A constructivist perspective on information-processing theories of mathematical activity
A distinction is made between weak and strong research programs in cognitive science, the latter being characterized by an emphasis on the development of runnable computer programs. The paper focuses on the strong research program and initially considers situations in which it claims to have advanced our understanding of mathematical activity. It is concluded that the program's characterization of students as environmentally driven systems leads to: (a) a treatment of mathematical activity in isolated, narrow, formal domains; (b) a failure to deal with relevance, common sense, and context, and (c) a separation of conceptual thought from sensory-motor action. Taken together, these conclusions imply a failure to deal adequately with the issue of mathematical meaning. In general, the program's primary focus appears to be on programmable mechanisms rather than fundamental problems of mathematical cognition. The purview of the discussion is then widened to consider the strong program's difficulties in dealing with social interaction, intellectual communities, and the hidden curriculum. It is noted that instructional implications derived from this program typically involve the organization of mathematical stimuli that make explicit or salient the relevant properties of a propositional mathematical environment. Finally, it is argued that some members of the strong program have recently acknowledged that it has limitations. The possibility of a rapprochement in which the strong program is supplanted by a form of social constructivism is discussed.