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History and Epistemology in mathematical Education From
play school to university
Proceedings of the Third European Summer University held in Louvain-la-Neuve
(15/07/1999-18/07/1999) and Leuven (18/07/1999-21/07/1999).
In 1999 the Third European Summer University took place in Louvain-la-Neuve and in Leuven (Belgium), after being held for the first time in Montpellier (France, 1993) and the second time in Braga (Portugal, 1996).
Since the very beginning the European Summer University turned out to be a highly qualified place for meeting, comparison, study and international debate on the history and epistemology in mathematical education. The Proceedings, which have been quite rightly published in memory of unforgettable John Fauvel, lately deceased, are gathered in two volumes of about 950 pages altogether.
The first volume contains the texts of
In reading the Proceedings, what immediately strikes the reader
is the extraordinary wealth and variety of themes, together with the profundity
of analysis and the topicality of didactic suggestions which seem to point to
some sort of universal, cross-cultural difficulties met by teachers from different
countries and cultures.
It is not by sheer chance that introducing his plenary lecture Can mathematics education learn from history?, John Fauvel wrote: "I’m drawing today particularly on the British experience because that’s what I know best, but I would like to think any ideas and insights we may be able to glean from the situation are transferable elsewhere and make sense in other national contexts too"; whereas later on, he ironically urges the teachers to take on an almost political view of their being teachers: "One thing we can learn from international gatherings, such as the European Summer University, is how we foot-soldiers in the educational trenches can learn to control our officer class, the politicians, better"; finally, he passes on to examine rapidly what contributions England got from other countries, such as Danes, French, Italians, Dutch, Portuguese, Swiss, Belgians, and what, in turn, she can contribute to them.
This give-and-take is indeed the fil rouge, the thin line running
through and enlivening the European Summer University, and the Proceedings
themselves.
Another feature worth noting is the fact that they all harmonize on the concepts
of "word" and "vision". For instance, Evelyn Barbin states
she intends "d’examiner, a partir de texte historiques, quelques visuels
du texte mathématiques". In fact, "Un texte mathématique
est un texte qui se lit, où se tient un discours, mais le texte mathématique
est aussi un texte qui se regarde, où des traces sur le papier demandent
et font compréhension, qu’il s’agissent de figures géométriques,
de lettres ou de symboles".
Therefore in order to understand it is essential to see, then to touch, to experience, to travel, to know as Marjolein Kool writes: "During the Middles Ages merchants went along the houses with their basket with goods….They needed to change money in many different ways, because each city had its own money system…. The job of the merchant and his calculations became more and more complicated…Many arithmetic books were written in the vernacular…" Here is an excellent example of investigation into a historical setting, which though pertaining to Holland, can be quite similar to other European countries in the Middle Ages; in short, it can provide useful ideas to integrate into a serious teaching proposal.
Nicolas Rouche resumes the geometrical discourse by pointing out how "les
déterminismes géométriques reconnus dans l’action
(par exemple le dessin aux instruments) fournissent des inférences, dont
les premières sont évidentes, relèvent de "la lumière
naturelle". Et celles-ci ensuite se combinent en inférences non
évidentes, en preuves discoursives."
Finally, Constantinos Tzanakis, using an effective, chiastic play on words
- Mathematical Physics vs Physical Mathematics – describes a few excellent
and very deep concepts in which he analyses the subtle and seminal links between
mathematics and physics. Moreover, he offers a concrete and very detailed teaching
experience, the Bernoulli’s ‘brachistochrone problem’
Some of the themes dealt with in the plenary lectures appear to be tackled
by other authors from different standpoints in the numerous lectures that occupy
most of vol. 1. Generally speaking, geometry seems to prevail in constant, lucky
revival, as compared to the recent past; other topics are present such as mathematical
analysis, applied mathematics to physics and economics, chaos theory, stochastic
processes and arithmetic, the Queen of mathematics, according to Gauss. Also
in this case the reader will realise how often, more or less overtly, stress
is laid on the language – reporting and dramatising, for instance –
being of vital importance to mathematics epistemology and education.
The second volume is entirely devoted to the workshops, and consequently it
appears to be more didactic in its connotation. A few works in this volume are
particularly rich in annexes, which is a feature worth noting. Also in this
book there is a wide range of topics and teachers will find a mine of ready-to-use
information, suggestions and useful ideas for classroom teaching.
The proceedings of the third European Summer University
held in Louvain-La-Neuve, Leuven (1999) have been just published. They cost
30 Euros.
Enquiries to
radelet@fyma.ucl.ac.be
(Patricia Radelet de Grave)
Edition Bernoulli, Université de Louvain
2 chemin du Cyclotron,
B-1348 Louvain-la-Neuve, Belgique
In conclusion, the Proceedings of the Third European Summer
University are no doubt a very useful, almost indispensable tool for anyone
willing to follow the international debate on the themes of epistemology and
history of mathematics, in a teaching perspective. At any rate, the two volumes,
which are also rich in lovely pictures, are excellent and interesting to read
and offer opportunity to stimulate useful discussion among teachers.
Contact addresses
P. Radelet, Institut de physique théorique (FYMA), Université
catholique de Louvain, chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgique
Radelet@fyma.ucl.ac.be
D. Janssens, Acad. Lerarenopleiding Wiskunde, Katholieke Universiteit Leuven,
Celestijnenlaan 200B, B-3001 Heverlee, België
Dirk.Janssens@wis.keuleuven.ac.be
Giuliano Testa
Vicenza - Italy
Gerdes, P., Geometry from Africa: Mathematical and educational explorations
The Mathematical Association of America, 1999, Pp 210 (1999) ISBN 0-88385-715-4
Reviewed by John D. Barrow (Director Millennium Mathematics Project,
Cambridge University) for PLUS Magazine (June 2001).
This beautifully illustrated book by the world's leading authority on African
mathematics provides us with a wide-ranging introduction to mathematical intuition
in sub-Saharan African cultures. The full review can be found at
http://plus.maths.org/issue15/reviews/book2/
Paulus Gerdes
Maputo, Mozambique
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