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

TL;DR

This paper investigates how lower secondary students in France validate circular proofs, revealing their understanding and misconceptions, and offers a new explanation based on the distinction between operative and theoretical statuses of propositions.

Contribution

It challenges existing interpretations of students' acceptance of circular proofs and introduces a new framework based on the distinction between operative and theoretical propositions.

Findings

Students recognize the conclusion as a reformulation but struggle with the proof's conditions.

The study refutes Miyazaki et al.'s interpretation of circular proof validation.

A new didactical approach is proposed based on the distinction of proposition statuses.

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.