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

Abstract

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 know/edge space. It is assumed that if 2 subsets K and K' of questions are assumed to be states in a knowledge space X, then K U K is also assumed to be a state in Ji. 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

Learning to Think: Cognitive Mechanisms of Knowledge Transfer

“…9 More generally, computer-collected student performance and learning data have been used to evaluate cognitive models and to select among alternative models (e.g., Anderson, 1993;Ohlsson & Mitrovic, 2007). Automated methods have been developed to search for a best-fi tting cognitive model either purely from performance data, collected at a single time (e.g., Falmagne, Koppen, Villano, Doignon, & Johannesen , 1990), or from learning data, collected across multiple times (e.g., Cen et al, 2006). We should also be alert for decomposition opportunities, where the transfer may not be at the whole problem level (problem schemas), but at the level of intermediate steps (step schemas or knowledge components).…”

Bayesian Student Models Based on Item to Item Knowledge Structures

“…Millán, de-la Cruz, & Suárez, 2000;Vomlel, 2004;VanLehn et al, 1998). In contrast to hierarchical structures, item to item structures build structures among observable knowledge item themselves (Falmagne, Koppen, Villano, Doignon, & Johannesen, 1990;Doignon & Falmagne, 1999), bypassing concept links. A number of researchers have worked on the problem of building student models within this framework (Dowling & Hockemeyer, 2001;Kambouri, Koppen, Villano, & Falmagne, 1994).…”

Qualitative Discovery in Medical Databases

“…Although thisformalismcouldnot fully represent allpossible knowledge states, it capturesa largepartof the constraints on the ordering among KU and can be usedforthepurposeofautomaticknowledgeassessment [3], [7]. The data used to induceimplication networksformedical diagnosis consists of a setof attributes which arecontinuous variables.…”