Knowledge space

In mathematical psychology, a knowledge space is a combinatorial structure describing the possible states of knowledge of a human learner. To form a knowledge space, one models a domain of knowledge as a set of concepts, and a feasible state of knowledge as a subset of that set containing the concepts known or knowable by some individual. Typically, not all subsets are feasible, due to prerequisite relations among the concepts. The knowledge space is the family of all the feasible subsets. Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and Jean-Claude Falmagne and have since been studied by many other researchers. They also form the basis for two computerized tutoring systems, RATH and ALEKS.

It is possible to interpret a knowledge space as a special form of a restricted latent class model.

Some basic definitions used in the knowledge space approach -

An important subclass of knowledge spaces, the well-graded knowledge spaces or learning spaces, can be defined as satisfying two additional mathematical axioms:

A set family satisfying these two axioms forms a mathematical structure known as an antimatroid.

In practice, there exist several methods to construct knowledge spaces. The most frequently used method is querying experts. There exist several querying algorithms that allow one or several experts to construct a knowledge space by answering a sequence of simple questions.

Another method is to construct the knowledge space by explorative data analysis (for example by Item tree analysis) from data. A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.

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