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The Historical Roots of the Limit Notion: Cognitive Development and the Development of Representation Registers

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Abstract

The influence upon didactics of visualization and of verbal and symbolic expressions of the main infinitesimal methods and, in particular, the importance of that influence for the correct characterization of concepts are well known. In this paper, different ideas and expressions of infinitesimal methods in history and in mathematics education are investigated, with particular reference to the limit notion. Historical development of representation registers can lead to a parallel development of the notion in students’ minds, and this should make it possible to design new ways to overcome some obstacles and to develop students’ ability to use and to coordinate different registers; however, explaining the problems encountered by mathematicians in history (who inhabited different paradigms, with different social-knowledge structures and different beliefs) does not necessarily help students with their difficulties. My main contribution resides in showing that the dynamic and the static ideas of limit arc encompassed by different semiotic registers.

Sommaire exécutif

L’analyse exhaustive d’un éventuel parallélisme entre l’histoire et la croissance cognitive exigerait une théorie spécifique de la connaissance qui permettrait de comparer la croissance de la connaissance chez les étudiants et le développement historique des concepts. De plus, il serait nécessaire d’ajouter quelques remarques sur l’efficacité et les limites d’un tel parallélisme, liées principalement à des paradigmes différents, associés à des structures sociales de connaissances et de croyances différentes, et caractérisant différentes étapes dans le développement historique des concepts. Notre principal objectif est moins ambitieux: il consiste à montrer que, du point de vue éducatif, les notions dynamiques et statiques de limites, telles que formulées à différents moments de l’histoire, font partie de registres sémiotiques différents.

L’utilisation des registres de représentation comme outil pour analyser les aspects historique et éducatif de la notion de limite est certes une piste intéressante à suivre, mais la question sur laquelle il faut se pencher est celle des liens qui pourraient exister entre les processus phylogénétiques et les processus ontogénétiques.

Le problème du passage du discret au continuum est avant tout culturel, et les questions historiques sont importantes si on veut l’analyser: elles ouvrent la possibilité de mettre au point de nouvelles façons de surmonter certains obstacles et de développer chez les étudiants des habiletés qui leur permettent d’utiliser et de coordonner différents registres. Il peut être efficace de se servir d’exemples tirés de l’histoire des mathématiques pour introduire certains concepts fondamentaux, par exemple les notions statique et dynamique de limite en référence aux registres sémiotiques utilisés; cela permet d’une part de faire une intéressante analyse a priori des difficultés des étudiants, et d’autre part de créer de nouveaux moyens de surmonter les obstacles traditionnels. Cependant, le fait d’expliquer les problèmes délicats qu’ont dû affronter les scientifiques au cours de l’histoire n’aide pas nécessairement les étudiants à surmonter leurs difficultés, car les mathématiciens du passé habitaient simplement des paradigmes différents, avec des croyances et des structures sociales de savoirs différents. De plus, il est important de tenir compte du fait qu’il n’y a pas qu’un seul registre pour un type donné: la nature même d’un registre dépend de la communauté des pratiques dont il est question. Les exemples tirés de l’histoire doivent donc être utilisés de façon contrôlée si on vise le plein apprentissage, par exemple en évaluant les détails empiriques du travail avec les étudiants.

Plusieurs questions demeurent encore ouvertes: que dire, par exemple, de l’importance de lire les sources primaires? Et quel est le rôle des enseignants? Les didacticiens des mathématiques ont-ils la responsabilité de former les enseignants qui devraient être sensibilisés à la question de ce parallélisme? Comment cela contribuerait-il à la formation des enseignants? Des recherches ultérieures pourront se pencher sur ces questions, de façon à déterminer clairement quelles sont les catégories de personnes (étudiants, enseignants, enseignants des mathématiques, chercheurs en didactique des mathématiques) qui devraient tenir compte de la question traitée plus haut dans leur pratique et de quelle façon.

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  1. Department of Mathematics and Computer Science, University of Udine, Italy

    Giorgio T. Bagni

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Bagni, G.T. The Historical Roots of the Limit Notion: Cognitive Development and the Development of Representation Registers. Can J Sci Math Techn 5, 453–468 (2005). https://doi.org/10.1080/14926150509556675

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