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Working as mathematics teacher educators at the meta-level (to the focus of the teachers on developing their teaching).

In this chapter, we reflect on our learning as university mathematics teacher educators (MTEs) from our different perspectives, expert through to novice. We elaborate on some of the principles behind the design of the teacher education course at the University of Bristol, illustrated through four stories. The course design makes explicit use of a reflecting cycle, from: describing experiences; to, identifying issues; to, committing to next actions. This cycle informs all of our teaching sessions at university and parallels how we learn as MTEs. One commonality, across our own learning and what we offer our prospective teachers, is the importance of an explicit metacommenting about awarenesses of mathematics and awarenesses of teaching mathematics. A further commonality is the importance of a "second person" perspective, or empathic "other" in the learning process. The prospective teachers are learning about mathematics and its teaching and we, as MTEs, are learning about them.

This chapter explores different ways in which prospective teachers are considered to be active constructors of their knowledge embedded in a variety of social environments. A dialogue between two theoretical perspectives, enactivism and professional noticing, is used to try to understand prospective teachers' learning about teaching mathematics and, in so doing, illustrates the learning of university mathematics teacher educators. Prospective teachers' learning (discourse and actions) and different types of practice from both theoretical perspectives are linked considering different forms of attention and labelling.

The world is in a period of rapid change. Climate chaos is leading to floods, droughts and exacerbating inequality. In this article, we ask, in such a context, what is the role of mathematics education and, in particular, what can mathematics teacher educators (MTEs) do in supporting teachers to respond to global challenges, and the associated injustices, within their mathematics classrooms? We review work done within critical mathematics education and develop a model for reflection on the role of the MTE. We report on work taking place in one university in the UK where, since 2015, MTEs have been encouraging teachers to address global issues in their classrooms. One role which has emerged for an MTE is in supporting the translation of scientific work on climate change into classroom tasks. In England, the spaces for such work are being squeezed in an education system increasingly subject to conservative political forces. We conclude the article by considering possibilities for MTEs to provoke a generic questioning of assumptions, which we argue is possible even within such constraints.