by t h e World H e a l t h O r g a n i z a t i o n (WHO) a n d t h e U n i t e d N a t i o n s Development Programme (UNDP)

.

The p u r p o s e of t h e p r e s e n t p a p e r i s t o d i s c u s s s e v e r a l m o d e l s f o r t h e t r a n s m i s s i o n o f s c h i s t o s o m i a s i s . The p a p e r i s o f a r e v i e w c h a r a c t e r . H y p o t h e s e s , m e t h o d o l o g y , and model p r o p e r t i e s a r e d e s c r i b e d w h i l e ref e r e n c e s a r e made t o t h e l i t e r a t u r e f o r f u r t h e r d e t a i l s and f o r mathematical d e r i v a t i o n s . I n f e c t i o n i n a community o c c u r s a s a r e s u l t o f v a r i o u s b i o l o g i c a l and e n v i r o n m e n t a l i n f l u e n c e s . T h e models a l l o w a s t u d y o f i n f e c t i o n l e v e l s a n d o f e f f i c i e n c e s o f a c t i o n s aimed a t c o n t r o l o r e r a d i c a t i o n . A l o n g - t e r m g o a l o f t h e development o f s c h i s t o s o m i a s i s models i s t o

a c h i e v e s u c h a h i q h d e g r e e of r e a l i s m t h a t t h e m o d e l s c a n b e o f u s e i n t h e p l a n n i n g o f c o n t r o l o r e r a d i c a t i o n programs. T h i s r e q u i r e s Ill

I12

I. NASELL

t h a t a l l mechanisms t h a t a r e e s s e n t i a l f o r t h e t r a n s m i s s i o n o f t h e i n f e c t i o n be i n c o r p o r a t e d i n t o t h e model s t r u c t u r e . W e i n t r o d u c e t o o l s and c r i t e r i a t h a t a l l o w one t o judge i f a mechanism h a s a n e s s e n t i a l o r a n o n - e s s e n t i a l i n f l u e n c e on t h e epidemiology and cont r o l of t h e i n f e c t i o n . Many o f t h e p a r a m e t e r s t h a t d e t e r m i n e t h e p a r a s i t e p o p u l a t i o n e x h i b i t s u b s t a n t i a l t e m p o r a l and s p a t i a l v a r i a b i l i t y . I t i s t h e r e f o r e n o t v e r y m e a n i n g f u l t o demand a c c u r a t e q u a n t i t a t i v e p r e d i c t i o n s from t h e models. I n s t e a d , t h e emphasis t h r o u g h o u t t h e p a p e r i s on q u a l i t a t i v e a s p e c t s and on order-of-magnitude

q u e s t i o n s . The s e a r c h i s f o r models

t h a t show q u a l i t a t i v e agreement w i t h o b s e r v e d d a t a . Only when s u c h agreement h a s been r e a c h e d i s i t m e a n i n g f u l t o p r o c e e d w i t h model refinements of a q u a n t i t a t i v e n a t u r e . A b r i e f d e s c r i p t i o n of t h e l i f e c y c l e o f t h e c a u s a t i v e p a r a s i t e

( s c h i s t o s o m e ) f o l l o w s . The l i f e c y c l e i s i m p o r t a n t s i n c e i t s s t r u c t u r e i s r e f l e c t e d i n t h e m a t h e m a t i c a l models. Human b e i n g s s e r v e a s d e f i n i t i v e h o s t s and c e r t a i n s p e c i e s o f f r e s h w a t e r s n a i l s s e r v e a s i n t e r m e d i a t e h o s t s f o r t h e s c h i s t o s o m e s . Sexua l m a t i n g between male and female s c h i s t o s o m e s t a k e s p l a c e i n human b e i n g s , w h i l e t h e r e p r o d u c t i o n i s a s e x u a l i n s n a i l s . The t r a n s m i s s i o n o f t h e i n f e c t i o n t a k e s p l a c e m a i n l y i n t r o p i c a l v i l l a g e s where i t is a i d e d by p o o r h y g i e n i c c o n d i t i o n s . Some o f t h e f e r t i l i z e d e g g s l a i d by mated female p a r a s i t e s l e a v e the body o f t h e h o s t w i t h t h e body w a s t e s . They h a t c h i f and when t h e y r e a c h f r e s h w a t e r . The r e s u l t i n g small larvae, c a l l e d miracidia, a r e infectious t o c e r t a i n species of s n a i l s . I n f e c t e d s n a i l s r e l e a s e swarms o f a second t y p e o f l a r v a e , c a l l e d c e r c a r i a e , i n t o t h e w a t e r . Human b e i n g s become i n f e c t e d by merely i n s e r t i n g t h e i r hands i n t o w a t e r c o n t a i n i n g c e r c a r i a e , o r by wading i n i t w i t h b a r e f e e t o r by any o t h e r d i r e c t body c o n t a c t wi.th such water. Any a c t i o n t a k e n t o c o n t r o l o r e r a d i c a t e t h e i n f e c t i o n i n a community c a n b e viewed a s a n i n t e r r u p t i o n a t s o m e p o i n t o f t h e p a r a s i t e ' s l i f e c y c l e . This viewpoint i s reflected i n t h e mathematical t r e a t m e n t of such a c t i o n s .

An a u t h o r i t a t i v e d i s c u s s i o n of t h e epidemiology of s c h i s t o s o m i a s i s

MATHEMATICAL MODELS OF SCHISTOSOMIASIS

113

i s g i v e n by J o r d a n and Webbe ( 1 9 6 9 ) . E x c e l l e n t r e v i e w s o f m a t h e m a t i c a l models o f s c h i s t o s o m i a s i s a r e g i v e n by F i n e ( 1 9 7 6 ) and Cohen (1977). 2 . The B a s i c Model

The t r a n s m i s s i o n of s c h i s t o s o m i a s i s i n a community c a n b e viewed a s an e c o l o g i c a l problem. The p o p u l a t i o n s o f t h e l i f e c y c l e s t a g e s o f t h e s c h i s t o s o m e s are d e t e r m i n e d by b i o l o g i c a l , e n v i r o n m e n t a l , h y g i e n i c , and b e h a v i o u r a l f a c t o r s . A main t a s k o f t h e models i s t o r e f l e c t the corresponding relationships.

W e proceed h e r e t o e s t a b l i s h a b a s i c

model f o r t h e t r a n s m i s s i o n of s c h i s t o s o m i a s i s . All o t h e r models t o he d e s c r i b e d c a n b e viewed a s v a r i a t i o n s o r e x t e n s i o n s of t h e b a s i c mod e l . F o r f u r t h e r d e t a i l s , see N d s e l l and H i r s c h ( 1 9 7 3 ) . The d e f i n i t i v e and i n t e r m e d i a t e h o s t p o p u l a t i o n s are assumed t o b e c o n s t a n t i n t i m e . T h e i r s i z e s a r e d e n o t e d by N, and N 2 ,

respectively.

The r a t e o f e g g l a y i n g by a mated f e m a l e s c h i s t o s o m e i s d e n o t e d by X I and t h e r a t e o f s h e d d i n g o f c e r c a r i a e by an i n f e c t e d s n a i l i s X 2 .

,

The

p r o b a b i l i t y f o r a c e r c a r i a of i n f e c t i n g a g i v e n human b e i n g i s p , , and t h e p r o b a b i l i t y t h a t a n e g g w i l l l e a d t o i n f e c t i o n o f a g i v e n s n a i l i s p2. (This implies t h a t the success p r o b a b i l i t i e s of c e r c a r i a e and e g g s a r e p l N ,

and p2N2, r e s p e c t i v e l y . )

s c h i s t o s o m e i s c o n s t a n t and e q u a l t o p l ,

The d e a t h r a t e p e r m a t u r e

and t h e r e c o v e r y r a t e f o r

i n f e c t e d s n a i l s i s c o n s t a n t and e q u a l t o p 2 . Thus a t o t a l o f e i g h t p a r a m e t e r s h a v e b e e n i n t r o d u c e d . They are assumed t o be s t r i c t l y pos i t i v e t o avoid t r i v i a l s i t u a t i o n s . I t i s p r o f i t a b l e f o r the succeeding discussion t o introduce a trans-

f o r m a t i o n o f t h e e i g h t - d i m e n s i o n a l p a r a m e t e r s p a c e o n t o a two-dimens i o n a l one by w r i t i n g (1)

T I = P1X2N2/”1

and T 2 = p2XlN1/p2

The q u a n t i t i e s T,

.

(2)

and T 2 a r e s e e n t o b e d i m e n s i o n l e s s . They a r e re-

f e r r e d t o a s t r a n s m i s s i o n f a c t o r s . They c a n b e viewed a s e p i d e m i o l o g i c a l c o u n t e r p a r t s t o R e y n o l d ’ s number i n hydrodynamic t h e o r y . I t w i l l be s e e n i n what f o l l o w s t h a t t h e y s u f f i c e t o d e s c r i b e many o f t h e i m p o r t a n t p r o p e r t i e s o f t h e model. The b a s i c m a t h e m a t i c a l model c o n s i s t s o f a c o l l e c t i o n o f M a r k o v c h a i n s

I.

114

where F ( k ) ( t ) ( M ( k ) ( t ) )

NASELL

s t a n d s f o r t h e number o f female ( m a l e ) p a r a -

s i t e s i n d e f i n i t i v e h o s t k a t t i m e t , and S ( t ) s t a n d s f o r t h e number

of i n f e c t e d s n a i l s a t t i m e t . Each o f t h e p r o c e s s e s F ( k ) , M ( k )

i s an

immigration-death p r o c e s s w i t h i m m i g r a t i o n r a t e a t t i m e t g i v e n by

and c o n s t a n t d e a t h r a t e 11,

p e r i n d i v i d u a l . Mating between a v a i l a b l e

male and f e m a l e p a r a s i t e s i s assumed t o t a k e p l a c e monogamously w i t h o u t d e l a y . The number o f p a i r e d f e m a l e p a r a s i t e s i n h o s t k a t t i m e t is P ( ~ ()t ) = min ( M ( ~ () t ), F ( ~ () t ) ) .

The p r o c e s s S i s an i n f e c t i o n - r e c o v e r y p r o c e s s . I f S ( t ) = i , t h e n t h e infection rate a t t i m e t is N.

and t h e r e c o v e r y r a t e p e r i n f e c t e d s n a i l i s p 2 . The model whose h y p o t h e s e s a r e d e s c r i b e d above i s a h y b r i d model. The two s t a g e s of t h e p a r a s i t e ' s l i f e c y c l e s p e n t i n d e f i n i t i v e and i n t e r m e d i a t e h o s t s a r e modelled by s t o c h a s t i c p r o c e s s e s whose i n f l u e n c e s on e a c h o t h e r a r e d e t e r m i n i s t i c and g i v e n by e x p e c t a t i o n s . The hypot h e s e s can b e used t o d e r i v e a s y s t e m o f d i f f e r e n t i a l e q u a t i o n s s a t i s f i e d by t h e e x p e c t a t i o n s a p p e a r i n g i n t h e i m m i g r a t i o n and i n f e c t i o n r a t e s . T h i s s y s t e m t a k e s a s i m p l e form i n t h e s p e c i a l case when t h e ( 0 ) a r e i d e n t i c a l and g i v e n i n i t i a l d i s t r i b u t i o n s of F ( k ) ( 0 ) and 1 by P o i s s o n d i s t r i b u t i o n s w i t h common e x p e c t a t i o n 2 w o . By w r i t i n g

w(t)

=

E [ F ( k ) ( t ) + M ( k ) ( t )] f o r t h e e x p e c t e d number o f p a r a s i t e s p e r

d e f i n i t i v e h o s t ( " t h e worm l o a d " ) and y ( t ) = p r o p o r t i o n o f i n f e c t e d s n a i l s , one f i n d s

EIS(t)l N2

f o r the expected

MATHEMATICAL MODELS OF SCHISTOSOMIASIS

115

i s t h e e x p e c t e d number o f p a i r e d p a r a s i t e s p e r h o s t when t h e e x p e c t e d number o f p a r a s i t e s i s w. The case w i t h a r b i t r a r y i n i t i a l d i s t r i b u t i o n s of F ( k ) ( 0 ) and M ( k ) ( 0 ) i s t r e a t e d i n N a s e l l a n d H i r s c h ( 1 9 7 3 ) . The e m p h a s i s i n what f o l l o w s i s on t h e a s y m p t o t i c b e h a v i o u r o f t h e s o l u t i o n o f i n i t i a l v a l u e problem ( 3 )

-

( 6 ) . I t c a n b e shown t h a t t h e

g e n e r a l f e a t u r e s o f t h i s b e h a v i o u r d o n o t depend on t h e i n i t i a l d i s t r i b u t i o n s o f F ( k ) ( 0 ) and M ( k ) ( 0 ) . Thus c o n s i d e r a t i o n o f t h e s p e c i a l c a s e r e p r e s e n t e d by t h e a b o v e s y s t e m o f e q u a t i o n s d o e s n o t c o n s t i t u t e a

loss o f g e n e r a l i t y . A b i f u r c a t i o n phenomenon c a n b e e s t a b l i s h e d f o r t h e i n i t i a l v a l u e

-

problem ( 3 )

( 6 ) . The a p p r o p r i a t e p a r a m e t e r s p a c e i s t h e p o s i t i v e

- T p l a n e . T h e r e e x i s t s a t h r e s h o l d f u n c t i o n TI 1 2 t h a t s e r v e s t o d e f i n e two s u b s e t s F1 and F2 of t h e p a r a m e t e r s p a c e ,

q u a d r a n t of t h e T

defined a s follows:

The t o p o l o g i c a l s t r u c t u r e o f t h e s e t o f a l l s o l u t i o n s o f

( 3 1 , (4) is

changed a s t h e p a r a m e t e r v e c t o r ( T I , T 2 ) i s moved a c r o s s t h e boundary between t h e s e t s F T

1

and F 2 ' i . e .

a c r o s s t h e t h r e s h o l d f u n c t i o n where

1 -- TI ( T 2 ) . The number o f c r i t i c a l p o i n t s of t h e s y s t e m o f e q u a t i o n s

(31,

( 4 ) i s one when t h e p a r a m e t e r v e c t o r l i e s i n t h e s e t F 1 , b u t i t

i n c r e a s e s t o t h r e e i f t h e p a r a m e t e r v e c t o r i s moved i n t o t h e s e t F 2 . The p o i n t ( 0 , O ) i s a c r i t i c a l p o i n t o f (31, ( 4 ) f o r a l l p a r a m e t e r v e c t o r s . The two a d d i t i o n a l c r i t i c a l p o i n t s t h a t a p p e a r when t h e parameter vector lies i n F

2 are

1 = (B1 ( T I r T 2 ) I B1 ( T I , T 2 ) / T 1 ) and z 2 = ( B 2 ( T1 , T 2 ) , B 2 ( T 1 , T 2 ) / T 1 ) , w h e r e , a s i n d i c a t e d by t h e n o t a t i o n , t h e c o o r d i n a t e s a r e g i v e n by functions B

1

and B 2 o f t h e t r a n s m i s s i o n f a c t o r s T I and T 2 . The c r i t i -

I. NASELL

116

c a l p o i n t z1 i s u n s t a b l e , w h i l e t h e c r i t i c a l p o i n t s (0,O)

and z 2 a r e

a s y m p t o t i c a l l y s t a b l e . One c a n show t h a t t h e s o l u t i o n o f i n i t i a l v a l u e problem ( 3 )

-

(6)

f o r a n y admissible i n i t i a l v a l u e s (wo20, O0) approaches a

and f o r a n y a d m i s s i b l e p a r a m e t e r v a l u e s ( T I > O

c r i t i c a l p o i n t o f ( 3 ) , * ( 4 ) as t i m e t + W e c o n c l u d e from t h i s t h a t i f t h e p a r a m e t e r v e c t o r l i e s i n F, , t h e n t h e s o l u t i o n of ( 3 ) 6) a p p r o a c h e s t h e c r i t i c a l p o i n t ( 0 , O ) as t + m f o r a l l admissible i n i 03.

-

t i a l v a l u e s . On t h e o t h e r h a n d , i f t h e p a r a m e t e r v e c t o r l i e s i n F 2 ' t h e n t h e asymptotic behaviour o f t h e s o l u t i o n i s n o t independen of

t h e i n i t i a l v a l u e s . The s e t o f a l l admissible i n i t i a l v a l u e s c o n t a i n s

t w o s u b s e t s , o n e c o n t a i n i n g t h e p o i n t (0,O) z 2 , such t h a t t h e s o l u t i o n of

(3)

-

and the o t h e r containing

( 6 ) f o r any i n i t i a l v a l u e s i n

e i t h e r o f the t w o s u b s e t s a p p r o a c h e s t h e a s y m p t o t i c a l l y s t a b l e c r i t i -

c a l p o i n t belonging t o t h a t s u b s e t as t n o t e d domains o f a t t r a c t i o n o f ( 0 , O ) s o l u t i o n of

(3)

-

--t

m.

The two s u b s e t s a r e d e -

and o f z 2 , r e s p e c t i v e l y . The

( 6 ) f o r any i n i t i a l v a l u e on t h e boundary between

t h e t w o domains o f a t t r a c t i o n a p p r o a c h e s t h e u n s t a b l e c r i t i c a l p o i n t z1

as t

+ m.

es z1 as t

The s e t o f i n i t i a l v a l u e s f o r which t h e s o l u t i o n a p p r o a c h -

+ m

h a s m e a s u r e 0 . On t h e b a s i s of t h i s r e s u l t w e r u l e o u t

t h e p r a c t i c a l o c c u r r e n c e of any s o l u t i o n t h a t u l t i m a t e l y approaches the critical point z,.

T h i s p o i n t i s r e f e r r e d t o as a " b r e a k p o i n t " .

I t s importance l i e s i n i t s p r o p e r t y o f l y i n g on t h e boundary between

t h e t w o domains o f a t t r a c t i o n . A d i s c u s s i o n o f t h r e s h o l d s and b r e a k p o i n t s i n a more g e n e r a l s e t t i n g i s g i v e n by May ( 1 9 7 7 a ) . The a s y m p t o t i c v a l u e s o f w ( t ) and y ( t ) as t

--t

m

correspond t o t h e

s t e a d y s t a t e e x p e c t e d w o r m l o a d p e r d e f i n i t i v e h o s t and t h e s t e a d y

s t a t e e x p e c t e d s n a i l p r e v a l e n c e , r e s p e c t i v e l y , t h a t r e s u l t from a long t e r m exposure of t h e p a r a s i t e population t o c o n s t a n t b i o l o g i c a l , e n v i r o n m e n t a l , b e h a v i o u r a l , and h y g i e n i c i n f l u e n c e s . T h e s e i n f l u e n c e s a r e m e a s u r e d by t h e t w o t r a n s m i s s i o n f a c t o r s T,

and T 2 .

The m a t h e m a t i c a l b i f u r c a t i o n phenomenon d e s c r i b e d a b o v e h a s a n e p i d e m i o l o g i c a l c o u n t e r p a r t i n a t h r e s h o l d phenomenon. E r a d i c a t i o n o f a n i n f e c t i o n t h a t h a s e s t a b l i s h e d i t s e l f i n a community c a n be a c h i e v e d by o n e o f t w o c o n c e p t u a l l y d i f f e r e n t methods. Method number o n e con-

s i s t s i n p e r m a n e n t l y r e d u c i n g t h e v a l u e s o f the t r a n s m i s s i o n f a c t o r s

t o l i e below t h e t h r e s h o l d f u n c t i o n . Method number t w o c o n s i s t s i n momentarily reducing w ( t )and/or y ( t ) t o s u f f i c i e n t l y l o w v a l u e s i n t h e domain o f a t t r a c t i o n of ( 0 , O ) . Method number o n e h a s a c l e a r s u p e r i o r i t y o v e r method number t w o s i n c e i t l e a d s t o a n e r a d i c a t i o n

I I7

MATHEMATICAL MODELS OF SCHISTOSOMIASIS

t h a t i s g l o b a l l y s t a b l e : Any amount o f i n f e c t i o n i n t r o d u c e d i n t o a community c h a r a c t e r i z e d by a p a r a m e t e r v e c t o r i n t h e s e t F1 w i l l u l t i m a t e l y d i e o u t o f i t s own. The same c o n c l u s i o n d o e s n o t h o l d f o r an e r a d i c a t i o n t h a t h a s been a c h i e v e d t h r o u g h method number two. The critical point (0,O)

i s a s y m p t o t i c a l l y s t a b l e , b u t i t s domain o f

a t t r a c t i o n may b e s m a l l . Even a r a t h e r s m a l l amount o f i m p o r t e d i n f e c t i o n may i n t h i s case l e a d t o t h e i n f e c t i o n e s t a b l i s h i n g i t s e l f a t t h e h i g h endemic l e v e l c o r r e s p o n d i n g t o t h e c r i t i c a l p o i n t z 2 . Condit i o n s a f f e c t i n g e r a d i c a t i o n a c c o r d i n g t o method number two a r e d i s cussed i n Ndsell (1976b). I f e i t h e r o f t h e two t r a n s m i s s i o n f a c t o r s i s h e l d f i x e d , t h e n t h e rel a t i o n between t h e s t e a d y s t a t e e x p e c t e d worm l o a d p e r d e f i n i t i v e h o s t and t h e o t h e r t r a n s m i s s i o n f a c t o r c a n b e s e e n t o h a v e t h e same q u a l i t a t i v e f e a t u r e s a s RenP Thom's f o l d c a t a s t r o p h y . The amount o f p a r a m e t e r m o d i f i c a t i o n t h a t i s r e q u i r e d i n o r d e r t o a c h i e v e e r a d i c a t i o n by method number o n e i s r e f e r r e d t o a s t h e eradi*c a t i o n e f f o r t . I t i s measured by t h e r a t i o o f t h e p r o d u c t o f t h e two t r a n s m i s s i o n f a c t o r s b e f o r e and a f t e r t h e p a r a m e t e r m o d i f i c a t i o n . I f f o r example e r a d i c a t i o n i s a c h i e v e d by r e d u c t i o n o f t h e s i z e o f t h e s n a i l p o p u l a t i o n , t h e n t h e e r a d i c a t i o n e f f o r t e q u a l s t h e f a c t o r by which t h e s n a i l p o p u l a t i o n s i z e must be r e d u c e d i n o r d e r t o a c h i e v e e r a d i c a t i o n . The p r e - a c t i o n d e n o t e d by T

and T 2 ,

values of t h e transmission f a c t o r s a r e

and t h e p o s t - a c t i o n v a l u e s by T I 1 and T 2 1 . Na-

1 t u r a l a s s u m p t i o n s a r e t h a t ( T I , T 2 ) E F2, T I 1 -

TI

= TI ( T 2 1

5

TI

,

T2,

5

T2,

1 . Thus t h e e r a d i c a t i o n e f f o r t e q u a l s T2

and i s t h e r e f o r e a f u n c t i o n o f t h e p o s t - a c t i o n

transmission f a c t o r

I t c a n be shown t h a t yT ( y ) i s a n i n c r e a s i n g f u n c t i o n o f 1 y . The e r a d i c a t i o n e f f o r t i s t h e r e f o r e minimized by t a k i n g T21 a s

value T21.

l a r g e a s p o s s i b l e . Thus, minimum e r a d i c a t i o n e f f o r t i s a c h i e v e d by r e d u c i n g T7 t o i t s t h r e s h o l d v a l u e w i t h f i x e d T 2 . The minimum e r a d i c a t i o n e f f o r t i s g i v e n by El ( T , , T 2 )

=

T2

,= -

I

~

T2T1 ( T 2 )

( T I r T 2 ) E'F2

I

(11)

TI ( T 2 )

and depends o n l y on t h e p r e - c o n t r o l

v a l u e s TI and T 2 o f t h e two t r a n s -

118

I.

NASELL

mission f a c t o r s . Maximum e r a d i c a t i o n e f f o r t i s a c h i e v e d by r e d u c i n g T 2 t o i t s t h r e s h o l d v a l u e with f i x e d T,.

where

T2

The maximum e r a d i c a t i o n e f f o r t i s g i v e n by

i s t h e i n v e r s e of

?f1 '

The r a t i o o f maximum t o minimum e r a d i c a t i o n e f f o r t i s a measure o f t h e r e l a t i v e advantage i n a c h i e v i n g e r a d i c a t i o n through r e d u c t i o n o f TI

i n s t e a d o f through r e d u c t i o n o f T 2 . T h i s advantage i s n o t of p r a c -

t i c a l i m p o r t a n c e i f t h e r a t i o i s c l o s e t o o n e . The r a t i o c a n b e shown t o t a k e v a l u e s i n t h e i n t e r v a l ( 1 , E, ( T , , T 2 ) ) . Thus, t h e r e l a t i v e a d v a n t a g e of e r a d i c a t i n g s c h i s t o s o m i a s i s t h r o u g h r e d u c t i o n o f TI i n s t e a d of T 2 i s a l w a y s s m a l l i f t h e minimum e r a d i c a t i o n e f f o r t i s s m a l l , b u t may b e a p p r e c i a b l e i f t h e minimum e r a d i c a t i o n e f f o r t i s l a r g e . W e u s e t h e minimum e r a d i c a t i o n e f f o r t a s a m e a s u r e o f t h e r e s i l i e n c e

o f a s c h i s t o s o m i a s i s i n f e c t i o n i n a community; c f t h e d i s c u s s i o n by H o l l i n g (1973)

.

The e f f i c i e n c i e s o f v a r i o u s c o n t r o l a c t i o n s c a n be measured by f u n c t i o n s d e f i n e d on t h e s p a c e where t h e two t r a n s m i s s i o n f a c t o r s t a k e t h e i r v a l u e s . We c o n s i d e r c o n t r o l a c t i o n s w i t h s m a l l r e l a t i v e c h a n g e s i n t h e t r a n s m i s s i o n f a c t o r s . The r e s u l t i n g r e d u c t i o n i n s t e a d y s t a t e e x p e c t e d worm l o a d p e r d e f i n i t i v e h o s t i s used t o m e a s u r e t h e e f f e c t o f t h e c o n t r o l a c t i o n . Two f u n c t i o n s C1 and C 2 a r e d e f i n e d on t h e s e t F2 by s e t t i n g

Then, C 1 ( T , , T 2 )

i s a m e a s u r e o f t h e e f f i c i e n c y o f c o n t r o l of t h e

s t e a d y s t a t e e x p e c t e d worm l o a d p e r d e f i n i t i v e h o s t t h r o u g h a r e d u c t i o n of T, w i t h f i x e d T 2 , w h i l e C 2 ( T , r T 2 ) m e a s u r e s t h e e f f i c i e n c y o f c o n t r o l o f the s a m e epidemiological q u a n t i t y through a r e d u c t i o n o f T2 w i t h f i x e d T I .

The r a t i o o f t h e two c o n t r o l e f f i c i e n c i e s

( T ,T ) i s l a r g e r t h a n one t h r o u g h o u t t h e s u b s e t F2 o f 1 2 1 2 p a r a m e t e r s p a c e where i t i s d e f i n e d . T h i s l a t t e r r e s u l t l e a d s t o t h e

C, ( T , T 2 ) / C

q u e s t i o n how much l a r g e r t h a n o n e t h e c o n t r o l e f f i c i e n c y r a t i o i s .

MATHEMATICAL MODELS OF SCHISTOSOMIASIS

119

By r e l a t i n g i t t o t h e minimum e r a d i c a t i o n e f f o r t o n e f i n d s t h a t t h e c o n t r o l e f f i c i e n c y r a t i o i s l a r g e r t h a n t h e minimum e r a d i c a t i o n e f f o r t t h r o u g h o u t . t h e domain o f d e f i n i t i o n F2. T h i s r e s u l t l e a d s t o t h e conc l u s i o n t h a t t h e r e l a t i v e a d v a n t a g e i n c o n t r o l l i n g t h e worm l o a d t h r o u g h r e d u c t i o n o f TI i n s t e a d o f T 2 i s a p p r e c i a b l e i f t h e r e s i l i e n c e of t h e i n f e c t i o n i s high.

