Abstract
Part of the international reflection on the use of history in mathematics teaching consists in a quest of frameworks and models suitable for empirical studies. Following this demand, this paper explores the way Balacheff’s cKȼ model, a model taken from the didactics of mathematics, can be used in the analysis of learning at student level. In the first part of this paper, Balacheff’s cKȼ model (conceptions, knowledge, ȼoncepts) is shortly presented, and in the second part, the relationship between the epistemological background of the model and the use of the history of mathematics is explored in order to show a possible suitableness. The third part addresses an example of a school activity (about ancient Indian geometry) in which the model is applied and the historical issues clarified. Questioning the role of problems both in the cKȼ model and in the use of history, the last part shows how a study at the students’ conception level enlightens the way in which historical elements can interact with contemporary mathematical learning.
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Notes
“To have the same object with respect to a conception Ca” sets an equivalence relation among conceptions. Let us now claim the existence of a conception Cμ more general than any other conception to which it can be compared; this seems to be an abstract declaration, but pragmatically, it corresponds to a piece of a mathematical theory. (Balacheff, 2013)
“Le rôle que nous donnons à l’histoire des sciences est. également épistémologique et philosophique. L’histoire telle que nous la concevons est. une histoire des idées, des concepts et des modèles, donc à la fois une histoire et une philosophie des sciences telles qu’elles ont été conçues dans la tradition française du XXe siècle, d’Alexandre Koyré à Georges Canguilhem.” (Morange, 2008, pp. 18–19)
“Si l’on veut un carré, une méthode est. de prendre une corde de longueur égale au carré donné, faire des noeuds aux deux extrémités et une marque en son milieu. On trace la ligne et on plante un piquet en son milieu. On fixe les deux noeuds au piquet et on trace un cercle avec la marque. Deux piquets sont plantés aux deux extrémités du diamètre. Un noeud étant fixé à l’est., on trace un cercle avec l’autre; la même chose à l’ouest. Le second diamètre est. obtenu des points d’intersection de ces deux; on plante deux piquets aux deux extrémités du diamètre. Avec deux noeuds fixés à l’est., on trace un cercle avec la marque; on fait la même chose au sud, à l’ouest et au nord. Les points d’intersection donnent le carré.” (Delire, 2016, pp. 75–77 and pp. 217–220)
Online, a film recorded in 1975 by Robert Gardner and J.F. Staal for The Film Study Center at Harvard University (distributed by Documentary Educational Resources) shows an Indian perpetuating the constructions of altars: http://www.der.org/films/altar-of-fire.html.
“un rond normal” she says in French.
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de Vittori, T. Analyzing the use of history in mathematics education: issues and challenges around Balacheff’s cKȼ model. Educ Stud Math 99, 125–136 (2018). https://doi.org/10.1007/s10649-018-9831-6
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DOI: https://doi.org/10.1007/s10649-018-9831-6