2.2. Principled Design of Cognitive Tutors

The design of PAT was also guided by theoretical principles. As a cognitive tutor (Anderson, et. al, 1995), PAT has the defining feature of containing a psychological model of the cognitive processes behind successful and near-successful student performance. Based on the ACT theory, this cognitive model is written as a system of if-then production rules that are capable of generating the multitude of solution steps and mis-steps typical of students. The cognitive model is the basis for two student modeling techniques: model tracing and knowledge tracing. Model tracing is used to monitor student's progress through a problem solution (see Anderson, Boyle, Corbett, & Lewis, 1990). This tracing is done in the background by matching student actions to those the model might generate. The tutor is mostly silent. However, when help is needed, the tutor knows where the student is and can provide hints that are individualized to the student's particular approach to the problem. Knowledge tracing is used to monitor students' learning from problem to problem (see Corbett & Anderson, 1992). A Bayesian estimation procedure identifies students' strengths and weaknesses relative to the production rules in the cognitive model. This assessment information is used to individualize problem selection and optimally pace students through the curriculum.

PAT's cognitive model and general design is the consequence of basic research on mathematical cognition. Our previous research has shown that students have informal inductive routes to mathematical knowledge that often precede formal instruction in the deductive use of symbols (Koedinger & Anderson, 1990; 1991). Thus, contrary to popular belief, students can perform better on algebra word problems under certain circumstances than on the equivalent algebraic equations (Koedinger & Tabachneck, 1995). We applied such results in early experiments with PAT where we showed that students learned more from a theory-inspired "inductive-support" version of the tutor than from a "textbook" version based on a popular Algebra text (Koedinger & Anderson, 1996).