3 . External Infection The b a s i c model d e a l s w i t h t r a n s m i s s i o n o f s c h i s t o s o m i a s i s i n an i s o l a t e d community. The model i n N B s e l l ( 1 9 7 5 ) allows f o r a c o n t i n u o u s e x p o s u r e o f t h e community t o e x t e r n a l i n f e c t i o n . Two p o s i t i v e parame-

t e r s , 6 and 6 2 , a r e i n t r o d u c e d t h a t m e a s u r e t h e amount of e x t e r n a l 1 i n f e c t i o n o f human b e i n g s and s n a i l s , r e s p e c t i v e l y . The p a r a m e t e r 6 1 i s t h e i n c r e a s e i n t h e p r o p o r t i o n o f i n f e c t e d s n a i l s which h a s t h e same e f f e c t on the r a t e o f i m m i g r a t i o n o f p a r a s i t e s i n t o human b e i n g s a s t h e e x t e r n a l i n f e c t i o n , and 6 2 i s t h e i n c r e a s e i n t h e number o f p a i r e d f e m a l e p a r a s i t e s p e r h o s t which h a s t h e same e f f e c t on t h e i n f e c t i o n r a t e of s n a i l s a s the e x t e r n a l i n f e c t i o n . The p a r a m e t e r s p a c e f o r t h i s model i s g i v e n by t h e s p a c e where T 1 , T 2 ,

61,

and 6 2 t a k e t h e i r v a l u e s . E r a d i c a t i o n o f t h e i n f e c t i o n i s p o s s i b l e

o n l y f o r 6 1 = 6 2 = 0 , i . e . i n t h e case d e a l t w i t h by t h e b a s i c model. The r e l a t i o n between t h e s t e a d y s t a t e e x p e c t e d worm l o a d p e r d e f i n i t i v e h o s t and t h e p a r a m e t e r s T

61,

f o r fixed values of T2, 6 2 , has

t h e same q u a l i t a t i v e f e a t u r e s a s Ren6 Thom's c u s p c a t a s t r o p h y . For l a r g e v a l u e s of A 1

,

t h e s t e a d y s t a t e worm l o a d i s a m o n o t o n i c a l l y i n -

c r e a s i n g f u n c t i o n o f T 1 , a n d i t i s u n i q u e l y d e t e r m i n e d by t h e i n i t i a l v a l u e s . For s m a l l e r v a l u e s o f 6 1 ,

t h e s t e a d y s t a t e worm l o a d i s

u n i q u e l y d e t e r m i n e d by t h e i n i t i a l v a l u e s f o r s m a l l and l a r g e v a l u e s o f T1, b u t t h e r e i s an i n t e r m e d i a t e r a n g e of T1-values f o r which t h e r e e x i s t two a s y m p t o t i c a l l y s t a b l e c r i t i c a l p o i n t s and s u c h t h a t t h e s t e a d y s t a t e worm l o a d h e r e d e p e n d s on t h e i n i t i a l v a l u e s . I n t h i s r a n g e t h e two c o n c e p t u a l l y d i f f e r e n t methods f o r e r a d i c a t on d i s c u s s e d above f o r t h e b a s i c model have t h e i r c o u n t e r p a r t s i n two d i f f e r e n t methods f o r l a r g e r e d u c t i o n s o f t h e endemic l e v e l i n t h e community. The r a n g e o f T 1 - v a l u e s f o r which method number t w o

rnomen-

t a r y r e d u c t i o n i n s t a t e s p a c e ) i s a v a i l a b l e i s , however, l i m i t e d t o t h e i n t e r m & d i a t e r a n g e mentioned a b o v e . Method number o n e ( p e r m a n e n t r e d u c t i o n i n p a r a m e t e r s p a c e ) i s always a v a i l a b l e , and i s t h e o n l y method a t o u r d i s p o s a l when T,

l i e s above t h e i n t e r m e d i a t e r a n g e . The

a n a l y s i s of e x t e r n a l i n f e c t i o n h a s given a d d i t i o n a l r e a s o n f o r pre-

I. NASELL

I20

f e r r i n g method number one o v e r method number two. I n o t e a d i s a g r e e ment w i t h May's

( 1 9 7 7 a ) c o n c l u s i o n t h a t method number t w o h a s a n ad-

v a n t a g e o v e r method number o n e . 4 . S n a i l Latency A model a c c o u n t i n g f o r l a t e n c y i n t h e i n f e c t i o n among t h e s n a i l s and

f o r d i f f e r e n t i a l mortality i n the s n a i l population is d e a l t with i n N d s e l l ( 1 9 7 6 a ) . The model i s s t o c h a s t i c f o r t h e p h a s e i n t h e d e f i n i t i v e h o s t and d e t e r m i n i s t i c f o r the p h a s e i n t h e i n t e r m e d i a t e h o s t . The s n a i l s are c l a s s i f i e d i n t o f o u r c a t e g o r i e s : s u s c e p t i b l e ? e x p o s e d , i n f e c t i v e ? and r e c o v e r e d . The d e a t h rates o f t h e s n a i l s are a l l o w e d t o be d i f f e r e n t i n a l l f o u r c a t e g o r i e s . The rates a t which e x p o s e d s n a i l s become i n f e c t i v e and i n f e c t i v e s n a i l s r e c o v e r a r e assumed cons t a n t . A l l newborn s n a i l s a r e s u s c e p t i b l e and a l l r e c o v e r e d s n a i l s a r e immune. The model e x h i b i t s a t h r e s h o l d phenomenon s i m i l a r t o t h e o n e f o r t h e b a s i c model. The a n a l y s i s i s f a c i l i t a t e d by t h e i n t r o d u c t i o n o f two t r a n s m i s s i o n f a c t o r s t h a t r e s e m b l e t h o s e o f t h e b a s i c model. The d e f i n i t i o n s o f t h e t r a n s m i s s i o n f a c t o r s show t h a t u n d e r t h e h y p o t h e s e s o f l a t e n c y and d i f f e r e n t i a l m o r t a l i t y o n e c a n i d e n t i f y t h e number o f s n a i l s t h a t are " e f f e c t i v e " i n t r a n s m i t t i n g schistosome i n f e c t i o n s . I n t h e s p e c i a l c a s e o f no r e c o v e r y and no d i f f e r e n t i a l m o r t a l i t y ( s e e N d s e l l (1977) ) t h e p r o p o r t i o n of s n a i l s t h a t are " e f f e c t i v e " i n t r a n s m i t t i n g schistosome i n f e c t i o n s i s e x a c t l y e q u a l t o t h e p r o p o r t i o n o f e x p o s e d s n a i l s t h a t s u r v i v e t o become i n f e c t i v e i n s t e a d y s t a t e . 5 . Modes o f Mating The models d e s c r i b e d so f a r h a v e assumed monogamous m a t i n g between male and f e m a l e p a r a s i t e s i n each d e f i n i t i v e h o s t . Models b a s e d on t h e a l t e r n a t i v e h y p o t h e s e s o f p r o m i s c u o u s m a t i n g and o f p a r t h e n o g e n e t i c e g g l a y i n g a r e d e a l t w i t h i n N d s e l l ( 1 9 7 8 a ) . Comparisons between t h e t h r e e models a r e made t o s t u d y t h e f o l l o w i n g t w o q u e s t i o n s : 1) I s i t i m p o r t a n t t o know t h e mode o f m a t i n g between t h e p a r a s i t e s , i . e . t o f i n d o u t i f t h e m a t i n g i s monogamous o r p r o m i s c u o u s ? 2 1 Is i t a c c e p t a b l e t o s i m p l i f y t h e m a t h e m a t i c a l t r e a t m e n t o f s c h i s t o -

s o m i a s i s by u s i n g a model f o r p a r t h e n o g e n e t i c e g g l a y i n g , a l t h o u g h m a t i n g i s known t o be n e c e s s a r y t o p r o d u c e f e r t i l i z e d e g g s ? The f i r s t q u e s t i o n i s p a r a s i t o l o g i c a l .

The knowledge a b o u t t h e m a t i n g

p a t t e r n o f s c h i s t o s o m e s i s i n c o m p l e t e ? and d i r e c t o b s e r v a t i o n o f l i v e s c h i s t o s o m e s i n t h e i r h a b i t a t ( b l o o d v e s s e l s o f human b e i n g s ) i s

MATHEMATICAL MODELS OF SCHISTOSOMIASIS

121

p r a c t i c a l l y i m p o s s i b l e . The s e c o n d q u e s t i o n c o n c e r n s t h e a c c e p t a b i l i t y o f a c e r t a i n a p p r o x i m a t i o n t h a t h a s b e e n made i n t h e f o r m u l a t i o n o f m a t h e m a t i c a l models f o r s c h i s t o s o m i a s i s and t h a t l e a d s t o c o n s i d e r a b l e mathematical s i m p l i f i c a t i o n s . I n t h e c o m p a r i s o n between t h e t h r e e models t h e r e a p p e a r s t h e need f o r c r i t e r i a t o j u d g e when c o n c l u s i o n s o f two models a r e s i m i l a r and when t h e y a r e n o t . The c r i t e r i a t o be a d o p t e d r e f l e c t t h e d e s i r e t o d e a l w i t h q u a l i t a t i v e a s p e c t s and t o b e c o n c e r n e d w i t h o r d e r - o f - m a g n i t u d e changes. The r e s i l i e n c e s o f two models a r e compared by a s t u d y o f t h e r a t i o o f t h e c o r r e s p o n d i n g r e s i l i e n c e m e a s u r e s on t h a t subset o f p a r a m e t e r s p a c e where b o t h r e s i l i e n c e m e a s u r e s a r e d e f i n e d , i . e .

f o r parameter

v a l u e s t h a t l i e above t h e l a r g e r o f t h e two t h r e s h o l d f u n c t i o n s . The d i f f e r e n c e between t h e two models i s s a i d t o b e n o n e s s e n t i a l w i t h reg a r d t o t h e r e s i l i e n c e s i f t h e r a t i o of t h e r e s i l i e n c e s i s c l o s e t o o n e f o r a l l p a r a m e t e r v a l u e s where t h e r a t i o i s d e f i n e d ; o t h e r w i s e t h e d i f f e r e n c e i s e s s e n t i a l . Two models c a n a l s o be compared w i t h reg a r d t o a number o f o t h e r e n t i t i e s , s u c h a s e p i d e m i o l o g i c a l q u a n t i t i e s and c o n t r o l e f f i c i e n c i e s . Each s u c h c o m p a r i s o n i s made by s t u d y i n g t h e r a t i o o f c o r r e s p o n d i n g e n t i t i e s on t h e s u b s e t o f p a r a m e t e r s p a c e where b o t h a r e d e f i n e d . I n e s t a b l i s h i n g comparison c r i t e r i a w e r e c o g n i z e t h e p o s s i b i l i t y f o r s u c h r a t i o s t o d e v i a t e c o n s i d e r a b l y from one n e a r t h e l a r g e r o f t h e two t h r e s h o l d f u n c t i o n s . I n o r d e r f o r t h e d i f f e r e n c e between t h e two models t o be c a l l e d n o n e s s e n t i a l w i t h r e g a r d t o t h e p a r t i c u l a r e n t i t y w e r e q u i r e t h a t t h e d i f f e r e n c e between t h e models w i t h r e g a r d t o t h e r e s i l i e n c e s b e n o n e s s e n t i a l and i n a d d i t i o n t h a t t h e r a t i o o f c o r r e s p o n d i n g e n t i t i e s be c l o s e t o one f o r a l l p a r a m e t e r v a l u e s where t h e r e s i l i e n c e ( o f e i t h e r model) i s h i g h ; o t h e r w i s e t h e difference is essential. The c r i t e r i a d e s c r i b e d a b o v e t o j u d g e i f d i f f e r e n c e s b e t w e e n two mod e l s a r e n o n e s s e n t i a l o r e s s e n t i a l may a p p e a r v a g u e s i n c e t h e y i n v o l v e r a t i o v a l u e s "close t o o n e " a n d " h i g h r e s i l i e n c e " . The i n t e n t h e r e i s t o d e a l w i t h orders of m a g n i t u d e ; a r a t i o i s close t o o n e i f i t l i e s i n the i n t e r v a l ( 1 / 3 ,

3)

,

and a r e s i l i e n c e i s h i g h i f i t e x c e e d s t e n .

The r e s u l t s d e s c r i b e d below a r e n o t s e n s i t i v e t o s u b s t a n t i a l c h a n g e s i n t h e s e numbers. I n g o i n g f r o m monogamous m a t i n g v i a p r o m i s c u o u s m a t i n g t o p a r t h e n o -

I. NASELL

122

g e n e t i c e g g l a y i n g , one d e a l s w i t h p a r a s i t e s w i t h p r o g r e s s i v e l y more e z f i c i e n t r e p r o d u c t i v e s y s t e m s , and w i t h i n f e c t i o n s w i t h c o r r e s p o n d i n g l y h i g h e r r e s i l i e n c e s . The r a t i o o f t h e r e s i l i e n c e i n t h e promiscuo u s t o t h a t i n t h e monogamous c a s e i s a bounded f u n c t i o n o f T2 whose values a r e r e s t r i c t e d t o t h e i n t e r v a l (1 , 1 .320). r a t i o s between worm l o a d s , l o a d s o f e g g - l a y e r s ,

Furthermore, t h e

and c o n t r o l e f f i c i e n -

c i e s a r e a l l c l o s e t o one f o r i n f e c t i o n s w i t h moderate t o h i g h res i l i e n c e . W e c o n c l u d e t h e r e f o r e t h a t t h e d i f f e r e n c e between monogamous and promiscuous m a t i n g i s n o n e s s e n t i a l w i t h r e g a r d t o r e s i l i e n c e , e p i d e m i o l o g i c a l q u a n t i t i e s and c o n t r o l e f f i c i e n c i e s . I n contrast, t h e r a t i o of the r e s i l i e n c e i n the parthenogenetic case t o t h a t i n e i t h e r o f t h e mating c a s e s i s an unbounded f u n c t i o n o f T 2 . Furthermore,

t h e r a t i o s between l o a d s of e g g - l a y e r s ,

and between con-

t r o l e f f i c i e n c i e s , c a n become l a r g e even when t h e r e s i l i e n c e i s h i g h . T h u s t h e r e i s 'an e s s e n t i a l d i f f e r e n c e between p a r t h e n o g e n e t i c egg-

l a y i n g and e i t h e r mode o f m a t i n g . The two q u e s t i o n s p o s e d a t t h e b e g i n n i n g o f t h i s s e c t i o n c a n t h e r e f o r e be responded t o a s f o l l o w s :

1) I t i s n o t o f e s s e n t i a l i m p o r t a n c e t o s t u d y t h e m a t i n g p a t t e r n o f l i v e s c h i s t o s o m e s i n such d e t a i l t h a t t h e p r e s e n t u n c e r t a i n t y o f monogamy v e r s u s p r o m i s c u i t y i s r e s o l v e d . 2 ) I t i s n o t a c c e p t a b l e t o i g n o r e t h e m a t i n g between male and female

p a r a s i t e s i n e s t a b l i s h i n g h y p o t h e s e s f o r a s c h i s t o s o m i a s i s model. 6 . Immunity

The l e v e l o f a s c h i s t o s o m i a s i s i n f e c t i o n i n a community i s s t r o n g l y i n f l u e n c e d by d e n s i t y d e p e n d e n t f a c t o r s t h a t a p p e a r i n t h e v a r i o u s l i f e c y c l e s t a g e s of t h e c a u s a t i v e p a r a s i t e . The b a s i c model, and t h e v a r i a t i o n s o f i t d i s c u s s e d above, c o n t a i n s o n l y one d e n s i t y d e p e n d e n t f a c t o r : The r a t e o f c e r c a r i a l s h e d d i n g o f an i n f e c t e d s n a i l i s i n d e p e n d e n t of t h e number o f i n f e c t i o n s t h a t t h e s n a i l h a s r e c e i v e d . An immune r e a c t i o n among human b e i n g s t o s c h i s t o s o m e i n f e c t i o n s would l e a d t o an a d d i t i o n a l d e n s i t y d e p e n d e n t f a c t o r . The e x i s t e n c e o f s u c h a n immune r e a c t i o n i s c o n t r o v e r s i a l . Mathematical m o d e l l i n g may n o t r e s o l v e t h e e x i s t i n g c o n t r o v e r s y , b u t it may l e a d t o f u r t h e r i n s i g h t by d e v e l o p i n g t h e e p i d e m i o l o g i c a l and c o n t r o l - r e l a t e d c o n s e q u e n c e s o f a n immune r e a c t i o n among human b e i n g s . A model b a s e d on h y p o t h e s e s t h a t g i v e one i n t e r p r e t a t i o n o f t h e mecha-

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123

n i s m of s o - c a l l e d c o n c o m i t a n t immunity i s p r e s e n t e d by N d s e l l ( 1 9 7 8 b ) . C o n c o m i t a n t immunity d e v e l o p s g r a d u a l l y and i s o n l y p a r t i a l . I t i s c a u s e d by m a t u r e p a r a s i t e s , i t d o e s n o t d e s t r o y a d u l t worms o r p r e v e n t them from p r o d u c i n g e g g s , b u t i t t e n d s t o d e s t r o y i n v a d i n g s c h i s t o s o mula. I n t h e h y p o t h e s e s o f t h e model, t h e immune l e v e l o f any h o s t i s s e t e q u a l t o t h e number o f l i v e p a r a s i t e s . The e x i s t e n c e o f a c o n s t a n t

i s p o s t u l a t e d w i t h t h e p r o p e r t y t h a t t h e r a t e of i m m i g r a t i o n o f new p a r a s i t e s i s p r o p o r t i o n a l t o t h e d i f f e r e n c e between

s a t u r a t i o n l e v e l S,

t h e s a t u r a t i o n l e v e l and t h e immune l e v e l o f t h e h o s t . M a t i n g between male and f e m a l e s c h i s t o s o m e s i s assumed t o b e p r o m i s c u o u s . An i m p o r t a n t reason f o r t h i s assumption i s t h a t it l e a d s t o a mathematically manageable model. The same d o e s n o t s e e m t o h o l d f o r a model b a s e d on t h e a l t e r n a t i v e h y p o t h e s i s o f monogamous m a t i n g . The model c a n b e viewed a s an e x t e n s i o n o f t h e b a s i c model w i t h p r o m i s c u o u s m a t i n g . The l a t t e r model i s r e c a p t u r e d a s a s p e c i a l c a s e i f t h e s a t u r a t i o n l e v e l S,

100

--t

m.

-

R e a l i s t i c values of t h e s a t u r a t i o n l e v e l a r e of the order

1000.

An i n c r e a s e i n t h e s a t u r a t i o n l e v e l S1 c o r r e s p o n d s t o a w e a k e r immune r e a c t i o n and a h i g h e r r e s i l i e n c e o f t h e i n f e c t i o n . The r a t i o o f t h e r e s i l i e n c e i n t h e case S1

--t

m

t o t h e r e s i l i e n c e i n t h e c a s e S1 = 1 0 0

i s a f u n c t i o n of t h e t r a n s m i s s i o n f a c t o r T 2 . T h i s f u n c t i o n t a k e s v a l u e s i n t h e i n t e r v a l (1 .005,

1 . 1 0 6 ) . Thus w e c o n c l u d e t h a t a c c o u n t -

i n g f o r t h e immune r e a c t i o n among human b e i n g s i n t h e way d o n e i n t h i s model l e a d s t o o n l y a s m a l l and i n s i g n i f i c a n t d e c r e a s e i n t h e r e s i l i ence of t h e i n f e c t i o n . The h y p o t h e s e s o f t h e model imply t h a t t h e worm l o a d p e r d e f i n i t i v e h o s t i s bounded by t h e s a t u r a t i o n l e v e l S 1 . The r a t i o o f t h e worm l o a d i n t h e c a s e S1 --t m t o t h e worm l o a d f o r f i n i t e S1 i s , f o r cons t a n t r e s i l i e n c e i n t h e l a t t e r case, a n unbounded f u n c t i o n o f T I . Thus, t h e r e i s a n e s s e n t i a l d i f f e r e n c e between t h e t w o cases w i t h reg a r d t o t h e w o r m l o a d . S i m i l a r l y , one f i n d s a n e s s e n t i a l d i f f e r e n c e w i t h r e g a r d t o t h e e f f i c i e n c y of c o n t r o l o f t h e worm l o a d t h r o u g h red u c t i o n o f T I . These e s s e n t i a l d i f f e r e n c e s s u f f i c e t o draw t h e conc l u s i o n t h a t i t i s i m p o r t a n t t o r e c o g n i z e t h e immune r e a c t i o n o f human b e i n g s i n any s c h i s t o s o m i a s i s model.

I. NASELL

I24

7. Aggregation The s t e a d y s t a t e d i s t r i b u t i o n s o f p a r a s i t e s p e r h o s t a r e P o i s s o n f o r a l l models d e s c r i b e d above e x c e p t f o r t h e model w i t h immunity where t h e d i s t r i b u t i o n s a r e b i n o m i a l . These d i s t r i b u t i o n s r e s u l t from t h e h y p o t h e s i s o f homogeneous e x p o s u r e o f a l l human b e i n g s t o i d e n t i c a l environments. This h y p o t h e s i s i s c l e a r l y n o t s a t i s f i e d i n p r a c t i c e . Furthermore, t h e r e i s e m p i r i c a l evidence t h a t s u g g e s t s t h a t t h e s t e a d y

s t a t e d i s t r i b u t i o n s are o v e r d i s p e r s e d . May (1977b) h a s r e p l a c e d t h e s t e a d y s t a t e P o i s s o n d i s t r i b u t i o n s o f p a r a s i t e s by n e g a t i v e b i n o m i a l d i s t r i b u t i o n s whose d e g r e e o f a g g r e g a t i o n a r e measured by a p o s i t i v e p a r a m e t e r k ; a l a r g e d e g r e e o f a g g r e g a t i o n c o r r e s p o n d s t o a s m a l l val u e o f k . May d i s t i n g u i s h e s between t h e c a s e s when t h e two s e x e s o f p a r a s i t e s a r e d i s t r i b u t e d t o g e t h e r o r s e p a r a t e l y . The s e x e s a r e s a i d t o be d i s t r i b u t e d t o g e t h e r when t h e t o t a l number o f worms p e r h o s t f o l l o w s a n e g a t i v e b i n o m i a l d i s t r i b u t i o n ( w i t h mean m and a g g r e g a t i o n p a r a m e t e r k ) , and each i n d i v i d u a l worm i s male o r female w i t h proba1

b i l i t y 2 . The s e x e s a r e s a i d t o be d i s t r i b u t e d s e p a r a t e l y when male and female worms a r e i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d , e a c h i n a n e g a t i v e b i n o m i a l d i s t r i b u t i o n ( w i t h mean m/2

and a g g r e g a t i o n p a r a -

meter k). The r e s u l t s c o r r e s p o n d i n g t o P o i s s o n d i s t r i b u t i o n s a r e t h e l i m i t i n g case of e i t h e r type (sexes d i s t r i b u t e d together o r separatel y ) o b t a i n e d a s t h e a g g r e g a t i o n d e c r e a s e s toward z e r o (k

+ m).

I n t e r m e d i a t e r e s u l t s show t h a t t h e p r o p o r t i o n o f f e m a l e s mated, f o r g i v e n mean worm l o a d m ,

increases a s the aggregation is increased

when t h e sexes a r e d i s t r i b u t e d t o g e t h e r , b u t d e c r e a s e s a s t h e a g g r e g a t i o n i s i n c r e a s e d when t h e s e x e s a r e d i s t r i b u t e d s e p a r a t e l y . The t h r e s h o l d f u n c t i o n i s found t o d e c r e a s e a s t h e a g g r e g a t i o n i s i n c r e a s e d i f t h e two s e x e s a r e d i s t r i b u t e d t o g e t h e r . T h i s means t h a t t h e r e s i l i e n c e of the i n f e c t i o n increases. Q u a l i t a t i v e l y s i m i l a r r e s u l t s

a r e r e p o r t e d by Barbour ( 1 9 7 8 ) . An i n c r e a s e i n a g g r e g a t i o n i s found t o have t h e o p p o s i t e e f f e c t on t h e t h r e s h o l d if t h e two s e x e s a r e d i s t r i buted separately. I n v e s t i g a t i o n s a r e also made i n May (197713) of t h e i n f l u e n c e of s n a i l l a t e n c y and o f promiscuous mating. These i n v e s t i g a t i o n s a r e more gener a l t h a n t h o s e d i s c u s s e d above i n S e c t i o n s 4 and 5 , s i n c e t h e y a r e c a r r i e d through f o r an a r b i t r a r y v a l u e o f t h e a g g r e g a t i o n parameter k . On t h e o t h e r hand, May d o e s n o t e x t e n d h i s a n a l y s i s t o i n c l u d e a t r e a t m e n t o f c o n t r o l e f f i c i e n c i e s o r of e r a d i c a t i o n e f f o r t s .

MATHEMATICAL MODELS OF SCHISTOSOMIASIS

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8 . C o n c l u d i n g Remarks The p r e v a l e n c e o f s c h i s t o s o m i a s i s i s h i g h i n many c o m m u n i t i e s i n t h i r d w o r l d c o u n t r i e s . The s o l u t i o n o f t h e a c c o m p a n y i n g p u b l i c h e a l t h p r o blems r e q u i r e s a c o n c e n t r a t e d r e s e a r c h a p p r o a c h o f a m u l t i d i s c i p l i n a r y n a t u r e . The review o f t h i s p a p e r shows how m a t h e m a t i c s c a n c o n t r i b u t e t o t h i s r e s e a r c h t a s k a n d i n d i c a t e s t h e power i n h e r e n t i n t h e q u a l i t a t i v e a n a l y s i s . The d e v e l o p m e n t o f r e a l i s t i c m o d e l s r e q u i r e s c l o s e c o o p e r a t i o n between mathematicians and r e p r e s e n t a t i v e s o f epidemiology and o t h e r r e l e v a n t d i s c i p l i n e s . M a t h e m a t i c a l work t h a t a s p i r e s t o h a v e e p i d e m i o l o g i c a l i m p o r t a n c e c a n n o t be c a r r i e d out u n l e s s m a t h e m a t i c i a n s and e p i d e m i o l o g i s t s are w i l l i n g t o t r e s p a s s t h e t r a d i t i o n a l l i m i t s o f t h e i r s c i e n t i f i c domains. 9 . Acknowledgement T h i s work h a s b e e n s u p p o r t e d by t h e S w e d i s h N a t u r a l S c i e n c e R e s e a r c h Council. 1 0 . References 111 A . D .

B a r b o u r ( 1 9 7 8 ) : B r e a k p o i n t phenomena, a n d t h e i n i t i a l s t a g e s

o f t h e t r a n s m i s s i o n o f b i l h a r z i a , Bull. I n t . S t a t . I n s t . , ( i n press) 121 J . E .

Cohen ( 1 9 7 7 ) : M a t h e m a t i c a l m o d e l s of s c h i s t o s o m i a s i s ,

Ann. Rev. E c o l . S y s t . , 1 3 1 P.E.M.

4 7 , I1

. 8 , 209-33.

F i n e ( 1 9 7 6 ) : M a t h e m a t i c a l Models of S c h i s t o s o m i a s i s ,

P r o c . Workshop B e l l a g i o , I t a l y , Clark Foundation,

9-14 May. New York: Edna McConnell

58 p p .

141 C . S . H o l l i n g ( 1 9 7 3 ) : R e s i l i e n c e and s t a b i l i t y o f e c o l o g i c a l s y s t e m s , Ann. Rev. E c o l . S y s t . , 4 , 1-23.

I 51

P . J o r d a n a n d G . Webbe ( 1 9 6 9 ) : Human S c h i s t o s o m i a s i s , Thomas, Springfield, Ill.

161 R.M.

May ( 1 9 7 7 a ) : T h r e s h o l d s a n d b r e a k p o i n t s i n e c o s y s t e m s w i t h a

m u l t i p l i c i t y of s t a b l e states, Nature,

171 R.M.

2 6 9 , 471-7.

May (197713): T o g e t h e r n e s s among s c h i s t o s o m e s :

t h e d y n a m i c s o f t h e i n f e c t i o n , Math. B i o s c i . ,

I t s e f f e c t s on

3 5 , 301-43.

181 I . Nssell ( 1 9 7 5 ) : S c h i s t o s o m i a s i s i n a community w i t h e x t e r n a l i n f e c t i o n , P r o c . 8 t h I n t . B i o m e t r i c Conf., E d i t u r a A c a d e m i e i R e p u b l i c i i S o c i a l i s t e Romania, 1 2 3 - 3 1 . 191 I . N d s e l l ( 1 9 7 6 a ) : A h y b r i d model of s c h i s t o s o m i a s i s w i t h s n a i l l a t e n c y , T h e o r . P o p u l . B i o l . , 1 0 , 47-69.

I. NASELL

126

1101 I. Ndsell ( 1 9 7 6 b ) : On eradication of schistosomiasis, Theor. Popul. Biol., 10, 1 3 3 - 4 4 . 1111 I. NAsell ( 1 9 7 7 ) : On transmission and control of schistosomiasis, with comments on Macdonald's model, Theor. P o p u l . B i o l . , 12, 335-65. 1121 I . Ndsell ( 1 9 7 8 a ) : Mating models for schistosomes, J. Math. Biol.

(in press). 1131 I. Ndsell ( 1 9 7 8 b ) : Schistosomiasis with concomitant immunity, B u l l . Int. Stat, Inst., 47, I1 (in press). 1141 I. Ndsell and W.M. Hirsch ( 1 9 7 3 ) : The transmission dynamics of schistosomiasis, C o r n . Pure Appl. Math., 26, 395-453.

DeveZoping Mathematics i n Third WorZd Countries, M.E.A. EZ Tom led.) 0 North-Holland Publishing Company, 1979

DEVELOPING MATHEMATICS

Rene Thom

I h a v e t o t h a n k t h e o r g a n i z e r s of t h i s C o n g r e s s f o r h a v i n g p r o v i d e d

m e w i t h t h i s u n i q u e o p p o r t u n i t y t o become a c q u a i n t e d w i t h t h i s problem of M a t h e m a t i c s and Underdevelopment.

I must s a y t h a t I d o

n o t f e e l p a r t i c u l a r l y c o m p e t e n t t o d e a l w i t h t h e s e m a t t e r s . The o b s e r v a t i o n s I w a n t t o g i v e you o n t h a t o c c a s i o n h a v e d e v e l o p e d f o r t h e m o s t p a r t d u r i n g my a t t e n d a n c e a t t h e C o n g r e s s . They a r e n o m o r e t h a n s i m p l e m i n d e d , common s e n s e r e f l e x i o n s s u g g e s t e d t o

m e by t h e p r o b l e m s d e a l t w i t h , and t h e a t t i t u d e s o f many o f t h e p r o t a g o n i s t i n t h e Congress i t s e l f . T h e r e a r e b a s i c a l l y two o p p o s i t e d a n g e r s t h r e a t e n i n g t h e mathema-

t i c a l d e v e l o p m e n t i n a n a t i o n o n t h e v e r g e of s c i e n t i f i c - d e v e l o p ment : t h e d a n g e r o f u n d e r e s t i m a t i n g i t s own p o s s i b i l i t i e s , a n d t h e d a n g e r o f o v e r e s t i m a t i n g them.

1) The d a n g e r s of u n d e r e s t i m a t i n g c e n t e r a r o u n d t h e i d e a , f r e q u e n t l y e x p r e s s e d i n t h e C o n g r e s s , and e v e n i n i t s f i n a l r e s o l u t i o n s , t h a t mathematical i n s t i t u t i o n s of a n academic n a t u r e ,

in

s u c h m a t h e m a t i c a l l y young n a t i o n s s h o u l d d e a l w i t h u r g e n t p r o blems connected w i t h economical development (or/and) p r e s e n t u n d e r d e v e l o p m e n t and f u t u r e e x p a n s i o n . Whereas s u c h a n a t t i t u d e p r o c e e d s f r o m a n u n d o u b t e d l y commendable c o n c e r n f o r s o c i a l u s e f u l n e s s I am a f r a i d i t may l e a d t o t h e h i g h l y d e b a t a b l e s t a n d p o i n t t h a t academic i n s t i t u t i o n s i n s u c h n a t i o n s a r e of

a d e c i d e d l y d i f f e r e n t n a t u r e t h a n i n d e v e l o p e d c o u n t r i e s , and possibly

( a l t h o u g h t h i s w o n ' t b e o f f i c i a l l y r e c o g n i z e d ) of

lower s c i e n t i f i c s t a t u s . I t h i n k t h a t m a t h e m a t i c i a n s s h o u l d s t i c k t o t h e p o i n t t h a t t h e i r s c i e n c e is u n i v e r s a l ,

t h a t is t o

s a y t h a t i t h a s u n i v e r s a l v a l i d i t y , and t h a t t h e r e i s no p o i n t

127

128

R. THOM

i n t r y i n g t o e s t a b l i s h a m a t h e m a t i c s f o r t h e poor a s opposed t o a m a t h e m a t i c s f o r t h e r i c h . Of c o u r s e , s u c h a d e v e l o p i n g n a t i o n

w i l l n o t be a b l e t o o f f e r , a t t h e beginning, t h e tremendous v a r i e t i e s of c o u r s e s , o r s e m i n a r s you may f i n d i n p l a c e s l i k e Cambridge (England or M a s s . ) , o r P a r i s . But p r o v i d e d t h e t e a c h i n g f i r m l y s t i c k s t o t h e t r a d i t i o n a l c u r r i c u l u m o f h i g h e r mathema-

t i c a l s t u d i e s , l e a d i n g t o C a l c u l u s , t h e n t h e p o s s i b i l i t y of d e t e c t i n g good e l e m e n t s among t h e p o p u l a t i o n i s p r e s e r v e d , and

i t may p r o v i d e t h e n e c e s s a r y background f o r a p p l i c a t i o n s of m a t h e m a t i c a l and e n g i n e e r i n g s t u d i e s . One s h o u l d a i m towards b u i l d i n g u p t h e common s t e m of m a t h e m a t i c s , which I w a n t t o d e s c r i b e l a t e r ; o n e s t a r t s from i t s r o o t s , c l i m b i n g p r o g r e s s i v e l y t o w a r d s i t s b r a n c h e s w h e n m o r e andmore s p e c i a l i z e d p e o p l e a r e formed. Of c o u r s e w e meet h e r e w i t h t h e t r i c k y p r o b l e m t o c o r r e l a -

t e m a t h e m a t i c a l b u i l d i n g u p w i t h t h e g e n e r a l r i s i n g u p of l i t t e r a c y i n t h e p o p u l a t i o n . Whereas of c o u r s e n o t h i n g p r o h i b i t s t h e f o r m a t i o n of a h i g h l y d e v e l o p e d s c i e n t i f i c e l i t e among a p o p u l a t i o n which r e m a i n s i l l i t e r a t e i n i t s overwhelming m a j o r i t y , t h e d a n g e r s of s u c h a s i t u a t i o n a r e o b v i o u s : p o l i t i c a l i n s t a b i l i t y , f r a i l i t y of t h e b a s e s of s c i e n t i f i c c u l t u r e . I t i s n e c e s s a r y

t o walk hand i n hand w i t h t h e g e n e r a l s p r e a d of l i t e r a c y i n t h e population. Here of c o u r s e , o n e meets w i t h t h e p r o b l e m o f m o t i v a t i n g t h e

d e v e l o p m e n t of s c i e n t i f i c , and i n p a r t i c u l a r m a t h e m a t i c a l s t u d i e s , among t h e g e n e r a l p o p u l a t i o n . T h e r e h a s b e e n a t r e n d i n t h e Cong r e s s t o j u s t i f y mathematics through i t s immediate u s e f u l n e s s f o r p r a t i c a l , n a t i o n a l purposes.

