How Should a Theory of Learning and Cognition Inform Instruction
We have developed a set of computer-based cognitive tutors (Anderson, Corbett, Koedinger, & Pelletier, 1995; Koedinger, Anderson, Hadley, & Mark, 1997) for high school mathematics that have been effective in improving learning. These tutors interact with students at the grain size of about 20 seconds during which the student might transform an equation or calculate an angle and the tutor will respond to that. The tutors are able to interact with the student at this grain size on the basis of a cognitive model that can solve these problems in the various ways students are supposed to be able to solve the problems. Compared to most computer-based instruction this is a rather fine grain size. However, compared to current theories of human cognition including our own ACT-R (Anderson & Lebiere, 1998, in press) this is a very gross level of interaction. Temporally, ACT-R postulates primitive acts of cognition taking under 1 second and often as little as 50 msec. Component-wise the theory postulates separate visual, memory, imaginal, and motor components to a student action like transforming an equation. While our focus is on ACT-R, its grain size is typical of modern theories of cognition. The question that this paper will broach is whether there is a role for a theory of cognition at this grain size in impacting something like mastery of high-school algebra. Although not offering any definitive answers to the questions we will review some of our research on eye movements that has attempted to track a finer temporal grain size in mathematical problem solving and we will review some of our research on brain imaging that has attempted to track different cognitive components in mathematical problem solving.