Mathematics Teachers' Subtle, Complex Disciplinary Knowledge

What mathematical competencies must a teacher have to teach the subject well? This has proven difficult to investigate (1). A current view is that teachers' knowledge of mathematics “remains inert in the classroom unless accompanied by a rich repertoire of mathematical knowledge and skills relating directly to the curriculum, instruction, and student learning” (2). Unfortunately, there is no consensus on which “knowledge and skills” might activate teachers' inert knowledge. Two perspectives prevail, neither with a research base that enables strong claims about practice. The majority of current studies focus on explicit knowledge of curriculum content and instructional strategies. Such knowledge might be assessed directly through observation, interview, or written test (2), with a parallel research emphasis on the formal contents of teacher education programs [e.g., (3)]. A second school of thought, presented here, is that the most important competencies tend to be tacit, like skills involved in playing concert piano, learned but not necessarily available to consciousness.