Que r\'ev\`ele l'activit\'e de validation de d\'emonstration circulaire sur la compr\'ehension de d\'emonstration

Abstract

This communication contributes to research on proof validation as a lens for uncovering didactical phenomena related to proof and proving. We revisit the puzzling case of lower secondary students in France who validate circular proofs. That is proofs in which the conclusion is used within a step of the proof itself. While these 12--13-year-old students accurately identify the final statement of such proofs as a reformulation of the conclusion of the claim, they encounter difficulties in interpreting how the condition of the claim is taken up within the proof. Our analysis challenges Miyazaki and al.'s interpretation of this phenomenon, which attributes students' acceptation of the validity of circular reasoning to a misunderstanding of modus ponens when there are two in a row. Instead, we propose an alternative explanation grounded in the distinction between the operative and theoretical statuses of mathematical propositions. This distinction provides both a theoretical rationale for the invalidity of circular proofs and a didactical tool for examining students' argumentative activity when they attempt to reject such proofs.