I cannot but f e e l t h a t t h i s

j u s t i f i c a t i o n i s wrong. Few p r o b l e m s i n a d m i n i s t r a t i o n ( e x c e p t v e r y t e c h n i c a l o n e s , l i k e c o n s t r u c t i o n of highways o r dams o r f a c t o r i e s ) need more t h a n f a i r l y s i m p l e m a t h e m a t i c s . You c a n n o t e x p e c t t o r a i s e t h e g e n e r a l m a t h e m a t i c a l l e v e l of a c o u n t r y j u s t by a l l u d i n g t o t h e few e x p e r t jobs needed f o r t h e e c o n o m i c a l p l a n i f i c a t i o n , or d e v e l o p m e n t . You c a n n o t promise e v e r y p r o s p e c t i v e math s t u d e n t t h a t h e w i l l g e t a j o b i n t h e g o v e r n m e n t a l a g e n c i e s , working a t n a t i o n a l d e v e l o p m e n t . Here I t h i n k t h e mathe-

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m a t i c i a n ' s a t t i t u d e should be clear : Mathematics i s needed, because w i t h o u t maths it i s i m p o s s i b l e t o u n d e r s t a n d t h e world

we l i v e in. I n a s much a s t h e l o c a l g o v e r n m e n t s become a w a r e of t h i s s i t u a t i o n , t h e y w i l l d o t h e i r b e s t t o r a i s e t h e l e v e l of m a t h e m a t i c a l knowledge among t h e c i t i z e n s . Of c o u r s e t h e p r o c e s s w i l l be s l o w , and no i m m e d i a t e s u c c e s s may b e e x p e c t e d . Those c o u n t r i e s ,

like

I n d i a , which had a t r a d i t i o n a l e l i t e d e v o t e d t o p u r e s t u d i e s , are, i n t h a t respect,

f a r b e t t e r off t h a n o t h e r s , some i n b l a c k

A f r i c a , who w e r e d e s c r i b e d now a s h a v i n g no m a t h e m a t i c i a n s poss e s s i n g a h i g h e r l e v e l t h a n a M a s t e r ' s D e g r e e . F o r s u c h coun-

t r i e s , t h e b u i l d i n g u p p r o c e s s w i l l h a v e t o be l o n g and p a i n s t a k i n g , t i l l t h e c o u n t r y g e t s t o t h e p o s i t i o n of h a v i n g r e a c h e d t h e c r i t i c a l mass where a s e l f s u s t a i n e d a c t i v i t y may s t a r t . I n s u c h a p r o c e s s t h e h e l p of d e v e l o p e d c o u n t r i e s , e i t h e r by s e n d i n g t e a c h e r s home, or s e n d i n g s t u d e n t s a b r o a d , may h a v e a d e c i sive influence. Here i t i s p e r h a p s t i m e t o i s s u e a word of w a r n i n g a b o u t some

t e m p t a t i o n s t h a t may o c c u r d u r i n g t h i s s t a g e . Namely t h e p r o j e c t of u s i n g " s h o r t c u t s " i n math e d u c a t i o n may a p p e a r a s a v e r y s e d u c t i v e o n e . I t i s known t h a t d e v e l o p e d c o u n t r i e s f r e q u e n t l y h a v e a v e r y r i g i d , r o u t i n i z e d t y p e of m a t h e m a t i c a l e d u c a t i o n , e s p e c i a l l y i f t h i s system is t i e d (as it i s i n France) w i t h n a t i o n a l c o m p e t i t i v e e x a m i n a t i o n s . Hence i t may a p p e a r v e r y i n t e r e s t i n g n o t t o c o p y i n a s e r v i l e way t h e e x i s t i n g s y s t e m s of t h e d e v e l o p e d c o u n t r i e s , and i m m e d i a t e l y , o p t f o r b r a n d new

s y s t e m s and c u r r i c u l a . Here comes i n t o p l a y t h e m o d e r n i s t t e m p t a t i o n . Many s p e c i a l i s t s i n m a t h e m a t i c a l pedagogy may come

t o t h e g r o w i n g n a t i o n s , and p r o m i s e them t h e w o n d e r f u l f o r m u l a w h i c h would a l l o w them t o a c q u i r e i n a v e r y r a p i d way t h e m o s t r e c e n t d a t a i n m a t h e m a t i c a l s t u d i e s . Many d e v e l o p e d c o u n t r i e s , i n f a c t , f e l l i n t o t h i s t e m p t a t i o n , and s t a r t e d a f a r r e a c h i n g o v e r h a u l of t h e i r t e a c h i n g s y s t e m . The b a s i c i d e a i s a l w a y s t o t e a c h m a t h e m a t i c a l " s t r u c t u r e s " a t a v e r y young a g e , t h u s a l lowing t h e p u p i l ( l a t e r t h e s t u d e n t ) t o understand immediately

130

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t h e n a t u r e of t h e m a t h e m a t i c a l e n t i t i e s h e h a s t o l e a r n . I h a v e explained elsewhere what i s i m p o r t a n t ,

*

how t h i s p o i n t of v i e w i s f a l l a c i o u s . F o r

i n d e a l i n g w i t h mathematical o b j e c t s , i s n o t

what t h e y a r e , b u t how o n e u s e s t h e m . Hence, i n t h e l o n g r u n a good r o u t i n e t r a i n i n g , i s i n g e n e r a l b e t t e r t h a n a p r e m a t u r e a b s t r a c t d e f i n i t i o n . T h e r e i s n o r o y a l way t o m a t h e m a t i c s , and q u i t e c e r t a i n l y , w h a t e v e r t h e methods u s e d , a good p r o p o r t i o n of s t u d e n t s w i l l f a i l t o g r a s p t h e s e n o t i o n s ( n o t by l a c k of i n t e l l i g e n c e , b u t by l a c k of i n t e r e s t ) . Hence o r g a n i z e r s of s u c h p r o g r a m s s h o u l d b e warned n o t t o g i v e u p t h e t r a d i t i o n a l subj e c t s of m a t h e m a t i c a l t e a c h i n g : l e a r n i n g b y h e a r t , m e n t a l

a r i t h m e t i c , elementary geometry,

...

i n f a v o u r of t h e more

modern s u b j e c t s , l i k e s e t t h e o r y , l i n e a r a l g e b r a , a b s t r a c t a l g e b r a . T h i s i s , I b e l i e v e , a d a n g e r o u s i l l u s i o n , and t h e r e s p o n s i b l e a g e n c i e s s h o u l d b e v e r y c a u t i o u s i n s u c h moves : t h e common s t e m t o a l l m a t h e m a t i c a l knowledge d i d n o t v a r y s o much i n i t s r o o t a s a r e s u l t of r e c e n t p r o g r e s s . I n p a r t i c u l a r , e l e m e n t a r y t e a c h i n g h a s no r e a s o n t o b e a f f e c t e d by t h i s l a s t c e n t u r y ' s d i s c o v e r i e s i n mathematics. 2 ) L e t u s p a s s now t o t h e s e c o n d p o i n t , t h e d a n g e r of o v e r e s t i m a -

t i n g o u r own c a p a b i l i t i e s . L e t u s t h e n a d m i t o p t i m i s t i c a l l y t h a t t h e c r i t i c a l s i z e h a s been r e a c h e d ,

t h a t a few l o c a l m a t h e m a t i -

c i a n s d i d s u c h good work a s t o g e t i n t e r n a t i o n a l r e c o g n i t i o n . Then a new t e m p t a t i o n may a r i s e : t o d e v e l o p i n a n e x t e n s i v e way t h i s p r i m a r y f i e l d where t h e s e i n i t i a l s u c c e s s e s were o b t a i n e d . I t i s t o be e x p e c t e d t h a t t h i s i n i t i a l b r e a k t h r o u g h w i l l o c c u r

i n a m a t h e m a t i c a l b r a n c h of a somewhat p e r i p h e r a l i n t e r e s t : f o r it i s e a s i e r t o g e t r e s u l t s of i n t e r n a t i o n a l l e v e l i n a non-fashionable

d i s c i p l i n e where c o m p e t i t i o n i s n o t t o o h e a v y ,

and which d o e s n o t a t t r a c t t h e a t t e n t i o n of t o p m a t h e m a t i c i a n s . Of

c o u r s e s u c h a s u c c e s s , i n a n u p t o now m a t h e m a t i c a l l y s t e r i -

l e n a t i o n , d e s e r v e s f u l l a p p r o v a l . But t h e danger i s t h a t t h e s e

l o c a l m a t h e m a t i c i a n s , h a v i n g a c h i e v e d t h i s s u c c e s s may o b t a i n

a v e r y s t r o n g l o c a l p o s i t i o n and u s e t h e i r i n f l u e n c e t o i n c r e a s e t h e s h a r e d e v o t e d t o t h e i r own s p e c i a l i t y . A s t h e y a r e p r a c t i c a l -

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l y i n a p o s i t i o n o f monopoly, t h e y may f i n d n o o p p o s i t i o n t o t h e i r growth.

(May I r e c a l l , i n t h a t r e s p e c t ,

t h e i m p o r t a n c e of

graph t h e o r y i n t h e South E a s t Asian Mathematics Department). I t s h o u l d b e t h e d u t y of t h e l o c a l a u t h o r i t i e s t o remedy t h i s

s i t u a t i o n ; h e r e o n e f i n d s a g a i n t h e d a n g e r of o v e r s p e c i a l i z a t i o n w h i c h was d e n o u n c e d s e v e r a l t i m e s i n t h i s C o n g r e s s . T h e r e

i s n o d o u b t t h a t , w i t h r e s p e c t t o t h e g e n e r a l i n c r e a s e of mathematical c u l t u r e i n h i s country a mathematician should be Let m e say only t h a t c u r r i c u l a i n d e v e l o p e d c o u n t r i e s a r e n o t a r b i t r a r i l y

a b l e t o g e t o v e r h i s own p r e j u d i c e s a n d b i a s e s .

f i x e d . T h e r e i s a p r a c t i c a l c o n s e n s u s o n w h a t a s t u d e n t who wants t o s t u d y e x a c t s c i e n c e s (Physics, Chemistry) has t o know i n M a t h e m a t i c s . Namely,

the e s s e n t i a l notions i n Calculus

( D e r i v a t i v e s , I n t e g r a l s , d i f f e r e n t i a l e q u a t i o n s , some P . D . E ) ; t h e w h o l e c u r r i c u l u m i n p r i m a r y and s e c o n d a r y t e a c h i n g s h o u l d b e aimed a t t h e a c q u i s i t i o n of t h e s e b a s i c n o t i o n s ( t h e common

s t e m ) . T o t h a t end m a t h e m a t i c a l l y p r o p e r f u n d a m e n t a l n o t i o n s , l i k e l i n e a r a l g e b r a , a l g e b r a i c geometry have t o be s u b o r d i n a t e d . The s i t u a t i o n i s l e s s c l e a r i n t h e s e c o n d a r y t e a c h i n g , w h e r e c o n t r o v e r s y o n E u c l i d i a n Geometry i s s t i l l r a g i n g . Here t h e t r o u b l e comes f r o m t h e f a c t t h a t w e h a v e t o t a k e i n t o a c c o u n t p u p i l s who, f o r t h e m o s t p a r t w o n ' t c o n t i n u e t h e i r s t u d i e s

t o t h e C o l l e g e l e v e l , and w i l l e n t e r a t t h e a g e o f 1 6 i n t o p r o f e s s i o n a l l i f e . What k i n d o f m a t h e m a t i c s s h o u l d b e t a u g h t

t o s u c h s t u d e n t s ? E s p e c i a l l y t o t h o s e who w i l l l a t e r t a k e non-scientific

d i s c i p l i n e s l i k e Medicine, Legal s t u d i e s , b e l l e s -

l e t t r e s . . . I d o b e l i e v e t h a t f o r s u c h p u p i l s some k n o w l e d g e of c l a s s i c a l E u c l i d i a n Geometry u s e l e s s a s i t may a p p e a r , i s v e r y a p p r o p r i a t e f o r extending t h e i r c u l t u r a l background, b e c a u s e i t g i v e s them t h i s v e r y i m p o r t a n t e x a m p l e of g e o m e t r i c r e a s o n i n g , which t h e y w o n ' t h a v e o c c a s i o n t o l e a r n l a t e r i n l i f e . A t t h e u p p e r l e v e l of C o l l e g e s t u d i e s , m a t h e m a t i c a l s p e c i a l i z a t i o n s h o u l d n o t b e too h a s t y , e v e n a t t h e c o s t of i n i t i a l l y i m p e d i n g t h e f o r m a t i o n of i n d i g e n o u s m a t h e m a t i c i a n s . Summarizing : i t i s better t o d e l a y y o u r s t a r t t h a n t o i n f l a t e your department i n a g e n e r a l l y b i a s e d o r i e n t a t i o n .

132

R. THOM

L e t m e end f i n a l l y , by a g e n e r a l remark a b o u t t h e r o l e of

mathematics i n developing n a t i o n s . A s explained e a r l i e r ,

I have

t h e f e e l i n g t h a t t h e s t a n d p o i n t d e f e n d e d by many s p e a k e r s a t t h e m e e t i n g on t h a t m a t t e r i s somewhat d a n g e r o u s . Many d i d e x p r e s s t h e need f o r m a t h e m a t i c i a n s t o stress by p u b l i c r e l a t i o n methods t o t h e g e n e r a l c o l l e c t i v i t y t h e i m p o r t a n c e of mathemat i c s f o r t h e g e n e r a l d e v e l o p m e n t of t h e c o u n t r y . F i r s t , o n e h a s t o b e c a r e f u l a b o u t t h e u s e of mass media methods : t h e y e a s i l y g e t o u t o f c o n t r o l ( I a m s p e a k i n g h e r e by p e r s o n a l e x p e r i e n c e ) , and t h e y may lead you t o p o s i t i o n s , or engagements which l a t e r may p r o v e f o r you d i f f i c u l t t o s u s t a i n . Second, I b e l i e v e t h a t t h e case f o r m a t h e m a t i c s i s s u f f i c i e n t l y s t r o n g by i t s e l f and o b v i o u s f o r a n y u n b i a s e d o b s e r v e r t h a t a r t i f i c i a l mass media c a m p a i g n i n g may a p p e a r q u i t e u n n e c e s s a r y . Hence one s h o u l d n o t promote m a t h e m a t i c a l d e v e l o p m e n t by s t r e s s i n g t h e l o c a l i m p o r t a n c e o f m a t h e m a t i c s f o r t h e n e e d s of t h e c o u n t r y , b u t r a t h e r by a p p e a l i n g t o i t s u n i v e r s a l i m p o r t a n c e a s a t o o l f o r understanding r e a l i t y , a s a p r e r e q u i s i t e f o r p r a c t i c a l l y any scientific theoretization.

I t h i n k t h a t i n o r d e r t o promote i n

a n a t i o n t h e i n t e r e s t f o r m a t h e m a t i c s , o n e of t h e b e s t ways is to insist that local authorities require that a l l Civil

S e r v i c e a p p l i c a n t s h a v e a good l e v e l i n m a t h e m a t i c a l and s c i e n t i f i c knowledge, much h i g h e r , i n f a c t t h a n what i s p r o f e s s i o n a l l y n e e d e d . T h i s t r i c k h a s s u c c e e d e d (and i t s t i l l w o r k s ) i n a n o l d c o u n t r y l i k e F r a n c e , where t h e p r e s t i g e of m a t h e m a t i c s among t h e g e n e r a l p o p u l a t i o n i s f u n d a m e n t a l l y d u e t o t h e f a c t t h a t a h i g h l e v e l of m a t h e m a t i c a l t r a i n i n g i s needed t o e n t e r s u c h p r e s t i g i o u s s c h o o l s a s t h e E c o l e P o l y t e c h n i q u e (when

l a t e r i n t h e i r c a r e e r s t h e a l u m n i no l o n g e r need i t

... )

I

s t r o n g l y s u g g e s t , t h a t i t i s by h a v i n g s u c h r u l e s e n f o r c e d on t h e p e o p l e e n t e r i n g a d m i n i s t r a t i v e p o s t s , a t a l l l e v e l s , from postman t o e n g i n e e r s and t e c h n i c i a n s ) , t h a t t h e g e n e r a l l e v e l

of m a t h e m a t i c s i n a c o u n t r y c o u l d b e t h e m o s t e a s i l y r a i s e d . T h i s would be more r e a l i s t i c and f i n a l l y more e f f i c i e n t t h a n p r o m i s i n g t h e would-be m a t h e m a t i c i a n s g o v e r n m e n t a l jobs which would be

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anyway t o o s m a l l i n number t o a l l o w f o r a g e n e r a l s c i e n t i f i c t e a c h i n g among t h e p o p u l a t i o n .

*

See f o r i n s t a n c e : R.

Thom. Modern Math. An E d u c a t i o n a l a n d P h i l o s o p h i c a l E r r o r American S c i e n t i s t , V o l .

59, N6,

1 9 7 1 , p p . 695-99

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Developing Mathematics in Third World Countries, M.E.A. E l Tom l e d . ) 0 North-Holland Publishing Company, 1979

COMPUTERS,

MATHEMATICS A N D APPLICATIONS

Jacques-Louis L I O N S COLLEGE DE FRANCE PARIS

INTRODUCTION

!4e would l i k e , i n t h i s l e c t u r e , t o make some s i m p l e remarks, o f a v e r y i n t r o d u c t o r y n a t u r e and o f an elementary c h a r a c t e r , on t h e c o n n e c t i o n s between t h e s o - c a l l e d " P w e " and " A p p Z i e d " Matherni?tics and t o b r i e f l y i n d i c a t e how t h e Computers have d r a s t i c a l l y chanqed ( s i n c e , say, 1950) t h e s t r u c t u r e , t h e l e v e l and t h e n a t u r e o f t h e r e l a t i o n s h i p s between Mathematical Sciences and t h e o t h e r d i s c i p l i n e s and t h e whole human a c t i v i t y . These f a c t s (I) have two consequences :

(1) t h i s c o n s i d e r a b l e (and r e c e n t ) i n c r e a s e o f t h e r o l e o f mathematical methods i n t h e whole human a c t i v i t y (and I t h i n k t h a t what we have seen i s o n l y t h e v e r y b e g i n n i n q o f an even more d r a s t i c chanqe) i m p l i e s t h a t q r e a t e f f o r t s s h o u l d be devoted t o make t h e mathematical t o o l s a v a i l a b l e , o r t o make them a d j u s t e d t o p a r t i c u l a r s i t u a t i o n s which can l o c a l l y a r i s e : t h i s is t h e probi'eni o f Mathcmut?kal Education, a m a t t e r we s h a l l n o t c o n s i d e r h e r e ( 2 ) ; ( 2 ) i t becomes a v e r y i m p o r t a n t problem t o o r o a n i z e mathematical a c t i v i t i e s i n a f l e x i b l e and d e c e n t r a l i z e d manner, so as t o adapt t o l o c a l problems, w i t h o u t l o o s i n g t h e g e n e r a l view o f t h e proqresses o f t h e Mathematical Sciences ; and i t i s a l s o

1 ( ) These a r e indeed f a c t s , and n o t q e n e r a l i d e a s based on how t h i n q s s h o u l d be ! F o r i n s t a n c e , t h e I n t e r n a t i o n a l Conoress o f Wathematicians q a t h e r s e v e r y f o u r y e a r s s e v e r a l thousand o f mathematicians - and t h i s i s o n l y a s m a l l p r o p o r t i o n o f t h e Mat h e m a t i c a l Community (due t o economic d i f f i c u l t i e s t o a t t e n d these Congresses). New mathematical S o c i e t i e s a r e o r q a n i z i n o themselves a l l o v e r t h e w o r l d , some o f them b e i n q a l s o o r o a n i z e d on a Reaional b a s i s , such as t h e A f r i c a n Mathematical S o c i e t y , t h e South East A s i a n Mathematical S o c i e t y . New s o c i e t i e s c r e a t e themselves i n t h e purpose o f o r q a n i z i n o l i n k s between Mathematics and o t h e r d i s c i p l i n e s . New count r i e s a r e b e c m i n q members o f I.M.U. ( I n t e r n a t i o n a l Mathematical Union) and Societ i e s such as I . F . I . P . ( I n t e r n a t i o n a l F e d e r a t i o n f o r I n f o r m a t i o n Processing) o r I.F.A.C. ( I n t e r n a t i o n a l F e d e r a t i o n o f A u t c m a t i c C o n t r o l ) , o r q a n i z e i n t e r n a t i o n a l meetinqs t h e s i z e o f which i s becominn c l o s e t o t h a t of t h e I n t e r n a t i o n a l Congress o f Mathematicians - and t h i s w i t h o u t speakinq o f t h e s e v e r a l l a r g e and i m p o r t a n t s o c i e t i e s t h a t e x i s t i n S t a t i s t i c s and i n O p e r a t i o n a l Research. ( l ) Several l e c t u r e s w i l l address t h i s problem d u r i n o t h i s Symposium. We r e f e r i n p a r t i c u l a r t o t h e l e c t u r e s o f P r o f e s s o r s Ashour, d'Ambrosio and Nebres.

I35

136

J-L.

LIONS

v e r y i m p o r t a n t t o p r o v i d e each c o u n t r y w i t h t h e o p p o r t u n i t y o f havinq teams w o r k i n g t o o e t h e r on A p p l i c a t i o n s ; t h i s i s t h e problem of t h e o rg a n i za t i o n o f MathematicaZ Research, in connection w i t h Human needs. T h i s problem w i l l be c o n s i d e r e d i n s e v e r a l l e c t u r e s p r e s e n t e d i n t h i s Symposium. We s h a l l p r e s e n t here some s i m p l e remarks i n t h i s c o n n e c t i o n ; we do n o t o f f e r " s o l u t i o n s " ; we o n l y hope t o s t i m u l a t e d i s c u s s i o n s and, may be, t o propose some s i m p l e i d e a s which would n o t be t o o d i f f i c u l t t o implement and which we t h i n k should prove useful ( c f . Section 5 o f t h e l e c t u r e ) . The p l a n o f t h e l e c t u r e i s as f o l l o w s :

1. On t h e "Pure" and t h e " A p p l i e d " t o p i c s . 2. Connections between t o p i c s . 3. C r i t i c a l "mass" and how t o move on a q i v e n " v e r t i c a l " , 4. Choice o f t o p i c s . 5 . Sane p e r s p e c t i v e s . 6 . F i n a l remarks.

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1. ON THE "PURE" AND THE "APPLIED" TOPICS

A t y p i c a l m a t t e r which i s o f t e n r a i s e d when d i s c u s s i n g on t h e development o f Science, and, i n p a r t i c u l a r , on t h e development o f Mathematics, i s whether one s h o u l d emphasize t h e "pure" aspects o f t h e s c i e n c e , o f whether one s h o u l d g i v e some k i n d o f p r i o r i t y t o t h e " a p p l i e d " aspects o f i t . And i t i s q u i t e u s u a l t o see mathematic a l t o p i c s be " d i v i d e d " i n "Pure" and " A p p l i e d " . As f a r as t h e t o p i c s a r e concerned, we t h i n k t h a t t h i s i s a somewhat meaningless debate. Indeed each t o p i c has ( a t l e a s t i n Mathematical Sciences) a " v e r t i c a 2 " development, g o i n q f r o m t h e p u r e t o t h e a p p l i e d , o r f r o m t h e a p p l i e d t o t h e pure, o r from t h e around t o t h e s k y (I).

...

L e t us q i v e some s i m p l e examples. A t o p i c l i k e ProbabiZity Theory aoes f r o m t h e a p p l i c a t i o n s such as f i l t e r i n g , r e a l t i m e i m p l e m e n t a t i o n o f c o n t r o l d e v i c e s i n presence o f n o i s e , t o v e r y a b s t r a c t quest i o n s such as g e n e r a l approaches t o P o t e n t i a l Theory.

A r e p u t e d l y " a p p l i e d " t o p i c l i k e "ln,formntics" ( o r Computer S c i e n c e s ) , and more s p e c i f i c a l l y a t o p i c such as programninq lanquaqes, qoes f r o m p a r t i c u l a r a p p l i c a t i o n s t o q u i t e a b s t r a c t approaches r e l a t e d t o qeneral a l q e b r a and t o l o g i c . L e t us remark h e r e some methods, some when computers a r e computers and o n l y

t h a t t h i s p i c t u r e would radicaZ2y change w i t h o u t computers : i d e a s , which would be " p u r e " w i t h o u t computers become " a p p l i e d " a v a i l a b l e ; one can implement these methods w i t h t h e use o f w i t h t h e use o f computers.

Keepinq i n mind t h i s " v e r t i c a l " development o f a q i v e n t o p i c , we t h i n k i t i s q u i t e c l e a r t h a t a r e a s o n a b l e approach t o t h e problem o f r e s e a r c h on a g i v e n t o p i c i s more t o o r q a n i z e t h i n g s on a q i v e n " v e r t i c a l " ( t o p i c ) , on how to move ox t h e vert i c a l , t h a n t o e x c l u d e a t o p i c on t h e f a l s e p r e t e x t t h a t i t i s t o o " p u r e " o r t o o " a p p l i e d " ( 2 ) . T h i s apDroach a l s o a l l o w s t o t a k e i n t o account t h e already e x i s t i n g situation a t a o i v e n p l a c e and a t a q i v e n t i m e .

2. CONNECTIONS BETWEEN TOPICS

One f a c t , which i s w e l l known, b u t t h a t s h o u l d be emphasized a g a i n and a q a i n , i s t h a t a l l t h e " v e r t i c a l s " ( i n t h e t e r m i n o l o g y we i n t r o d u c e d i n S e c t i o n l ) , a r e rel a t e d t o q e t h e r i n a more o r l e s s s t r o n q way. T h i s i s t h e " h o r i z o n t a l " a s p e c t o f

1

( ) We do n o t p u t a h i e r a r c h y o f v a l u e ( a n o r d e r r e l a t i o n ! ) on t h e v e r t i c a l development. The word " v e r t i c a l " i s used t o f i x i d e a s ! 2

( ) I t remains t r u e t h a t , at a given t i m e , and depending on t h e eechnoZoqy a v a i l a bZe, some t o p i c s have more connections w i t h t h e a p p l i c a t i o n s t h a n o t h e r s . IJe s h a l l return t o this point.

J-L.

138

LIONS

t h e Mathematical Sciences. O f course a l l connections a re not known ! ! I t i s i n f a c t an i m p o r t a n t p a r t ( t h e most i m p o r t a n t one ? ) o f t h e development o f Science t o see "connections" where t h e y a r e ! ! ( l ) . T h i s w e l l known remark shows t h a t t h e choice of t o p i c s ( * ) i s n o t as c r i t i c a l as i t c o u l d l o o k a t f i r s t s i q h t . And - a l l t h i s i s obvious - t h e c h o i c e o f t o p i c s should t a k e i n t o account what already e x i s t s at a g i v e n p l a c e , b u t t h i s l a s t p o i n t i s n o t as s i m p l e as i t l o o k s , as we a r e now proceedinq t o see.

3 . CRITICAL "MASS" AND HOM TO MOVE ON A G I V E N "VERTICAL"

A prcliminary remark i s t h e f o l l o w i n q : i f i t i s decided t o work on a g i v e n t o p i c

w i t h t h e aim t o c o n s i d e r t h e t h e o r e t i c a l aspects as w e l l as t h e a p p l i c a t i o n s , a t m r n ( 3 ) i s needed. T h i s i s t h e n o t i o n of " c r i t i c a l mass" ; w i t h o u t a team, one aspect w i l l always be p r i v i l i q i e d w i t h r e s p e c t t o t h e o t h e r s , and we do n o t t h i n k t h a t an harmonious development w i l l be achieved. I n most cases a computer w i 7 1 a l s o be needed ( 4 ) . We p e r f e c t l y r e a l i z e t h a t t h i s need o f h a v i n q teams (even i f small teams) i s an extremely s e r i o u s d i f f i c u l t y . The o n l y t h i n g we can do a t t h i s s t a g e i s : ( i ) t o emphasize t h a t , w i t h o u t t h e o p p o r t u n i t y t o o r q a n i z e teams, t h e i d e a o f hav i n q s r i o u s a p p l i c a t i o n s implemented, t o q e t h e r w i t h a s e r i o u s t h e o r e t i c a l t h i n k i n q ( ) , i s an i l l u s i o n ;

5

( i i ) t o make some remarks on t h e way t h e s e teams c o u l d work, assuminq t h e y e x i s t . A t t h i s staqe, t h e main q u e s t i o n s a r e questions o f i n f o rm a t i o n :

( i ) how t o have i n f o r m a t i o n on t h e " l o c a l " problems a r i s i n q i n t h e I n d u s t r y and i n t h e Economy ; ( i i ) how t o have i n f o r m a t i o n on t h e " s t a t e o f t h e a r t " i n a g i v e n t o p i c . Q u e s t i h lil i s n o t as i n n o c e n t as i t l o o k s a t f i r s t s i q h t . I n o r d e r t o r e a l l y q r a s p what a r e t h e s c i e n t i f i c and, i n p a r t i c u l a r , t h e MathematicaZ Froblems which

( .1) J u s t t o m e n t i o n a few examples o f connections r e c e n t l y discovered, l e t us ment i o n t h e connections between some q u e s t i o n s o f A l q e b r a i c o r o f A n a l y t i c Geometry w i t h non l i n e a r P a r t i a l D i f f e r e n t i a l Equations a r i s i n q i n Physics, o r t h e connect i o n s between problems i n O p e r a t i o n a l Research and Management and i n P a r t i a l D i f f e r e n t i a l Equations. ( 2 ) A t l e a s t on a w o r l d b a s i s . I t i s o f course a n o t h e r m a t t e r a t t h e l o c a l l e v e l , w i t h a few mathematicians a t t h e same p l a c e . We s h a l l r e t u r n t o t h i s p o i n t .

3

( ) The s i z e o f these teams depends on t h e problems and on t h e l o c a l s i t u a t i o n . They do n o t have t o be t o o l a r q e , o t h e r w i s e t h e y would l o o s e a d a p t i v i t y . 4 ( ) L e t us emphasize t h a t t h e c o s t o f computers i s movinq down. 5 ( ) Which i s a l s o i n d i s p e n s a b l e f o r teaching t h e r e s u l t s o f t h e a p p l i c a t i o n s t o younqer s t u d e n t s .

