Item to Skills Mapping: Deriving a Conjunctive Q-matrix from Data

Uncovering which skills are determining the success to questions and exercises is a fundamental task in ITS. This task is notoriously difficult because most exercise and question items involve multiple skills, and because skills modeling may involve subtle concepts and abilities. Means to derive this mapping from test results data are highly desirable. They would provide objective and reproductible evidence of item to skills mapping that can either help validate predefine skills models, or give guidance to define such models. However, the progress towards this end has been relatively elusive, in particular for a conjunctive skills model, where all required skills of an item must be mastered to obtain a success. We extend a technique based on Non-negative Matrix Factorization, that was previously shown successful for single skill items, to construct a conjunctive item to skills mapping from test data with multiple skills per item. Using simulated student test data, the technique is shown to yield reliable mapping for items involving one or two skills from a set of six skills.