Modeling learning in knowledge space theory through bivariate Markov processes

Bivariate Markov processes (BMPs) described by Ephraim and Mark (2012) consist of a pair of stochastic processes in the continuous time, one observable and the other latent, that are jointly Markov. In the present article, the navigation behavior and the learning process of a user of a web-based tutoring system are jointly modeled as BMPs constrained by assumptions that are coherent with the concepts of competence-based knowledge space theory. Such constraints are expressed as formal assumptions about the web-based system and about the nature of the learning process. Scenarios are considered where the observed process is the navigation of an individual through the pages of an intelligent tutoring system, whereas the latent learning process consists of transitions among states in a competence structure. The approach seems to be rather general and flexible in modeling learning scenarios with different assumptions. As an example, BMP models are developed for some exemplary scenarios. Maximum likelihood parameter estimation via expectation–maximization algorithm is presented. The results of a simulation study showed that the parameter values are well-recovered by the estimation algorithm. The results of the application of a bivariate Markov model to the real data of students navigating the intelligent tutoring system Stat-Knowlab showed that the proposed approach provides useful insight into students’ learning processes.