We first distinguish between the school mathematics tradition typically established in textbook-based classrooms and the inquiry mathematics tradition established in classrooms where instruction is compatible with constructivism. We then focus on the inquiry mathematics tradition and consider the theoretical and pragmatic tensions inherent in the view that mathematical learning is both a process of active individual construction and a process of acculturation. Particular attention is given to the ways in which both constructivist and sociocultural theorists address this issue. Finally, we discuss the development of instructional activities for inquiry mathematics classrooms.

The primary purpose of this chapter is to clarify the basic tenets of activity theory and constructivism, and to compare and contras instructional approaches developed within these global theoretical perspectives. This issue is worthy of discussion in that research and development programs derived from these two perspectives are both vigorous. For example, the work of sociocultural theorists conducted within the activity theory tradition has become increasingly influential in the United States in recent years. One paradigmatic group of studies conducted by Lave (1988), Newman, Griffin, and Cole (1089). and Scribner (1984) has related arithmetical computation to more encompassing social activities such as shopping in a supermarket, packing crates in a dairy, and completing worksheets in school. Taken together, these analyses demonstrate powerfully the need to consider broader social and cultural processes when accounting for children’s development of mathematic cal competeuce.

Key words: mathematics education, activity theory, mathematical activity, mathematical practice, instructional development.