In this chapter, a critique of direct instruction is followed by a theoretical discussion of constructivism, and by a consideration of what constructivism means to a classroom teacher. A model of instruction is proposed with six components: the promotion of student autonomy, the development of reflective processes, the construction of case histories, the identification and negotiation of tentative solution paths, the retracing and group discussion of the paths, and the adherence to the intent of the materials. Examples of each component are provided.

Excerpt: Recent years have seen two large-scale efforts at improving the curricular goals and pedagogical methods of school mathematics by placing greater emphasis on student experience, on good analytical thinking, and on creativity. The first of these was proclaimed (incorrectly) to have been a failure. Will our present-day sophistication, as represented by today’s constructivist perspective, mean that the second attempt will prove any more successful?

To claim that one’s theory of knowing is true, in the traditional sense of representing a state or feature of an experiencer-independent world, would be perjury for a radical constructivist. One of the central points of the theory is precisely that this kind of “truth,” can never be claimed for the knowledge (or any piece of it) that human reason produces. To mark this radical departure, I have in the last few years taken to calling my orientation a theory of knowing rather than a “theory of knowledge.” One of the consequences of such an appraisal, however, must be that one does not persist in arguing against it as though it were or purported to be a traditional theory of knowledge. Another consequence is that constructivism needs to be radical and must explain that one can, indeed, manage without the traditional notion of Truth. That this task is possible, may become more plausible if I trace the sources of some of the ideas that made the enterprise seem desirable.

Key words: philosophy, radical constructivism, education, experience

Excerpt: What is the best way to characterize the body of knowledge that we call mathematics? How do children and adults learn mathematics most effectively? How can we best study their learning processes, and assess the outcomes of learning? Can meaningful learning be consistently distinguished from nonmeaningful or rote learning? What constitutes effective mathematics teaching, and how can elementary and secondary school teachers be enabled to provide it?

Constructivism is a popular position today not only in mathematics education but in developmental psychology, theories of the family, human sexuality, psychology of gender, and even computer technology. It is also the center of considerable controversy in mathematics education. In a spirit of support for what constructivists are trying to accomplish, I want to discuss some strengths and weaknesses in the position. In particular, I will suggest that constructivism is not a strong epistemological position despite its adherents’ claims. Indeed it might best be offered as a post-epistemological perspective.