The most influential and widely accepted philosophical perspective in mathematics education today is constructivism. This view, which holds that individuals construct their own knowledge, can be traced back to Piaget and beyond. While it takes many forms, at its simplest, it sees the learner as an active participant, not as a blank slate upon which we write or as an empty vessel which we fill. In this view, cognition is considered adaptive, in the sense that it tends to organize experiences so they "fit" with a person's previously constructed knowledge. As a consequence, both researchers and teachers ask, "What is going on in students' minds when . . . ?", rather than speaking of behavioral outcomes and asking, "Which stimulus will elicit a desired response?"

The term "constructivism" often designates this view of how people learn, and constructivist teaching often simply means taking students' views and background into account so as to engender active, meaningful learning. However, constructivism comes in a variety of "flavors." There is a "moderate" version, compatible with the way most mathematicians see mathematics, and a social constructivist version, inspired by the work of Vygotsky, which takes into account sociocultural perspectives. There is the radical constructivism of von Glasersfeld, and beyond that, the sociology of scientific knowledge (SSK), which replaces the idea of truth with that of utility. We will describe these views and place them along a (increasingly radical) continuum from the above "plain vanilla" version, which almost everyone in mathematics education today accepts and tries to act on, through the radical view, to the very relativistic SSK.

Adopting the "plain vanilla" view that students construct (or reconstruct) knowledge for themselves does not prescribe a single "constructivist way of teaching." It does, however, suggest that lecturing is likely to be less effective, than more active approaches such as cooperative group learning, and as a result, students are likely to make weak constructions. Ways of teaching that require students, not only to be more active, but to reflect on their work, are likely to encourage them to make strong constructions that result in increased conceptual knowledge and more connections. We will indicate some general principles this constructivist view entails.

John Selden, MERC, Box 2781, Cookeville, TN 38502
Annie Selden, Tennessee Technological University, Cookeville, TN 38505

As a supplement to this presentation, see also "Kinds of Constructivism: An Annotated Bibliography."