In mathematical psychology and education theory, a knowledge space is a combinatorial structure used to formulate mathematical models describing the progression of a human learner.[1] Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and Jean-Claude Falmagne,[2] and remain in extensive use in the education theory.[3][4] Modern applications include two computerized tutoring systems, ALEKS[5] and the defunct RATH.[6]
Formally, a knowledge space assumes that a domain of knowledge is a collection of concepts or skills, each of which must be eventually mastered. Not all concepts are interchangeable; some require other concepts as prerequisites. Conversely, competency at one skill may ease the acquisition of another through similarity. A knowledge space marks out which collections of skills are feasible: they can be learned without mastering any other skills. Under reasonable assumptions, the collection of feasible competencies forms the mathematical structure known as an antimatroid.
Researchers and educators usually explore the structure of a discipline's knowledge space as a latent class model.[7]
Knowledge Space Theory attempts to address shortcomings of standardized testing when used in educational psychometry. Common tests, such as the SAT and ACT, compress a student's knowledge into a very small range of ordinal ranks, in the process effacing the conceptual dependencies between questions. Consequently, the tests cannot distinguish between true understanding and guesses, nor can they identify a student's particular weaknesses, only the general proportion of skills mastered. The goal of knowledge space theory is to provide a language by which exams can communicate[8]
Knowledge Space Theory-based models presume that an educational subject S can be modeled as a finite set Q of concepts, skills, or topics. Each feasible state of knowledge about S is then a subset of Q; the set of all such feasible states is K. The precise term for the information (Q, K) depends on the extent to which K satisfies certain axioms:
In educational terms, any feasible body of knowledge can be learned one concept at a time.If S∈K, then there exists x∈S such that S\{x}∈K
The more contentful axioms associated with quasi-ordinal and well-graded knowledge spaces each imply that the knowledge space forms a well-understood (and heavily studied) mathematical structure:
In either case, the mathematical structure implies that set inclusion defines partial order on K, interpretable as a educational prerequirement: if a(⪯)b in this partial order, then a must be learned before b.
The prerequisite partial order does not uniquely identify a curriculum; some concepts may lead to a variety of other possible topics. But the covering relation associated with the prerequisite partial does control curricular structure: if students know a before a lesson and b immediately after, then b must cover a in the partial order. In such a circumstance, the new topics covered between a and b constitute the outer fringe of a ("what the student was ready to learn") and the inner fringe of b ("what the student just 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.[9][10][11]
Another method is to construct the knowledge space by explorative data analysis (for example by item tree analysis) from data.[12][13] A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.[14]