Developing specialised ways of seeing as mathematics teacher educators

This chapter is about the interrelated expertise of mathematics teachers and mathematics teacher educators (MTEs), written by three MTEs who all teach or have taught on the same initial teacher education programme for secondary mathematics teachers in the southwest of England. The chapter tells a ‘then and now’ story focusing on the developing expertise of one of the authors, Tracy, by drawing on data from her PhD studies as a novice MTE (then) and from a recent research project working with three in-service mathematics teachers (now). In the first section, we use extracts from Tracy’s PhD to set the scene of the novice MTE in search of conviction and to introduce some of key ideas informing our conceptualisation of expertise. In section two, we draw on concepts from enactivism, including the link between perception and action and Rosch’s basic-level categories, to set up expertise as specialised ways of seeing linked to actions. In the third section, we present a recent dialogue between Tracy and three in-service mathematics teachers, to demonstrate their interrelated ways of seeing and to show how Tracy’s ways of seeing as an MTE have developed over time. Throughout this story of developing expertise, we focus on the way in which one principle, there is not one way of teaching mathematics, becomes a way of seeing for Tracy as an MTE, supported by the other two authors, Laurinda and Alf, and how this principle is a joint purpose for them in their ways of working with mathematics teachers.