The Composition Effect: Conjuntive or Compensatory? An Analysis of Multi-Skill Math Questions in ITS

Multi skill scenarios are common place in real world problems and Intelligent Tutoring System questions alike, however, system designers have often relied on ad-hoc methods for modeling the composition of multiple skills. There are two common approaches to determining the probability of correct for a multi skill question: a conjunctive approach, which assumes that all skills must be known or a compensatory approach which assumes that the strength of one skill can compensate for the weakness of another skill. We compare the conjunctive model to a learned compositional function and find that the learned function quite nearly converges to the conjunctive function. We can confidently report that system designers can implement the AND gate to represent the composition function quite accurately. Cognitive modelers may be interested in the small compensatory effect that is present. We use a static Bayesian network to model the two hypotheses and use the expectationmaximization algorithm to learn the parameters of the models.