Abstract
Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to do mathematical problems and to think about mathematical relationships. In this article we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra, and calculus, then on to the formalism of axiomatic mathematics. This evidence is taken from a number of research studies recently conducted for doctoral dissertations at the University of Warwick by students from the United States, Malaysia, Cyprus, and Brazil, with data collected in the United States, Malaysia, and the United Kingdom. All the studies form part of a broad investigation into why some students succeed, while others fail.
Résumé
Les symboles jouent un rôle central entre les processus à accomplir et les concepts auxquels on doit penser. Ils nous permettent à la fois de résoudre des problèmes mathématiques et de réfléchir aux relations mathématiques. Dans cet article, nous examinons les discontinuités qui apparaissent le long du chemin d’apprentissage emprunté par différents élèves et qui entraînent une divergence entre la pensée conceptuelle et la pensée procédurale. Nous fournirons des exemples tirés de plusieurs contextes différents illustrant l’acquisition des symboles par l’intermédiaire de l’arithmétique, de l’algèbre et du calcul différentiel et intégral, pour ensuite passer au formalisme des mathématiques axiomatiques. Ces exemples proviennent d’un certain nombre d’études menées récemment à l’Université de Warwick par des doctorants américains, malaisiens, chypriotes et brésiliens, et qui portent sur des données recueillies aux États‐Unis, en Malaisie et au Royaume‐Uni. Toutes ces études s’inscrivent dans le cadre d’une vaste recherche visant à déterminer les raisons pour lesquelles certains élèves réussissent alors que d’autres échouent.
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Tall, D., Gray, E., Ali, M.B. et al. Symbols and the Bifurcation Between Procedural and Conceptual Thinking. Can J Sci Math Techn 1, 81–104 (2001). https://doi.org/10.1080/14926150109556452
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DOI: https://doi.org/10.1080/14926150109556452