From Pseudo-Objects in Dynamic Explorations to Proof by Contradiction

Proof by contradiction presents various difficulties for students relating especially to the formulation and interpretation of a negation, the managing of impossible mathematical objects, and the acceptability of the validity of the statement once a contradiction has been reached from its negation. This article discusses how a Dynamic Geometry Environment (DGE) can contribute to students’ argumentation processes when trying to explain contradictions. Four cases are presented and analysed, involving students from high school, as well as undergraduate and graduate students. The approach of the analyses makes use of a symbolic logical chain and the notion of pseudo-object. Such analyses lead to a hypothesis, that experiencing a pseudo-object during an exploration can foster DGE-supported processes of argumentation culminating in geometrical proofs by contradiction, while the lack of experience of a pseudo-object may hinder such processes. If this hypothesis is confirmed by further studies, we foresee important didactical implications since it sheds light on the transition from students’ DGE-based argumentations to proofs by contradiction.