Abstract planning and perceptual chunks: Elements of expertise in geometry

Abstract

We present a new model of skilled performance in geometry proof problem solving called the Diagram Configuration model (DC). While previous models plan proofs in a step-by-step fashion, we observed that experts plan at a more abstract level: They focus on the key steps and skip the less important ones. DC models this abstract planning behavior by parsing geometry problem diagrams into perceptual chunks, called diagram configurations, which cue relevant schematic knowledge. We provide verbal protocol evidence that DC's schemas correspond with the step-skipping inferences experts make in their initial planning. We compare DC with other models of geometry expertise and then, in the final section, we discuss more general implications of our research. DC's reasoning has important similarities with Larkin's (1988) display-based reasoning approach and Johnson-Laird's (1983) mental model approach. DC's perceptually based schemas are a step towards a unified explanation of (1) experts' superior problem-solving effectiveness, (2) experts' superior problem-state memory, and (3) experts' ability, in certain domains, to solve relatively simple problems by pure forward inferencing. We also argue that the particular and efficient knowledge organization of DC challenges current theories of skill acquisition as it presents an end-state of learning that is difficult to explain within such theories. Finally, we discuss the implications of DC for geometry instruction.