Theory of learning mathematics with computer-assisted instruction: Hierarchical and nonhierarchical options

The learning of arithmetic problems is assumed to be a Markov process involving conditioning of a set of k subskills, each consisting of one or more productions. An axiom set is provided, with the choice between two options for one axiom controlling which of two models results. Model 1, which assumes that every subskill is attempted on every occurrence of a given problem, is a nonhierarchical model. Model 2, which assumes that every subskill in the temporal sequence for a problem is attempted until one is failed, is a somewhat unconventional hierarchical model: It is hierarchical in the sense that conditioningor chance success at one level is prerequisite to performance of the next subskill (next level) in the problem. As the value of the guessing parameter, g, declines, Model 2 leads to less efficient learning than Model 1.