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a r i s e i n g i v e n a p p l i c a t i o n s , one needs w o r k i n g i n c o l l a b o r a t i o n w i t h Engineers o r w i t h Managers t h a t a l r e a d y have some e x p e r t i s e i n Mathematics and i n Computer S c i ence - and t h i s i m p l i e s i n t u r n t h a t g r e a t concern and q r e a t c a r e s h o u l d be t a k e n t o t e a c h t o Enqineers and t o Economics p l a n n e r s and t o Managers, t h e most fundament a l t o o l s o f some p a r t s o f Mathematics and o f Computer Science, and a l s o t o l e a r n f r o m them what t h e i r main problems a r e ( 1 ) . Q u e s t i o n ( i i ) i s , f o r t h e time b e i n g , n o t s o l v e d , b u t i t seems t o US t h a t one c o u l d o r q a n i z e , on an i n t e r n a t i o n a l b a s i s , s e v e r a l ways t o answer t h i s t y p e o f q u e s t i o n . We r e t u r n t o t h i s p o i n t i n S e c t i o n 5 . T h e r e f o r e , a t t h i s s t a q e o f t h e d i s c u s s i o n , we assume t h a t t h e r e i s a w i s h t o develop a v e r t i c a l approach o f a g i v e n t o p i c , t h i s t o p i c b e i n q chosen on t h e b a s i s o f t h e l o c a l i n f o r m a t i o n a v a i l a b l e and o f t h e " s t a t e o f t h e a r t " . A very s e r i o u s probZem rernnins. I t i s connected w i t h t h e coreer o f a mathematician, and i n p a r t i c u l a r of a young mathematician. Suppose t h a t a younq Ph.D ( o r e q u i v a l e n t ) wishes t o move "up" o r "down" ( l e t us say once more t h a t t h i s t e r m i n o l o q y i s f o r c l a r i t y o f e x p o s i t i o n and t h a t i s n o t connected w i t h a h i e r a r c h y o f v a l u e s ! ) . Not o n l y he s h o u l d be f r e e t o do t h a t , b u t he s h o u l d n o t be p e n a l i z e d by t h i s c h o i c e ; i n o t h e r words, i t seems t o us t h a t t h e p r o b a b i l i t y o f h a v i n g a r e g u l a r c a r e e r by moving one way o r t h e o t h e r s h o u l d be t h e same - and t h e s e f a c t s should be k e p t i n niind by t h e e x p e r t s ( s h o u l d t h e y be chosen on a l o c a l b a s i s , o r on a r e q i o n a l o r i n t e r n a t i o n a l b a s i s ) who a r p usun7ly consulted f o r promotiens. We r e a l i z e t h a t t h i s i s a v e r y d e l i c a t e p o i n t , t h a t c o u l d l e a d t o m i s t a k e s , t o e r r o r s i n a p p r e c i a t i o n s . IJe o n l y w i s h t o make a v e r y p r e l i m i n a r y a t t e m p t i n t h i s d i r e c t i o n . The mathematical community has t o f a c e t h i s k i n d o f problem, t h a t should o f course, be s t u d i e d much more d e e p l y .

4. C H O I C E OF TOPICS We supDose now t h a t r e l i a b l e i n f o r m a t i o n i s a v a i l a b l e , b o t h l o c a l , r e g i o n a l , and i n t e r n a t i o n a l , and t h a t ( s m a l l ) teams can work i n a qood environment, i n connect i o n w i t h l o c a l needs i n t h e I n d u s t r y and i n t h e Economy ( i f t h i s i s t h e ZocaZ choice ! ) . What would happen ? I t i s v e r y l i k e l y t h a t a number o f t o p i c s w i l l be chosen a g a i n and a g a i n ; t h e y a r e these t o p i c s t h a t - a t t h e p r e s e n t time - have the l a r q e s t connections w i t h applications ; they are :

-

S t a t i s t i c s and A p p l i e d P r o b a b i l i t y , A p p l i e d F u n c t i o n a l A n a l y s i s , Numerical A n a l y s i s and F i n i t e Elements, O p e r a t i o n a l Research, P a r t i a l D i f f e r e n t i a l Equations o f Mathematical P h y s i c s , and a p p l i c a t i o n s i n hydrodynamics, s o i l mechanics, r o a d c o n s t r u c t i o n s , ..., Computer Sciences (proorammi ng 1anquaqes , o p e r a t i nq systems).

L e t us f i r s t remark t h a t t h e r e i s n o t h i n q more one c o u l d do a g a i n s t such c h o i c e s , when l o c a l l y made. B u t l e t us add t h a t such " u n i f o r m " l o o k i n q c h o i c e s would n o t be as bad as t h e y seem f o r t h e f o l l o w i n q reasons : ( i ) these c h o i c e s would be in parnlZel w i t h o t h e r topics t h a t alrendy e x i s t i n a given place ; ( i f ) an harmonious " v e r t i c a l " development should be enrouraged ;

( I ) C f . a l s o i n t h i s r e s p e c t : " T r a i n i n q o f Enqineers, T e c h n o l o q i s t s and Technic i a n s i n Developing C o u n t r i e s " , Report prepared by COSTED, March 1976.

140

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LIONS

( i i i ) t h e " horizontal" connections ( c f . S e c t i o n 2 ) should be t a k e n i n t o account a s soon a s i t i s p o s s i b l e .

5. SOuE PERSPECTIVES

On t h e b a s i s o f t h e above remarks, one can draw some o f t h e p o s s i b l e avenues f o r t h e promotion o f Mathematical Methods and o f Mathematical A p p l i c a t i o n s . One c o u l d t h i n k o f o r o a n i z i n o - may be a t t h e l e v e l o f UNESCO v e r t i c a l development of a o i v e n t o p i c .

-

i n f o r m a t i o n on t h e

T h i s i n f o r m a t i o n c o u l d be o i v e n i n t h e f o r m o f s m a l l b o o k l e t s q i v i n g examples of a p p l i c a t i o n s o f t h e techniques i n v o l v e d , and a s h o r t b i b l i o q r a p h y . These b o o k l e t s c o u l d be made, under t h e c o o r d i n a t i o n o f UNESCO, by teams o f e x p e r t s , i n c o l l a b o r a t i o n w i t h Reqional UNESCO Centers and a l s o i n c o l l a b o r a t i o n w i t h v a r i o u s committ e e s o f t h e I n t e r n a t i o n a l Mathematical Union ( I .M.U.) (1). They should be e a s i l y a v a i l a b l e and t h e y should be presented i n a way t h a t c o u l d be e v o l u t i v e and adaptive ( s o as t o t a k e i n t o account t h e new i d e a s , and t o make c o r r e c t i o n s when mistakes a r e found o r when some methods become o b s o l e t e ) . These b o o k l e t s should a l s o c o n t a i n some i n f o r m a t i o n on t h e " h o r i z o n t a l " connections, b e t ween t o p i c s

.

I t a l s o seems t o us - we r e a l i z e t h i s i s a v e r y d i f f i c u l t p o i n t , t o be handled w i t h t h a t when e x p e r t s a r e c o n s u l t e d f o r oromotions o f younq mathematiqreat care c i a n s , t h e y c o u l d be asked t o base t h e i r judqment and a d v i c e n o t o n l y on t h e r e search papers, b u t a l s o on t h e qeneral c o n t r i b u t i o n t o t e a c h i n g , and, o r , t o t h e development and, o r , t h e i m p l e m e n t a t i o n o f a p p l i c a t i o n s .

-

T h i s approach would h e l p i n t h e d i f f i c u l t problem o f h a v i n q an harmonious development o f Mathematics, w i t h a qood e q u i l i b r i u m between t h e fundamental aspects and t h e more c o n c r e t e a p p l i c a t i o n s . A c t u a l l y t h e importance and t h e r o l e o f t h i s e q u i l i b r i u m should be emphasized i n t h e Education. I n t h i s r e s p e c t , we a l s o w i s h t o add t h a t t r a i n i n g c o u r s e s , o r g a n i z e d on a r e g i o n a l b a s i s , s h o u l d prove useful ( 7 ) . These t r a i n i n q courses c o u l d be o r g a n i z e d i n c o n n e c t i o n w i t h t h e UNESCO Reqional Centers and these c e n t e r s could, i n t u r n , be connected w i t h t h e o r g a n i z a t i o n o f Ph.D. proqrams ( w h i c h i s an i m p o r t a n t s t e p , i n p a r t i c u l a r i f t h e r e i s a l o c a l need and aoreement f o r o r q a n i z i n q a team on a g i v e n topic)

.

( 1) I t c o u l d a l s o be u s e f u l t o c o o r d i n a t e , o r a t l e a s t , t o e s t a b l i s h s y s t e m a t i c exchanqes o f i n f o r m a t i o n between d i f f e r e n t bodies w h i c h a r e concerned w i t h these nroblems. We have mentioned UNESCfl and IiyU ; we would l i k e t o a l s o m e n t i o n t h e e f f o r t s made by IFAC and by I F I P . 2 ( ) T h i s A u t h o r was p r i v i l e g i e d t o a t t e n d such a t r a i n i n q course, o r q a n i z e d i n May 1977 i n Penanq ( M a l a y s i a ) , under t h e chairmanship o f P r o f e s s o r Tan Wanq Senq, and w i t h t h e h e l p o f l o c a l sources. under t h e ausnices o f UNESCO and o f S.E.A.V.S.,

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fi. FINAL REMARKS

As we a l r e a d y emphasized elsewhere ( 1) stahiZity o f a qroup w o r k i n g on a g i v e n t o p i c i s a v e r y c r u c i a l element. One has indeed t o r e a l i z e t h a t t h e o r q a n i z a t i o n o f teams which a r e a b l e t o qrasp t h e whole p i c t u r e of a q i v e n problem and t o t r e a t t h e a e n e r a l t h e o r e t i c 1 aspects as we21 as t h e i m p l e m e n t a t i o n o f t h e a p p l i c a t i o n s , takes several years ( ) . Therefore s t a b i l i t y i s very important ; l o c a l incentives should p r e v e n t b r a i n d r a i n and l o c a l , r e q i o n a l and i n t e r n a t i o n a l s u p p o r t (when r e q u i r e d ) s h o u l d be p r o v i d e d a l s o on a s t a b l e b a s i s .

h

The e f f o r t s needed f o r p r o v i d i n q such a s u p p o r t a r e a b s o l u t e l y j u s t i f i e d by t h e fundamental r o l e o f Mathematics. The r o l e o f Mathematics as " b a s i c " s c i e n c e has been r e c o q n i z e d s i n c e more t h a n two thousand y e a r s . W i t h o u t d i m i n i s h i n g a t a l l t h i s a l r e a d y fundamental r o l e , t h e adv e n t o f computers has d r a m a t i c a l l y chanqed t h e s i t u a t i o n ; mathematical t e c h n i q u e s p r e v i o u s l y o f no use e x c e p t by a v e r y few e x p e r t s became o f common use by l a r g e numbers o f s c i e n t i s t s o f v a r i o u s d i s c i p l i n e s ; mathematical models, a l l o w i n g f o r q u a n t i t a t i v e r e s u l t s , and used i n d e c i s i o n s c i e n c e s , became o f s y s t e m a t i c use. I n t h i s way, and as we a l r e a d y s a i d , i n a v e r y s h o r t p e r i o d o f t i m e , mathematical methods (and o f t e n non t r i v i a l ones) e n t e r e d Economics, Medicine, Management, e t c . A f a n t a s t i c network o f p l a c e s where v a r i o u s s c i e n t i s t s , e n q i n e e r s , manaqers, meet and work t o q e t h e r , has been c r e a t e d by t h e computinq c e n t e r s a l l o v e r t h e w o r l d and by networks o f computers ( 3 ) . I n t h i s way, Mathematics a r e one o f t h e keys t o development.

1 ( ) L e c t u r e p r e s e n t e d a t t h e Conqress " C u l t u r e e t DPveloppement", f o r t h e seven-

t i e t h b i r t h d a y o f P r e s i d e n t L. Senqhor ; t h e s c i e n t i f i c p a r t o f t h e Congress was o r q a n i z e d by P r o f e s s o r Hoqbe-Nlend, Dakar, Sept. 1976.

( * ) C f . i n t h i s r e s p e c t , t h e l e c t u r e o f P r o f e s s o r E. Rofmnn, i n t h i s Symposium, about a f r u i t f u l e x p e r i e n c e achieved between h i s w o r k i n q qroup o f R o s a r i o (Argent i n a ) and LABDRIA ( I R I A ) , France.

( 3 ) A s y s t e m a t i c study, w i t h s p e c i f i c i m p l e m e n t a t i o n s , o f t h e use o f computer n e t works i n c o u n t r i e s which a r e d e v e l o p i n a computer s c i e n c e s i s b e i n g made by D r . K . Dang-Quoc, Computinq Center o f Pharo, U n i v e r s i t y d ' A i x - M a r s e i l l e , 53, Bd. Charles Livon, 13007 M a r s e i l l e .

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Developing Mathematics i n Third World Countries, M.E.A. E l Tom led.) Q North-Holland Publishing Company, 1979

MATHEMATICS RESEARCH I N THIRD WORLD COUNTRIES: PITFALLS AND OPPORTUNITIES*

S. Shahshahani The remarks I w i l l make i n t h i s paper a r e n o t presumed t o a p p l y t o a l l t h e c o u n t r i e s l a b e l e d ' T h i r d - W o r l d ' ; t h e y a r e o b s e r v a t i o n s based on my p e r c e p t i o n o f t h e c u r r e n t s t a t e and t h e p o s s i b l e f u t u r e d i r e c t i o n s o f mathematics r e s e a r c h

i n Iran.

However, I f e e l t h a t t h e s a l i e n t f e a t u r e s o f t h e s i t u a t i o n a r e p r e v -

a l e n t t o v a r y i n g degrees i n many o t h e r c o u n t r i e s .

The ' p i t f a l l s ' and t h e

' o p p o r t u n i t i e s ' I w i s h t o speak o f a r e i n h e r e n t t o t h e s t r u g g l e t o f o s t e r r e s p e c t a b l e s c i e n t i f i c a c t i v i t y i n an environment where no such modern t r a d i t i o n i s i n existence.

As such, t h e y a r e p e r f e c t l y common t o e v e r y endeavor where .a

n a t i o n o f t h e T h i r d World undertakes t o ' i m p o r t ' a h i g h l y developed i n s t i t u t i o n o r i n d u s t r y w i t h o u t indigenous r o o t s .

The ' o p p o r t u n i t i e s ' p r e s e n t themselves i n

t h e f o r m o f a v a i l a b l e e x p e r i e n c e and e x p e r t i s e t h e T h i r d - W o r l d n a t i o n can draw upon, w h i l e t h e ' p i t f a l l s ' g e n e r a l l y l i e i n a b l i n d a d a p t a t i o n o f s u p e r f i c i a l structures. Even i n an a c t i v i t y as harmless as mathematics,

lack o f foresight i n i t s

development can r e s u l t i n c o s t l y long-range damage which cannot be e a s i l y r e c t i fied.

There i s no d e a r t h o f examples, even i n h i g h l y developed c o u n t r i e s , where

misguided p l a n n i n g and emphasis backed by o f f i c i a l s a n c t i o n (be i t a government o r a p r o f e s s i o n a l o r g a n i z a t i o n ) has s e t back t h e e v o l u t i o n of a d i s c i p l i n e by decades.

The danger i s t h e more r e a l where u n i v e r s i t y budgets and r e s e a r c h funds

a r e c e n t r a l l y c o n t r o l l e d , and o p p o r t u n i t i e s f o r t h e development o f d i v e r s e and competing p o l i c i e s do n o t e x i s t . The Case f o r Mathematics Research Perhaps more t h a n o t h e r s c i e n t i f i c workers, mathematicians a r e c o n s t a n t l y c a l l e d upon t o j u s t i f y t h e i r s u p p o r t by s o c i e t y .

Where s c a r c i t y o f funds and/or

keen c o m p e t i t i o n f o r p r i o r i t i e s p r e v a i l , t h i s can t a k e on a f a m i l i a r l y h o s t i l e ring.

I n p a r t i c u l a r , i n a d e v e l o p i n g c o u n t r y , t h e government may j u s t i f i a b l y

want t o know what mathematics r e s e a r c h can c o n t r i b u t e t o a c h i e v i n g t h e g o a l s

o f development.

There i s , I b e l i e v e , no answer t o such i n q u i s i t i o n s t h a t w i l l be

*The views expressed h e r e a r e t h o s e o f t h e a u t h o r ; t h e y do n o t n e c e s s a r i l y r e p r e s e n t t h e o f f i c i a l views o f t h e s c i e n t i f i c a u t h o r i t i e s o f t h e Government o f Iran.

143

144

S. SHAHSHAHANI

a b s o l u t e l y s a t i s f a c t o r y t o everyone. systems.

U l t i m a t e l y , i t becomes a q u e s t i o n o f v a l u e

One can e a s i l y imagine a Spartan s t a t e w i t h l i t t l e t a s t e f o r any k i n d

o f 'philosophy.'

To t a k e a d i f f e r e n t s i t u a t i o n , i n a c o u n t r y undergoing t h e

phase o f development where t h e achievement o f a minimum m a t e r i a l s t a n d a r d o f l i v i n g i s t h e overwhelming g o a l , b a s i c r e s e a r c h does n o t u s u a l l y e n j o y h i g h p r i o r i t y . I t i s a l s o hopeless t o argue f o r a s p e c i a l t y p e o f a p p l i e d mathematics research,

p a r t i c u l a r l y s u i t e d t o t h e s t a t e o f underdevelopment. t h e hands o f panacea d e a l e r s .

T h i s would be p l a y i n g i n t o

The s t a t e o f development of mathematics t o d a y i s

such t h a t t h e d i r e c t t r a n s f o r m a t i o n o i bona f i d e r e s e a r c h i n t o GNP can t a k e p l a c e e s s e n t i a l l y o n l y through i n t e r a c t i o n with h i g h l y sophisticated technology o r w i t h i n a h i g h l y complex economy. What t h e n i s t h e r a t i o n a l e f o r t h e s u p p o r t o f mathematics r e s e a r c h i n t h e T h i r d World?

Depending on t h e s t a t e o f development o f a T h i r d World c o u n t r y and

t h e e x t e n t o f long-range p l a n n i n g , one can argue w i t h honesty about t h e n e c e s s i t y o f h a v i n g a c t i v e mathematical minds a v a i l a b l e once a minimum l e v e l o f i n d u s t r i a l i z a t i o n has t a k e n p l a c e .

Even i n a w o r l d composed o f competing Spartan s t a t e s ,

t h e war machines w i l l r e q u i r e c o n s t a n t u p g r a d i n g f o r which c e r t a i n t y p e s o f mathematics r e s e a r c h i s i n d i s p e n s a b l e .

I n n o v a t i v e technology, r e g a r d l e s s o f i t s aims,

g i v e s r i s e t o new problems demanding mathematical i n v e s t i g a t i o n . On a more i d e a l i s t i c plane, one can c i t e a n o t h e r reason f o r t h e s u p p o r t o f mathematics r e s e a r c h .

T h i s , however, presupposes an acceptance o f a v a l u e system

which h o l d s man's c u l t u r a l h e r i t a g e i n g r e a t r e s p e c t .

While no p u r i s t i n t h e

t r a d i t i o n o f G. H. Hardy, I f e e l t h a t t h o s e who make w o r t h y c o n t r i b u t i o n s t o one o f t h e most t i m e l e s s t r e a s u r e s o f mankind efforts.

need n o t be a p o l o g e t i c f o r t h e i r

I f a goal o f development i s t h e i n t e l l e c t u a l advancement o f t h e popu-

l a c e and an enhanced awareness o f t h e u n i v e r s e , investment i n mathematics i s indeed q u i t e p r o f i t a b l e . With t h i s o u t l o o k , t h e u n i v e r s a l n a t u r e of mathematics ( p u r e o r a p p l i e d ) makes i t s s u p p o r t a v a l i d p o l i c y i r r e s p e c t i v e o f geographical boundaries o r t h e degree o f development.

One should a v o i d e x c e s s i v e c l a i m s f a r t h e b e n e f i t s and

t h e ' r e l e v a n c e ' o f a p p l i e d mathematics research; t h e d i r e c t impact o f such r e s e a r c h i s l i k e l y t o be p r o p o r t i o n a l t o t h e degree of development.

A similar

s i t u a t i o n e x i s t s i n o t h e r a p p l i e d f i e l d s , such as e n g i n e e r i n g , where t h e most u r g e n t l y needed s e r v i c e s i n t h e T h i r d World a r e a t a l e v e l n o t p r o p e r l y a s s o c i a t e d w i t h h i g h - l e v e l research. Ph.D.

I n I r a n , f o r example, a n i m p r e s s i v e number o f

l e v e l engineers, t r a i n e d f o r t h e s o p h i s t i c a t e d t e c h n o l o g y o f t h e West, gen-

e r a l l y f i n d themselves c o n f i n e d t o u n i v e r s i t i e s , where t h e i r impact on t h e c o u n t r y ' s i n d u s t r y i s a t best i n d i r e c t .

A t t h e same t i m e , a c h r o n i c shortage o f

t e c h n i c i a n s and p r a c t i c a l e n g i n e e r s reduces t h e p r o d u c t i v i t y and t h e e f f i c i e n c y o f the industry.

MATHEMATICS RESEARCH IN THIRD WORLD COUNTRIES

145

Some Lessons f r o m t h e Developed World Once t h e p r o m o t i o n o f mathematics has been accepted as a w o r t h w h i l e endeavor, i t becomes o u r t a s k t o d e v i s e p r o d u c t i v e mechanisms f o r t h e growth o f valuable research.

I t w i l l be h i g h l y i n s t r u c t i v e t o l o o k a t some r e c e n t e x p e r i -

ence of t h e developed w o r l d i n t h i s domain.

.

on:

Specifically,

I wish t o dwell b r i e f l y

( 1 ) some f e a t u r e s of t h e development of mathematics i n t h e

U.S. i n t h e p o s t -

S p u t n i k e r a , and ( 2 ) t h e changing a t t i t u d e s i n t h e developed w o r l d toward t h e n a t u r e o f mathematics and mathematics r e s e a r c h .

My p e r s o n a l exposure t o t h e

U.S.

scene may be a f a c t o r i n t h e s e l e c t i o n o f t h e f i r s t t o p i c , b u t t h e c h o i c e i s b a s i c a l l y m o t i v a t e d by two o t h e r reasons.

One i s t h a t t h e U n i t e d S t a t e s w i t n e s s e d

a phenomenal growth o f mathematics r e s e a r c h i n t h e l a s t two decades, as evidenced by t h e s t a g g e r i n g volume o f p u b l i c a t i o n s by American and A m e r i c a - t r a i n e d mathem a t i c i a n s and t h e p a r a l l e l i n c r e a s e i n t h e number o f Ph.D.'s.

T h i s success s t o r y

was t a i n t e d , however, i n t h e l a s t decade by t h e trauma o f unemployed o r i m p r o p e r l y employed Ph.D.'s.

I t i s c e r t a i n l y w o r t h w h i l e t o understand t h e U.S.

phenomenon

and t o i n v e s t i g a t e i t s r e l e v a n c e t o t h e e f f o r t s t o promote r e s e a r c h i n o t h e r countries.

The second reason i s t h a t i n p a r t s o f t h e T h i r d World, ' f u t u r i s t i c '

planners tend t o f o l l o w t h e patterns evolved i n the

U.S., o f t e n w i t h o u t a c r i t i c a l

e v a l u a t i o n o f t h e i n t r i n s i c m e r i t s o f such systems o r t h e i r s u i t a b i l i t y f o r adaptation.

T h i s i s c e r t a i n l y t r u e i n I r a n where a v e r s i o n o f t h e ' p u b l i s h - o r - p e r i s h '

p h i l o s o p h y i s b e i n g i n s t i t u t e d i n U n i v e r s i t y p r o m o t i o n p o l i c i e s , Ph.D.

programs

a l o n g American l i n e s a r e b e i n g planned, and v e r y l i t t l e s e r i o u s a t t e n t i o n t o a l t e r n a t i v e academic systems such as those of France o r t h e S o v i e t Union i s g i v e n . The American e x p e r i e n c e o f t h e l a s t t w e n t y y e a r s has generated c o n s i d e r a b l e debate and c o n t r o v e r s y which I need n o t reproduce here.

While I do n o t w i s h t o

downgrade t h e many remarkable achievements of American mathematics, I have chosen t o d i s c u s s below two o f t h e n e g a t i v e f e a t u r e s o f t h e s i t u a t i o n i n t h e

U.S. (Canada

can a l s o be i n c l u d e d ) which w i l l be r e l e v a n t t o f u t u r e d i s c u s s i o n s : ( 1 ) There i s a s u p p l y o f young Ph.D.'s, o s t e n s i b l y t r a i n e d t o do mathematics r e s e a r c h , whose number f a r exceeds t h e apparent demand f o r t h e i r t a l e n t s . The 'Ph.D.

e x p l o s i o n ' i s g e n e r a l l y t r a c e d t o i n a c c u r a t e p r e d i c t i o n s o f demand i n

t h e e a r l y p o s t - S p u t n i k p e r i o d and t h e subsequent expansion o f r e s e a r c h - o r i e n t e d graduate s c h o o l s .

F u r t h e r , t h e m a j o r i t y o f t h e young Ph.D.'s a r e t r a i n e d i n one

o r a n o t h e r d i s c i p l i n e o f ' p u r e ' mathematics, and a r e deemed u n s u i t a b l e f o r employment o u t s i d e t h e academic community.

The i m p l i c a t i o n s o f t h i s s i t u a t i o n f o r

o t h e r c o u n t r i e s w i s h i n g t o embark on a c r a s h program o f d e v e l o p i n g mathematics is s i g n i f i c a n t , and w i l l be e x p l o r e d l a t e r in t h e paper.

(2)

There i s g r e a t d i s p a r i t y i n t h e q u a l i t y o f e d u c a t i o n and r e s e a r c h o f

American Ph.D.'s.

N o t w i t h s t a n d i n g t h e g l o r i o u s c l a i m s made f o r t h e f o r m a l course

r e q u i r e m e n t system o f t h e most American g r a d u a t e s c h o o l s , i t seems t h a t a t many

146

S. SHAHSHAHANI

o f t h e l o w e r l e v e l Ph.D. g r a n t i n g i n s t i t u t i o n s , t h e course work does l i t t l e b u t t o compensate f o r a w o e f u l l y inadequate undergraduate background.

Following

course work a t such a graduate school, t h e Ph.D. c a n d i d a t e i s t h e n groomed f o r e n t r y i n t o t h e Wonderful World o f Paper P u b l i s h i n g (a.k.a.,

' r e s e a r c h ' ) , by

b e i n g t r a i n e d i n t h e g e n t l e a r t o f g e n e r a l i z a t i o n i n narrow o f f - s h o o t s o f t h e f i e l d s o f mathematics o f t e n c o n s i d e r e d 'dead' i n t h e h i g h e r echelons o f t h e mathematical community.

T r a g i c a l l y , t h e process a c t u a l l y works.

There a r e many

people who a r e v e r y good a t t h i s , and a glimpse a t t h e many dozens o f mathematics j o u r n a l s would prove i t .

I do n o t mean t o i m p l y t h a t t h i s phenomenon i s l i m i t e d t o American mathematics.

By no means.

B u t i n America, w i t h t h e penchant f o r mass p r o d u c t i o n

and a t r a d i t i o n o f a n t i - e l i t i s m , t h e c o n d i t i o n has reached p a r t i c u l a r l y w a s t e f u l and grotesque extremes. a r e American-educated,

I have proposed t o my c o l l e a g u e s i n I r a n , most o f whom t h a t b e f o r e we g i v e ' p u b l i s h i n g ' a p o s i t i o n o f p r e -

eminence i n o u r h i r i n g and promotion p o l i c i e s , we must ensure t h a t a reasonable number o f o u r Ph.D.-holding

f a c u l t y members can pass a s i m p l e l i t e r a c y t e s t .

The t e s t I have proposed i s t h a t t h e d o c t o r a t e - h o l d i n g person be a b l e t o read and a c q u i r e some u n d e r s t a n d i n g o f a t l e a s t one o f some 40 one-hour addresses

I suspect t h a t an analo-

g i v e n a t t h e p a s t two I n t e r n a t i o n a l Congress meetings.

gous t e s t would meet l i t t l e r e s i s t a n c e among p h y s i c i s t s , chemists, o r b i o l o g i s t s , where, r e g a r d l e s s o f s p e c i a l i z a t i o n , c o n s i d e r a b l e consensus about t h e mainstreams of r e s e a r c h seems t o e x i s t .

I t w i l l be c l a i m e d t h a t m a t h e m a t i c s , w i t h i t s l e n g t h y

h i s t o r y , i s t o o broad a f i e l d and t o o t e c h n i c a l a d i s c i p l i n e f o r t h e k i n d o f t e s t proposed.

A l t h o u g h t h e r e i s some t r u t h t o t h i s c l a i m , t h e apparent v a s t b r e a d t h

and t h e m u l t i p l e f r a g m e n t a t i o n o f mathematics i s a h i g h l y a r t i f i c i a l andexaggera t e d s t a t e , p a r t i a l l y an u n f o r t u n a t e l e g a c y of t h e f o r m a l i s t t r a d i t i o n .

This

b r i n g s me t o t h e second o b s e r v a t i o n , c o n c e r n i n g t h e changing a t t i t u d e s i n t h e developed w o r l d among mathematicians. The f o r m a l i s t t r e n d o f t h e f i r s t h a l f o f t h i s c e n t u r y , which was instrument a l i n u s e f u l c l a r i f i c a t i o n s and a b s t r a c t i o n s i n mathematics, was a l s o a c o n t r i b u t i n g f a c t o r t o t h e breakdown o f a u n i v e r s a l v a l u e system i n t h i s f i e l d and t h e o n s e t o f a b i z a r r e d e m o c r a t i z a t i o n o f r e s e a r c h .

I t a l s o ruptured the

t r a d i t i o n a l t i e s o f mathematics w i t h t h e sciences and c u l t i v a t e d i s o l a t i o n i s t tendencies.

One tends t o f o r g e t these days t h a t t h e e x p l i c i t d i s t i n c t i o n between

t h e n o t i o n s o f 'mathematical v a l i d i t y ' and ' p h y s i c a l r e a l i t y ' a r e o f such r e c e n t o r i g i n t h a t even Gauss was h e s i t a n t t o make p u b l i c h i s h i g h l y - d e v e l o p e d excurs i o n s i n t o non-Euclidean geometry.

C e r t a i n l y t h e advent o f f o r m a l i s m was a g r e a t

l i b e r a t i n g f o r c e i n mathematics, b u t t h e vacuum l e f t by t h e d e p a r t u r e o f p h y s i c a l r e a l i t y as t h e u l t i m a t e 'superego' ushered i n a c r i s i s o f values and a democratiz a t i o n o f mathematical o u t l o o k b o r d e r i n g on anarchy.

