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Entre arithmétique et géométrie discrète, une étude épistémologique et didactique du théorème de Bézout et du théorème de Pick

Abstract : This thesis studies the problem of changing registers in mathematics education. More specifically,we have chosen to study the registers of the continuous and the discrete with interactions in thefields of arithmetic and geometry.This thesis shows, in particular, that "classic" adidactic / didactic situations do not allow suchinteractions to be implemented.We have shown, moreover, that there is a pervasiveness of the continuous in the conceptions of thestudents and even a resistance to consider the discreet. Our experiments were carried out withundergraduate mathematics students and trainers.Our first engineering deals with the study of whole points of a line of the plane. It highlighted theobstacle to recognizing a geometric characterization of the solutions of the Bézout equation(existence and exhaustiveness).This shows that in order to overcome this obstacle of changing registers, it is necessary to propose amore “open” type of situation concerning an epistemologically consistent mathematical problem.In this thesis, we studied the possibility of devolving a change in arithmetic / geometry register inthe context of "Research Situation for the Class". This is one of the objectives of our secondengineering covering the area of whole vertex polygons (with reference to Pick's theorem).Two pre-experiments made it possible to define the conditions for taking into account the discreteregister for a question relating to geometry.We have built a final experiment taking these conditions into account.The didactic analysis of the situation on Pick allows us to affirm that, on the one hand, the SiRCmodel is suitable for the engineering of situations of change of registers. On the other hand, it alsoshows that arithmetic and geometry are relevant mathematical domains for register interactions andwork on proof and reasoning.Among the conditions for proper devolution of registry changes, the nature of the question plays anessential role. We chose in engineering on the Pick problem to ask to search for a "method" or"formula" without specifying the variables and registers concerned.Our experience has shown that this type of question has enabled the development of many strategiesidentified in the mathematical analysis of the problem.
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Submitted on : Monday, October 19, 2020 - 4:18:07 PM
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  • HAL Id : tel-02971653, version 1



Sinaly Dissa. Entre arithmétique et géométrie discrète, une étude épistémologique et didactique du théorème de Bézout et du théorème de Pick. Histoire et perspectives sur les mathématiques [math.HO]. Université Grenoble Alpes [2020-..]; Université des Sciences Techniques et Technologiques de Bamako (Mali), 2020. Français. ⟨NNT : 2020GRALM008⟩. ⟨tel-02971653⟩



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