Forms of Proof and Proving in the Classroom

This chapter discusses forms of proof and proving in the learning and teaching of mathematics, including different representations used in proof production, different ways of arguing mathematically, different degrees of rigour in proving, and multiple proofs of the same statement. First, we focus on external forms of proof. We report research on students’ and teachers’ beliefs about visual aspects of proving and discuss the importance of visibility and transparency in mathematical arguments, particularly those using visualisation. We highlight the pedagogical potential of proving activities involving visualisation and reflect on its limitations. Next, we discuss the importance of various mathematical, pedagogical, and cognitive aspects of different forms of proof in multiple-proof tasks. We then examine which forms of proof might support students’ transition from empirical arguments to general proofs, using examples from the history of mathematics and discussing the roles of operative and generic proofs. We conclude by indicating potential future research agendas.