Research on the mathematics teacher educator (MTE) is growing based on recent publications (Chorney et al., 2025), but as Goos (2025) noted in a commentary paper for the Journal of Mathematics Teachers Education special issue (January 2025) on Researching the expertise of mathematics teacher educators in initial teacher education settings,The apparent complexity of the nature of the MTE knowledge (Chapman, 2021) and the need for broader acceptance by the field of methodologies such as self-based methodologies (Chapman et al., 2020; Suazo-Flores et al., 2025) and more rigorous approaches to these methods could contribute to both the continued underrepresentation of research and the need for further research in this area. It is also important for us to understand the expertise of MTEs in terms of the knowledge and skills that are necessary for exemplary teacher education to produce and support teachers to help their students to develop robust understandings of mathematics and thinking skills to flourish in a rapidly changing, unpredictable world. While we promote the importance of systematic research on MTEs, in Part 3 of this book we take a different approach.