| **ABSTRACT:**This article considers how we learn to become mathematicians and how we learn to become teachers of mathematics. There is an initial consideration of what makes mathematics a unique domain: it is the nature of mathematical proof, as explanation, that makes the doing of mathematics different to the doings in other domains. Becoming a mathematician therefore means learning an emotional orientation towards finding proofs convincing and explanatory. For students to learn such an orientation, the article proposes a mode of “proof-based teaching” in which only mathematical proofs are offered and accepted as explanations. Learning to teach in a proof-based manner implies developing an orientation towards proof, as a teacher. There is some evidence that, for teachers, engaging in classroom activities around proving is generative in terms of an orientation towards doing more proof-based teaching. | |