Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs

• Used a sociocultural perspective to examine the role of transactive reasoning in whole-class discourse on proof.
• Examined shifts in students’ knowledge about proofs as seen in their proof schemes and use of strategic knowledge.
• Examined how transactive reasoning supports negotiation of ideas in the construction of proofs.
• Classroom discourse that fosters students’ transactive reasoning supports their capacity to build mathematical arguments.

This study uses a sociocultural perspective to examine the role of transactive reasoning in, whole-class discourse as undergraduate students learn to construct mathematical proofs. The research, setting is an undergraduate mathematics course with 30 participants. Data are whole-class transcripts, of lessons focused on developing mathematical proofs and students’ written assessments on proofs. Transcript data are analyzed for (1) shifts in students’ knowledge about proofs; (2) the nature of, transactive reasoning (Berkowitz, Gibbs, & Broughton, 1980) in whole class discourse, including how it occurred and, indications that students appropriated transactive reasoning as a practice of discourse; and (3) how, transactive reasoning supported students’ active constructions of proofs and understanding of proof. Results indicate that classroom discourse that helps students appropriate transactive reasoning as a habit of interaction supports their capacity to build arguments about increasingly complex, mathematical ideas and, as such, has positive implications for their learning of proof.