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Teaching is everywhere, yet much of what is involved in teaching remains hidden, with comprehensive theories lacking. These challenges serve as the backdrop for research on mathematics teaching and the work of Thematic Working Group 19. To make progress, the group has used four domains to organize and consider research on teaching. This paper reviews the contributions and issues that arose in the group at CERME13. We elaborate on how the domains stimulated discussions of the meaning of teaching across papers, and we provide reflections and implications for future work. 

Stakeholders agree that the mathematical education of teachers needs to focus on mathematical knowledge for teaching, but the practice-based nature of this knowledge poses challenges for mathematics teacher educators — for understanding it, developing tasks that maintain its integrity in practice, and teaching it to teachers in ways that meaningfully support their learning to teach. We know little, however, about how mathematics teacher educators conceptualize the teaching that knowledge is to support. Our analysis reveals that thinking develops from a view of teaching as straightforward, where aspects can be treated in isolation, to a view of it as requiring focused attention while maintaining mutual regard for the whole. This difference has implications for how mathematics teacher educators understand specialized mathematical knowledge and for how to support their understanding and teaching of it. 

This introduction for TWG19 offers a brief history of the group and describes past challenges the group has experienced when discussing papers — seeing papers as related and as contributing to a common effort. These challenges led us as TWG19 team leaders to develop three initiatives to support communication among researchers who work in different contexts with different purposes. The initiatives are presented and used to discuss the papers. We conclude with implications for the future. 

Researchers conceptualize mathematical knowledge for teaching in different ways, but a coherent approach to the mathematical education of teachers requires teacher educators’ understanding to be robust and shared. At present, we know little about how teacher educators interpret and operationalize this important domain. Our analysis of interview data indicates two sites of divergence in teacher educators’ understanding. Some view this knowledge as a resource for the mathematical work of teaching, treating it as distant from actual practice, whereas others view it as a slice of the dynamic and situational work. Also, some view the mathematical work of teaching, and its knowledge demands, as detached from particulars of students and schooling, while others view this work as inseparable from student identities and the larger environments within which instruction occurs, thus integrating regard for equity. Implications are discussed.