Mathematical microworlds and intelligent computer-assisted instruction

My work with microworlds has been to design computerized environments that allow two foci: conceptual development and mathematical problem solving. The theory behind that approach has been elaborated elsewhere (Thompson, 1985a). In this chapter I will only briefly touch upon theoretical motivations, devoting the majority of the discussion to what I mean by a mathematical microworld, how one works, and to issues of designing microworld environments. The discussion in this chapter is framed by the research program in which I have been engaged over the past four years. Figure 5.1 outlines the principal components of that program. It shows that remarks given here about the design of software to be used in mathematics teaching and learning are not given in isolation. Rather, they are informed by results, conceptions, and metaphors from investigations of cognitive processes of mathematical comprehension and problem solving, prescriptions for cognitive objectives of instruction, and analyses of mathematical content (Dreyfus & Thompson, 1985; Thompson, 1985a). Each of these, in turn, is informed by knowledge gained through research and development of software for teaching and learning mathematics (Thompson, 1985b, 198Sc; Thompson & Dreyfus, in press). I do not mean to say that one must accept the research program outlined in Fig. 5.1 to design mathematical microworlds. Rather, I mean only to say that issues of design are at heart theoretical, and as an aid to communication it helps to make explicit one's theoretical perspective from the outset.