O f course, n o t even t h e

most w i l d - e y e d o f a n a r c h i s t s e v e r advocated ' l o g i c a l v a l i d i t y ' as t h e u l t i m a t e

147

MATHEMATICS RESEARCH IN THIRD WORLD COUNTRIES

Judge--a random c h a i n o f t a u t o l o g i e s has n e v e r made i n t e r e s t i n g mathematics. But i t i s p r e c i s e l y i n d i s t i n g u i s h i n g t h e ' i n t e r e s t i n g ' f r o m t h e ' d u l l ' and t h e 'profound' from t h e ' s u p e r f i c i a l ' t h a t our c r i s i s l i e s .

Probably e v e r y w o r k i n g

mathematician, a n c i e n t o r modern, has been a P l a t o n i s t s e a r c h i n g a w o r l d o f ideas as r e a l s u b j e c t i v e l y as t h e p h y s i c a l r e a l i t y and i n c l u d i n g t h e l a t t e r .

In

p r e - f o r m a l i s t t i m e s , t h e g r e a t m a j o r i t y o f mathematicians searched a h a n d f u l o f f a m i l i a r t e r r i t o r i e s i n t h i s P l a t o n i c w o r l d w h i c h had s t r o n g h i s t o r i c a l l i n k s t o t h e realm o f t h e ' r e a l . '

F o r m a l i s t l i b e r a t i o n h e r a l d e d an age o f e x t e n s i v e ,

a l b e i t indiscriminate, gropings i n t h i s world o f ideas.

As i n t h e a r t s , con-

sensus and t i m e have become t h e s o l e j u d g e s of t h e works o f mathematics.

While

a g r e a t degree o f consensus i n t h e e v a l u a t i o n o f r e s e a r c h was n e v e r l o s t t o mathematicians, t h e m a t t e r o f e s t a b l i s h i n g ' o b j e c t i v e ' c r i t e r i a became i n c r e a s i n g l y e l u s i v e as t h e p r o l i f e r a t i o n o f r e s e a r c h and p u b l i s h i n g c o n t i n u e d .

The

problem was f u r t h e r compounded by t h e need t o e x p l a i n d e c i s i o n s o f h i r i n g , p r o motion, and reward, based on r e s e a r c h achievement, t o non-mathematical a d m i n i s trators. While a complete r e t u r n t o p r e - f o r m a l i s t a t t i t u d e s i s perhaps n e i t h e r poss i b l e n o r d e s i r a b l e , a movement t o r e s t o r e balance t o mathematics has been s t e a d i l y g a i n i n g ground i n t h e l a s t decade.

The b e s t o f mathematics r e s e a r c h

today m a n i f e s t s renewed r e s p e c t f o r t h e s u b j e c t ' s i n t u i t i v e and h i s t o r i c a l r o o t s , b o t h w i t h i n and o u t s i d e t h e f i e l d . o r 'ugly,'

A p p l i e d q u e s t i o n s , once d i s d a i n e d as ' d i r t y '

a r e b e i n g welcomed by e r s t w h i l e p u r i s t s as sources o f b e a u t i f u l new

mathematics, and an i n f u s i o n o f modern methods has b r o u g h t l o n g - n e g l e c t e d f i e l d s i n t o t h e f o r e f r o n t o f mathematics r e s e a r c h .

Furthermore, t h e s t r i k i n g c o n v e r -

gence o f seemingly u n r e l a t e d f i e l d s o f mathematics, which a c t u a l l y s t a r t e d e a r l i e r , c h a r a c t e r i z e s much o f t h e b e s t r e s e a r c h done t o d a y and p r o v i d e s a s t r o n g u n i f y i n g force. I t i s t h e s e more r e c e n t t r e n d s i n mathematics t h a t p r e s e n t t h e g r e a t e s t c h a l l e n g e i n d e v e l o p i n g mathematics r e s e a r c h i n t h e T h i r d World.

W i t h o u t them,

we w i l l be d e v e l o p i n g second-class mathematics o f a foregone e r a ; t h e i r i n c l u s i o n , on t h e o t h e r hand, i s a d i f f i c u l t t a s k r e q u i r i n g awareness and competence. It i s t o o t e m p t i n g , e s p e c i a l l y i n t h e T h i r d World where d i r e c t c o n t a c t w i t h t h e

most a c t i v e c i r c l e s o f mathematics r e s e a r c h i s o f t e n l a c k i n g , t o f o l l o w ' t h e Dath o f l e a s t r e s i s t a n c e ' t h a t von Neumann has warned us about. P i t f a l l s and O p p o r t u n i t i e s To g e t back t o t h e main theme o f t h i s paper, I now w i s h t o p o i n t o u t how a s i m p l i s t i c s t i c k - a n d - c a r r o t p o l i c y o f t h e encouragement o f mathematics r e s e a r c h

w i l l n o t o n l y be o f l i t t l e consequence i n e l e v a t i n g t h e l e v e l o f r e s e a r c h i n T h i r d World c o u n t r i e s o f t h e predicament o f I r a n , b u t i t c o u l d a c t u a l l y c a r r y u n d e s i r a b l e consequences.

To be s p e c i f i c , some o f t h e p r o p o s a l s o f t e n p u t f o r t h

148

S.

SHAHSHAHANI

f o r advancing r e s e a r c h i n I r a n a r e : (i)

Research achievement as evidenced by p u b l i c a t i o n s i n i n t e r n a t i o n a l j o u r n a l s should be a m a j o r c o n s i d e r a t i o n i n promotion and t e n u r e policies o f universities.

(ii)

I n d i v i d u a l s w i t h o u t s t a n d i n g r e s e a r c h accomplishment s h o u l d be rewarded w i t h g r a n t s , membership i n an Academy, e t c .

(iii)

D o c t o r a t e - l e v e l programs o f e d u c a t i o n i n mathematics should be i n s t i t u t e d t o enhance t h e r e s e a r c h atmosphere.

( A t t h e present

t i m e , M a s t e r ' s degree i n mathematics i s o f f e r e d . ) (iv)

One o r more r e s e a r c h i n s t i t u t e s s h o u l d be e s t a b l i s h e d w h e r e i n t h e members would pursue r e s e a r c h unencumbered by t e a c h i n g and administrative duties.

While, on paper, t h e s e p r o p o s a l s seem as time-honored,

common-sensical

inducements t o research, t h e y prove t o be l u d i c r o u s l y premature once t h e mathe m a t i c a l environment i n t o which t h e y a r e t o be i n t r o d u c e d i s s c r u t i n i z e d . S u f f i c e i t t o s a y t h a t , d e s p i t e a c e r t a i n q u a n t i t y o f paper p r o d u c t i o n , c e r t a i n l y no more than a h a n d f u l o f some 50-100 d o c t o r a t e - h o l d i n g mathematicians i n I r a n would pass t h e 'minimum l i t e r a c y t e s t ' proposed e a r l i e r . c o n t r a s t w i t h t h e s i t u a t i o n i n t h e U.S. ably well.

Not o n l y d i d t h e U.S.

Note t h e

where t h e same s t r a t e g y worked admir-

a l r e a d y possess a s t r o n g mathematical

t r a d i t i o n b u i l t around f i g u r e s l i k e G. 0. B i r k h o f f , N. Wiener, and S. L e f s c h e t z , t h e i n f l u x o f World War refugees a l s o i n j e c t e d v i t a l i t y t o t h e mathematical environment. There i s no doubt i n my mind t h a t t h e i m p l e m e n t a t i o n o f such p o l i c i e s

w i l l l e a d t o a p r o l i f e r a t i o n o f p u b l i c a t i o n s ( t o which t h e c h e e r f u l a d m i n i s t r a t o r s w i l l p o i n t as t h e s i g n o f ' s u c c e s s ' of t h e p o l i c i e s ) .

But w i l l t h e y l e a d

t o ( a ) t h e g e n e r a l enhancement o f t h e q u a l i t a t i v e l e v e l o f competence and s o p h i s t i c a t i o n i n mathematics, o r ( b ) genuine c o n t r i b u t i o n s o f v a l u e t o o u r science?

L e t me s p e c u l a t e on what t h e l i k e l y outcomes would be:

( 1 ) Out o f t h e n e c e s s i t y f o r s u r v i v a l and r e s p e c t i n t h e community, t h e mathematicians w i l l b u s i l y engage i n t h e p r o d u c t i o n o f t h e t y p e o f p u b l i c a t i o n s t h e y can most e a s i l y produce.

Since t h e l e v e l o f mathematical e d u c a t i o n i s

r a t h e r low, t h e p u b l i c a t i o n s a r e n o t l i k e l y t o c o n t r i b u t e much t o t h e advancement o f mathematics.

They w i l l s i m p l y exacerbate an a l r e a d y a l a r m i n g p u b l i c a t i o n

p o l 1u t i o n .

( 2 ) L i t t l e t i m e w i l l be l e f t f o r t h e l u x u r y o f d e v e l o p i n g o n e ' s awareness and knowledge i n mathematics ( i s t h i s n o t what r e s e a r c h i s a l l a b o u t ? ) .

It

w i l l be t o o r i s k y t o work on d i f f i c u l t problems o f genuine i n t e r e s t .

(3) The f i e l d s o f mathematics l i k e l y t o develop i n t h e c o u n t r y w i l l be t h e h i g h l y i s o l a t e d narrow s p e c i a l t i e s t h a t r e q u i r e l i t t l e b r e a d t h o f knowledge

MATHEMATICS RESEARCH IN THIRD WORLD COUNTRIES

I49

or awareness of historical roots. We shall re-live the extremes of formalism. (4) Mathematicians adapted to this environment will have little ability t o adjust to changing conditions. Nor will they be able to respond to applied problems of urgency, should they arise. ( 5 ) Premature institution of doctorate-level program will lead to an increase in the number of poorly-qualified Ph.D.'s. As the capacity for the absorption o f mathematicians in the economy o f a Third World country is quite limited, an early saturation of positions may have a fatal effect on the natural qualitative and quantitative growth of mathematics. (6) An 'elite' class of most successful paper-writers will develop which will dominate the educational and research policies of the country. The elite will steer the direction of the development o f mathematics in ways that will best serve to uphold their own undeserved status. All this may sound overly exaggerated and pessimistic, especially to those of the developed world. But the realities o f the role of education in many countries o f the Third World have taught us a valuable lesson. For decades, the main product of the institution o f education in the underdeveloped world has been the creation of a ruling elite, while education has contributed little to meaningful development. We already have too many Ph.D.'s; what we now need are Research Institutes and Academies to properly continue our mimicry! What is to be done? What are the 'opportunities'? Of course none of the proposals (i)-(iv) indicated above is intrinsically bad. One simply has to adjust these to the realities of the situation, keeping in mind that a genuinely strong tradition will probably take years to develop. National policies can best serve to raise the general level of competence among mathematicians. Great traditions in mathematics began as schools formed around exceptional individuals. Such schools cannot be created by legislation or decree, and there are no instant formulas for promoting genuine research. A detailed list of sensible and implementable steps to raise the general level of mathematics will depend greatly on the particular predicament of the country in question, but a few general statements may be made. Instead of requiring published papers, the universities may demand the continuing education of their mathematicians until a satisfactory level of competence is reached. Sabbaticals and summer leaves may be used for this purpose. An upgrading o f undergraduate education is another crucial and practical task. Undergraduate education should include areas of the potential applications of mathematics, and joint programs with physics, engineering, and other related areas are desirable. Once a strong undergraduate program has taken root, graduate-level education with the possibility of a doctorate degree may be implemented. Graduate schools should emphasize broad education and a 'minimum literacy requirement,'

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or very soon the isolationist-escapist tendencies will creep in. We cannot afford such waste. In the Third World, the training of broadly educated people is specially crucial, for here the educated individual carries a heavier share of national responsibility than his counterpart in the developed world. Department of Mathematics and Computer Science Arya-Mehr University of Technology P.O. Box 3406, Tehran, Iran Department of Mathematics University of California Berkeley, CA 94720, USA

DeveZoping Mathematics i n Third World Comtries, M.E.A. E l Tom led.) 0 North-Holland Publishing Company, 1979

ORGANISING n'lAT1IEnTATICAL RESEARCTI I K DEVELOPIKG COUNTRIES V. S . N a r a s i m h a n

Tata I n s t i t u t e of F u n d a m e n t a l R e s e a r c h , IIorni n h a b h a Road C o l a b a , B o m b a y 400005, India T h e O r g a n i s i n g C o m m i t t e e of t h e C o n f e r e n c e h a s s u g g e s t e d t h a t i t would b e of i n t e r e s t t o know t h e e x p e r i e n c e g a i n e d i n t a c k l i n g p r o b l e m s of planning, o r g a n i s a t i o n a n d m a n a g e m e n t of c r e a t i v e a n d p r o d u c t i v e m a t h e r n a t i r a l a c t i v i t i e s

i n t h i r d world c o u n t r i e s .

I s h a l l begin by giving a b r i e f a c c o u n t of t h e e x p e r i e n c e

i n India, e s p e c i a l l y a t t h e T a t a I n s t i t u t e of F u n d a m e n t a l R e s e a r c h ( T I F R ) , i n organising mathematical research.

An e f f o r t in o r g a n i s i n g m a t h e m a t i c a l r e s e a r c h

w a s m a d e i n T I F R i n t h e e a r l y fifties.

At t h a t t i m e t h e r e w a s only a v e r y s m a l l

n u m b e r of good m a t h e m a t i c i a n s i n I n d i a , w o r k i n g i n i s o l a t i o n in a few u n i v e r s i t i e s ; t h e r e w e r e h a r d l y a n y e x p e r t s in m o s t of t h e d o m a i n s which w e r e i n t h e m a i n s t r e a m of m a t h e m a t i c s . A s m a l l n u m b e r of u n i v e r s i t i e s w e r e a b l e t o g i v e a r e a s o n a b l y good e d u c a t i o n i n a few s u b j e c t s , w h i l e t e a c h i n g a t t h e s a m e t i m e a n outdated c u r r i c u l u m to t h e n e g l e c t of s e v e r a l b a s i c t o p i c s . T h e r e w a s no d e a r t h of t a l e n t e d s t u d e n t s f a s c i n a t e d b y m a t h e m a t i c s , p o s s i b l y d u e t o t h e Indian t r a d i t i o n i n m a t h e m a t i c s . Rut t h e y w e r e i n s u f f i c i e n t l y p r e p a r e d to g o i n t o r e s e a r c h a n d t h e r e w a s no s e r i o u s effort m a d e to p r o v i d e t h e m with o p p o r t u n i t i e s t o c o m e i n t o c o n t a c t with high l e v e l m a t h e m a t i c s and to c h a n n e l i s e t h e i r talents.

More

often t h e i r t a l e n t s w e r e d i v e r t e d t o w a r d s t r i v i a l p r o b l e m s . A r o u n d 1950, Professor K. C h a n d r a s e k h a r a n i n i t i a t e d a t T I F R a c a r e f u l l y planned p r o g r a m m e f o r o r g a n i s i n g m a t h e m a t i c a l r e s e a r c h .

Students,

w h o had c o m p l e t e d t h e i r Yaster's d e g r e e , w e r e s e l e c t e d o n t h e b a s i s of a n interview.

D u r i n g t h e f i r s t y e a r of t h e i r s t a y t h e y w e r e given c o u r s e s i n b a s i c

m a t h e m a t i c s , u s u a l l y i n A l g e b r a , A n a l y s i s and Topology; t h e s e c o u r s e s w e r e c o m p u l s o r y for a l l s t u d e n t s , independent of t h e i r e v e n t u a l s p e c i a l i s a t i o n . L a t e r 151

M. S. NARASIMHAN

152

t h e y w e r e i n t r o du ced t o m o r e ad v an ced t o p i c s by m e a n s of l e c t u r e s by staff and v isi t i n g m a t h e m a t i c i a n s .

S i n ce t h e r e w e r e no e x p e r t s iq India a t tha t t i m e in

s e v e r a l i m p o r t a n t f i el d s , f o r e m o s t e x p e r t s in t h e s e f i e l d s w e r e invited f r o m a b r o a d t o g i v e l e c t u r e s a t t h e I n s t i t u t e.

Nu me rous s e m i n a r s o r g a n i s e d by t h e

s t u d e n t s , c o m b i ned with i n f o r mal c o n t a c t s between s t u d e n t s a m o n g t h e m s e l v e s and t h e i r t e a c h e r s , helped t h e m a c q u i r e d e e p e r insight into t h e i r fie lds. Stude nts w e r e e n c o u r a g e d to t a k e u p new fields: often the y found t h e i r own p r o b l e m s and worked on t h e m independently.

In a few y e a r s t h i s p r o g r a m m e p r o v e d to b e a n

immense success, On t h e b a s i s of t h i s e x p e r i e n c e I would m a k e t h e following points on t h e o r g a n i sa t i o n of r e s e a r c h in developing c o u n t r i e s .

I b e l i e v e tha t the y m a y b e

valid i n t h e context of s e v e r a l developing c o u n t r i e s . 1.

I n s t e a d of t r y i n g to i m p r o v e t h e l e v e l of r e s e a r c h uniform ly in all i n s t i -

tu t i o n s a t t h e s a m e t i m e , o n e should f i r s t c o n c e n t r a t e t h e e ffort in a v e r y s m a l l n u m b e r of i n s t i t ut i o n s , s a y one o r two.

T h i s would he lp to p r o v i d e s t r o n g initia l

m o m e n t u m a n d to build u p a v i ab l e g r o u p of m a t h e m a t i c i a n s ,

It is p r e f e r a b l e

th a t t h i s e f f o r t is m a d e i n a new i n s t i t u t i o n (which could b e a p a r t of a u n i v e r s i t y ) s p e c i f i c a l l y c r e a t e d f o r t h i s p u r p o s e a s , i n t ha t c a s e , one ne e d not b e bogged down by t h e c o n s t r a i n t s a n d inhibiting t r a d i t i o n s i m p o s e d by t h e e x i s t i n g s e t up. One h o p e s t h a t e v en t u al l y t h e m a t h e m a t i c i a n s f o r m e d a t t h e s e c e n t r e s would b e a b l e to b r i n g u p t h e g e n e r a l l ev el . 2.

It is i m p o r t a n t to g i v e an i n t e n s i v e p r e l i m i n a r y t r a i n i n g to r e s e a r c h

s t u d e n t s i n s o m e of t h e b a s i c s u b j e c t s i n o r d e r to m a k e up for t h e de fic ie nc ie s a n d g a p s i n b a s i c knowIedge d u e to t h e e x i s t i n g de fe c tive e duc a tiona l s y s t e m .

A b r o a d - b a s e d t r a i n i n g would s a f e g u a r d a g a i n s t n a r r o w s p e c i a l i s a t i o n , which is one of t h e g r a v e d e f e c t s in t h e r e s e a r c h s e t up in t h e s e c o u n t r i e s , As o v e r a l l

ORGANISING MATHEMATICAL RESEARCH IN THIRD WORLD COUNTRIES

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s t a n d a r d s improve, the t i m e devoted to t h i s p a r t of training can be curtailed. 3.

After this period of preliminary training, the students should b e i n t r o -

duced to subjects in the forefront of r e s e a r c h , e i t h e r by m e a n s of l e c t u r e s by s t a f f o r by student s e m i n a r s . In subjects where t h e r e is no e x p e r t i s e in the country, the co-operation of foreign e x p e r t s should b e sought for t h i s purpose, 4.

It is n e c e s s a r y t o allow freedom f o r students to choose t h e i r own field

of specialisation within a n overall framework of a r e a s chosen for development and t o encourage t h e m to take up r e s e a r c h in important fields in which t h e r e i s no e x p e r t i s e in the country,

One of the r e a s o n s for stagnation in r e s e a r c h i n

developing countries is the perpetuation of outdated r e s e a r c h in fields with no relevance to the p r e s e n t day mathematics, against "soft r e s e a r c h " in "modern" fields.

At the s a m e t i m e one should guard

For t h i s purpose, during the period

of preliminary training (and even a f t e r w a r d s ) the students should b e helped t o cultivate good taste and a feeling f o r standards. 5.

I t is v e r y e s s e n t i a l t o c r e a t e good a t m o s p h e r e and working conditions

and t o provide adequate facilities and financial support.

A p a r t f r o m the obvious

facilities like a good l i b r a r y , i t is e s s e n t i a l to provide m e a n s f o r maintaining continuous contacts with o t h e r mathematicians within the country and abroad. Fortunately the financial outlay n e c e s s a r y for setting up a r e s e a r c h institution in mathematics is comparatively s m a l l . 6.

up.

One cannot overemphasize the necessity for a flexible adminstrative set One of the obstacles in fostering r e s e a r c h in developing countries is the

set of antiquated r u l e s and regulations which make i t difficult for the academic staff to function effectively.

7.

While concentrating on developing f i r s t rate r e s e a r c h , such an i n s t i -

tution should, at the s a m e t i m e , attempt to i m p r o v e the level of university

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education.

NARASIMHAN

T h i s could be done by means of s u m m e r schools and by offering

visiting positions t o enable t e a c h e r s and r e s e a r c h s c h o l a r s to spend a period of time devoted exclusively for study and r e s e a r c h . 8.

Thought should b e given for obtaining suitable positions for t h e mathe-

maticians formed in these institutions, While a certain number of t h e s e could continue to work there, it is imperative t o place the o t h e r s in positions where they can function effectively, pursuing t h e i r own r e s e a r c h and improving the general level of mathematics. 9.

One method to improve r e s e a r c h standards, which is s o m e t i m e s

advocated, is t o select s o m e good students and t o send them t o mathematically advanced countries for training and r e s e a r c h , T h i s method h a s certain drawbacks; s u c c e s s rate is not likely to be high and the successful s c h o l a r s may not r e t u r n . If it is found n e c e s s a r y to send s o m e students abroad for r e s e a r c h in certain topics, t h i s is t o b e done only a f t e r they have acquired a certain amount of maturity and clearly defined i n t e r e s t s , by undergoing initial training in the

country.

A mathematician, who h a s taken the first s t e p s in r e s e a r c h in an institution in one’s own country, will have a s e n s e of attachment and loyalty to the institution, which is n e c e s s a r y i f one is not to b e tempted away by the attractions of a developed country. In any case, it is n e c e s s a r y t o have certain institutions i n the country providing facilities f o r mathematicians who r e t u r n after a period of training abroad. 10.

In appropriate c a s e s , one could set up c e n t r e s f o r r e s e a r c h which wilt

s e r v e s e v e r a l countries in a given region.

I understand that t h e r e are a l r e a d y

s o m e initiatives in this direction, In m y view, such c e n t r e s should not confine themselves to the organisation of s u m m e r schools and s e m i n a r s and to a

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v i s i t o r s ' programme: t h e i r usefulness would b e tremendously i n c r e a s e d i f they have a p r o g r a m m e for training students f o r r e s e a r c h o v e r an extended period and for initiating them into r e s e a r c h , enlisting f o r t h i s purpose the s e r v i c e s of s o m e able and devoted mathematicians on a long t e r m b a s i s , Another point I would like t o touch upon is the desirability of making good mathematical material easily a c c e s s i b l e through the preparation and publication

of l e c t u r e notes, proceedings of s u m m e r schools, etc.

T h i s activity could a l s o

b e of s o m e economic value for developing countries in t h e context of the c u r r e n t cost of mathematical l i t e r a t u r e ,

Preparation of the l e c t u r e notes by students

would enable them to obtain a b e t t e r g r a s p of the subject and would, a t the s a m e time, provide s o m e training to them in mathematical exposition, T h e vision, dynamism and the administrative ability of the p e r s o n s running the p r o g r a m m e and the enthusiasm and talent of the students will undoubtedly play a predominant r o l e in the s u c c e s s of any plan of the type suggested above. It is an exciting and challenging t a s k to produce mathematicians of international level in developing countries, I a m s u r e that this can b e done by nurturing the talent available in t h e s e countries and giving i t a s e n s e of direction.

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Developing Mathematics i n Third World Countries, M.E.A. E l Tom (ed.) 0 North-Holland Publishing Company, 1979

par Le P r o f e s s e u r H.

HOGBE-NLLWD

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P r e s i d e n t de 1'Union Math6matique A f r i c a i n e

IIWROWCTION L a s i t u a t i o n math@matique a c t u e l l e de l ' i l f r i q u e e s t une s i t u a t i o n a y a n t l e s deux c a r a c t e r i s t i q u e s s u i v a n t e s : grand r e t a r d s c i e n t i f i q u e e t grand Bveil s c i e n t i f ique. Primo : C ' e s t une s i t i i a t i o n de grand r e t a r d s c i e n t i f i q u e l o r s q u ' o n compnre 1 ' A f r i q u e mathematique aux a u t r e s c o n t i n e n t s , notamment 1'Europe e t 1'Amerique du Nord. Ce r e t a r d s e m a n i f e s t e c o n c r 6 t e n e n t p a r d i v e r s a s p e c t s : f a i b l e s s e du noqbre de math@n&iciens n a t i o n a u x q u a l i f i e s (Docteurs en Mathemitiques e t P H-uction mathematique de f a i b l e quant i t 6 p a r r a p p o r t 5 13 p r o d u c t i o n mondizle, f a i b l e s s e d e s aoxee_n_gmat6riels m i s 5 l a d i s p o s i t i o n d e s matht5maticiens (locaux r 6 d u i t s , b i b l i o t h e q u e s m?th&mati-* ques t r 6 s embryonnaires, b u d g e t s p r o p r e s aux rnathematiques gdneralement ink r d e s c h a r g e s d'enseignernent. e x i s t a n t s ...)

-

Secundo : C ' e s t une s i t u a t i o n de grand Bveil s c i e n t i f i a u e c a r a c t 6 r i s 6 e oar une p r i s e de c o n s c i e n c e de p l u s en p l u s n e t t e du r a l e capit a l oue l e s mathematiciens peuvent j o u e r dnns l e p r o c e s s u s d e d6veloppement de 1 ' A f r i q u e . Cet Bveil s e m a n i f e s t e p a r l e s s 6 r i e u x e f f o r t s f o u r n i s p a r l e s gouverneaents d e s p a y s a f r i c a i n s e t l e s mathBmaticiens de c e s pays pour comb l e r r a p i d e n e n t l e i e + a r d s e c u l a i r e de 1 ' A f r i q u e dzns l e domaine d e s mqth6mst i q u e s , notnmment p a r l e fenforcement de l a c o o p e r a t i o n mathematique au n i veau n a t i o n a l , r e g i o n a l e t i n t e r n a t i o n a l . L a n a i s s a n c e de m i o n M a t h h a t i oue A f r i c a i n e e s t l ' e x p r e s s i o n l a p l u s c o n c r 6 t e de c e t @ v e i l . Le p r e s e n t r n o p o r t s e r a d i v i s 6 en deux u a r t i e s

I

Dans l a premi&re p i r t i e nous f e r o n s une s y n t h h s e d e s renseignements f o u r n i s p a r d i v e r s dBpzrtements de mathematiques s u r l a b a s e du q u e s t i o n n a i r e de 1'uL'E:SCcI e t d e s e n q u l t e s menees d i r e c t e r n e n t p a r 1'Union Mathkmatique A f r i caine. Dans l a seconde p a r t i e nous f e r o n s quelques E r o p o s i t i o n s p o u r l l a c t i o n f u t u r e de 1'UhESCO dnns l e domaine d e s mnthernatiques e n Afrique.

En annexe, nou8 p u b l i o n s d e s i n d i c a t i o n s c h i f f r 6 e s s u r l e nombre a p p r o x i m a t i f d ' n f r i c a i n s t i t u l a i r e s a l u n D o c t o r a t ou PH.D en mathgrnatiques.

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E x t r a i t d ' u n r a p p o r t p r 6 s e n t 6 A l'UNESC0, en A v r i l 1978.

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I58

I

- SYNTHESE DES RENSEIGN'EWXJTS .WTHEIATIQUES

1) L o c a l i s a t i o n : En g6n6ra1, l a v i e mathematique en Afrique a u niveau nupOrieur s e concent r e dans d e s dBpartements de math6matiques de c e r t a i n e # f a c u l t B s (fRCUltbf3 f a c u l t ' s ou i n s t i t u t s charges d e s q u e s t i o n s d ' g d u c a t i o n , fad'ingdnieurs c u l t e s d e s sc!ences C e t t e l o c a l i s a t i o n d e s s e r v i c e s mathematiques d a n s d i v e r s e 8 f a c u l t 6 3 a pour i n c o n v e n i e n t mnjeur de n e p a s f a v o r i s e r l a c o o p e r a t i o n e n t r e mathematiciens d ' u n e mame U n i v e r s i t 6 ; c e q u i a f o r t i o r i ne f a v o r i s e plis l a coap0r-tion n a t i o n a l e ou r e g i o n a l e .

. .e

Cependant, dans c c r t a i n e s u n i v e r s i t & s , comme l ' U n i v e r s i t 6 de Xhartourn (Soudnn) 1' U n i v e r s i te d'Abid jCw( g a t e d f I v o i r e ) 1 'Universi t 6 de Dakar ( S b n Q a l j , 1 I U n i v e r s i t Q d'Ibadan (NigGria), de s 6 r i e u x e f f o r t s o n t Et6 r6cernment r h a l i s d s pour I n c r e a t i o n L I I n s t i t u t s d e msthematiques, coordonn a n t l ' e n s e m b l e d e s a c t i v i t b s rnath6matiques de 1V n i v e r s i t B .

,

2 ) Ecluipernent

I

En gGnhral, l e s looaux r e s e r v e s aux mathematiques en Afrique s o n t t r o p B t r o i t s e t ne t i e n n e n t p a s compte d e s p e r s p e c t i v e s d ' 8 v o l u a t i o n r a p i d e du p e r s o n n e l o n s e i g n a n t - c h e r c h e u r e t d e s Btudiants. En moyenne l a s u r f a c e t o t s l e d e s locaux occupBs p a r l e s DBp?xtements d e mathematiques a f r i c a i n s e s t de 750 8, 1000 M2.

-

I1 f a u d r a i t cependant n o t e r c e r t a i n s e f f o r t s s 6 r i e u x pour d o t e r l e s math&matiques d e locaux a d e q u a t g p a r exemple & Ibadan, N a i r o b i , Abidj'an... En gGn&ral, il n l y a p a s en Afrique de v e r i t a b l e b i b l i o t h e q u e s p 6 c i a l i s d e C6nBralement, une b i b l i o t h e q u e u n i v e r s i t a i r e c e n t r a l e c o n c e n t r e t o u s l e s ouvrages e t t o u t e s l e s r e v u e s de t o u t e s l e s d i s c i p l i n e s e t p a r f o i s , comme a u Cameroun, c e t t e b i b l i o t h 6 q u e e s t l a mame que c e l l e d e s B t u d i m t s . C e t t e p r a t i q u e a de s 6 r i e u x i n c o n v e n i e n t s pour l e s mathematic i e n s c a r l a b i b l i o t h e q u e e s t l e l a b o r a t o i r e du m a t h h a t i c i e n , il y p a s s e une tr&sgrande p : . a t i e de son temps de r e c h e r c h e s e t dIBtudes, beaucoup p l u s que l e p h y s i c i e n , l e c h i m i s t e , l e b i o l o g i s t e ou t o u t a u t r e s a v a n t exp e r i m e n t a l i s t e . Dans c e s b i b l i o t h e q u e s GGnerales, il y a en g e n e r a l peu dlouvrages e t de r e v u e s math&natiques, Sr p e i n e une d i z a i n e de r e v u e s dans p l u s i e u r s cas.

en mathematiques.

