Key ideas in a proof: The case of the irrationality of √2

摘要

The notion of key idea is a crucial component of mathematical explanation and a promising tool in the teaching of proof. This study investigates how undergraduate mathematics students interpret the notion of key idea in proof and how they identify key ideas in a particular proof of the irrationality of √2. Drawn from a questionnaire and students' work samples, the findings show that the students varied in their understanding of what “key idea” might mean. These variations stemmed in part from the different interpretations the students gave to the notion of key idea and in part from why a key idea is judged to be one. To help students succeed in proof construction and reconstruction, the findings suggest that mathematics instructors need to actively intervene to draw students' attention to features of proofs that are candidates for key ideas.