Towards a competence-based polytomous knowledge structure theory

The present article lays out the foundations of an axiomatic theory of attribute maps, an extension of skill maps to polytomous knowledge structure theory. A deterministic relationship between the available attributes and the observable item responses is established by means of two mappings denoted attribute map and item–response function. The attribute map assigns to each item–response pair the set of attributes that are instrumental for observing that particular response to the item. The item–response function assigns to each set of attributes the set of item responses that, according to the attribute map, can be obtained with those attributes. The proposed approach is shown to be rather general and capable of handling a multitude of polytomous items that can be encountered in practice. Examples are provided that cover the analysis of responses on Likert scales, responses awarded partial credits, and the investigation of misconceptions.