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The accepted framing of mathematics pedagogical content knowledge (PCK) as part of mathematical knowledge for teaching has centered on the question: What mathematical reasoning, insight, understanding, and skills are required for a person to teach elementary mathematics? Many have worked to address this question in K-8 teaching. Yet, there remains a call for examples and theory in the context of teachers with greater mathematical preparation and older students with varied and complex experiences in learning mathematics. In this theory development report we offer background and examples for an extended model of PCK – as the interplay among conceptually-rich mathematical understandings, experience in and of teaching, and multiple culturally-mediated classroom interactions.

The paper presents the guiding ideas behind our culturally responsive approach to teacher professional development and an overview of how those tenets inform, tacitly and directly, our efforts to realize the promise of the National Council of Teachers of Mathematics' five Process Standards. A review of the primary obstacles teachers face in implementing these standards in their own teaching and learning is followed by a description of the design elements in a university-based professional development program. Our goal is to provide an example of the foundations upon which an evolving approach to culturally responsive professional development planning has grown. We discuss research on what constitutes effective teacher professional development while noting the paucity of programs that embrace recognized needs. We do not give a prescription for effective teacher development. Instead, we speak as teacher-educators about the necessary philosophical and self-evaluative underpinnings to effective professional development and our approach to creating an environment where it is safe to leave the isolation of forced autonomy and it is valued to be reflective about community, mathematical activity, and intellectual engagement.

The purpose of this article is twofold: to give a sense of the current lay of the land in the preparation of mathematics graduate student teaching assistants (TAs) and to describe the collegiate mathematics education research base informing the next generation of college mathematics instructor preparation. We anchor discussion in three common types of TA preparation programs, each represented in one of the quotes above. Notably, the first quote represents a sink or swim experience that is becoming rare in US PhD-granting mathematics departments. Preparation of TAs for their instructional roles has blossomed in the last twenty years. The second quote is representative of current practices in many departments. The third quote illustrates the activities in innovative departments that are already implementing best practices suggested by research in collegiate mathematics education: sustained professional growth about teaching and learning.To highlight the challenges and benefits of spending time paying attention to teaching, we provide information from postsecondary and related secondary-level educational research of several types. This includes basic and applied educational research that identifies good instructional practices, examines experiences TAs bring with them to teaching, provides frameworks for the structuring of TA preparation, and gives insight into the kinds of mathematics-specific and teaching-specific knowledge that needs to be developed among TAs. And, once a program for supporting TAs to learn and grow as instructors is put in place, evaluation research explores the implementation of efforts to improve TAs' teaching and the related impacts on undergraduate student learning. We close with promising practices and sketch anticipations for the future of the field of related research.Though not common thirty years ago, today most doctorate-and master's-granting institutions provide some kind of TA preparation for teaching [4a]. The content for this professional development often comes from mathematicians offering their collective wisdom from practical