A SHORT PRESENTATION
The participants in this forum were experienced mathematicians, each one from a different country. Ronnie Brown represented Great Britain, Manuel Chaves Portugal, Michele Emmer Italy, Mike Field who had also worked in Great Britain and Australia, represented the United States, while Konrad Polthier represented Germany.
They told us about some of their experiences attempting to popularize and teach mathematics using various means, in particular through art.
This part of the Colloquium was perhaps a first attempt at establishing common ground on the interplay of Art and Science in education, espacially mathematical education.
Henri Poincaré and Hermann Weyl were among the deepest mathematical thinkers of the last two centuries. They were quite convinced that the main goal of education in mathematics was the formation of the mind. I quote from Hermann Weyl famous book “Space, Time, Matter” : “It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony” (p. 23 of the Dover English edition). Quoting Henri Poincaré (La Valeur de la Science : “Le but principal de l’enseignement des mathématiques est de développer certaines facultés de l’esprit”.
I will sum up what I believe are the advantages of a well constructed and guided mathematical education. The formation and development of the faculties of analysis and synthesis. The formation and the development of the faculties of observation, reasoning, and intuition. The formation and development of a feeling for intellectual beauty. Classical elementary geometry is perhaps the best tool by which theses aims can be achieved. Indeed, it gives birth to a large diversity of strange and amazing properties, which can be proved in a few organised and well written sentences. In this way, it stimulates the activity of the mind and contributes to the unfolding of all the previous qualities.
Among the tools which nowadays can be used to improve teaching, the conscious use of Art seems to be new. Here I use the term “Art”, in its widest sense to include all the forms it can take.
One form is literary art which is definitely missing from the standard teaching books. Formulae and basic language are used : they are insipid and not appealing for a young mind who is impregnated with emotional functions and realism. For such a mind, the abstract discourse does not make sense and can be repelling. We meet here the general tendency of blind modern pedagogy which is to insert the latest discoveries and methods of professionals into introductory courses. We should not forget that children do not have the experienced mind of professionals, and that instruction is an ontogenetic process.
The use of visual art (through fixed objects or animations) is yet in infancy. A valuable, though superficial, use consists in showing beautiful visualisations. They have the ability to give a kind of physical status to abstract objects, and give them some consistency so that the general public can get a better idea of the matter on which mathematicians are working. They do have a power of attraction due to their originality and strangeness. This in turn can stimulate curiosity, due to the strong aesthetic qualities of these visualizations. This power of attraction, inviting the onlooker to look repeatedly at these representations of mathematical objects, induces a familiarity with the objects, and so may help in the understanding of what lies behind them. They can also help others to understand some the aspects of mathematical beauty championed by many professional mathematicians.
A less trivial use of art consists in systematically looking at the mathematics which have inspired, or which may inspire, the realisation of beautiful real objects - some of them being real works of art. Teaching mathematics through art can be useful both in secondary schools and in schools of plastic or musical art. In this regard, although a little has been done, a huge amount of work is before us.
The speakers at this forum have successfully begun to open some doors. However, in order to get a positive result, there are some essential pre-conditions : an open-minded scientific community, flexible administrative rules, professors dominating all the aspects of their subjects. With these environment, it should be possible to design and create new curricula allowing for the teaching of new, non-traditional traditional, mathematical topics. This could have a major impact on the development of the spirit, and on the acquisition of mathematical knowledge.
We are once more facing the tricky problem of the content of mathematical education : given our aims, what do we have to teach and which programs is it better to set up ? We do not forget that additional difficulties arise from the diversity of the audiences we have to sensitize, inform, and teach.
Art can be used at many different levels. In each case with a common goal of fostering intellectual curiosity in a relaxed and stimulating atmosphere, in parallel with the development of an aesthetic appreciation of beauty.