L'dquipernent p r o p r e en moyens de r e p r o d u c t i o n s e r k d u i t B une mwhine 8, photocopies e t d e s s t e n c i l s . Les moyens de c a l c u l s o n t r e d u i t s b un t C r m i n a l d ' o r d i n a t e u r dana un t r h s p e t i t nombre de c a s , & r i e n du t o u t dans p l u s i e u r s cas. Un c e n t r e de c a l c u l p r o p r e B l I U n i v e r s i t 6 e t d i s t i n c t du Departement de math6mRtiques e x i s t e dans c e r t a i n e s u n i v e r s i t e s a f r i c a i n e s , notamment l e s grandes u n i v e r s i t e s du NigBria.

LA SITUATION ACTUELLE ET LES POTENTTALITES MATHEMATIQUES DE L'AFRIQUE

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3 ) Effectifs d'ensei~ants-chercheurs. En gEnGral, il y a un manque g r a v e d ' e n s e i g n m t s en mathematiques dnns l ' e n s e i g n e m e n t s u p g r i e u r en A f r i q u e , manque q u i se rnanifeste p a r d e tr&s lourdes c h a r g e s hebdomadaires d 'enseignement ( 1 0 h e u r e s en moyenne) e t p a r l e f t i t que l e s e n s e i g n a n t s , non Docteurs, s o n t en g r a n d e m a j o r i t 6 respons a b l e s d e s c o u r s m a g i s t r a u x e t d e s examens. Les c h a r g e s d'enseignement s o n t d ' a u t a n t p l u s l o u r d e s que l e s mathematiciens s o n t s o l l i c i t e s d a n s t o u t e s l e s a u t r e s d i s c i p l i n e s s c i e n t i f i o u e s , c e q u i 61F3ve considerablement l e nomb r e d ' 6 t u d i a n t s en mathemstiques ; s u r t o u t d > n s l a premiere annEe d ' e t u d e s scientifiques. Le nombre d e Docte_ur-s ou Ph. D e s t tr&s i n & g a l rjlun ddparteinent B un a u t r e d,?ns un rntme p'vs, comma l e Nigeria p a r exernple, ou d'un pays B un a u t r r . On p e u t compter e n v i r o n 500 Docteurs ou Ph. D rnathernatiquea a o t u e l l e r n e n t en A f r i q u e avec I n -plus grande c o n c e n t r a t i o n en SfEr'pte e t a u Nig6ria. Lea meillcurs dQpartements de mathematiques en A f r i q u e , du p o i n t de vue du nombre de Docteurs Nationyux ( e n t r e 10 e t 30 Docteurs ou Ph. D ) s o n t actuellement l e s suivants : Au Nigeria : Ibndan, Lagos, Nsukka, Ahmadu B e l l o , I f e , Benin-City

En Egypte

: Le C a i r e , A s s i u t , El-Azhar,

Ain-Shams

Au Cameroun : Yaounde En Cdte d ' I v o i r e : Abidjan Au Ghana : U n i v e r s i t y o f Legon, U n i v e r s i t e de Kumasi

En A l g 6 r i e t U n i v e r s i t 6 d ' b l g e r Au Maroc : U n i v e r s i t e d e Rabat

En T u n i s i e : U n i v e r s i t 6 de Tunis

En E t h i o p i e : U n i v e r s i t 6 d'Addis-Abeba Au Kenya

I

U n i v e r s i t e de Nairobi

Au SdnLgal r U n i v e r s i t e de Dakar Au Soudan : U n i v e r s i t y o f Khartoum

Au ZaPre : U n i v e r s i t 6 de Kinshasa

I1 y a d e s pays a f r i c a i n s q u i n l o n t e n c o r e aucun n a t i o n a l Docteur ou Ph.D

...

en mathematiques, p a r exemple, l e Tchad, 1'Empire Centre-Africain, Gambie., l e Gabon, etc.

la

4) P r i n c i p a u x th6mes de r e c h e r c h e exist'ants En g e n e r a l l e s math6maticiens a f r i c a i n s p o u r s u i v e n t d e s t r a v a u x de r e c h e r che sup d e s thBmes p r o l o n g e s n t l e u r s t r a v a u x de D o c t o r a t commenc6s dans d e s U n i v e r s i t e s d'Europe e t d'AmBrique, d ' o h une t r k s v a s t e gamme d e domaines d e r e c h e r c h e s mathernatiques r e p r e s e n t & en Afrique.

160

H. HOGBE-NLEND

Les domaines l e s mieux r e p r e s e n t e s a c t u e l l e m e n t s o n t notamment l e s suivants t

- MBcaniques

(mecanique d e s f l u i d e s , mecanique guantique, micanique du s o l i d e ) 6 l a s t i c i t B . . .)

- Analyse numdrique - Statistiques

- Equations

d i f f e r e n t i e l l e s e t applications diverses

- Equations

aux d e r i v e e s p a r t i e l l e s

- Education

math6matique

- Analyse f o n c t i o n n e l l e e t harinonique - Topologie e t GBorn6trie d i f f e r e n t i e l l e s En g e n e r a l , il n ' y a p a s de g r o u p e de r e c h e r c h e s f o n c t i o n n n n t r 6 g u l i 6 r e rnent a u t o u r d l u n theme donne dans l e s dPp3rtements, oauf quelques c a s e x c e p t i o n n e l s . Les c h e r c h e u r s s o n t en g e n e r a l i s o l e s dans l e u r s p r o p r e s departements, p a r manque de s p b c i a l i s t e s d m s l e m6me dom-*ine ou d e s dom-iines v o i s i n s . D'oh n 6 c e s s i t 6 de r e n f o r c e r l a coopCration s c i e n t i f i q u e r e g i o n a l e e t sous r e g i o n a l e , s e u l e c a p a b l e de c r d e r d e s v e r i t a b l e s ,TOW pes de recherche. 5)

P u b l i c a t i o n s mathematisues

Dans l e s departernents de rnathematiques a f r i c a i n s , l e s p r i n c i p a l e s p u b l i c a t i o n s s o n t l e s p u b l i c a t i o n s roneotyp6es d e cours. Les r e s u l t a t e o r i g i naux de r e c h e r c h e de math6maticiens a f r i c a i n s s o n t p u b l i e s , s n i t dans d e s Annales d e F a c u l t e s d e S c i e n c e s , s o i t d i n s d e s r e v u e s i n t e r n a t i o n n l e s . I1 e x i s t e une revue mathematique de 1 l U n i v e r s i t E du Z a l r e p a r a i s s m t deux f o i s p a r an aveo une f o r t e empreinte d e s math6maticiens roumains t r a v a i l l a n t & Kinshasa. S i g n e l o n s a u o s i l a r e v u e " C i t n c i a s Mat6maticas'publi6e p a r 1' U n i v e r s i t e de Lourenqo Marques (Mozambique) dni, 1 l e d e r n i e r volume (1974/1975) e s t paru en AoQt 1977.

6)

Stages de recherche P l u s i e u r s math6maticiens a f r i c a i n s o n t la p o s s i b i l i t e d l e f f e c t u e r d e s s t a g e s dans d e s c e n t r e s europeens e t a m s r i c a i n s s o i t "ns l e c a d r e d ' a c c o r d s i n t e r - u n i v e r s i t a i r e s ou inter-gouvernementaux s o i t l e cadre d'annbes s a b b a t i q u e s p o u r l e s Anglophones. On ne n o t e pratiquement p-8 de s t a g e d l u n pays a f r i c n i n 5, un a u t r e .

7)

Conferenciers e x t 6 r i e u r s Les departements de mathematiques a f r i c a i n s r e p o i v e n t quelques confdrenc i e r s e x t e r i e u r s vennnt principalernent d'Europe e t d'brnerique.

LA SITUATION ACTUELLE ET LES POTENTIALITES MATHEMATIQUES DE L'AFRIQUE

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Dins l e s u n i v e r s i t b s anylophones, l e s y s t h e d'exarniwiteurs e x t e r n e s

f k x t e r n : ]1 examii!arS') permet . c e r t c i n s 4.Ipnrte::;ents de r e c e v o i r l e u r s colli.gues a f r i c a i n s d ' a u t r e s prays. Dnns l ' b f r i q u e francophone, Te programme d'6chnnges de p - o f e s s e u r s , f i n a n c e p a r 1'RUPXLF ( A s s o c i a t i o n d e s U n i v e r s i t e s p n r t i e l l e m e n t ou e n t i h - e m e n t de l m g u e f r a n p a i s e ) f a v o r i s e considkrablement I n c i r c u l a t i o n d e s mnthkmaticiens a f r i c a i n s d ' u n e univ e r s i t i ! francophone B une a u t r e .

8) P r i n c i p a u x b e s o i n s d e s departeinents de mathernatinues a f r i c a i n s Les p r i n c i 9 3 u x b e s o i n s s i g n z l e s s o n t les s u i v a n t s

t

a ) b e s o i n s en p e r s o n n e l e n s e i c n a n t : on s o u h n i t e a v o i r d e s math8rnnticiens d e q u a l i t i ! p o u r d e s s e j o u r s r e l a tivement longs (1 B 2 'ansf ; m e i s les mir;oioi?s d e 3 s e m i n e s ii q u e l o u e s m o i s s o n t a u s s i t r b s u t i l e s notamment pour l e s e n s e i g n e a e n t s d e h a u t n i veau ( 3 6 c y c l e , p o s t - g r a d u a t i o n ) . b ) LEsoins en documentation :

notaminent a c h a t d e l i n e s e t abonneinent I? d i v e r s journaux m a t h h a t i ques. c ) b o u r s e s de s t a g e s : pour p e r r n e t t r e aux math6maticiens a f r i c - z i n s de f a i r e d e s s t a s e a d e r e cherche de c o u r t e d u r e e ou d e s s t a g e s de forrntttion de longue d u r e e d m s d e s c e n t r e s mathkmatiques developp6s j d ) boursesde_voyages : pour p e r m e t t r e aux r n a t h h a t i c i e n s a f r i c a i n s de p a r t i c i p e r b d e s conSrBs e t c o l l o q u e s math6matiques i n t e r n a t i o n n u x . e ) Publications : l e s math6maticiens a f r i c a i n s s o u h a i t e n t vivement a v o i r l a p o s s i b i l i t 6 de p u b l i e r rapidement l e u r s t r a v a u x rnathernatiaues o r i g i n a u x .

f ) E x t e n s i o n d e s l o c m x r k s e r v e s aux msthgmatiques : Toute a i d e f i n a n c i a r e de auelque f o n d a t i o n Etrangiire s e r a i t l a bienvenue.

162

H. HOGBE-NLEND

11

- PROPOSITIONS POUR L ' A C T I O N

FUTU'IEDE

L'UNESCO POUR LES K4TBTIQUES EN AFiIQVE L ' a c t i o n f u t u r e de 1lUNESCO dans l e dom-tine d e s mathematiques en A f r i q u e d o i t v i s e r B r e n f o r c e r l a c o o p 6 r a t i o n mathematique r d g i o n n l e e t sous r e g i o n a l e . I1 e x i s t e une c e n t a i n e environ de dhpartetnents de mathematiques en A f r i q u e d l i n p o r t m c e tr&sv a x i a b l e 3 il e s t pratiquement i m p o s s i b l e de v i s e r 5 f a i r e de chacun de c e s departements un dCpsrtement de n i v e a u i n t e r n a t i o n a l . Les deux i n s t r u m e n t s fondamentaux de l a c o o p 6 r a t i c n mathematique i n t e r - a f r i c a i n e s o n t , 3. n o t r e a v i s I

1

- La creation

2

- La p u b l i c a t i o n d ' u n

e t l ' a n i m n t i c n d * u n ou p l u s i e u r s c e n t r e s r e g i o n a u x de mathtimatiques au s e r v i c e du dbveloppement. j o u r n a l mathematique a f r i c a i n de n i v e a u i n t e r n a t i o n a l d e s t i n e ,i encourager l a p r o d u c t i o n mathematlque o r i g i n a l e en Afrique.

Un r a p p o r t s p e c i a l s e r a c o n s a c r e au Y k m t r e A f r i c a i n de mathkrnatiques pour le developpementll. Un j o u r n a l rnnth6nntj oue a f r i c a i n "AFRIKA EIATEMATIKA" e s t actuellernent e n c o u r s du p u b l i c a t i o n p a r 1IUnion MathEmatique A f r i c n i n e , avec l e concours f i n a n c i e r de l'UNESC0, notamnent.Le premier nurnEro s o r t i r a en MA1 1978.

DEPAR TE':' ENT DE MATHEMATIQUES

DEI'ARTE'!EKT DE KATHE:dAl'IQUES

U n i v e r s i t i . de Bordezux I

F a c u l t 6 des Sciences

33405 TALENCE

YAOUNDE

FRANCE

CAMEROUN

LA S I T U A T I O N ACTUELLE E T L E S P O T E N T I A L I T E S MATHEMATIQUES DE L ' A F R I Q U E

A N N E X E I NOhlBRE A P P R O X I M A T I F D ' A F R I C A I N S T I T U L A I R E S

-D ' U N

DOCTORAT OU PH.D

EN bL4TXEiL4TIQUES

OU D ' U N D I P L O N E E Q U I V A L E 3 T (Total I ALGERIE

30

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110

LYBIE MAROC

25

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20

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5

11

- AFRIQUE

OCCIDENTALE

BEN I N -

8

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0

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5 5

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9

KAUR I T A N I E

2

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0

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25

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0

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5

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BURUNDL

90

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15

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3

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- AE'RIQUE

11

7

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6

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1

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2

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0

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25

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0

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3

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I11

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5 1

(EhiPIRE)

TCHAD

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0

0 10

163

164

H. HOGBE-NLEND

IV

BOTSWANA

1

COMORES

0

ETHIDPIE

11

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10

LESOTHO

1

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6

MALAWI

3

MAURICE (Ile)

o

MOX.NUIQIZ

8

- AFRIQLJE ORIEILTALE ET AUSTRALE S W CiiELLES 2 ~-

-SOUDAN SWAZILAND

18 0

SOMAL I E

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7 4

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8

Developing Mathematics i n Third World Countries, M.E.A. E l Tom led.) @ North-Holland Publishing Company, 1979

COOPERACION INTERNACIONAL : UNA EXPERIENCIA Y ALGUNAS REFLEXIONES

Edmundo ROFMAN (Univ. Rosario - Argentina) (Univ. Paris - Francia)

Antes de presentar 10s aspectos mis s a l i e n t e s de l a cooperaci6n que se e s t a b l e c i 6 en 1970 e n t r e e l LABORIA (IRIA) de Francia y e l I n s t i t u t o de Matemdtica “Beppo Levi” de Rosario, Argentina, s e a en cuanto a l a s c a r a c t e r f s t i c a s y naturaleza de l a s acciones emprendidas como de 10s resultados ya alcanzados, me parece conveniente d e s c r i b i r cu6les eran l a s condiciones en que nos halldbamos cuando nos propusimos impulsar, con ayuda e x t e r i o r , nuestro proyecto ligado a1 desarollo de l a s . aplicaciones de l a matemdtica. Pienso que de e s t a manera s e r a mas f 6 c i l a n a l i z a r s i e s t a experiencia puede, con o sin modificaciones, r e s u l t a r o no u‘til para o t r o s Institutos. Para ayudar e s t a descripci6n seiialar6 muy brevemente c u i l e s son, en general y cualquiera sea e l pays de que se t r a t e , l a s fuentes de cooperaci6n externa y l a s d i f e rentes situaciones i n t e r n a s que pueden presentarse. La cooperaci6n externa puede provenir de : a) 10s pafses vecinos (en e l marc0 o no de una c o o p e r a c i h regional) ; b ) 10s paises con gran d e s a r r o l l o c i e n t i f i c o en l a s d i s c i p l i n a s que i nteresen s e r desarol ladas ; c ) l a s Organizaciones Internacionales, sea mundiales o regionales.

E n cuanto a l a situacio’n i n t e r n a puede suceder que 1 ) e l pars haya establecido y tenga en ejecuci6n una planificacio’n de SU desarrollo s o c i o - e c o n h i c o y l a necesidades de formaci6n correspondiente. En e s t e caso e l e s t u d i o contendra (a1 menos) l i n e a s generales para l a p o l f t i c a de des a r r o l l o de l a matemati a sea como ciencia fundamental como en sus aspectos de ciencia de transrFerenciaf*). 2 ) no e x i s t e l a situacio’n descripta en e l punto a n t e r i o r . En ese caso tendremos : 2.1) que l a ccmunidad matemdtica o p a r t e de e l l a hayan definido una p o l f t i c a . (Esta situaci6n s e presenta frecuentemente. De acuerdo a l a naturaleza de t a l p o l i t i c a y de quienes l a s implementan su ejecucio’n puede t r a d u c i r s e tanto en u n proceso e s t a b l e 0, por e l c o n t r a r i o , en uno i n e s t a b l e , generador de c o n f l i c t o s ) .

( * ) No se me escapan l a s d i f i c u l t a d e s que cualquier t i p o de pays debe e n f r e n t a r pa-

r a e s t a b l e c e r t a l p l a n i f i c a c i h ; pero a n a l i z a r l a s s e r f a s a l i r n o s del tema. Interesantes reflexiones sobre e s t e problerna s e encuentran en e l documento de R. S a i n t Paul y J. Fabre enviado a l a Reuni6n sobre l a Enseiianza Post-Univers i t a r i a en l a s Ciencias Ba’sicas, UNESCO-CEPES, Bucarest, Junio 1978.

165

166

E. ROFMAN

2 . 2 ) que l a seleccidn y soste'n de un proyecto dependa exclusivamente

de 10s esfuerzos individuales o locales a 61 dispensado.

Un hltimo pare'ntesis para indicar caracteristicas de l a formaci6n que exhibfan 10s

integrantes del drea matemitica de nuestra Facultad cuando canenzd, en 1969, l a acci6n destinada a constituir un grupo interesado en Aplicaciones de l a Matemitica. Hab-fa dos doctores (con t i t u l o obtenido en Roma - I t a l i a ) y varios licenciados (graduados en Buenos Aires - Argentina) ; pricticamente todos e l l o s contaban previamente con e l t i t u l o de ingeniero y h a b i a n desarollado una prolongada y valiosa l a b o r docente en las lineas tradicionales de matemitica para las carreras de IngeHabfa, adeni,er$a y/o Ciencias Econo'micas y/o Estadrstica y/o Bioqufmica, e t c . . mas, un grupo de ingenieros que, sin haber cursado regularmente otros estudios, habian agregado a una s6lida preparacih c l i s i c a otros conocimientos por 10s que estaban particularmente interesados (entre e l l o s i n v e s t i g a c i h operativa y cmputac i b ) . Resta agregar, para completar l a imagen. que desde un par de aiios atras se habia creado en nuestra Facultad l a carrera de licenciado en Matemgticas de l a que, hasta esa fecha, se habia dictado e l ciclo b$sico debiendo cmenzar, en ese aiio, el de algunos cursos del ciclo superior. La culminacidn de esta carrera exigia, como etapa final del ciclo superior, el cursado de tres materias de especializaci6n. E l grupo de alumnos que inici6 esta carrera era particularmente calificado pues estaba en gran parte constituido por quienes venian desarollando en el 'area matemetica tareas docentes auxiliares.

.

Paso, ahora

si, a1 motivo

central de e s t e t r a b a j o :

En l a fecha indicada (1969) impulse la c r e a c i h de un grupo de estudio, de cuya d i r e c c i h me ocupe, sobre temas de Contr6l y Evaluacih. Nos propusimos analizar aspectos ligados a esos problemas, profundizando), discutiendo temas de analisis funcional , a'nal i si s num6ri co, informa'ti ca, estadi s t i c a , investigaci6n operativa , econmca, e t c . . , aprovechando que entre 10s integrantes del grupo habia quienes, sobre una formacion matematica inicial , se habian posteriormente interesado en algunas de l a s citadas direcciones. Tenfamos como primera meta l a de poder ofrecer algunos cursos de posgrado y tambien l a de realizar tareas de apoyo a otras dreas de l a Universidad 0, inclusive, alguna asesorqa exterior a ese ambiente.

.

Esas ambiciones se materializaron en 1970. Se digtaron cursos en nuestro Instituto, en e l Uto. de Posgrado de l a Facultad y s e brindo asistencia en dos problemas concretos a1 Dto. de Metalurgia de l a Comisidn Nacional de Energfa Atomica. Nuestro trabajo de estudio se vio' reforzado en esos dos aiios por las v i s i t a s ( 3 semanas, u n mes y medio y dos meses) de 10s profesores G. Stampacchia, P. Kree y U. Mosco cuyas actividades, brindadas a toda el area matemztica y en particular a 10s alumnos del ciclo superior de l a licenciatura, resultaron sumamente beneficiosas para nuestro proyecto. Debo seiialar que l a concreci6n de t a l e s v i s i t a s requirid enorme esfuerzo para vencer serias dificultades. Las de tipo econ6m'ico se superaron u t i l i zando, sucesivamente, fondos de l a O . E . A . , de nuestra Facultad y del Consejo de Investigaciones Cientfficas y Te'cnicas ; volver a recurrir a $sas fuentes no resultaba posible inmediatamente y estar al tanto de eso ayudara a comprender nuestros pasos posteriores. En Octubre de 1970 durante una brevisima v i s i t a de medio d i a a nuestra Facultad, e l Profesor Lions mantuvo una reuni6n con todo el cuerpo docente del Brea matema'tica. Se plantearon inquietudes, ambiciones y se es_cucharon propuestas y orientaciones para diferentes casos. El Instituto de Matematicas, en cuyo ambito actuaba el grupo de estudio a1 que hice antes referencia, establecio en dicha ocasi'on el acuerdo de cooperaci6n con e l IRIA sobre cuyos detalles me ocupare' enseguida, tras resumir l a situaci6n en esa fecha :

i ) can0 cooperaci6n externa solo podfarnos aspirar a l a seiialada en el p u n t o b) ; i i ) en el caso interno l a seiialada como 2 . 2 ) ;

COOPERACIOiJ INTERNACIONAL

167

i i i ) e x i s t f a un grupo de alrededor diez jo'venes (en su mayorfa alumnos del c i c l o superior de l a l i c e n c i a t u r a ) con Clara ambicibn de progreso y vocacio'n hacia l a s aplicaciones ; i v ) nuestro grupo o f r e c i a un proyecto en e j e c u c i h , con algunos o b j e t i vos ya alcanzados y habiendo u t i l i z a d o prscticamente a1 maximo nuest r o s esfuerzos y recursos locales (punto 6 s t e que consider0 fundamental poder e x h i b i r ) . Con e l Profesor Lions, que representaba a1 IRIA, qued6 convenido que en e l transcurso de cinco afios recibiriamos, una vez por aiio y durante un mes como mdximo, l a v i s i t a de un miembro de ese I n s t i t u t o . El tema que s e r v i r i a de h i l o conductor s e r i a e l de Optimizacih y Control. Los cinco v i s i t a n t e s r e a l i z a r i a n , fundamentalmente, un c i -

clo de cursos intensivos que tenfan como objectivo f i n a l p o s i b i l i t a r una acentuada actualizaci6n sobre temas ligados a e s e argumento. El nombre del v i s i t a n t e , tema del curso y 10s conocimientos que resultaba conveniente poseer para u n mejor provecho del mismo (dato 6 s t e que autom5ticamente brindaba motivo para nuestro t r a b a j o de preparacio'n) s e conocerian alrededor de diez meses antes de la, v i s i t a . Previa a l a decisi6n de qui6n s e r i a e l v i s i t a n t e consecutivo s e evaluaria e l u'ltimo curso realizado y l a s preferencias y tendencias e x i s t e n t e s a1 momento. Hoy debe d e c i r s e que todos 10s nombres y temas fueron as{ decididos, de comu'n acuerdo, e n t r e ambos responsables de l a cooperacion. Los cursos s e realizaron durante e l period0 de vacaciones u n i v e r s i t a r i a s de invierno, l o que permiti6 a 10s p a r t i c i p a n t e s una dedicacidn prscticamente t o t a l a 10s

mismos. Fueron particularmente intensivos y estuvieron as7 progrmados durante sus cuatro semanas de duraci6n : 10s lunes, 4 horas por l a t a r d e ; 10s martes, mie'rcol e s y jueves, 3 horas e f e c t i v a s por l a maiiana. Los p a r t i c i p a n t e s ( * ) , formando d i s t i n t o s grupos, trabajaban por l a s t a r d e s sobre e l material propuesto, contando con l a colaboracio'n del Profesor v i s i t a n t e , quien permanecia con dicho objeto en e l I n s t i t u t o hasta l a s 17,30 horas. Para d a r una idea del volumen de temas d e s a r r o l lado, 10s profesores incluyeron en dicho mes e l material que l e s significaba e l dictado de dos cursos anuales en sus respectivos I n s t i t u t o s de Francia. La preparaci6n de esos cursos intensivos permiti; a t r e s de 10s v i s i t a n t e s ordenar un mater i a l que d i 6 lugar a l i b r o s sobre esos temas, como e l l o s mismos lo han reconocido en 10s prblogos de e s a s obras en 10s que c i t a n nuestro I n s t i t u t o de Rosario como e l s i t i o donde expusieron inicialmente dicho m a t e r i a l . Por nuestra p a r t e , redactados por algunos p a r t i c i p a n t e s a1 c i c l o y bajo mi responsabilidad s e publicaron todos esos cursos en una s e r i e especial : "Cuadernos del I n s t i t u t o de Matematicas" creada para p o s i b i l i t a r una difusi'on, en c a s t e l l a n o , de temas sobre 10s que l a b i b l i o g r a f i a r e s u l t a b a de o t r a manera, de muy d i f i c i l acceso. Los profesores v i s i t a n t e s , d i r e c t o r e s c i e n t i f i c o s en sus respectivas especialidades dentro del IRIA fueron, sucesivamente, desde 1971 hasta 1975 : P. Bernhard, P. Faurre, A. Bensoussan, R. Glowinski y A. Bensoussan. El costo de sus boletos de avio'n f u e absorbido por e l Ministerio de Relaciones Exteriores de Francia mientras que 10s honorarios fueron cubiertos por fondos de nuestro proyecto. Comenzamos brindando a s e s o r f a a otros I n s t i t u t o s u Organismos que, C M O compensacion, tomaron a su cargo dicho gastos. Posteriormente ( i el movimiento s e demuestra andando ! ) contamos con subsidios de l a UNESCO y de l a O . E . A . No hubo necesidad de r e q u e r i r ayuda financiera, destinada a 10s v i s i t a n t e s , de nuestra Facul tad. Los profesores v i s i t a n t e s realizaron una breve labor complementaria (conferencias)

en o t r a s Universidades y, a p a r t i r del t e r c e r curso, agregaron l a importante t a r e a ,

( * ) Deb0 s e c a l a r que siempre se cont6 con concurrentes ajenos a nuestro g r u p o de

trabajo. La difusidn y e l anuncio de 10s cursos del c i c l o permiti6 a o t r a s Universidades del pafs e inclusive de pafses l i m r t r o f e s ( B r a s i l , Chile, Paraguay, Uruguay) enviar p a r t i c i p a n t e s .

168

de proponer e impulsar temas de investigaciSn. Este aspect0 tuvo rapido d e s a r r o l l o y permitio ( j permite aun ! ) l a obtencion de u n conjunto de resultados, algunos altamente s i g n i f i c a t i v o s , que han sido motivo de publicaciones en r e v i s t a s especializadas. Varios miembros del grupo de trabajo (en e l que s e fueron incorporando algunos de 10s que s e graduaron C M O 1 icenciados u t i l i z a n d o C M O materias de e s p e c i a l i z a c i h 10s cursos del c i c l o ) participamos activamente en e l Autumn Course on Mathematical and Numerical Methods in Fluid Dynamics desarrollado en e l ultimo t r i m e s t r e de 1973 en e l I.C.T.P. de T r i e s t e ( C M O tambien en l a s t a r e a s preparatorias destinadas a u n gran grupo de participantes latinoamericanos). Posteriormente, gracias a una f l u i d a relaci6n con e l Servicio de Cooperaci6n Tecnica de l a s Embajada de Francia, s e lograron t r e s becas destinadas a completar doctorados en Francia as7 cotno o t r a s dos ligadas a t a r e a s de perfeccionamiento en aplicaciones i n d u s t r i a l e s .

En resumen, a1 te'rmino de l a s actividades durante 10s cinco aiios que e l acuerdo prevefa ( y en 10s cuales hubo oportunidad de i n c l u i r o t r a s e s p e c i a l e s , can0 c o r t a s v i s i t a s de profesores que mucho significaron en nuestra labor) l a situacion alcanzada fue l a siguiente :

- E n e l plano de l a formaci6n y l a enseiianza

: Un n h e r o de alrededor 15 estudiantes obtuvieron su diploma de licenciados en matema'ticas utilizando l a repetici6n de 10s cursos del c i c l o como materias de especializaci6n. Varios de e l l o s , como tambien o t r o s docentes ligados a1 grupo de t r a b a j o , se fueron encargando sucesivamente del dictado de dichos cursos t a n t o en nuestro I n s t i t u t o como en o t r a s Facultades y Centros del Pafs, como consecuencia de acuerdos de a s i s t e n c i a que llegaron a i n c l u i r o t r o s 10 I n s t i t u t o s de o t r a s t a n t a s ciudades argentinas.

- En el plano de l a a s i s t e n c i a "a terceros" ( I n d u s t r i a s y Organismos Nacionales) s e realizaron en v a r i a s ocasiones tareas de asesoria en importantes proyectos, con l a prolongada participacihn de miembros del grupo de trabajo.

-

En e l plano de l a investigacibn : Una parte del grupo tom6 e s t e camino y continu'a en e s t a t a r e a . A 10s resultados obtenidos por quienes ya habian alcanzado un grado acadgmico debe sefialarse que como consecuencia de t a l actividad en l a invest i g a c i h , 4 integrantes del grupo habr6n obtenido en e l presente afio su doctorado (t,res de "Troisieme cycle" y otro " d ' E t a t " ) . La obtenciih de e s t o s diplomas s e r d solo una etapa en l a dedicaciBn que e s t o s miembros del grupo brindan a l a s t a reas de investigaci6n.

Para terminar e s t a presentaciijn deseo seiialar que t r a s el acuerdo de cinco allos a que he hecho r e f e r e n c i a , l a c o o p e r a c i h e n t r e ambos I n s t i t u t o s continu6 ( y continu'a) bajo v a r i a s o t r a s formas mas tradicionales que no creo j u s t i f i q u e n detenerme en e l l a s .

Developing Mathematics i n Third World Countries, M.E.A. E l Tom (ed.) 0 North-Holland Publishing Company, 1979

DEVELOPMENT OF MATHEMATICS I N SOUTHEAST ASIA: THE EXPERIENCE OF THE SOUTHEAST ASIAN MATHEMATICAL SOCIETY Lee Peng Yee ( S i n g a p o r e )

1.

H i s t o r i c a l background

T h i s i s a r e p o r t on r e g i o n a l c o o p e r a t i o n i n t h e f i e l d of m a t h e m a t i c s e d u c a t i o n and research i n Southeast Asia.

I t c o v e r s m a i n l y Asean c o u n t r i e s ( P h i l i p p i n e s , Indo-

n e s i a , T h a i l a n d , M a l a y s i a and S i n g a p o r e ) and Hong Kong.

