Two Theories of "Theory" in Mathematics Education: Using Kuhn and Lakatos to Examine Four Foundational Issues.

Meaningful inquiry is always guided by a theory. The theory may be a refined, highly predictive calculus, as it is in physics, or it may be a rough, tentative collection of hunches, as it often is in education. When a mathematics educator studies the effects of lax and restrictive learning environments on children of different anxiety levels, she presumably has a theory that relates achievement to both anxiety and the structure of the learning environment. Or when a cognitive psychologist examines classification and sedation tasks in the learning of early number concepts, the psychologist most likely has a hunch as to how these tasks are related. Or, when a doctoral candidate designs an experiment in which children are taught several different problem solving heuristics, she presumably has a theory that predicts which of these treatments will be the most effective.