Utilisation de la géométrie dynamique avec de futurs enseignants de mathématiques au secondaire pour repenser le développement du raisonnement

Technological tools are ubiquitous in society and their use is growing in the classroom. In mathematics education, these tools have been assigned roles that have evolved over the years: support, visualization, content enrichment. The use of technological tools in education comes with potential pragmatic and epistemic contributions, but also has limitations and risks. We must therefore examine at the activity level the role technology should play. Mathematical reasoning is one of the competencies aimed by school (MELS, 2006) and future teachers seem to place less emphasis on validation and proving processes as components of this reasoning (Mary, 1999). We hypothesize that a sequence of activities showing the complementarity of the proving processes with explorations leveraging technology could help future teachers better understand these issues.   This research is based on didactical engineering to develop and validate a sequence of activities with GeoGebra software. The sequence of activities has been designed to: introduce pre-service secondary mathematics teachers to dynamic geometry software and give them the opportunity to see mathematical activities using technology that aim at developing mathematical reasoning and proof. The theoretical framework on which this research is based integrates elements of the anthropological theory of the didactic (Chevallard, 1992, 1998, 2003) and of the instrumental approach (Vérillon and Rabardel, 1995; Trouche, 2000, 2003, 2007; Guin and Trouche, 2002). Some ideas on robust and soft constructions (Soury-Lavergne, 2011), the distinction between figure and drawing (Laborde and Capponi, 1994) and the deductive network (Tanguay, 2006) served as benchmarks in the construction of the sequence of activities.   This research took place at a Quebec university during the winter of 2016, in a geometry didactics course for pre-service secondary mathematics teachers. A preliminary questionnaire was given to the participants to capture their prior knowledge of programs, technological tools and conceptions about mathematics teaching and learning. Subsequently, the students experienced the sequence of activities and had to decide on the knowledge involved in each activity, the relevance of its use with high school students, and the perceived adaptations necessary for its realization (if considered). Written traces of their work have been kept as well as a diary as the sequence unfolds.   By triangulating the various data collected, it was found that the sequence, while contributing to the instrumentation of the participants with regard to the software used, had, for some of them, an impact on their vision of the development of mathematical reasoning in mathematics education at secondary level. The analysis of the data highlighted the place given to the reasoning by the future teachers, the reasonings mobilized by the students in the various activities and also signs of the instrumental geneses inducing these reasonings and accompanying them. Following the analysis of these data and the findings that follow, modifications are proposed to improve the sequence of activities.

Keyphrases: GeoGebra, Dynamic geometry software, Reasoning, Levels of proof, Deductive network, Proof, Instrumental approach, Anthropological approach