Tutoring concepts, percepts, and rules in Geometry problem-solving
Kenneth R. Koedinger · 1991
This thesis is a mixture of basic and applied Cognitive Science research focussed on understanding the underlying nature of a complex skill and translating this understanding into improved instruction. The basic research provides a cognitive model of the intuitive and informal proof planning abilities of skilled geometers. This model is implemented as a computer program, an expert system that can efficiently plan proof solutions. Previous models characterize geometry problem solving as heuristic search through a problem space of formal geometry rules--a method that originates from the step-by-step approach implied by two-column proof examples in traditional textbooks. In contrast, skilled geometers do not plan proofs in this step-by-step, rule-based fashion. Employing the methodology of verbal protocol analysis, I show that skilled subjects often skip formal steps while planning. Initially, they only consider the key ideas and leave the details for last. The knowledge that affords this efficient and informal planning is not organized around geometry rules as in previous models, but instead around categories of perceptual objects (percepts). Attached to these percepts are clusters of conceptual knowledge that indicate properties and sufficiency conditions relevant to making proof inferences. The cognitive model incorporating this knowledge is both more efficient than previous models and a better match to the human data. This model ties together a number of empirical results on the nature of human expertise and supports an inductive explanation of skill acquisition that is in contrast with the deductive approach typical of some dominant theories of skill acquisition. The goal of the applied research is to see if this improved understanding of a complex skill can be translated into improved instruction. I built an Intelligent Tutoring System (ITS), called ANGLE, to test this idea. ANGLE was built following specific guidelines (described within) for translating the underlying cognitive model into design specifications for the computer interface and tutoring component. For example, ANGLE's interface reifies the perceptual categories in the cognitive model and, thus, provides students with a novel notation with which to think about geometry. I present a preliminary evaluation of ANGLE comparing it with a previous ITS for geometry.