A multi-faceted framework for identifying students’ understanding of the generality requirement of proof

Abstract

The generality requirement, or the requirement that a proof must demonstrate a claim to be true for all cases within its domain, represents one of the most important, yet challenging aspects of proof for students to understand. This article presents a multi-faceted framework for identifying aspects of students' work that have the potential to provide insight into their understanding of the generality requirement. I use a single student's work across different snapshots in time to illustrate her understanding of the generality requirement as evidenced by the justifications, examples, variables, and diagrams present within her written and oral work when proving mathematical claims and her rationales when evaluating whether provided arguments were proofs. Analysis of the student's work across multiple categories at a single moment in time shows that students can exhibit understanding of the generality requirement for some categories while demonstrating limited or unclear evidence of understanding for others. By looking across the framework dimensions and types of tasks, researchers can identify patterns in student understanding and specific aspects that could be addressed through future instruction.