### Book: De grands défis mathématiques: d’Euclide à Condorcet

## De grands défis mathématiques: d’Euclide à Condorcet

[On major challenges of mathematics: from Euclid to Condorcet]

Evelyne Barbin (ed.), Paris: Vuibert; Adapt-Snes, 2010, 176pp, ISBN 9782356560100;9782311000191

The output of IREM colleagues is admirable and enviable. This latest volume, under the editorship of Evelyne Barbin, provides nine incidents or problems that stimulated mathematical responses. The purpose of the descriptions here is to provide original material that can be used in the classroom of upper secondary schools, that is, in those years immediately preceding university entrance. The claim is slightly misleading in that the article by Evelyne Barbin describes work with third year university students preparing to teach and the historical time frame of the articles extends to before Euclid and after Condorcet. The work is divided into four sections: measurement of magnitudes, representing magnitudes, probability, approximations to curves.

However entertaining and stimulating, I often feel that French studies linking history to pedagogy would not appeal to the Anglo-Saxon reader and I even wonder how much they might be taken up in the French classroom, save for the enthusiasts who have written these pieces. It is therefore encouraging to find, alongside suggested exercises based on historical texts, or at least inspired by them, some illustrations of work done by students themselves. Thus Dominique Tournès from Réunion has examples of students’ constructions of solution curves of differential equations, using Euler’s method, which they did following a guided reading of an extract of *Institutionum calculi integralis* (1768). From Patrick Guyot of Bourgogne we have an account of students working on the problem of inscribing a square inside a triangle which includes images of the students’ initial naïve attempts. Indeed, it is clear that in almost all cases the writers are telling us of material they themselves have used.

But this collection of articles is more than a description along the lines of a simple ‘these have worked for us’. Each piece comes with an introduction of the mathematical, historical and educational context and some offer further reflections and all have source bibliographies. There are some nice discoveries for me. Leibniz does not appear among the founders of probability calculus but Renaud Chorlay has used a letter from him describing the expected outcomes of a simple game of dice which is easy to read and has the pedagogic advantage that his assumptions are wrong. He fails to count both (3,2) and (2,3) as distinct ways of obtaining 5, something which students can easily correct. Gérard Harmon treats early approaches to Bayes’ Theorem by Condorcet (1805) and then Lacroix (1816) both considering possible outcomes of taking black or white balls from an urn. I also enjoyed reading the background to modern digital type fonts and the use of Bézier curves which extended Loïc Le Corre’s lesson on Dürer’s geometric representation of fonts.

The educational context – defending the choice of material – makes us aware of French concerns that would not trouble the British secondary mathematics teacher, more accustomed to a pragmatic sloppiness of approach to teaching mathematics, particularly with regard to notation. *Vive la différence!*

**Chris Weeks,**

Great Britain

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