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In knowledge structure theory (KST), an individual's knowledge state represents the items that the individual can completely solve. Based on the differences in individuals' latent cognitive competence, polytomous knowledge states can be used to partially represent individuals to solve items. This paper explores the construction of polytomous knowledge states and polytomous knowledge structures on a polytomous knowledge domain Q × L . A quasi-ordinal polytomous knowledge space and a polytomous knowledge space can be induced by two different entail relations, respectively. When the polytomous knowledge structure (Q , L , K) on Q × L is determined, accurately evaluating an individual's polytomous knowledge state is the key to providing learning guidance and taking teaching remedial measures for the individual. Therefore, we study the basic assessment procedure for a given polytomous knowledge structure, and a concrete example is designed to illustrate the method presented in this paper. [ABSTRACT FROM AUTHOR]

In knowledge structure theory (KST), an individual’s knowledge state represents the items that the individual can completely solve. Based on the differences in individuals’ latent cognitive competence, polytomous knowledge states can be used to partially represent individuals to solve items. This paper explores the construction of polytomous knowledge states and polytomous knowledge structures on a polytomous knowledge domain Q×L. A quasi-ordinal polytomous knowledge space and a polytomous knowledge space can be induced by two different entail relations, respectively. When the polytomous knowledge structure (Q,L,K) on Q×L is determined, accurately evaluating an individual’s polytomous knowledge state is the key to providing learning guidance and taking teaching remedial measures for the individual. Therefore, we study the basic assessment procedure for a given polytomous knowledge structure, and a concrete example is designed to illustrate the method presented in this paper.

Heller (2021) generalized quasi‐ordinal knowledge spaces to polytomous items. Inspired by this paper, we propose CD‐polytomous knowledge space and its polytomous surmise system. A Galois connection is established between the collection K of all polytomous knowledge structures and the collection F1 of particular polytomous attribute functions. The closed elements of the Galois connection are CD‐polytomous knowledge spaces in K and polytomous surmise functions in F1, respectively. With the help of these, this paper provides a characterization of the polytomous knowledge structure corresponding to the polytomous surmise function that is weakly factorial. Based on the finite sets of items and response values, these results generalize the previous approaches for polytomous knowledge spaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]