Entering a Practice of Mathematical Proof: On the Difficulty of Teaching Proofs

The concept of mathematical proof has received much attention from philosophers as well as from mathematics education researchers. There is widespread agreement on the importance of including proofs at all levels of mathematics education. This consensus is reflected in many national curriculum documents. However, a common experience among educators is that proofs are difficult for students to understand, let alone produce. In response to these problems, research on the teaching of proofs has focused on facilitating students’ understanding of proofs by, inter alia, distinguishing different roles that proofs may play in mathematics. The identification of the different roles of proofs has given educators a valuable tool when selecting proofs to include in their teaching. It has been suggested that proofs that have an explanatory role should be given priority over proofs that serve mainly to verify theorems. It has also been recommended that educators should focus on informal proofs rather than on formal proofs. The aim of this paper is to argue that research on the teaching of proofs can benefit from Wittgenstein’s perspective on mathematical practice and context sensitivity. Although much research on the teaching of mathematical proofs already incorporates a focus on mathematical practice, it is argued here that Wittgenstein’s remarks on mathematical proofs can provide new insights into the issue of students’ struggle with proofs. This is accomplished by viewing proving as a practice. To enter this practice involves becoming familiar with the different roles of proofs. The struggle of students is here not viewed as a struggle with particular roles of proofs but as an expression of the struggle to enter a new practice that includes learning a mode of verifying a proposition that is unique to this practice.