In this paper, we explore in more detail why knowing advanced mathematics might be beneficial for teachers, specifically in relation to their classroom practice. Rather than by listing courses or specific advanced topics, as though those were the agents of change, we do so by considering advanced mathematical content for teachers in terms of more general forms of knowledge. In particular, we identify five forms of knowledge of advanced mathematics for teaching: peripheral, evolutionary, axiomatic, logical, and inferential. These categories were derived from analysis of an extensive mapping process linking K-12 content to relevant advanced mathematics. We connect these forms of knowledge to particular practices that teachers engage in so as to clarify the perceived relations to classroom practices. We view such work as important to and productive for teacher educators, particularly in conceptualizing and structuring mathematics courses for teachers so that content that truly informs the work of K-12 teaching can be highlighted, and in a manner that facilitates teachers’ formation of those connections.
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