Skill Proficiency Model for Delineating Polytomous Knowledge Structures

A fuzzy skill map assigns a fuzzy set of skills to each problem, representing the level of proficiency required to solve it. A student is assumed to solve a problem if their competence meets the corresponding requirements; otherwise, the problem is regarded as unsolved. However, students whose proficiency falls short may still produce partial solutions rather than failing completely. Building on this observation, we extend the framework by identifying partial solutions under the same fuzzy skill map, thereby developing a skill proficiency model for delineating polytomous knowledge structures via L-fuzzy rough approximation operators. Unlike existing polytomous extensions that require reassigning fuzzy skill sets to each response of each problem, our approach builds directly on the original fuzzy skill map. Furthermore, by incorporating intuitionistic L-fuzzy approximation operators, we enhance the model to simultaneously capture both skill proficiency and misconceptions of fuzzy skills, where a fuzzy skill refers to a skill at a specific proficiency level. Finally, as a direct application of the skill proficiency model, we propose a method for providing personalized learning guidance that recommends which unmastered fuzzy skills should be learned next and which mastered fuzzy skills should be reinforced. We further provide illustrative examples to clarify how both the skill proficiency model and the corresponding learning guidance method can be used.