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In this paper, we analyze an investigative situation proposed to a class of 5th graders in a primary school. The situation is based on the following task: In a sale with group rates on a sliding scale, the students must find the lowest possible purchase price for a given number of tickets. A study of students' arguments made it possible to identify a large number of rhetorical forms. However, it turned out that one of the intrinsic features of the situation restricted the teacher's possibilities of making didactical use of the students' forms of reasoning and led him to try to support students' learning with "didactical reasons" rather than with "reasons for knowing".RÉSUMÉ. L'article analyse une situation de recherche proposée dans une classe de 5 ième année de primaire. Dans une vente par lotsà tarif dégressif, lesélèves doivent minimiser le prix d'achat pour une quantité donnée. L'étude des arguments des uns et des autres fait apparaître de nombreuses formes rhétoriques, mais une propriété intrinsèque de la situation va limiter les possibilités du professeur dans l'utilisation didactique des raisonnements deś elèves et va dissocier les raisons de savoir et les raisons didactiques utilisées.

MISE EN OEUVRE D'UN MODÈLE D'ANALYSE DES RAISONNEMENTS EN CLASSE DE MATHÉMATIQUES À L'ÉCOLE PRIMAIRE Cet article vise à établir la pertinence d'un modèle destiné à analyser les raisonnements produits dans des situations didactiques comportant une dimension recherche. Notre souhait est, d'une part de décrire précisément le modèle, afin de pouvoir mettre en évidence les éléments qui structurent l'analyse des situations ; d'autre part, de montrer qu'il est aussi utilisable dans des analyses de corpus, même dans un contexte éloigné de celui qui nous a servi initialement d'appui pour sa conception. Le modèle, élaboré à partir de la théorie des situations didactiques et de la sémiotique de C.S. Peirce, offre la possibilité d'analyser les raisonnements produits par les élèves et par l'enseignant dans des situations d'action, de formulation mais aussi en situation de validation. Nous l'utiliserons dans le cadre de l'étude approfondie d'une séquence, élaborée par G. Brousseau, visant à favoriser la pratique du raisonnement et à faire découvrir les règles du jeu de la preuve à des élèves de CM2. This paper aims at establishing relevance of a model to analyse reasoning processes in problem solving situations. We would like to describe our model of analysis on one hand in order to highlight the main elements which structure the analysis, on the other hand to show his interest to analyse lesson, even in a context further from this one used during his conception. This model, made mainly within the frameworks of the theory of didactical situations and Pierce' s semiotic theory, allows to characterize reasoning processes, elaborated by the pupils and the teacher, during situation of action, situation of formulation and also in situation of validation. We will exemplify this model to study reasoning processes elaborated during a lesson, produced by Brousseau, aiming at favouring reasoning process and at revealing the rules of the games of the proof to pupils, in primary school (pupils aged 10 to 11).

Our research focuses on the difficulties students encounter with the learning of calculus, considering that they have to cope with many more mathematical objects but also with new ways of reasoning – not only algebraic calculation, but also the practice of approximation, and a scaffolding way of using functions, limits, derivative, integrals, etc. to justify their answers. The semiotic facet of new objects, and the way to manage it, is also a source of great difficulties. In this article we establish that the model we built (Bloch & Gibel, 2011) is adequate to describe the work of University students who have to deal with the resolution of exercises about parametric curves and differential equations, even if this context is not an adidactical situation. In 2018, L2 students of Pau University were asked to solve little problems about limits, integral calculations or recurrence questions. They revealed difficulties to organize their knowledge and conclude about a limit, for instance. We give some examples of these troubles. We conclude for the necessity to implement adequate devices to help students better understand these 'new mathematics'. Calculus, students' understanding of mathematical signs and objects, reasoning processes, parametric curves, differential equations

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