The need for theories of conceptual learning and teaching of mathematics

One of the important challenges in mathematics education is the development of pedagogical theory that can guide mathematics instruction, instructional design, teacher education, and research in each of these areas. In this chapter, I begin with a brief review of the current state of the field with respect to pedagogical theory. I then offer a vision of what might constitute needed pedagogical theory. Such theory would include useful ways of describing the mechanisms of mathematics conceptual learning and, derived from the characterization of these mechanisms, design and instructional principles for fostering mathematics conceptual learning. Using the research program I am involved in as an example, I describe some of the methodological challenges in producing an empirical basis for such theory development.