THE READER has already met a variety of learning situations drawn together by a common set of ideas about what makes for effective learning. In this chapter we turn directly to these ideas and to the theoretical sources by which they are informed. Of these we focus on two: first, the Piagetian influence, and second, the influence of computational theory and artificial intelligence.
I have previously spoken of "Piagetian learning," the natural, spontaneous learning of people in interaction with their environment, and contrasted it with the curriculum-driven learning characteristic of traditional schools. But Piaget's contribution to my work has been much deeper, more theoretical and philosophical. In this chapter I will present a Piaget very different from the one most people have come to expect. There will be no talk of stages, no emphasis on what children at certain ages can or cannot learn to do. Rather I shall be concerned with Piaget the epistemologist, as his ideas have contributed toward the knowledge-based theory of learning that I have been describing, a theory that does not divorce the study of how mathematics is learned from the study of mathematics itself.
I think these epistemological aspects of Piaget's thought have been underplayed because up until now they offered no possibilities for action in the world of traditional education. But in a computerrich educational environment, the educational environment of the