Boolean Algebra Applied to Determination of Universal Set of Knowledge States

摘要

Diagnosing cognitive errors possessed by examinees can be considered as a patternclassification problem which is designed to classify a sequential input of stimuli into one of several predetermined groups. The sequential inputs in our context are item responses and the predetermined groups are various states of knowledge resulting from misconceptions or different degrees of incomplete knowledge in a domain. In this study, the foundations of a combinatorial algorithm that will provide the universal set of states of knowledge will be introduced. Each state of knowledge is represented by a list of can/cannot cognitive tasks and processes (called cognitively relevant attributes or latent variables) which are usually unobservable. A Boolean descriptive function will be introduced as a mapping between the attribute space spanned by latent attribute variables and the item response space spanned by item score variables. The Boolean descriptive function plays the role of uncovering the unobservable content of a black box. Once all the possible classes are retrieved explicitly and expressed by a set of ideal item response patterns which are described by a can/cannot list of latent attributes, the notion of bug distributions and statistical pattern classification techniques will enable us to diagnose students' states of knowledge accurately.