Abstract Building intelligent tutoring systems (ITSs) that are aware of the students’ 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), 196–204) 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 “polytomous skill map” (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 “cognitive rule system”) 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.

中文翻译：

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建模知识空间理论中的误解

摘要 构建能够意识到学生误解的智能辅导系统 (ITS) 一直是许多知识和学习计算机化评估方法的目标之一。在本文中，我们将知识空间理论 (KST) 扩展到错误和误解。所提议的方法完成并扩展了 J. Lukas (1997, Modellierung von Fehlkonzepten in einer algebraischen Wissensstruktur [Modeling misconceptions in an algebraic Knowledge Structure], Kognitionswissenschaft, 6(4), 196–204) 发起的关于信息系统应用的工作在 KST 中用于建模误解。该方法分为确定性和概率性部分。“多分技能图”（PSM）的概念代表了确定性部分的基石。建立 PSM 的特性，以确保一方面项目响应与另一方面由个人应用的（正确或错误的）认知规则之间的一致联系。提出了一种较弱的信息系统概念（称为“认知规则系统”）来表示认知规则之间的两种类型的依赖关系：优先级和兼容性。概率部分包括将基本局部独立模型扩展到两个以上的响应备选方案。该模型可用于知识诊断，也可用于对来自 PSM 的确定性假设进行经验测试。使用两种不同的 PSM 说明和讨论了该概率模型对 331 名大学生的反应的实证应用。