Knowledge Space Theory
 ALEKS (Assessment and LEarning in Knowledge Spaces) refers to ALEKS' theoretical basis in mathematical cognitive science known as Knowledge Space Theory.
Knowledge Space Theory applies concepts from Combinatorics and Probability Theory to the modeling and empirical description of particular fields of knowledge. Within this theory, a mathematical language has been developed to delineate the ways in which particular elements of knowledge (concepts in Algebra, for example) can be gathered to form distinct knowledge states of individuals.
This language enables the creation of computer algorithms for the construction and application of discipline-specific knowledge structures (known as "Knowledge Spaces"). For example, Arithmetic is regarded as a domain of roughly one hundred basic concepts, giving rise to a structure of approximately 40,000 empirically feasible knowledge states.
An adaptive assessment based upon this "Knowledge Structure," and employing Markovian1 procedures, asks the student a few questions (e.g.15-25 for Arithmetic), quickly yielding an exhaustive analysis of the student's knowledge.
See Knowledge Spaces by Jean-Paul Doignon and Jean-Claude Falmagne (Springer-Verlag 1999) for an authoritative development and recapitulation of the theory.
For a relatively accessible introduction to Knowledge Space Theory, see Falmagne, Koppen, Villano, Doignon & Johanessen: "Introduction to Knowledge Spaces: How to Build, Test, and Search Them" Psychological Review, 1990, Volume 97, pp. 201-224. A list of sample titles of relevant publications is available.
1. From the Russian mathematician A. A. Markov, 1856-1922, whose work contributed to the early developments of the theory of stochastic processes; see Doignon and Falmagne 1999 Chapters 10 and 11.
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