Exeter 1972
Ramesh Kapadia
at the
INTERNATIONAL CONGRESS on MATHEMATICAL EDUCATION.
Mathematics is perhaps unique in occupying a central position in the education system in all countries. Its language is universal, transcending national and geographical boundaries. And yet it is probably the worst taught of all subjects, invoking a sort of reverence from those who can't do mathematics towards those who can. An awe dating back many centuries to when mathematics, thelogy and superstition were intimately linked; when, to do an arithemetical operation required great skill and knowledge.
Thus it is most appropiate to have an international congress to discuss the difficulties and problems surrounding the teaching of mathematics; to compare and exchange different solutions; and to take account of the increasing applications of mathematics to many other disciplines. The first congress was held in Lyons in 1969, attracting over 600 participants. The 2nd International Congress on Mathematical Education was held in Exeter from the 29th of August to the 2nd of September 1972. There were over 1400 full members and 300 associate members; in fact, because of limited facilities, a number of people were turned away.
Sir James Lighthill, president of ICMI and chairman of the congress stressed its importance: "We can forsee in the twenty-first century a world with more complex logistic, economic, technological and environmental problems than ever before. Mathematical educators have a key role to play in the preparation of those young people who will be collectively involved in tackling these problems. This above all is the context in which I ask for the subject matter of our congress to be treated with high seriousness.
"The ancient city of Exeter and its fine modern University Campus, set among the warm beauty of the Devon countryside, will assure a perspective of continuity in which we may combine an historical consciousness of the importance of our ancient calling with recognition of the crucial significance for modern life, and with a warmth of good fellowship and of pleasure in meeting friends and colleagues from all over the world in such a setting."
The congress was officially opened on Tuesday (29th August) evening by the pro-Chancellor of Exeter University, who welcomed the participants to the West country. He admitted his own failure at mathematics and envied the ability of the congress members to understand and appreciate it. He hoped that the weather would do justice to the West country. In fact we had a full week of sunshine.
In his opening speech, Sir James Lighthill explained the organization of the congress. There were to be six plenary seasions in which a distinguished speaker would address the entire congress. The rest of the time was devoted to more informal activities including working groups, at which small groups of people would discuss in depth, specific problems in mathematical education. There were thirty-eight different groups covering the entire spectrum of mathematical education, ranging from 'Pre-School and Primary Mathematicas' to 'University Mathematics for the Specialist'.
The Congress also invited two distinguished guests to attend the conference. The Chairman of the programme committee wrote that. "... much of the programme is devoted to considering the effects of deeper insights into the nature of mathematical learning, the field in which one of our two distinguished guests, Jean Piaget, has made his unique contribution. The programme also takes account of the rapidly increasing use of mathematical structure and techniques in dealing with the many types of problems arising in our complex society. Our other distinguished guest, George Polya, in the leading exponent of the role of problem-solving in mathematical education."
Plenary Sessions
Professor David Hawkins (from the United States) emphasized the place of mathematics in the wider context of education as one of the major parts of Western culture, in a talk entitled 'Nature, Man and Mathematics.' Professor S. Sobolev (Union of Soviet Socialist Republics) gave an uninspiring account of reforms in Russian schools, while Professor H.Philip (Australia) discussed 'Mathematics in the Developing World: some Teaching-Learning Problems'.
Dr Edmund Leach. the renowned anthropologists gave a rather obscure lecture on 'The Concept of Time in Cross-Cultural Perspective'. It was brave of the organizing committee to invite a distinguished non-mathematician to address the congress but, sadly, there was little communication. Professor Hans Freudenthal (Holland) entertained the congress with slides to accompany his theme: 'What Groups mean in Mathematics and what they Should mean in,Mathematical Education'. Perhaps he should have said 'symmetries' not 'groups'. For he never mentioned Galois theory or infinite groups which do not really come under his claim that groups are important because the automorphisms of a structure form a group under composition. It is only symmetry groups which are studied in this way.
The most and challenging talk was given by Professor Rene Thom (France) on 'Modern Mathematics - does it Exist?' He was almost barracked as he made some rather controversial but highly interesting points. He is a distinguished mathematician who was awarded the Fields medal in 1958 (see MANIFOLD-8); his recent work has been on catastrophes and mathematical linguistics, fields which could only be related by a highly creative mind. (We hope to publish articles on cotastophe theory in the next MANIFOLD.)
