In didactics, we usually consider that learning geometry is linked to problem solving, as this makes the knowledge useful. Nevertheless, learning geometry also involves discovering ways of acting, of talking, of interpreting the problems, based on a social construction that is specific to geometry. Then, the question is to understand how these two dimensions of the geometrical activity are intertwined, and how they interact in the knowledge construction process. The role of language during the learning process seems to be crucial for this, so we try in this article to use a methodological tool, so called frequentation modes, to describe a pupil’s activity and its evolution. This aims at combining language analysis to the more usual ones about problem solving, and at studying the interactions between the many faces of the pupils’geometrical activity. At the end, it helps better understanding the mechanism of its evolution, due both to adaptation mechanism and social construct of knowledge.

In this article we examine how it is possible, in the teaching and learning of geometry, to bridge the gap between problems involving drawings and figures, which is essential to the learning of mathematical proof. More precisely, the way students' drawings perception has to evolve, from Iconic Visualization to Non-Iconic Visualization (Duval, 2005). We show that the Instrumental Deconstruction process is multifaceted and central in this evolution. We present a theoretical framework, in relation with an experiment based on a 3D Dynamic Geometry Environment. Based on a case study, we show that construction tasks with specific representations make the instrumental work play a key role in the learning of geometry.

Initial teacher education in geometry at primary school. The context of the research is initial primary school teacher education in geometry. We study and compare three implementations of a collectively designed training sequence by three trainers with distinct status and/or research interest. We highlight some variations in the knowledge that the trainers institutionalize, with a specific focus on figure visualization. The research background concerns the potential effects of those variations on teacher training.Résumé. Le contexte de notre recherche est celui de la formation initiale des professeurs des écoles en géométrie. Nous étudions les variations des savoirs institutionnalisés par trois formateurs aux profils différents, concernant notamment la visualisation en géométrie, sur la base d'une séquence conçue collectivement ; et le rôle de leur discours dans le processus d'institutionnalisation. En arrière-plan de ce travail se trouve la question des effets potentiels de ces variations sur la formation des enseignants.