Introduction to knowledge spaces: How to build, test, and search them.

This article gives a comprehensive description of a theory for the efficient assessment of knowledge. The essential concept is that the knowledge state of a subject with regard to a specified field of information can be represented by a particular subset of questions or problems that the subject is capable of solving. The family of all knowledge states forms the knowledge space. It is assumed that if 2 subsets K and K′ of questions are assumed to be states in a knowledge space ,then KU K′ is also assumed to be a state in K. Such a theory is consistent with the idea that at least some of the notions in the field may be acquired from different sets of prerequisites. Various aspects of the theory are discussed. In particular, the problem of constructing a knowledge space in practice is analyzed in detail. A first sketch of the knowledge space can be obtained by consulting expert teachers in the field. The mathematical theory necessary to render this consultation efficient is given. This preliminary construction can then be tested and refined on the basis of empirical data. To this end, a probabilistic version of the theory is developed, which is similar in spirit to some psychometric models, but it is grounded on the concept of a knowledge space rather than on that of skill or ability. An exemplary application of this probabilistic theory to a high school mathematics test is described, based on a sample of several hundred students. By standard likelihood ratio methods, it is shown how the preliminary knowledge space can be gradually refined, and the number of possible knowledge states substantially reduced. Two classes of Markovian knowledge assessment algorithms are outlined. Most of the results presented summarize previous articles published in various technical journals. The application of the probabilistic theory to the high school mathematics test is original to this article.