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, and ALEKS. It is possible to interpret a knowledge space as a special form of a restricted latent class model.
Origin
Knowledge Space Theory was motivated by the shortcomings of the psychometric approach to the assessment of competence like SAT and ACT. The theory was developed with an objective of designing automated procedures which -
Assessments based on KST are adaptive and can account for possible slips or guesses. KST aims to give a detailed assessment of student's knowledge state as opposed to a numerical mark in traditional assessments. More specifically, the result of a KST based assessment tells two things -
What the student can do and
What the student is ready to learn.
Basic concepts
Knowledge State
Precedence Relation
Knowledge Structure
The outer and inner fringes
Definitions
Some basic definitions used in the knowledge space approach -
A knowledge structure is called a knowledge space if it is closed under union, i.e. whenever.
A knowledge space is called a quasi-ordinal knowledge space if it is in addition closed under intersection, i.e. if implies. Closure under both unions and intersections gives the structure of a distributive lattice; Birkhoff's representation theorem for distributive lattices shows that there is a one-to-one correspondence between the set of all quasiorders on Q and the set of all quasi-ordinal knowledge spaces on Q. I.e., each quasi-ordinal knowledge space can be represented by a quasi-order and vice versa.
An important subclass of knowledge spaces, the well-graded knowledge spaces or learning spaces, can be defined as satisfying two additional mathematical axioms:
If and are both feasible subsets of concepts, then is also feasible. In educational terms: if it is possible for someone to know all the concepts in S, and someone else to know all the concepts in T, then we can posit the potential existence of a third person who combines the knowledge of both people.
If is a nonempty feasible subset of concepts, then there is some concept x in S such that is also feasible. In educational terms: any feasible state of knowledge can be reached by learning one concept at a time, for a finite set of concepts to be learned.
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 from data. A third method is to derive the knowledge space from an analysis of the problem solvingprocesses in the corresponding domain.
