Cognitive Science and Mathematics Education

By Alan H. Schoenfeld | Go to book overview

analyzing a single 1-hour videotape of a problem-solving session, and perhaps 2 or 3 years writing computer programs that "simulate" the behavior that appeared in that 1 hour of problem solving--must appear odd to someone looking from outside the discipline. A major goal of this chapter is to demonstrate that such apparently odd behavior can be both sensible and useful. More precisely, my goal is to explicate two main ideas: the idea of a "cognitive process analysis" at a very fine level of detail, and of a "constructivist perspective."


BACKGROUND: SOME ALTERNATIVE PERSPECTIVES AND APPROACHES

A basic assumption underlying work in cognitive science is that mental structures and cognitive processes (loosely speaking, "the things that take place in your head") are extremely rich and complex--but that such structures can be understood, and understanding them will yield significant insights into the ways that thinking and learning take place. Analyses in cognitive science tend to be very detailed. They focus on cognitive processes in an attempt to explain what produces "productive thinking." And because the studies are often carried out in tremendous depth, the number of "subjects" in those studies is often quite small.

The cognitive science perspective is best illustrated by some practical examples, in which the "cognitive approach" can be contrasted with the approaches suggested by more conventional methods. To establish a context for our discussion, we begin with a brief description of some of the learning theories, curricular approaches, and research methods that have had significant impact on American educational practice in this century. Having discussed these, we will turn in the next section to some examples of work in cognitive science that have implications for mathematics instruction.


Associationism

E. L. Thorndike's seminal book The Psychology of Arithmetic was published in 1922. Thorndike's learning theory was based on the notion of mental "bonds," or associations between sets of stimuli and the responses to them (e.g. "two plus two" as a stimulus and "four" as a response). According to the theory, bonds become stronger as a result of reinforcement or frequent use, weaker as a result of punishment, or decay as a result of infrequent use. The associationists proposed some general organizational principles for instruction, for example, the principle that bonds that "go together" should be taught together. Translated into

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