Professor Thom distinguished between two aspects of modern reforms in mathematical education: pedagogical renovation and modernization of the syllabus. He concentrated on the second aspect, claiming that there were genetic restraints as to what should or could be taught at any particular level. He derided the emphasis on structure in the teaching of mathematics because finer relations were very difficult to formalize. He suggested that rigour is not particularly important in mathematics, and that he would always choose meaning before rigour. Finally he claimed that geometry, not algebra, is the natural and perhaps irreplaceable intermediary step between language and mathematics. In language syntax is poor and semantics is rich; while in algebra, semantics is poor and syntax rich. Geometry, having good syntax and semantics is the natural intermediary. Some of his other points were lost in the simultaneous translations: Professor Thom spoke very fast making the tranalator's job quite difficult. We look forward to reading his lecture in the (inevitable) book on the proceedings of the congress. It might spark off a much-needed discussion and review of the rationale behind reforms.
Working Groups
There were so many different working groups that it is impossible to report on all of them. The main problem in all the groups was communication. Most groups were chauvinistic, using only English an the medium ot communication. Others tried to give a simultaneous.tranalation after each paragraph of the speaker's paper. Unfortunately this was very wearing and left no time for discussion Perhaps duplicated copies of translations of each paper is the best answer, but this requires a lot of extra work in administration.
The general format of each session was that one person opened the discussion by presenting a short paper. There were a number of interesting papers read in the working groups, but there were, inevitably, clashes when one had to choose between two or more speakers. A few people did concentrate on just one working group attending all its sessions; but most people flitted from one group to another.
There was a lively group discussing the teaching of geometry. Professor Pedoe advocated that the clock should be put back and projective geometry be brought back into the university syllabus once again. As far as school geometry is concerned it was decided to have a special conference to discuss its place in the syllabus. The group on the place of history in mathematics was quite fruitful, discussing how it may be profitably used in teaching. The problems of the mathematics undergraduate were aired. The group investigating the teaching of mathematics in developing countries came up with some resolutions to put before the entire congress as did a number of other groups. Space forbids me to describe all the groups.
Each country was invited to contribute a national presentation; the countries which did make displays were: Argentina, Australia. Eire, Ghana, India, Italy, Japan, Korea, Malawi, Poland, South Africa, United Arab Republic, United Kingdom. United States and West Germany. The biggest exhibition was the one from Britain, and it included a number of demonstration classes. There was a special exhibition for the School Mathematics Project and for the Nuffield Project. The American presentation contained an exhibition entitled Mathland, which proposed a total alternate school mathematics curriculum devised by the M.I.T. Logo group under Seymour Papert. They illustrated their ideas by showing how to use computers to enable children to 'easily master very much more sophisticated mathematical thinking ... through goal-directed activities'. Stripped of the jargon, there seemed to be little to support this claim.
In general the presentations consisted of exhibitions of books, materials and pupils' work. There were also examples of television programmes made to teach mathematics. Some countries also gave reports on recent developments.
A number of instructional mathematical films were shown independently of the presentations. There was also a display of visual aids for teaching mathematics and a large book exhibition.
The Future
After the final plenary session there was a report of the ICME business meeting. They elected a new executive committe; Professors layanaga (Japau), Suranyi (Hungary). Freudenthal (Holland), Pollak (U.S.A.), and Sobolev (U.S.S.R.). There will be eight ICME sponsored symposia in the next four years. These include two symposia on primary education and one on 'New aspects of applicable mathematics relevant to school level'. There will be more regional meetings, including two on the particular problems of developing countries (one in Nairobi and one in New Delhi). There will be a joint symposium with IFIP on the teaching of computer science. And also a symposium on the teaching of school geometry, in 1974.
A number of resolutions were also passed at the meeting. ICME resolved to help in disseminating relevant information on symposia, mathematical journals, competitions, etc. They also adopted some resolutions proposed by working groups. For example that 'all possible encouragement and assistance be given to developing countries to make changes to existing curricula so that the cultural backgrounds of people can be taken into account'.
The third international congress on mathematical education will be in 1976 in either Spain, Germany, the United States or the Netherlands. Meanwhile there is the international congress of mathematicians in 1974 at Vancouver.