Identifying discursive entry points in paired-novice discourse as a first step in penetrating the paradox of learning mathematical proof

• This study uses a commognitive research framework to examine the learning of mathematical proof.
• It presents a detailed analysis of paired discourse by undergraduate mathematics majors working on an introductory proof task.
• The findings point to places in the discourse, named discursive entry points, which hold significant potential for expert intervention.
• Identifying discursive entry points may be valuable in helping students to overcome the learning paradox related to mathematical proof.

In this article, I demonstrate the efficacy of using the commognitive research framework to study the learning of mathematical proof. To this end, I present a detailed analysis of an excerpt of paired discourse by undergraduate mathematics majors working on an introductory proof task. Additionally, I draw on the excerpt to illustrate that there exists in discourse between novice interlocutors natural opportunities, especially ripe with potential, in which experts could intervene to steer the discourse toward increasing mathematical sophistication. I have named these opportunities discursive entry points and suggest their identification as significant in addressing the learning paradox as it relates to mathematical proof. Overall, the findings illustrate how rich description of the nature of learning in paired discourse, a result of the commognitive research lens, can inform reform era instructional practices.