FORUM
A SHORT
PRESENTATION[1]
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[2][1] 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.