S i n c e Hong Kong o c c u p i e s

a u n i q u e p o s i t i o n , many problems d e s c r i b e d i n what f o l l o w s w i l l n o t a p p l y t o

Hong Kong. These n e i g h b o u r i n g c o u n t r i e s have v e r y d i f f e r e n t h i s t o r i c a l background though t h e y

are a l l i n t h e same t r o p i c s .

A l l o f them e x c e p t o n e , namely T h a i l a n d , were a t o n e

t i m e t h e c o l o n i e s o f some European powers.

M a l a y s i a and S i n g a p o r e were B r i t i s h ,

I n d o n e s i a Dutch, and t h e P h i l i p p i n e s S p a n i s h and l a t e r American.

Unavoidably t h e

educational systems of t h e s e c o u n t r i e s follow mostly those of t h e i r former coloni a l masters.

Though a f t e r independence a g r e a t d e a l h a s changed, t h e i m p r i n t i s

still there.

I n a way, w e are even more d i f f e r e n t now t h a n b e f o r e t h e c o l o n i a l

period.

It i s o n l y r e c e n t l y t h a t w e b e g i n t o g e t t o g e t h e r and now Asean (Associa-

t i o n o f S o u t h e a s t A s i a n N a t i o n s ) i s n o t o n l y working b u t a l s o known t o b e working.

We are d i f f e r e n t i n many ways. dary education.

Most o f u s h a v e t w e l v e y e a r s o f p r i m a r y and secon-

The F i l i p i n o s h a v e o n l y t e n .

F o r u n i v e r s i t y e d u c a t i o n , i n Malay-

s i a , S i n g a p o r e and Hong Kong i t i s t h r e e y e a r s l e a d i n g t o B.Sc. y e a r f o r t h e v e r y good s t u d e n t s ( c a l l e d h o n o u r s ) .

p l u s one e x t r a

I n Indonesia and t h e P h i l i p p i n e s

i t is g e n e r a l l y f o u r y e a r s whereas i n T h a i l a n d i t is f i v e .

F o r t h e number o f

u n i v e r s i t i e s , i n M a l a y s i a , S i n g a p o r e and Hong Kong o n e c a n c o u n t w i t h t h e f i n g e r s i n o n e hand.

But i t i s much more numerous i n t h e o t h e r c o u n t r i e s .

problems are o f t e n o f d i f f e r e n t n a t u r e .

is more advanced and i n what s e n s e .

Hence t h e

It i s v e r y d i f f i c u l t t o s a y which c o u n t r y

What c a n b e s a i d i s t h a t t h e y a r e i n v a r i o u s

s t a g e s o f development and t h e d i r e c t i o n s a l s o v a r y . An added d i f f e r e n c e i s t h e l a n g u a g e .

Though a t t h e r e g i o n a l c o n f e r e n c e s t h e

l i n g u a f r a n c a i s E n g l i s h , i n t h e i r own c o u n t r i e s t h e y a r e n o t t h e same. remember what a n I n d o n e s i a n p r o f e s s o r t o l d m e .

I always

H e s a i d t h a t when h e w a s a s t u d e n t

h e w a s t a u g h t m a t h e m a t i c s i n Dutch i n t h e f i r s r y e a r , E n g l i s h i n t h e second y e a r (by t h e Americans s i n c e t h e Dutch had l e f t ) , and i n Bahasa I n d o n e s i a i n t h e t h i r d year.

T h e r e h a v e been e f f o r t s t o t e a c h m a t h e m a t i c s i n n a t i o n a l l a n g u a g e s n o t o n l y

a t t h e p r i m a r y and s e c o n d a r y b u t a l s o , a t t h e u n i v e r s i t y level.

Then t h e r e are

problems o f a d e q u a t e t e a c h e r s , t e x t b o o k s and t h e n e c e s s a r y t e c h n i c a l terms i n

LEE PENG YEE

170

national languages. Indochina countries are different again.

They were French before.

used to include these countries until lately.

Our programme

It is our hope that we shall renew

our contact with Vietnam and the other Indochina countries in the near future. It suffices to say that there are many active research mathematicians in Ho Chi Minh City as well as Hanoi.

As stated above we shall confine our discussion mainly

to Asean countries. Furthermore, we shall consider the generality whenever possible.

Professor Nebres will consider in more detail the Philippines in a separate

paper [41. 2.

Present situation

Before we say anything, we must mention that the salary of teaching profession in some countries is generally low.

This fact sometimes nullifies other efforts in

improving the working condition of university teachers such as to have better training and research facilities. The teaching load can also be heavy (20 hours per week).

These and many other factors put together have made it difficult for

the universities to recruit sufficient numbers of Ph.D.'s into their faculty. Some stay overseas after their graduate training.

The few Ph.D.'s

that do come

back and stay are often promoted quickly to some administrative positions. They will then no longer teach mathematics. qualified teachers.

So

we are again with not having enough

Though it does not happen in all Asean countries, in the few

countries that this happens is enough to slow down the development of mathematics in these countries. Some of the trends and problems brought up by Dr. van Lint in his paper [l] apply also to Southeast Asia.

Instead of repeating what had been said there, I choose

to concentrate on more local problems. On the other hand, what I am going to say here is more organisational in nature than mathematical. Nearly all of the research mathematicians in Southeast Asia were either fully or partially trained overseas. The choice of their research topics were very often accidental rather than deliberate. They are sometimes accused of not doing something relevant. It is difficult to be directly relevant. Still the reluctance of some mathematicians to take part in more applied research can not easily be forgiven by the national planners. When they return, if they do return, they are then isolated and find it difficult to keep up their work.

In view of the promo-

tion system in the universfties and many other reasons, the research mathematicians are reluctant to change fields. Hence the research tends to be individual and the link is with their co-researchers overseas rather than among themselves. Furthermore, they are torn between research and service. At the end they give up. We cannot say that there is much insentive for research and even less for changing from pure to applied.

It is very difficult

to

break this. If there is a team

DEVELOPMENT OF MATHEMATICS IN SOUTHEAST ASIA

171

Furthermore, mathematics i s always at the

working together, it might be easier. bottom of the list for priority.

I think there should be no question as to whether we should do basic research or not.

I believe that we should

However this is low priority in all countries.

emphasize more on technology than pure science. At the same time I also believe that it is important to have a core of pure scientists without whom there can be no significant advances in applied science.

not-to-have but what is the proportion.

It is not a question of to-have-or-

That is to say, there should be more

engineers and fewer scientists. But it would be a great pity if we should go so far as to exclude basic science. Certainly we can learn a great deal from the developed countries. At the same time we can learn a great deal from each other too.

This fact is gradually being

recognized but not enough to change many old practices.

For example, the academic

exchange is normally with the advanced institutes in the developed countries. Spending one's sabbatical leave at a university i n the region i s not recognized and therefore not possible.

There are ASAIHL exchange schemes where ASAIHL stands

for the Association of Southeast Asian Institutions of Higher Learning.

But there

were few takers. Recently, there have been several exchanges between the Philippines and Singapore, and between Indonesia and Malaysia.

I do feel that many

universities in the region have the necessary facilities for staff training and sometimes it i s more efficient to have staff trained locally than to send them one by one overseas. 3.

The formation of SEAMS

The Southeast Asian Mathematical Society or in short SEAMS was formally established in 1972 with its inaugural meeting held from 24th to 28th July 1972 at Nanyang University, Singapore. This i s the result of a Southeast Asian tour by Professor Wong Yung Chow of Hong Kong University at the suggestion of ASAIHL. After the tour and with the support of the mathematicians in the region, Professor Wong formed an interim council which planned and eventually realized the formation of the Society known as SEAMS.

In fact, even before that there were already

two regional meetings held i n Vietnam, and one in Hong Kong. Since 1972 and for a period of six years, a total of 20 regional and national mathematical conferences were held (see Appendix I).

This is essential since these

meetings provide opportunities for the mathematicians to meet, to exchange views and research results, and to discuss their common problems.

We must know each

other first before we can talk about any cooperation. The Society publishes a quarterly Newqletter.

It has also published two director-

ies, one on the mathematical facilities, namely, library holdings of mathematical periodicals in each university,.and the other a list of mathematicians in

LEE PENG YEE

172

Southeast Asia. Recently, the Society published a Bulletin, The aim as stated in the editorial is to inform and to provide a forum. It carries mainly expository and survey articles. Others include research problems, research announcements, papers on mathematical education, reviews and abstracts. A Society monograph written by Wong Yung Chow on linear geometry is also being published in Hong Kong.

For other mathematical

periodicals published in the region, see Appendix 11. Now the Society is gradually being recognized locally and internationally. We have also moved one step further from organising general conferences to holding training courses and seminars on special topics.

A regional workshop in numerical

analysis and computer science was held in May 1977 in Penang sponsored by UNESCO. The latest was a workshop on automata and related topics with Professor Art0 Salomaa of Finland and Dr. James Kearns of AIT (Asian Institute of Technology, Bangkok) as guest speakers. The next workshop will be on combinatorics and graph theory to be held in Singapore in 1979. Itis adeliberate policy of the Society to hold meetings at different centres.

It

has proved to be very effective in stimulating activities in the country where the meeting is held.

The first such meeting was held in 1975 in Manila.

What happened

afterward had been retold in [ 3 ] . The Society has helped to achieve at least the following: 1)

to make the necessary international contact,

2)

to arrange the sharing of oversea visitors,

3 ) to organise and to sponsor regional and national mathematical meetings and workshops,

4) 5)

to publish a quarterly Newsletter and a half-yearly Bulletin, to generate and to support local activities in the individual member countries,

6) to facilitate the exchange of information among the mathematicians in the region, 7)

to increase the number of academic exchanges among the universities in the

region. 4.

Problems

There are many problems.

We have identified four of them [ Z ] .

1) What sort of undergraduate mathematics programme should we and can we have? At present, the undergraduate mathematics programme varies from institute to institute. . A recent trend is to include computing and operational research.

I

think this is good and desirable. However the core of pure mathematics should also stay. as skill.

It is like playing badminton.

One needs to be physically fit as well

While operational research provides us with skill, pure mathematics

trains our mind.

We cannot do without either of them.

DEVELOPMENT OF MATHEMATICS IN SOUTHEAST ASIA

173

I would not propose a common core syllabus for the region. This is not necessary. However, as mentioned by Professor Nurul of Indonesia [ 5 ] , a common core syllabus at the national level may make it easier for students who wish to continue their graduate study at another university.

I was told that Indonesia is planning for

that and the target date is 1980. Each institute will have to decide for themselves what to teach in the undergraduate programme.

I think one would agree that no programme is complete without a

certain amount of applied mathematics (classical or modern).

Also, the core of

pure mathematics should stay.

2)

How do we train and upgrade our staff?

Upgrading our present staff is as important as recruiting new ones. that training them locally is a viable proposition. on we shall stop sending any of them overseas.

trained locally. 3)

We have proved

I do not mean that from now

However the majority will be

In fact, we are working gradually toward this.

Which direction should we take in terms of mathematical research?

Each country may have its own preference. I think it is generally agreed that we should be selective and concentrate only on a few areas. team effort

so

We should also encourage

that eventually one or more research schools will be built up.

am not suggesting that everyone should be doing research in the same area.

I

But it

is a good policy to develop some good research schools in the region. Research is something that cannot be forced. However we can help to provide the correct environment. For example, the Society has sponsored a training course and several seminars on special topics and will sponsor more. tried to print research problems and research announcements.

The Bulletin has On the whole, we

should encourage research. However we should also allow those who like teaching to teach.

We should not create such a situation that one feels compelled to do

research.

Then we would have defeated our purpose. Nevertheless in a department

there should be people who are actively engaged in research.

4)

How do we make ourselves known to the public?

This is important. We need the support of the administrators and the public to carry out our programme.

There have been many activities promoting mathematics

in the region, such as mathematical competitions, exhibitions, public lectures and lecture tours to schools. The international conference on developing mathematics in the third world countries to be held i n Khartoum in 1978 will also help to promote our cause. 5)

Future plan

I think we have reached a point where we need more coordination of the many

174

LEE PENG YEE

regional and national activities in mathematics. At a council meeting of the Society in Manila in 1975, the idea of having a regional centre was first brought up.

It was decided a year later in another council meeting in Bandung that the

Society should try to set up such a centre. The centre will also serve as an agency to receive aid from outside if any.

It is envisaged that the centre will have among others the following functions: 1) to organize training courses, seminars on special topics and other related activities, 2)

to offer fellowships for the staff members of the universities in the region

to spend their study leave in the regional or other centres, 3)

to act as an information and service centre, including the publication of the

Bulletin and other materials. The choice of a location was a difficult one. At the end, it was agreed that there will be several centres with one of them serving as a coordinating centre.

We feel that we have achieved something collectively which we could not have done or would be difficult to do by oneself. Asean as a unit can certainly do much more than any conutry individually. What made it work, as Professor Nebres put it, is the existence of a group of mathematicians who are willing to work together for a common goal.

I think there are two other factors. One is that we all share

some common problems and have similar needs.

Another is that we are now much more

aware of regional coorperation and have common desire to work together. As it goes

on, we shall be more and more interdependent on each other. There will be

more uniformity among us, and the general standard will go up. In 1978, the following have already been committed. There will be a bi-annual meeting of SEAMS to be held in Bangkok.

It is hoped that it will generate the

same enthusiasm as it did in Manila in 1975 and later in Bandung in 1976. There will also be a conference on mathematical education to be held in Manila.

This is

the second conference in the region to be sponsored by an international body, and this time it w i l l be ICMI (International Commission on Mathematical Instruction). In order to prepare for the conference, the teachers in the Philippines have already benefited a great deal even before the convening of the meeting. The third is a mathematics symposium to be held in Kuala Lumpur and hosted by the National University of Malaysia. This series of mathematics symposiums started many years ago with the universities i n Malaysia and Singapore taking turns to host the occasion. It stopped for a while and was resumed again in 1976. This is probably the longest series of mathematics symposiums held i n Southeast Asia. In 1979, there will be a Franco-Southeast Asian mathematical conference. It has been proposed that the 1980 bi-annual meeting of SEAMS be held in Hong Kong. There will be more meetings to be planned later.

DEVELOPMENT OF MATHEMATICS IN SOUTHEAST ASIA

I75

I was asked once what we need now in Southeast Asia in the development of mathematics. As far as I can see, the following two are important and rather urgent: constant stimulants and fellowships. The former is needed

so

that the development

will carry on. At present we need more better qualified teachers. The fellowship scheme as proposed above is the shortest path to our destination. I am assuming of course that other development will still go on.

I have confidence that what we

hope for will come true one day. References 1.

J.H. van Lint, Mathematics education at university level, New Trends in Mathematics Teaching IV, Chapter 4 , UNESCO 1977.

2.

Lee Peng Yee, Mathematics education in Asean countries, SEAMS Newsletter No.15, April 1976.

3.

B.F. Nebres, Mathematics in the Philippines: beginnings and growth of the Mathematical Society of the Philippines, Southeast Asian Bulletin of Mathematics vol. 1 no. 1 pp.3-8 (1977).

4.

--- , Research

and graduate education in mathematics:

the Philippines experi-

ence, International conference on developing mathematics in the third world countries, Khartoum 1978. 5.

Nurul Muchlisah, Curriculum development and the role of mathematics departments in the universities in Indonesia (in Indonesian), L e e Kong Chian Institute of Mathematics and Computer Science Occasional paper No. 12, Nanyang University 1976.

6. Lee Peng Yee, A regional centre for mathematics? SEAMS Newsletter No. 19, April 1977. 7.

A proposal for regional coordinating centre in Southeast Asia for pure and applied mathematics 1977.

176

LEE PENG YEE

APPENDIX I A LIST OF MATHEMATICAL CONFERENCES IN SOUTHEAST ASIA 1972-1980 1972 FIRST BI-ANNUAL MEETING OF SEAMS July 24-28. Nanyang University, Singapore. 1973 FIRST NATIONAL MATHEMATICS CONGRESS (PHILIPPINES) May 30-June 2, Manila, Philippines. SEMINAR ON ABSTRACT ALGEBRA AND ITS APPLICATIONS June 25-29. Nanyang University. Singapore. MATHEMATICS SEMINAR December 17-20, Hong Kong 1974 SECOND NATIONAL MATHEMATICS CONGRESS (PHILIPPINES) May 5. Manila, Philippines. SEMINAR ON APPLICATIONS OF OPERATIONAL RESEARCH IN SINGAPORE June 14-15, Nanyang University, Singapore. SECOND BI-ANNUAL MEETING OF SEAMS July 8-13, Penang, Malaysia. MATHEMATICS SYMPOSIUM July 15-17, Universiti Malaya, Kuala Lumpur, Malaysia.

1975 FIRST MSP SUMMER INSTITUTE IN MATHEMATICS April 21-May 16, Ateneo de Manila University, Manila, Philippines. SEMINAR ON THE ART OF COUNTING November 17-19, Nanyang University, Singapore. 1976 SECOND MSP SUMMER INSTITUTE IN MATHEMATICS April 5-May 15, Ateneo de Manila University, Manila, Philippines. MATHEMATICAL SYMPOSIUM April 26-30, Universiti Malaya, Kuala Lumpur. Malaysia. SEMINAR ON OPERATIONAL RESEARCH IN DECISION MAKING July 12-13, RELC, Singapore. THIRD BI-ANNUAL MEETING OF SEAMS AND FIRST NATIONAL MATHEMATICS CONGRESS(IND0NESI.A) July 12-17, Institut Teknologi Bandung, Indonesia.

DEVELOPMENT OF MATHEMATICS IN SOUTHEAST ASIA

SEMINAR ON 1001 APPLICATIONS OF GRAPHS November 15-17, Nanyang University, Singapore. 1977 THIRD MSP SLIMMER INSTITUTE IN MATHEMATICS April

-

May, Ateneo de Manila University, Manila Philippines.

REGIONAL WORKSHOP IN NUMERICAL ANALYSIS AND COMPUTER SCIENCE (sponsored by UNESCO) May 3-14 Penang, Malaysia. MATHEMATICAL SYMPOSIUM May 30-June3, University of Singapore, Singapore. WORKSHOP ON AUTOMATA AND RELATED TOPICS November 14-18, Nanyang University, Singapore. SECOND NATIONAL MATHEMATICS CONGRESS (INDONESIA) November 28-30, Universitas Gajah Mada, Yogyakarta, Indonesia. 1978 MATHEMATICAL SYMPOSIUM May 24-26, Universiti Kebangsaan, Kuala Lumpur, Malaysia. SOUTHEAST ASIAN CONFERENCE ON MATHEMATICAL EDUCATION (Sponsored by ICMI) May 29-June 3 , Manila, Philippines. ILIGAN CONFERENCE ON MATHEMATICAL EDUCAITON June 4-10, Iligan, Philippines. FOURTH BI-ANNUAL MEETING OF SEAMS July 18-21, Chulalongkorn University, Bangkok, Thailand. 1979 FIRST FWCO-SOUTHEAST ASIAN MATHEMATICAL CONFERENCE May 14-25 (Workshops), 28-31 (Conference), Nanyang University, Singapore. 1980 FIFTH BI-ANNUAL MEETING OF SEAMS Hong Kong (Proposed).

I77

178

LEE PENG YEE

APPENDIX I1 A LIST OF MATHEMATICAL PERIODICALS PUBLISHED IN SOUTHEAST ASIA SOUTHEAST ASIAN BULLETIN OF MATHEMATICS Published by SEAMS Publications Committee, c/o Department of Mathematics, Nanyang University, Singapore 22, Republic of Singapore. NANTA MATHEMATICA Published by Lee Kong Chian Institute of Mathematics and Computer Science, Nanyang University, Singapore 22, Republic of Singapore. MATHEMATICAL MEDLEY Published by Singapore Mathematical Society, c / o Department of Mathematics, University o f Singapore, Singapore 10, Republic of Singapore. DISCOVERY Published by Mathematics Unit, Ministry o f Education, Kay Siang Road, Singapore 10, Republic of Singapore. MATHEMATICAL GARDEN Published by Nanyang University Mathematical Society. BULLETIN OF THE MALAYSIAN MATHEMATICAL SOCIETY Published by the Malaysian Mathematical Society, c/o Department of Mathematics, University of Malaya, Kuala Lumpur, Malaysia. MENEMUI MATEMATIK Published by the Malaysian Mathematical Society. MATIMYAS MATEMATIKA Published jointly by the University of the Philippines Science Education Center and the Ateneo de Manila University, c/o Department o f Mathematics, Ateneo de Manila University, P.O. Box 154, Manila, Philippines. MATHEMATICS MAGAZINE (in Thai) Published by the Mathematical Association o f Thailand, c/o Department of Mathematics, Chulalongkorn University, Bangkok 5, Thailand.

PART THREE

THE F I N A L R E P O R T

This Page Intentionally Left Blank

REPORT

FINAL

The International Conference o n Developing Mathematics in Third World Countries, meeting in Khartoum from March 6th to March 9 t h , 1978, sponsored by the University of Khartoum, Noting the contents of the invited and contributed talks; Taking into account the plenary discussions of the preliminary reports prepared by the Chairmen of the four working groups, Adopts, this 9 t h day of March 1978, this unified Final Report with a view to guiding the efforts of those responsible for the development of mathematics i n Third-World Countries.

I. S C H O O L 1. &sic

MATHEMATICS

Assumptions

Underlying the report and the recommendations of the Conference on School Mathematics are the following assumptions :

(i) (ii)

Third-World Countries differ among themselves. In most countries, the primary stage of education is terminal for most pupils.

(iii) The primary stage normally lasts for about 6 years. The age of the primary pupil need not be universally defined : the number of years of education is more significant. Mathematically, the primary level can only be defined by the level of attainment at the end of this stage (a broad meaning of "attainment" is intended h e r e , involving skill, understanding and other abilities). The secondary stage is a minority activity for a relatively small - but potentially influential and powerful

-

number of selected pupils. In this respect Third-World Countries are essentially different from developed countries and will remain s o for a long time.

181

182

THE FINAL REPORT

I n view of (ii) above, the Conference recommends :

That special attention be given to primary mathematical education. 2.

Primary Mathematical Education The discussion centred around five basic areas : objectives, curriculum, evaluation, language, and the teacher.

2.1

O b k c t i v e s of teaching mathematics at the primary level The Conference identified five objectives and noted that they are general and should be common to all countries. They are

(i)

Functional numeracy (including an understanding o f place value, decimal fractions and an appreciation of the size of the number).

how but (ti)

also

when

"Functional" implies not only knowing to perform an arithmetical operation.

Acquisition of certain mental attitudes to enable the development of problem-solving strategies.

(iii) Acquisition of techniques o f representing and interpreting data (numerical and otherwise). (iv)

Measurement, approximation and estimation.

(v)

Development of spatial concepts, and the ability to represent them e.g. scale drawings, maps.

It was felt that these objectives, while universally valid, are too general to be of immediate practical use to the teacher. Therefore the Conference recommends : That a further conference should be convened in order to define the objectives in more concrete and detailed terms which may, however, need to be attuned to local conditions. 2.2

Curriculum As for methodology the following were agreed : (i)

Pupils should learn mathematics as far as possible through active practical experience and with learning aids drawn from the environment. Generalizations and Structures should come out of pupils'experiences,

rather

than from formal assertions from the teacher. (ii)

Unifying concepts of mathematics should be emphasized.

183

THE FINAL REPORT

(iii) The spiral approach i n planning and implementing any curriculum is recommended. 2.3

Evaluation

(i)

Assessment of the individual p u p i l The Conference believed that a competitive examination for entry into secondary education is not adequate for the purposes o f assessing the attainment of the objectives of primary education; but i t recognised that questions of pupil evaluation pose deep educational and social problems which are not yet solved. The Conference therefore recommends : That further research needs to be undertaken, country .by country, into the problem o f the validity of various methods of pupil assessment.

(ii)

Assessment of the curriculum The Conference believed that refinement of the objectives in 2.1 above is a pre-requisite for proper curriculum evaluation to be undertaken.

2.4

Language The relationship between the learning of mathematics, and the language through which i t is learnt, is a fundamental issue on which more research needs to be done country by country. The following references will be found useful :

(i)

"Interactions between linguistics and mathematics education"; report of a CEDO-ICMI-UNESCO symposium, Nairobi 1974, published by UNESCO.

(ii)

"Language and the teaching o f Science and Mathematics, with special reference to Africa"; report of a CASME (Commonwealth Association of Science and Mathematics Education) seminar, Accra, 1975, published by the Commonwealth Secretariat, Marlborough House, London.

(iii) "Mathematique, Langues Africaines et Francaises", Seminaire de Niamey, 2 4

-

28 Janvier 1977, IREM (Institut

de Recherches e t Etudes Mathematique), Niamey, Niger.

Universite de

THE FINAL REPORT

184

2.5

The teacher The Conference noted that there are common problems to all primary school teachers in most Third-World Countries. Three such problems which often result in a lack o f dedication to the teaching profession are :

(i) (ii)

poor conditions of service; lack of self-esteem;

(iii) the unstimulating atmosphere o f the school. The Conference believed that i t is important to tackle these problems vigorously and that the teaching profession itself can do much t o improve the conditions under which its members work. 2 . 5 . 1 e n e r a l Recommendations The Conference recommends :

(i)

T O UNESCO : That a year (perhaps 1980

81) should be

designated as World Teachers'Year, to draw the attention of governments and society at large to the above problems.

(ii)

TO Ministries o f Education : That increased responsibility should he given to teachers in educational and curriculum planning s o as to help to enhance teachers' self-esteem, and to aid professional development.

2.5.2Recommendations o n the mathematics teacher The Conference recommends :

(i)

That there should be more coordination between the curriculum o f pre-service teacher training, and the curriculum in the schools.

(ii)

That there should be more emphasis o n in-service education, which should be s e e n as a continuous process and which should be undertaken b y teachers i n school-time.

(iii) That special national programmes should b e launched during the proposed World Teachers'Year. (iv)

That mathematicians and educationalists in universities should participate actively in both school curriculum development and the in-service education of teachers.

(v)

That a professional association of mathematics teachers should he established i n each country; this will assist

THE FINAL REPORT

t h e a t t a i n m e n t of

3.

2.5.1

185

( i ) above.

Secondary Education While i t i s b e l i e v e d

(i)

that

t h e r e can be o b j e c t i v e s f o r

p r i m a r y m a t h e m a t i c s m o r e o r l e s s common t o a l l c o u n t r i e s , t h i s i s not the case a t t h e secondary l e v e l ,

since the

proportions of children going i n t o secondary schools vary widely.

However,

a l l t h e recommendations

in

2.5

apply equally t o secondary education. (ii)

Mathematics c u r r i c u l a a t secondary-school be designed t o be r e l e v a n t of

level should

t o t h e needs of

that majority

t h e s t u d e n t s f o r whom t h i s s t a g e w i l l b e t e r m i n a l . A s

much c a r e s h o u l d b e g i v e n t o t h e d e v e l o p m e n t o f a p p r o p r i a t e mathematics curricul a f o r the technical

and

v o c a t i o n a l s e c t o r s a s f o r t h e academic s e c t o r . ( i i i ) While s e l e c t i o n f o r h i g h e r education i s i n e v i t a b l e a t t h e end of

the secondary level,

t h e s e l e c t i o n mechanism

should not be confused w i t h t h e e v a l u a t i o n of n a t i o n a l objectives f o r secondary education.

4.

Styles of curricula The C o n f e r e n c e c o n s i d e r e d t h a t t o r e g a r d " t r a d i t i o n a l "

mathe-

m a t i c s a n d "new" m a t h e m a t i c s a s a d i c h o t o m y i s n o t h e l p f u l t o a discussion of

t h e c u r r i c u l u m , f o r many d i f f e r i n g i n t e r p r e I n any c a s e ,

t a t i o n s a r e a s s o c i a t e d w i t h t h e s e two t e r m s . fine details of level,

the

the syllabus, especially a t the secondary

i s n o t t h e major problem : t h e proper p r e p a r a t i o n of

t e a c h e r s i s much m o r e i m p o r t a n t . T h e r e f o r e t h e C o n f e r e n c e recommends : That any major development of

school mathematics c u r r i c u l a

m u s t b e p r e c e d e d by a d e q u a t e p r e p a r a t i o n o f

the teachers.

The

p r o f e s s i o n a l development o f t h e t e a c h e r s and t h e improvement of

classroom conditions are higher p r i o r i t i e s

than f u r t h e r

changes of s y l l a b u s .

I1

1.

UNIVERSITY -

M A T H E M A T I C S INSTITUTIONS

Academic The C o n f e r e n c e n o t e d t h a t i t h a s become i n c r e a s i n g l y a p p a r e n t

186

THE FINAL REPORT

i n r e c e n t y e a r s t h a t m a t h e m a t i c s c a n make a s i g n i f i c a n t c o n tribution to the social,

l i f e and management s c i e n c e s a s i t

h a s done i n t h e p a s t i n t h e p h y s i c a l s c i e n c e s and e n g i n e e r i n g . T h e r e f o r e , r e c o g n i s i n g t h e importance of

t h e r o l e o f mathe-

m a t i c a l models i n e x p l a i n i n g and p r e d i c t i n g phenomena a r i s i n g i n t h e real world,

i t l a i d emphasis on t h e need f o r a deeper

involvement of mathematicians i n s o l v i n g t h e problems of their respective societies,

and on t h e c o n s e q u e n t i a l need

t o t r a i n mathematicians f o r these purposes. The C o n f e r e n c e t h e r e f o r e recommends : (i)

That mathematics courses a t the undergraduate l e v e l

(ii)

That every mathematics department should o f f e r courses

s h o u l d c o n t a i n many r e a l i s t i c a p p l i c a t i o n s .

which s e r i o u s l y and c o m p r e h e n s i v e l y t r e a t r e a l i s t i c problems and which emphasize model b u i l d i n g . .(iii) That,

i n designing courses, mathematics departments

should t r y t o s t r i k e a balance between the fundamentals and a p p l i c a t i o n s of mathematics a p p r o p r i a t e t o t h e i r country's needs. (iv)

That p r a c t i c a l training,

during t h e course of

their

u n d e r g r a d u a t e s t u d i e s may b e d e s i r a b l e f o r s t u d e n t s aiming a t i n d u s t r i a l o r governmental c a r e e r s .

2.

F a c u l t y deveIopment It i s usually t r u e t h a t those departments t h a t are f a i r l y s m a l l , do n o t have d o c t o r a l programmes,

and a r e somewhat

i s o l a t e d from major c e n t r e s o f s c h o l a r s h i p , need s p e c i a l a s s i s t a n c e and encouragement t o keep p r o f e s s i o n a l l y a l i v e and v i t a l .

Hence,

i t is important that postdoctoral a c t i v i t i e s

a re provided. T h e r e f o r e t h e C o n f e r e n c e recommends (i)

:

T h a t exchange programmes between f a c u l t i e s o f

developing

and d e v e l o p e d c o u n t r i e s as w e l l as f a c u l t i e s o f T h i r d World C o u n t r i e s s h o u l d b e e s t a b l i s h e d as f o l l o w s :

a.

M a t h e m a t i c i a n s who h a v e a t t a i n e d s u f f i c i e n t m a t u r i t y t o e n a b l e them t o d i s c u s s and i n t e r a c t w i t h mathem a t i c i a n s of o t h e r c o u n t r i e s . for short periods.

should be exchanged

THE FINAL REPORT

b.

187

Mathematicians who need to go abroad for research i n an area for which proper guides and facilities do not exist i n their own countries should be exchanged for longer periods.

