Integrating Euclidean rationality of proving with a dynamic approach to validation of statements: The role of continuity of transformations

International audienceDuring a long-term teaching experiment aimed at developing 10th grade students’ culture of theorems through a pathway in Euclidean plane geometry, some students’ autonomous reasoning moved towards non-Euclidean proofs based on continuity of transformation of geometric figures. Based on the use of existing analytical tools to analyze such episodes, the aim of this paper is to outline a wider scope for synthetic geometry in order to make it more suitable for students’ approach to the culture of theorems. Through the introduction of a continuity principle to legitimate such extension, the paper suggests how to exploit students’ potential in transformational reasoning, and to bridge the gap between synthetic geometry and analytic geometry rationalities in classroom work