Recent opinions of those interested in mathematics education have suggested that variations in the classroom culture and the nature of the patterns of interaction that occur between the teacher and the students create quite different settings for enhancing learning. Even earlier studies had pointed to the discrepancies between the intentions of teachers, their actual practice, and the latent learning of students (Bauersfeld, 1980; Holt, 1982; & Willis, 1977). In many ways, teachers have often unwittingly undermined their own goals by failing to realize that the consequences of their interaction are often quite different from their intentions. Descriptions of various mathematics classes commonly reveal them to be settings in which teachers still view their role as being responsible for only ensuring students' learn specified procedures for solving mathematical problems (Goodlad, 1983; Stodolsky, 1988). From this philosophy a long tradition of rule-bound school mathematics has continued. It has restrained the motivation for inventiveness for any student who enjoys both independent reasoning and collaboration of ideas. Children who engage in mathematical activity in interactive situations do learn mathematics in a manner that extends beyond the realms of memorized procedures.