Our overall objective in this paper is to share a few observations made and insights gained while conducting a recently completed teaching experiment. The experiment had a strong pragmatic emphasis in that we were responsible for the mathematics instruction of a second grade class (7 year-olds) for the entire school year. Thus, we had to accommodate a variety of institutionalized constraints. As an example, we agreed to address all of the school corporation’s objectives for second grade mathematics instruction. In addition, we were well aware that the school corporation administrators evaluated the project primarily in terms of mean gains on standardized achievement tests. Further, we had to be sensitive to parents’ concerns, particularly as their children’s participation in the project was entirely voluntary. Not surprising, these constraints profoundly influenced the ways in which we attempted to translate constructivism as a theory of knowing into practice. We were fortunate in that the classroom teacher, who had taught second grade mathematics “straight by the book” for the previous sixteen years, was a member of the project staff. Her practical wisdom and insights proved to be invaluable.

Key words: mathematics education, radical constructivism

Excerpt: Why, despite the many sources of difficulty, is successful communication of mathematics among individuals possible? Why is it that to apply mathematical ideas, one must inevitably choose one or more notations in which to materialize those ideas? And why the large variation in the ways that these notations support and/or constrain our thinking processes? In this chapter I will not presume to answer such questions, but rather will attempt to provide some means for others, hopefully including the reader, to gain insight into them. Central to this task will be to describe the twin mediating roles of notation systems, in mediating between what is normally regarded as “pure mathematics” and one’s experienced world, and in mediating communication processes among individuals. In so doing, I hope also to show how a representational framework for mathematical cognition and learning is consistent with constructivism.

In an epistemology where mathematics teaching is viewed as goal-directed interactive communication in a consensual domain of experience, mathematics learning is viewed as reflective abstraction in the context of scheme theory. In this view, mathematical knowledge is understood as coordinated schemes of action and operation. Consequently, research methodology has to be designed as a flexible, investigative tool. The constructivist teaching experiment is a technique that was designed to investigate children’s mathematical knowledge and how it might be learned in the context of mathematics teaching (Cobb & Steffe, 1983; Hunting, 1983; Steffe, 1984). In a teaching experiment, the role of the researcher changes from an observer who intends to establish objective scientific facts to an actor who intends to construct models that are relative to his or her own actions.