Regular ArticleStochastic Learning Paths in a Knowledge Structure
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AbstractThis paper presents a stochastic process describing the progress of subjects (for example, students learning a particular field) over a period of time. Typical data involve a fixed sample of subjects tested repeatedly. At the core of the model is a knowledge structure, that is, a possibly large collection Q of items, together with a family of its subsets representing the possible knowledge states. The basic prediction concerns the joint probabilities P(Rt1 = R1 , …, Rta = Rn ) of observing sets of correct responses R1, …, Rn at times t1 < · · · < tn (Thus, Rt1 , …, Rtn are jointly distributed random variables taking their values in 2Q, for any choice of the times t1 < · · · <.) The prediction is obtained by analyzing the possible learning paths of the subject through the knowledge structure, mapping out the visits of possible knowledge states at the times of the tests, and integrating over the possible learning rates of the subjects. The learning rate of the subjects, and the times required to master the items along a learning path, are assumed to be distributed gamma. These results elaborate, as a stochastic process, a model proposed earlier by the author.

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