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 stochastic processes 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 framework enables the creation of computer algorithms for the construction and application of discipline-specific knowledge structures (known as "Knowledge Spaces"). For example, Algebra 1 is regarded as a domain of approximately 350 basic concepts, giving rise to a structure of millions of empirically feasible knowledge states.
Despite the millions of knowledge states in a structure, an adaptive assessment based upon this "Knowledge Structure," employing Markovian1 procedures, can gauge a student's knowledge state in a relatively small number of questions (approximately 25-30).
Knowledge Space Theory is set forth authoritatively in Learning Spaces by Jean-Paul Doignon and Jean-Claude Falmagne (Springer-Verlag, 2011). This monograph is a revision and expansion of Knowledge Spaces (Springer-Verlag, 1999) and includes an examination of the mathematical basis for learning space theory and its applicability to various practical systems of knowledge assessment (such as ALEKS).
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 select list 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.