(ii)

That all basic facilities for research in mathematics should be provided i n every department.

(iii) That programmes should be established for advanced and "relevant" training of new and existing members of staff through study leave, short courses, seminars, and visits to centres of innovative curricula and instructional developments as well as to centres where interdisciplinary projects are being undertaken for society's problems. (iv)

That international funding agencies, such as UNESCO,. should be approached for help with such programmes.

3.

Organization

(i)

Every University should have its School of Mathematical, Sciences. Such a School will include s o m e , if not all, of

the disciplines such as Applied Mathematics, P u r e

Mathematics, Statistics, Computer Science, Operations Research and Mathematical Education. (ii)

Such a School, once established, should develop by extending its services to other Faculties such as Economics, Engineering and Science.

111. 1.

MATHEMATICS AND DEVELOPMENT

General The Conference examined a number of issues related to the importance of mathematical activities in the process of development of Third-World Countries, and noted some of the circumstances which have often prevented these activities from making effective contributions to the solution of particular national problems. The Conference therefore felt the need for a set of terms of reference for the problem of where mathematics takes its place i n a national society, i n order to avoid'conflicting

THE FINAL REPORT

188

p o s i t i o n s concerning t h e o r i e n t a t i o n of mathematical a c t i v i ties. 2.

Mathematical objectives The C o n f e r e n c e was o f

t h e opinion t h a t mathematics must be

c o n s i d e r e d w i t h t h r e e g e n e r a l o b j e c t i v e s i n mind

:

(The o r d e r i n which t h e s e o b j e c t i v e s a r e l i s t e d d o e s n o t s i g n i f y any r e l a t i v e importance o r p r i o r i t y ) (i)

The development o f a c o u n t r y i n v o l v e s r e l a t e d c u l t u r a l and m a t e r i a l a c t i v i t i e s . Mathematics belongs t o t h e c l a s s of

acti-

intellectual

v i t i e s which t o g e t h e r determine t h e c u l t u r a l c h a r a c t e r o f a c o u n t r y a n d i s o n t h e same l e v e l a s t h e A r t s a n d t h e H u m a n i t i e s . The v a l u e o f m a t h e m a t i c a l r e s e a r c h d e p e n d s a s much a s o n i t s q u a l i t y , s a l standards,

a s judged by univer-

a s on i t s o r i e n t a t i o n o r r e l e v a n c e t o

national needs. (ii)

Self-determination i n s c i e n t i f i c and technological m a t t e r s i s becoming i n c r e a s i n g l y i m p o r t a n t i n t h e development process of

any c o u n t r y .

This can o n l y be

a c h i e v e d t h r o u g h t h e e x i s t e n c e o f a wide and s t r o n g m a t h e m a t i c a l community. ( i i i ) T h e r e a r e a r e a s where t h e u s e of m a t h e m a t i c a l s k i l l s can p l a y an important r o l e i n t h e s o l u t i o n of

concrete

and immediate problems i n t h e n a t i o n a l development. S e v e r a l c l a s s e s o f p r o b l e m s were m e n t i o n e d d u r i n g t h e Conference

-

t i o n growth, others

-

such as health,

economic planning,

a g r i c u l t u r a l development,

where i t i s e v i d e n t t h a t

popula-

e n e r g y , and

t h e a p p l i c a t i o n of

mathematics can have immediate b e n e f i t s .

N o t w i t h s t a n d i n g t h e s e o b j e c t i v e s , and t h e m a n i f e s t f a c t t h a t mathematics has an important r o l e i n t h e development of

a

country, the mathematical a c t i v i t y i n developing countries should not,

i n general,

concern i t s e l f only with those

problems d i r e c t l y connected with immediate n a t i o n a l needs

:

i n d e e d i t may o f t e n h a p p e n t h a t i n t e r e s t i n t h e s t u d y o f n a t i o n a l problems i s g r e a t e r i n i n d i v i d u a l s o r groups from the developed countries.

An a n a l y s i s

of

t h i s s i t u a t i o n suggests

THE FINAL REPORT

189

the following : (i)

The 'critical mass' has not yet been achieved i n most developing countries as regards the mathematical establishment, and hence there are not enough trained people to permit systematic efforts in dealing with concrete problems.

(ii)

There is a divorce between the mathematical establishment and the decision-making bodies which makes i t difficult to bring together those w h o have the information and the ability to identify real-life problems and those who can study them i n a scientific way.

(iii) Academic institutions tend not to give adequate recognition to work directed to the solution of real-life problems and hence do not encourage the participation of

(iv)

mathematicians in their study and solution.

Formal education of mathematicians does not usually involve exposure to concrete problems and tends to promote interest in areas which have n o direct relevance to the needs of the country concerned.

(v)

Lack of adequate leadership in the national mathematical community makes i t difficult to overcome these problems and to provide a sense of purpose.

3.

Recommendations As possible ways of successfully attacking the above-mentioned causes, the Conference recommends :

(i)

That mathematicians should be encouraged to take the initiative i n seeking closer cooperation with the decision-making bodies in an institutional w a y , i n order to involve themselves i n the process o f formulating national priorities and to initiate joint interdisciplinary actions.

(ii)

That this involvement needs high-quality scientists and therefore the production o f high-quality mathematicians is of great importance.

(iii) That academic institutions must be advised about the risks of following the same patterns as developed

a

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190

c o u n t r i e s i n t h e o r g a n i z a t i o n and e v a l u a t i o n o f of mathematicians.

t h e work

The e v a l u a t i o n s h o u l d t a k e i n t o

account a l l contributions,

pure,

a p p l i e d and e d u c a t i o -

nal, while maintaining universal standards i n original research. (iv)

T h a t f o r m a l e d u c a t i o n o f m a t h e m a t i c i a n s must b e modif i e d i n o r d e r t o expose young s t u d e n t s t o t h e t y p e of work

-

such as consulting,

f o r example

-

which has a

b e a r i n g on c o n c r e t e problems.

IV.

1.

MATHEMATICS POLICY & INTERNATIONAL COOPERATION

General considerations. W h i l e r e c o g n i z i n g common g o a l s a n d c o n s t r a i n t s f o r m a t h e m a t i c s policy i n developing countries, d i f f e r e n c e s i n t h e s t a g e s of

the Conference noted t h a t the

development of t h e s e variou's

c o u n t r i e s must be k e p t i n mind. T h e r e f o r e t h e C o n f e r e n c e recommends (i)

:

That l o c a l mathematicians convene r e g u l a r l y

to discuss

general mathematics p o l i c y i n t h e i r c o u n t r i e s ,

as a

f i r s t s t e p t o w a r d s t h e c o o r d i n a t i o n o f e f f o r t s on a n international scale. (ii)

T h a t i n i t i a l e f f o r t s m u s t be u n d e r t a k e n by e x i s t i n g groups i n developing c o u n t r i e s t o c r e a t e favourable conditions t o absorb external co-operation.

Experience

s h o w s t h a t maximum b e n e f i t i s m o s t l i k e l y w h e n l o c a l e f f o r t s h a v e p r e p a r e d t h e g r o u n d f o r i t . When l o c a l m a t h e m a t i c i a n s i d e n t i f y t h e i r n e e d s and p r i o r i t i e s , t r a i n people t o a c e r t a i n degree of m a t u r i t y ,

and

external

a i d can be sought t o h e l p t h e achievement of e f f e c t i v e n a t i o n a l p r i o r i t i e s and g o a l s . T h i s would a v o i d d i s s i p a t i o n of e f f o r t and would have p o s i t i v e e f f e c t s i n t h e choice o f i n s t i t u t i o n s and i n d i v i d u a l s w i t h which t o co-operate.

These c o n s i d e r a t i o n s apply e s p e c i a l l y

t o the sending of

s t u d e n t s abroad and the i n v i t i n g of

foreign mathematicians. ( i i i ) That,

i n recognition of

the fact that individual trai-

n i n g abroad does not always produce t h e d e s i r e d results,

s p e c i a l e f f o r t s s h o u l d b e made b y l o c a l i n s t i t u -

THE FINAL REPORT

191

tions to establish training and graduate programmes with help from more developed centres. W h e n students are sent abroad they should be sent at a n advanced stage of their studies, with the specific purpose of working i n specified areas, for short periods o r as postdoctoral fellows. 2.

The International Centre for Pure and Applied Mathematics The Conference strongly supported the idea o f establishing an International Centre o f Pure and Applied Mathematics to respond to the needs o f Third-World Countries. The Conference therefore recommends :

(i)

That mathematicians from Third-World Countries should. be actively involved, from the initial stages, i n the planning of this Centre.

(ii)

That the functions of the Centre should include training, the provision of research facilities, and continuous assistance i n the development of mathematics in Third-World Countries.

(iii) That equally important is the creation of Regional Centres in order to complement the International Centre 3.

International Mathematics Union (IMU) The Conference urges Third-World Countries which are not members of IMU to apply for membership; and the Conference recommends to IMU : That an "International Commission o n Mathematics and Development", be established i n order to promote the development of mathematics and its applications i n Third-World Countries.

*********** RESOLUTION OF THANKS The participants i n the International Conference o n Developing Mathematics i n Third-World Countries (Khartoum, 6-9th March 1978) record their thanks and gratitude to the University of Khartoum, the organizing oommittee and its Chairman

Dr. M.E.A. El T o m for the excellent organization of the

192

THE FINAL REPORT

C o n f e r e n c e a n d t h e warm w e l c o m e a n d h o s p i t a l i t y e x t e n d e d t o them.

APPENDICES

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APPENDIX 1

:

List of p a r t i c i p a n t s

Australia DIESENDORF Mark : CSIRO Division o f Mathematics & Statistics, P.O.Box 1965 - Canberra City

-

A.C.T. 2601

Belgium DEDECKER Paul : Institut de Mathematique Pure et Appliquee, Ch. du Cyclotron 2 - B 1348 Louvain-La-Neuve Brazil D'AMBROSIO Ubiritan : Institute o f Mathematics, Statistics and Computer Science (IMECC) Campinas

-

-

Universidade Estadual de

13.100 Campinas - S.P.

NACHBIN Leopoldo : Av. Gilbert0 Amado 223

-

Barra da Tijuca ZC-18

Rio de Janeiro -RJ ~ Cameroon HOGBE-NLEND H. : U.E.R. de Mathematiques et d'Informatiques Universite de Bordeaux I - 33405 Talence and Department o f Mathematics, B.P. 812

-

-

-

France Yaounde

Canada DLAB Vlastimil : Dept. o f Mathematics

-

Carleton University

-

Ottawa KIS 5 B 6 RIBENBOIM Paulo : Dept. Mathematics - Queen's University

-

Kingston - Ontario

-Costa-Rica MONTERO Bernard0 : Escuela de Matematica Rica

-

Egypt ANIS Abdel-Azim : Mathematics Dept. Sham University Abbassia ASHOUR A.A.

-

Universidad de Costa

San Jose

-

Faculty of Sciences-Ein Cairo

: Department o f Mathematics

Cairo University

-

Faculty of Science

-

- Giza

MOHAMMED A/Karim Abu E l Hassan : Department o f Mathematical and Physical Sciences of Alexandria

-

-

Faculty o f Engineering

Alexandria

I95

-

University

196

APPENDIX 1

France

-

: I.R.I.A.

LIONS J . L .

court

-

Domaine de V o l u c e a u

ROFMAN Edmundo : U n i v e r s i t e d e P a r i s I X Marechal

-

-

de L a t t r e de T a s s i g n y

B.P.

105

Dauphine

-

-

Rocquen-

-

P l a c e du

75016 P a r i s

L a b o r i a - Domaine d e V o l u c e a u

TAN Wank S e n g : I . R . I . A . court

-

7 8 1 5 0 Le C h e s n a y

B.P.

105

-

-

Rocquen-

7 8 1 5 0 Le C h e s n a y

-

THOM R e n e : I n s t i t u t d e s H a u t e s E t u d e s S c i e n t i f i q u e s ( I H E S ) 91440 Bures S u r Y v e t t e India MISHRA R.S.

-

: Department of Mathematics

Banaras Hindu U n i v e r s i t y

-

: T a t a I n s t i t u t e of Fundamental Research

N A R A S I M H A N M.S.

B h a b h a Road

-

-

F a c u l t y of S c i e n c e

V a r a n a s i 221005

-

Homi

E m b a y 400005

Iran SHAHSHAHANI S .

: Department of Mathematics and Computer S c i e n c e

-

U n i v e r s i t y of Technology

Arya-Mehr

-

a t Department of Mathematics Berkeley,

Ca 9 4 7 2 0

-

Tehran ( P r e s e n t l y

University of California

U.S.A).

Iraq ALZOOBAEE O r a b i

: C o l l e g e of E d u c a t i o n

-

Baghdad U n i v e r s i t y

-

Baghdad AWADH A b d u l J a b b a r M . Basrah

-

: Mathematics Division

-

U n i v e r s i t y of

Basrah

MOHAMMED JAWAD S a a d A 1 D i n : The D i r e c t o r a t e o f C u r r i c u l u m a n d

Textbooks

-

Ministry of Education

-

Baghdad

Ivory Coast BAKTAVATSALOU M .

: Faculte des Sciences - Universite Nationale

de Cote d ' I v o i r e TOURE S .

-

Abidjan

: I n s t i t u t de Recherches Mathematiques, d'Abidjan

-

Universite

Abidjan

Italy

-

V i a Leon B a t t i s t a FRANCHETTI C a r l o : I n s t i t u t o d i M a t e m a t i c a 16132 Genova Battista Alberti 4

-

-

197

APPENDIX 1

-

LEVKO John : Collegio Internazionale del Gesu 45

-

Piazza del Gesu

Rome

00186

Jordan AL-TAKI Ahmad : Head o f Mathematics Section - Curriculum Dept. Ministry o f Education

-

-

Amman

-

NUSAYR Abdul Majid : Yarmouk University

P.O.Box 5 6 6

-

Irbid

NATSHEH Muhammad A. : University o f Jordan - Amman Kenya CALLEB B.O.

-

: Department of Mathematics

-

University of Nairobi

Nairobi NORRIS J. : Inspector o f Schools rate)

-

P.O.Box 3 0 4 2 6

-

Ministry o f Education (InspectoNairobi

Lebanon

-

ELABD Ibrahim : Dean, Faculty o f Engineering

Arab University

-

Beirut HASHIM Ismail : Beirut Arab University

-

B.O.Box 5 0 2 0

-

Beirut

Malawi BANDA Chiwackson Tobias : Maths. Inspector

-

P.I.B. 3 2 8

-

Ministry o f Education

Lilongwe 3

Malaysia LIM Chong-Keang : Department o f Mathematics

-

University o f Malaya-

Kuala Lumpur Tan Sin Leng : Department o f Mathematics

-

University o f Malaya

-

Kuala Lumpur Mexico ABREU, J . : IIMAS, UNAM, Mexico 2 0 , D.F.

GARZA Tomas : Director 20-726

-

I I M A S , UNAM, Apartado Postal

- Mexico 2 0 , D.F.

IMAZ C . : Departamento de Matematicas Educativas Investigacion del IPN Mexico 14 MARQUINA J .

-

-

Centro de

Apartado Postal 14-740

-

- D.F.

: Academic Section o f Mathematics

-

Escuela Nacional

de Estudios Profesionales "ZARAGOZA" Calzada Ignacio Zaragoza s / n

-

Col. Ejercito de Oriente - Mexico 9, D.F.

198

APPENDIX 1

-

RODITI Albert0 : Facultad de Ciencias

Dept. Matematicas

-

UNAM

-

Mexico 2 0 , D.F. Nigeria CHUKWU E.N. : University o f JOS

-

Private Mail Bag 2 0 3 4

-

HAWTHORNE William : Institute o f Education

- 30s

University o f Ibadan

-

Ibadan MADUNAGU Edwin Ikechukwu : Department o f Mathematics o f Calabar

-

-

University

Calabar

-

OHUCHE R. Ogbonna : Department of Education

University of Nigeria

Nsukka Philippines NEBRES Bienvenido F. : Department of Mathematics University

-

-

Ateneo de Manila

P.O.Box 1 5 4 - Manila

Poland GREN Jerzy : Central School o f Planning and Statistics Niepodleglosci 1 6 2

-

-

Al.

Warsaw

02-554

Saudi Arabia AL-SALEH Abdul-Aziz : Ministry o f Education, Riyadh

-

AL-SALMAN Salman : Ministry o f Education AL-YAMANI Jamil : Ministry o f Education

-

Riyadh Riyadh

SA'ID Abdel Rahman M. : Department o f Mathematics, College o f Engineering

-

King Abdulaziz University

AL-DAFFA Ali A. : College o f Science and Minerals

-

-

-

Jeddah

University o f Petroleum

Dhahran

Senegal BADJI Cherif : Departement de Mathematique Universite de Dakar

-

-

Faculte des Sciences

Dakar-Fann

Sierra Leone LABOR Adonis : Institute o f Education

-

Tower Hill

-

Free Town

Sudan ABDEL GHAFAR Kamal : Dept. of Mathematics, Faculty o f Science, Khartoum University

-

Khartoum

ABDEL RAHMAN Isam : Dept. o f Mathematics, Faculty o f Science Khartoum University

-

Khartoum

-

199

APPENDIX 1

ALAM Hassan Alam : Finance Branch, G.H.Q., People's Armed Forces Khartoum AYOUB H. : Dept. o f Mathematics, Faculty o f Engineering and Architecture BABIKER A.G.A.

-

-

Khartoum University

Khartoum

-

: Dept. o f Mathematics, Faculty of Science

-

Khartoum University

Khartoum

BABIKER Mohamed : Physics Dept., Faculty o f Science University

-

-

Khartoum

Khartoum

BABIKER Susan : Dept. o f Economics, Faculty of Economics and Social Studies

-

Khartoum University

BAKHIT Charles : University o f Juba, Box 82

-

Khartoum

- Juba

BAKRI Mirghani A. : Mathematics Dept.. Khartoum Polytechnic

-

Khartoum BESHIR S.A. : Telecommunications Corporation DUGGAL B.P.

:

-

Khartoum

Dept. o f Mathematics, Faculty o f Science

University

-

-

Khartoum

Khartoum

-

EL AFFENDI Mohamed A. : Computer Centre, Khartoum University

-Kha r t o urn EL AGIB El Tahir : Dept. o f Mathematics, Faculty o f Education,

-

Khartoum University

Khartoum

EL BULUK Malik : Electrical Engineering Dept., Faculty o f Engineering and Architecture Khartoum University

-

-

Kha r t oum EL DASISE M. : Ministry o f Education - Khartoum EL HUSSEIN Abubakr : Electrical Engineering Dept., Faculty o f Engineering and Architecture

-

-

Khartoum University

Kha r t o um -

-

EL MEKKI Osman M. : Dept. o f Mathematics, Faculty o f Science

-

Khartoum University

Khartoum

-

EL SANOUSI Salih : Dept. o f Mathematics, Faculty of Science

-

Khartoum University

Khartoum

EL TAHIR Mohamed A. : Computer Centre, Khartoum University

-

Khartoum EL TARAS

S.

:

Computer Centre

-

Khartoum University

-

Khartoum

-

200

APPENDIX 1

EL TAYEB Ibrahim A. : Dept. o f Mathematics, Faculty o f Science

-

Khartoum University EL TOM Mohamed E.A.

:

-

Khartoum

Dept. of Mathematics, Faculty of Engineering

and Architecture

-

Khartoum University

-

Khartoum

GAMIL Abdel Hamid Mohammed : Statistical Branch, G.H.Q., People's Armed Forces

-

Khartoum

GOSH B. : Department of Mathematics, Faculty of Engineering and Architecture

-

Khartoum University

-

Khartoum

-

HAG0 Ahmed M. : Mathematics Dept., Khartoum Polytechnic HASSAN Mohamed H.A.

:

Dept. of Mathematics, Faculty o f Science

-

Khartoum University

-

McBETH R.B.

-

: Computer Centre

-

Khartoum University Khartoum University

-

-

Khartoum Khartoum

-

: Dept.of Mathematics, Faculty o f Education

University

-

Khartoum

IZZELDIN Ahmed : Computer Centre MAQBOOL, U . S .

Khartoum

Khartoum

Khartoum

MOHAMMED Salah A. : Dept. of Mathematics, Faculty of Engineering and Architecture

-

Khartoum University

-

Khartoum

MEADON J. : Computer Centre

-

Khartoum University

-

Khartoum

-

MOSHARRAFA Abdel : Cairo University, Faculty of Science Branch MUGRA M.L.

-

Khartoum

Khartoum

: Faculty of Science, Dept. of Mathematics, Khartoum

University

-

Khartoum

MUSTAFA Mustafa A. : Dept. o f Mathematics, Faculty of Science Khartoum University

-

Khartoum

NASR EL DIN Adil : Dept. o f Mathematics, Faculty o f Engineering and Architecture

-

Khartoum University

NAWARI Mustafa : Computer Centre

-

-

Khartoum

Khartoum Univers ty

NOUREIN Abdel Wahab M. : Computer Centre

-

-

Khartoum

Khartoum University

-

Khartoum EL NUB1 Ahmed 0 . : Mathematics Dept., Khartoum Polytechnic

-

Kha r to um OSMAN Izzel Din M. : Computer Centre t oum -

-

Khartoum University

-

=-

20 1

APPENDIX 1

SAMPSON D. : Mathematics Dept. Khartoum Polytechnic SANDERS O.J.

:

-

Khartoum

Dept. o f Mathematics, Faculty o f Engineering and

-

Architecture

-

Khartoum University

Khartoum

SIR ELKHATIM Mohamed El Amin : Statistical Branch, G.H.Q.,

-

People's Armed Forces

Khartoum

TANGASAUI A. : Dept. o f Mathematics, Faculty of Education

-

Khartoum University VOGEL Linda : Computer Centre

-

-

Khartoum Khartoum University - Khartoum

YADAV N. : Dept. of Mathematics, Faculty of Engineering and Architecture, Khartoum University

-

Khartoum

Sweden NASELL Ingemar : Department of Mathematics - The Royal Institute

-

of Technology

S-100 4 4 Stockholm

Togo BAMAZI Bizongani : Directeur-adjoint de 1'Ecole des Sciences University d u Benin

-

B.P. 1515

-

-

Turkey MULLINS Charles : Mathematics Dept.

-

Bogazici University -

P.K.2 Bebek-Istanbul OZDEN Hayriye : University of Hacettepe Department of Mathematics

-

-

Faculty of Science

-

zeytepe-Ankara

Uganda KAMIZZI Abudu : The Inspectorate P.O.Box 3568

-

KARUHIJE Eric : The Inspectorate

-

P.O.Box 3568

-

Ministry of Education

-

-

Ministry o f Education

-

Kampala

Kampala

U.K. AHMED Idris A. : University o f Newcastle - Upon Tyne - Newcastle Upon Tyne BAKER John Edward : Faculty o f Mathematics - The Open University Walton Hall

-

Milton Keynes M K 7 6 A A

BROWN Aldric : School o f Mathematics Upon Tyne GORDON William : U . S . ne Road

-

-

-

T h e University

-

Newcastle

NEI 7RU Office o f Naval Research

London NWl 5TH

-

2 2 3 Old Marylebo-

202

APPENDIX

1

HAMZA El Sadig : University of Liverpool HIDDLESTON

-

Liverpool

-

Patricia : St. Margaret's School Edinburgh EHl6 5PJ

East Suffolk Road

-

-

-

MITTENTHAL Lothrop : European Research Office Road

-

223 Old Marylebone

London NW1 5TH

MODAWI Abdel Gadir : Mathematics Institute

-

Warwick University

-

Coventry CV4 7AL THWAITES Bryan : Westfield College (University of London)

-

Kidderpore Avenue

-

London, NW3 7 S T

-

WILSON B.J. : Schools and Further Education Department British Council - 10 Spring Gardens YAHIA Mahjoub : School of Mathematics

-

The

London SWIA 2BN

-

The University -

-

B.G.S.U.

Newcastle Upon Tyne U.S.A. GUPTA Arjun K .

: Dept. o f Mathematics

-

Bowling Green

-

Ohio 43403 -

-

SHAMMA Shawky : Massachusetts Institute o f Technology 37-361

-

Mail Stop

Cambridge, MA 02139

SUPPES Patrick : Institute for Mathematical Studies in the Social Sciences (IMSSS)

-

Ventural Hall

-

Stanford University

Stanford, Ca 94305 VOGELI Bruce R. : Department of Mathematics, Statistics and Computing in Education University

-

-

Teachers College

-

Columbia

New York, N Y 10027

STONE Marshall H. : 260 Lincoln Avenue

-

Amharest, MA 01002

West Indies GEARY James : School of Education - The University of the West Indies

-

St. Augustine

-

Trinidad

APPENDIX 2 :

T i t l e s of c o n t r i b u t e d p a p e r s

Appropriate mathematics

-

its definition and dissemination

J. BAKER Mathematics and the teacher (Film)

.

W M. HAWTHORNE Les sciences exactes e t s o n enseignement - apprentissage A. RODITI Recent attempts at school mathematics curriculum renewal in Englishspeaking West Africa R.O. OHUCHE Strategies in research and graduate education in Nigeria E.N. CHUKWU Mathematics teaching and research in India M.S. MISHRA Mathematics in the Turkish university C.W. MULLINS Mathematical modeling and its role in solving societal problems S.E. SHAMMA Statistics in mathematics institutions : teaching and research A.K. GUPTA Some considerations o n the role of a mathematics section in a graduate school i n third world countries J. MARQUINA

A mathematical model of the role o f storage in the utilization of wind power M. DIESENDORF Problems o f nonlinear elasticity and their importance for the developing technology B.O.

CALLEB

The problems o f mathematics along with the North-South dialogue between industrialized and developing countries P. DEDECKER Mathematical education and national development B.R. VOGELI

203

204

APPENDIX 2

D i l e m m a of r e s e a r c h i n m a t h e m a t i c s i n s m a l l u n d e r d e v e l o p e d A.E.

countries

NUSAYR

Organization of mathematical research in Canada

P. R I B E N B O I M An explicative model of the development o f mathematics in Costa Rica

B. M O N T E R O

APPENDIX 3 :

The C o n f e r e n c e C o m m i t t e e s , Working Groups and Chairmen

Organizing Committee: E. ELAGIB S . EL SANOUSI M.E.A.

EL TOM (Chairman)

K.A/GHAFFAR S .A.

MOHAMMED

Follow-up Committee: All members of the Organizing Committee A.A. ASHOUR (coopted) U. D'AMBROSIO

(coopted)

H. HOGBE-NLEND (coopted) B.F. NEBRES (coopted) E. ROFMAN (coopted) Working Groups: WG- 1

School Mathematics

Chairman

M. J. SA'ADELDIN

Secretary

A.G.A. BABIKER

WG-2

University Mathematics Institutions

Chairman

M.S. MISHRA

Secretary

M.A. MUSTAFA

WG- 3

Mathematics and Development

Chairman

T. GARZA

Secretary

I.A/RAHMAN

WG-4

Mathematics Policy and International Cooperation

Chairman

E. ROFMAN

Secretary

I.A. EL TAYEB

Chairmen of plenary sessions: R. Ogbonna OHUCHE Abdel-Azim ANIS M.E.A.

EL TOM 205

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APPENDIX 4 : ALLOCUTION DE CLOTURE D E M. L E P R O F E S S E U R HENRI HOGBE-NLEND P R E S I D E N T D E L'UNION MATHEMATIQUE AFRICAINE

M o n s i e u r l e P r C s i d e n t d u Colloque C h e r s coll'egues e t a m i s ,

Nous v o i c i a u x t e r m e s d e s t r a v a u x d u Colloque I n t e r n a t i o n a l s u r la p r o m o t i o n d e s m a t h 6 m a t i q u e s d a n s l e s p a y s e n v o i e de d e v e l o p p e m e n t . C e Colloque a connu un g r a n d succ'es a i n s i q u e I ' a t t e s t e l e r a p p o r t f i n a l q u e n o u s venons d'adopt er . Qu'il m e soit p e r m i s , a u nom de tous l e s participants, d ' a d r e s s e r nos s i n c k r e s r e m e r c i e m e n t s e t n o s s e n t i m e n t s d e profonde g r a t i t u d e B s o n E x c e l l e n c e l e G e n e r a l NIMEIRY, P r e s i d e n t d e l a RCpublique du Soudan e t B s o n g o u v e r n e m e n t s a n s l ' a i d e d e s q u e l s c e Colloque n ' a u r a i t p a s pu s e t e n i r . Q u ' i l m e s o i t p e r m i s d e m C m e , d ' a d r e s s e r tout p a r t i c u l i ' e r e m e n t nos c h a l e u r e u x r e m e r c i e m e n t s h Monsieur l e M i n i s t r e d e 1'Education Nationale, B Monsieur l e R e c t e u r d e l ' U n i v e r s i t 6 d e K h a r t o u m , 'a l a f a c u l t 6 d e s S c i e n c e s e t 'a 1 ' E c o l e MathCmatique d e K h a r t o u m p o u r t o u t c e q u ' i l s ont f a i t pour a s s u r e r l e p l e i n succ'es d e c e Colloque. C h e r s amis ; L e Colloque d e K h a r t o u m e s t 'a d i v e r s t i t r e s , un Colloque h i s t o r i q u e . I1 e s t h i s t o r i q u e d ' a b o r d p a r son th'eme : c ' e s t l e p r e m i e r Colloque d e c e g e n r e c o n s a c r k 'a 1'Ctude d e s v o i e s e t m o y e n s d e p r o m o t i o n r a p i d e d e s m a t h k m a t i q u e s dans tous l e s pays en voie de dgveloppement e t auquel participent d e s m a t h e m a t i c i e n s v e n a n t d e s q u a t r e p a r t i e s du m o n d e . I1 e s t e n s u i t e h i s t o r i q u e p a r s e s r C s u l t a t s qui n e m a n q u e r o n t p a s d ' i n f l u e n c e r c o n s i d e r a b l e m e n t d e n o m b r e u x travaux futurs. nos coll'egues m a t h e m a t i c i e n s s o u d a n a i s C e succ'es e s t dQ a v a n t tout a y a n t B l e u r tGte, l e D r . E l TOM. E n v o t r e n o m , j e l e u r a d r e s s e n o s c h a l e u r e u s e s f e l i c i t a t i o n s e t l e u r a s s u r e q u ' i l s ont a i n s i f a i t un g r a n d h o n n e u r 'a n o t r e continent, 1'Afrique.

Avant d e f i n i r , p e r m e t t e z moi de r e m e r c i e r n o s c o l l b g u e s m a t h C m a t i c i e n s Venus d e s a u t r e s c o n t i n e n t s , d ' E u r o p e , d e s A m k r i q u e s , d ' A s i e e t du Moyen O r i e n t , p o u r l e u r i m p o r t a n t e c o n t r i b u t i o n p o u r l e succ'es de c e Colloque. Nous l e u r a s s u r o n s d e nos s i n c ' e r e s s e n t i m e n t s d e c o o p e r a t i o n s c i e n t i f i q u e durable. J e p r o c l a m e s o l e n n e l l e m e n t c l o s l e Colloque I n t e r n a t i o n a l d e K h a r t o u m s u r la p r o m o t i o n d e s m a t h e m a t i q u e s d a n s l e s p a y s e n v o i e d e d e v e l o p p e m e n t . 207

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