A Correct Response Model in knowledge structure theory

Abstract

Abstract In knowledge space theory, the (latent) knowledge state of a student consists of the subset of test items that he masters in principle. Even at a given stage of apprenticeship, the student’s knowledge state may vary in a given collection of subsets. The collection of all possible states of all potential students forms a knowledge structure. In the modeling of student answer production, the probabilities governing the knowledge state are parameters. Moreover, for each item, two additional parameters capture the probabilities of careless errors and lucky guesses in the answers to the item. From all the latter parameters, the Correct Response Model (CRM) predicts the probability of a correct answer to any isolated item. From the same parameters, the Basic Local Independence Model (BLIM) predicts the probability of any possible pattern of correct responses. Here, a particular pattern records at a given time all the items to which a given student would provide correct answers. While general properties of the BLIM (such as identifiability of parameters) have been investigated, the simpler Correct Response Model still requires scrutiny. The present paper investigates the CRM as regards testability, identifiability and characterizability. It either provides explicit answers or points out serious difficulties garnered from various mathematical disciplines.