Theorem Justification and Acquisition in Dynamic Geometry: A Case of Proof by Contradiction

Theorem acquisition and deductive proof have always been core elements in the study and teaching of Euclidean geometry. The introduction of dynamic geometry environments,DGE (e.g., Cabri-Géomètre, Geometer's Sketchpad), into classrooms in the past decade has posed a challenge to this praxis. Student scan experiment through different dragging modalities on geometrical objects that they construct, and consequently infer properties(generalities, theorems) about the geometrical artefacts. Because of the inductive nature of the DGE, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical and epistemological concern. In this paper, we will describe and study a ‘Cabri proof by contradiction’ of a theorem on cyclic quadrilaterals given by a pair of 16 year-old students in a Hong Kong secondary school. We will discuss how their construction motivates a visual-cognitive scheme on `seeing' proof in DGE, and how this scheme could fit into the theoretical construct of cognitive unity of theorems proposed by Boero, Garuti and Mariotti(1996). The issue of a cognitive duality and its relation to visualization will be raised and discussed. Finally, we propose a possible perspective to bridge the experimental-theoretical gap in DGE by introducing the idea of a dynamic template as a visualizer to geometrical theorem justification and acquisition.