Modeling misconceptions in knowledge space theory

Building intelligent tutoring systems (ITSs) that are aware of the students\u2019 misconceptions has been one of the ambitions for many of the approaches to computerized assessment of knowledge and learning. In the present article we extend knowledge space theory (KST) to mistakes and misconceptions. The proposed approach completes and extends the work initiated by J. Lukas (1997, Modellierung von Fehlkonzepten in einer algebraischen Wissensstruktur [Modeling misconceptions in an algebraic knowledge structure], Kognitionswissenschaft, 6(4), 196\u2013204) on the application of information systems in KST for modeling misconceptions. The approach is divided into a deterministic and a probabilistic part. The notion of a \u201cpolytomous skill map\u201d (PSM) represents the cornerstone of the deterministic part. Properties of the PSM are established that assure a consistent link between item responses on the one hand and the (correct or buggy) cognitive rules applied by an individual on the other hand. A weaker notion of information system (named a \u201ccognitive rule system\u201d) is proposed for representing two types of dependencies among cognitive rules: precedence and compatibility. The probabilistic part consists of an extension of the basic local independence model to more than two response alternatives. This model can be used for knowledge diagnoses, as well as for empirically testing the deterministic assumptions from a PSM. An empirical application of this probabilistic model to the responses of 331 university students is illustrated and discussed using two different PSMs