A re-examination of the role of instructional materials in mathematics education

Two current phenomena, the prevalence of computer-based learning environments and renewed interest in "hands on" learning as a popular response to recommendations of national curriculum statements such as the NCTM Standards, occasion a revisitation of issues surrounding the use of physical materials. We discuss the efficacy of several manipulative materials in instructional settings, noting both their strengths and their limitations. Adopting a modeling perspective on the use of instructional materials in mathematics learning, we draw critical distinctions between internal models and external embodiments, and between teacher-intended meanings and the subjective nature of student interpretation. We explore the consequences of a narrow interpretation of active learning and caution against relying on a single manipulative to capture the richness and connectedness of mathematical ideas. As an alternative to asking students to discover mathematical ideas as perceived by adults, we endorse the perspective that materials should be used within a larger pedagogical framework in which students individually and collectively negotiate mathematical meaning. Remaining unanswered questions are posed as an agenda for future research. An earlier version of this paper was presented at the Sixteenth Annual Meeting of the Mathematics Education Research Group of Australia, July 1993. The authors would like to thank Les Steffe for helpful comments on a draft.