Suppose that is a set of problems and is a set of skills. A skill function assigns to each problem -- i.e. to each element of -- those sets of skills which are minimally sufficient to solve ; a problem function assigns to each set X of skills the set of problems which can be solved with these skills (a knowledge state). We explore the natural properties of such functions and show that these concepts are basically the same. Furthermore, we show that for every family of subsets of Q which includes the empty set and , there are a set of (abstract) skills and a problem function whose range is just . We also give a bound for the number of skills needed to generate a specific set of knowledge states, and discuss various ways to supply a set of knowledge states with an underlying skill theory. Finally, a procedure is described to determine a skill function using coverings in partial orders which is applied to set A of the Coloured Progressive Matrices test Raven (1965).

this report is one result of the research project iA System for Developing HypermediaBased Teaching Components of Intelligent Tutoring Systemsj supported by General Direction XII at the European Commission (Grant ERBCHBICT941599, HCM Research Fellowship) and by the University of Graz, Austria

This chapter develops an extension of Doignon and Falmagne’s knowledge structures theory by integrating it into a competence-performance conception. The aim is to show one possible way in which the purely behavioral and descriptive knowledge structures approach could be structurally enriched in order to account for the need of explanatory features for the empirically observed solution behavior. Performance is conceived as the observable solution behavior of a person on a set of domain-specific problems. Competence (ability, skills) is understood as a theoretical construct accounting for the performance. The basic concept is a mathematical structure termed a diagnostic, that creates a correspondence between the competence and the performance level. The concept of a union-stable diagnostic is defined as an elaboration of Doignon and Falmagne’s concept of a knowledge space. Conditions for the construction and several properties of union-stable diagnostics are presented. Finally, an empirical application of the competence-performance conception in a small knowledge domain is reported that shall illustrate some advantages of the introduced modeling approach.

The ELeGI project focuses on integrating technology-enhanced learning methodologies into a pedagogy-driven and service-oriented architecture based on Grid technology. It aims at a system that is capable of realising personalised, adaptive, and experiential learning. This requires to have available a framework that, on the one hand, allows for representing existing domain knowledge, and, on the other hand, provides a representation of the learner’s current state of knowledge. It is shown that a competence-based extension of Knowledge Space Theory provides a representation of the conceptual organisation of a domain that not only allows for an adaptive assessment of an individual’s knowledge, but also for generating personalised learning paths. The discussion of this framework emphasizes its application within an open distributed service model, and in the context of Virtual Scientific Experiments.

Abstract: Several approaches for formalising prerequisite structures on skills or competencies based on the psychological theory of knowledge space have been suggested and applied for adaptive eLearning. In this paper, we will discuss how these structures may be applied in skill management in a broader sense. After introducing some formal structures for prerequisite relationships between competencies, we will briefly present an example of an adaptive eLearning system based on this approach (APeLS). Several other aspects of the system which promise to be useful for advanced skill management are discussed. In the final part of this paper, we will discuss such broader applications of the model with respect to personal as well as to organisational skill management.

Abstract. In this paper, we suggest methods for the adaptive assessment and training of students. The main idea is to apply the theoretical concept of the neighbourhood of a knowledge state to adaptive training. We also describe improvements of existing assessment algorithms. These ideas are implemented in a multi–platform tutoring system shell. This shell has been applied to two different fields of knowledge. 1

Abstract: This paper describes a competence–based approach to adaptive e–learning based on knowledge space theory and its competence–performance extension. Assigning to each learning object sets of required and taught competencies, one can easily build a course adapting in each step to the learner’s current knowledge state. The competence assignments implicitly define a prerequisite structure on the set of competencies. At the same time, a decentralised storage of these assignments allows (i) for easy changes of learning objects and courses in dynamically changing domains and (ii) for reuse of adaptive material in different courses by implementing the competence assignments through slightly extended metadata standards. An example implementation will be briefly presented.

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For effective processes of knowledge assessment, teaching, information presentation and information use, the information should be presented on basis of a structure which is defined by prerequisite relationships (Which information do I have to know before I’m able to understand and integrate the newly presented information?). This kind of prerequisite relation for information units has to be determined and tested before using it. A non-numerical theory for representing these structures is already worked out, and several techniques have been suggested to obtain these structures, e.g. • mass data collection, • analysis of didactics and curricula, • querying experts on prerequisite relationships, competence and performance • analysis of demands

The possible worlds semantics is a fruitful approach used in Artificial Intelligence (AI) for both modelling as well as reasoning about knowledge in agent systems via modal logics. In this work our main idea is not to model/reason about knowledge but to provide a theoretical framework for knowledge assessment (KA) with the help of Monatague-Scott (MS) semantics of modal logic. In KA questions asked and answers collected are the central elements and knowledge notions will be defined from these (i.e., possible states of knowledge of subjects in a population with respect to a field of information).