Knowledge structures delineated by fuzzy skill maps

Skills represent the underlying cognitive abilities, which consist in methods, algorithms or tricks etc. Skills can be used to interpret individuals' knowledge states. A skill map assigns a subset of skills to each item q. The disjunctive model assumes that at least one of the assigned skills is acquired for the mastery of the item q, and the conjunctive model states that all of the assigned skills are necessary for the mastery of the item q. The latent cognitive abilities of different individuals are distinct. Different levels of proficiency in skills may be required to solve different items. It's natural to use a fuzzy skill map or a fuzzy skill multimap to represent the levels of proficiency in skills for solving items. A fuzzy skill map (Q,S,τ) assigns a fuzzy set of skills to each item q, which represents the level of proficiency in skills needed to solve the item. A fuzzy skill multimap (Q,S,μ) assigns a collection of fuzzy sets of skills to each item q, any fuzzy set C of skills in μ(q) can be viewed as an approach to solve the item q. A knowledge space and a simple closure space can be delineated by a fuzzy skill map via the disjunctive model and the conjunctive model respectively. A knowledge structure can be delineated via the competency model by a fuzzy skill multimap. Algorithms for delineating knowledge structures via the disjunctive model and the conjunctive model by a fuzzy skill map are only concerned with skills in S. Delineating the knowledge structure by a fuzzy skill multimap just depends on the competencies and the weak competencies. For any fuzzy skill map (Q,S,τ) or fuzzy skill multimap (Q,S,μ), the knowledge states delineated by some skills can be replaced by knowledge states delineated by some other skills. Having removed these skills, the relationship between items and skills can be described more concisely without changing the knowledge structure. A minimal set of skills can faithfully summarize the relationship between items and skills, which results in a substantial economy of storage in a computer memory and a high efficiency of building, testing and searching for knowledge structures.