Proceedings of the Annual Meeting of the Cognitive Science Society

We have been refining a cognitive model, written in ACT-R, of student performance in early algebra problem solving. "Early algebra" refers to a class of problems and competencies at the boundary between arithmetic and algebra. Our empirical studies in this domain establish a striking contrast between students' difficulties with symbolic algebra and their relative success with certain kinds of "intuitive" algebraic reasoning. To better understand this contrast, we analyzed student solutions to identify the strategies and errors exhibited and then set out to account for this detailed process data with the utility-based choice mechanism of ACT-R. Our first model contained production mles for explicitly selecting strategies and for making certain systematic errors or bugs. It provided a good quantitative fit to student performance data (R2=.90), however, it had two quahtative shortcomings: 1) the productions for strategy selection appeared to serve no computational purpose and 2) the model systematically underpredicted the frequency of non-trivial errors on more complex problems. We created a new model in which explicit strategy selection was eliminated (strategic behavior is emergent) and in which failure to fire a production (an implicit, non-buggy error) is an option at every model choice point. Compared to the first model, this model achieved an equivalent quantitative fit with fewer productions and without the systematic deviations from the error data. We consider the implications of implicit strategies and errors for instruction.