Récréations mathématiques, géométrie de situation... De nouveaux outils pour enseigner les mathématiques à la fin du XIXe siècle
Résumé
At the end of the XIXth century, some mathematicians like Charles-Ange Laisant (1841-1920) or Édouard Lucas (1842-1891) develop new ideas about teaching mathematics using innovative tools or original visualizations. These ideas are linked to their interest for recreational games viewed through a mathematical background and other specific problems including combinatory, number theory, algorithms and the “géométrie de situation”... They are studied especially in L'Arithmétique amusante (Lucas, 1895) or the Initiation mathématique (Laisant, 1906), two books addressed to educators in a large sense.
We propose to give some examples of these mathematical representations and these ingenious games, explicitly presented by their authors in a didactical scheme. Thus, we talk in particular about :
- “l'éventail mystérieux” proposed in order to introduce binary computation
- “le vol des grues” and other drawings proposed in order to give visual proofs of particular sums of integers
- a singular representation of decomposition of integers into prime numbers using paths in specific drawings
- graph of functions, especially linear ones, where pupils can see solutions of meetings problems
- the use of chessboards in combinatory and so on...
Moreover, several of these approaches could be presented nowadays to students of secondary schools. We discuss the role of visualizations through these experimentations and point out the connections between diagrams and symbolisms. We study the need of arousing curiosity through striking results in order to present new mathematical notions. We also underline the goals of representing processes in certain moments of mathematical learning and the generalizations enabled by many of these presentations, much more than simple recreational questions.