ABSTRACT
In mathematics education, there must be a theory of how people learn and it must be constructed in conjunction with the practice of regular classroom activity. This is the first dichotomy. This chapter considers two additional dichotomies: research and development, and beliefs and choices. It considers the content of a course and shows how people learn that content, or the instructional methods that people use in the classroom, and which teaching methods one should use. The chapter reviews how the learning process works. In the case of research and development, it is a mistake to treat them separately, and that mathematics educators should form syntheses. All people make choices about what they do in a classroom. Whether they are aware of it or not, these choices are to some extent determined by the beliefs that are part of their conception of how people learn mathematics and what mathematics is.
