Mathematics teachers across the Western world are faced with an expectation that they make significant change to their teaching, but repeated attempts have shown little embedded success. This paper draws on a longitudinal study of two apparently well-placed English mathematics departments attempting to make change aligned with both policy and internationally-valued "good practice." It suggests deep teacher change draws on a wide range of both social and affective characteristics, as well as sophisticated professional skills and knowledge. The study supports a construct of "(mathematics) teacher capacity for change" at both individual and group levels within teachers' "personal domains," synthesising the range of characteristics apparently needed by teachers in times of change. In particular, it argues for the development of dispositions for collaborative learning and of other learning-supportive affects. Such an approach has the potential to place teachers in a better position to respond to demanding aspirations.