Constructing polytomous knowledge structures from fuzzy skills.

The dichotomous knowledge structure theory was introduced to polytomous items, which is proved meaningful in representing individuals' partial mastery of items. Given that different proficiencies in skills may be required to solve polytomous items, fuzzy skills can be applied to represent latent cognitive abilities of individuals. Then we consider how to construct polytomous knowledge structures by fuzzy skills. To characterize relationships between polytomous items and fuzzy skills related to solving these items, we introduce the concept of fuzzy skill function on polytomous items. Given a set Q×L of polytomous items and a set S of skills, a fuzzy skill function (Q×L,S,σ) assigns each response category (q,l)∈Q×L with a family σ(q,l) of fuzzy sets on S, where each element in σ(q,l) represents an approach for solving q to the level l. Notice that there are two special kinds of fuzzy skill functions: one is disjunctive, while the other is conjunctive. The former delineates polytomous knowledge spaces, while the latter delineates polytomous closure spaces. Based on these facts, we design algorithms for delineating the two kinds of polytomous knowledge structures. For any fuzzy skill function on polytomous items, all competencies and all weak competencies are enough to delineate a polytomous knowledge structure. Thus, we design an algorithm for delineating polytomous knowledge structures via fuzzy skill functions on polytomous items. This algorithm only depends on competencies and weak competencies of fuzzy skill functions on polytomous items. Besides, we take “Addition and subtraction of fractions” as knowledge domain to illustrate the process of constructing polytomous knowledge structures from fuzzy skills.