Knowledge Space Theory

In the teaching of introductory statistics to first year students economics and business, the Maastricht University uses a blended learning environment that allows students to design a individualized environment by attuning available learning tools to personal preferences. The blended learning environment consists of tutorials based on the problem-based learning principle, lectures, independent learning and an electronic learning environment based upon knowledge space theory: ALEKS. Usage of only the first component is required; the usage of other components can be set according to individual preferences. In this study, we will focus on the intensity of the use of the electronic learning environment ALEKS and investigate the relationship between this and student learning dispositions in a cluster analytic study. Data of about 4650 students taking this course are used. This study aims to contribute on the topic ‘development of tools to improve students’learning of statistics’.

In the teaching of introductory statistics to first year students economics and business, the Maastricht University uses a blended learning environment that allows students to design a individualized environment by attuning available learning tools to personal preferences. The blended learning environment consists of tutorials based on the problem-based learning principle, lectures, independent learning and an electronic learning environment based upon knowledge space theory: ALEKS. Usage of only the first component is required; the usage of other components can be set according to individual preferences. In this study, we will focus on the intensity of the use of the electronic learning environment ALEKS and investigate the relationship between this and student learning dispositions in a cluster analytic study. Data of about 4650 students taking this course are used. This study aims to contribute on the topic ‘development of tools to improve students’learning of statistics’.

Elaborations of Competence-based Knowledge Space Theory (CbKST) incorporate skills that refer to the conceptual information of the domain as well as to the activities learners are expected to perform in this context. Thus, they are suggested as a formal knowledge representation framework that is able to take into account current activity-oriented pedagogical trends in designing effective Units of Learning (UoL). The broad array of required behaviour to be achieved by learners demands a search for instruments like taxonomies that allow for conceptualising activities, and consequently, skills and learning objectives. It is shown that the availability of such a taxonomy-based framework may be utilised in order to enhance the access and interface functionalities of learning systems. In particular, the selection of proper learning units and the delivery of effective feedback mechanisms on the teaching and learning progress are facilitated.

Intelligent Tutoring Systems are based on the knowledge-module that is holding the system's knowledge in a well-structured format. Considering the current state of the art knowledge-module representations, we lack a model that can represent evolving information. Representing evolving information is needed for those tutoring systems that are working with dynamically changing domains, e.g.: software science. In this paper a new combined model is presented that is based on the Ontology model and the fundamentals of Knowledge Space Theory. Our new model introduces the term of abstract time to be able to formulate an evolving knowledge graph. Our model also introduces the term of evoking-hooks that makes it possible to realize connections between external domain elements and our model's nodes.

In the teaching of introductory statistics to first year students economics and business, the Maastricht University uses a blended learning environment that allows students to design a individualized environment by attuning available learning tools to personal preferences. The blended learning environment consists of tutorials based on the problem-based learning principle, lectures, independent learning and an electronic learning environment based upon knowledge space theory: ALEKS. Usage of only the first component is required; the usage of other components can be set according to individual preferences. In this study, we will focus on the intensity of the use of the electronic learning environment ALEKS and investigate the relationship between this and student learning dispositions in a cluster analytic study. Data of about 4650 students taking this course are used. This study aims to contribute on the topic ' development of tools to improve students' learning of statistics' .

Partial orders can be represented by subsets of a given set ordered by inclusion. Special kinds of such set representations are investigated because they facilitate the exploration of properties of knowledge spaces. A new type of order relation is deÿned which is closely related to interval orders. These orders and the interval orders are investigated with respect to their parsimonious set representations. The theorems are applied to knowledge space theory.

This paper aims to point at possibilities of applying Mathematica within the knowl- edge space theory and similar areas of cognitive psychology by describing a case in which software requirements of a research project are to be met by designing a Math- ematica package - Knowledge Spaces Mathematica Package (KSMP). The initial and main goal of KSMP is to implement the concepts relevant for research on and devel- opment of querying on skills procedures. Nevertheless, the ultimate goal of KSMP is set to implement all essential knowledge space theory developments, and consequently not only to provide assistance in theoretical explorations, simulations and teaching, but also to serve as a basis for developing user-friendly Mathematica applications for a knowledge space theory applicant. The first part of the paper gives an overview and an introduction to the knowledge space theory. After the short overview of the re- search topics, the selected fundamental concepts currently covered by t...

During the last decades, learning has once again become a key topic. However, this time, not only for students and professors but also in political and economic contexts. One reason for this is that a high level of education and skills of nations, organizations, and individuals are considered both necessary and crucially competitive advantages in the present knowledge society and the globalized market. Therefore, obtaining a quality education is fundamental for all of us in today's competitive business world. In particular, adult learning within the maritime sector has been important for the success of this industry for ages. The question now is how to streamline and facilitate the learning process for the learners, the lecturers, the authors, and the learning institutions. TERP has taken on the challenge of improving this learning process by introducing Abooks, electronic textbooks based on principles of pedagogy (the science of learning), and andragogy (the science of learning focusing on adults) that adapt to the learner through artificial intelligence. Abooks also introduces the opportunity of utilizing immersive techniques. This is being developed in the AIM project; Adapting to the Individual through Machine learning, a research project led by the research department in TERP in collaboration with the University of Stavanger and the Norwegian Computing Center.

In 1985, Doignon and Falmagne introduced surmise relations for representing prerequisite relationships between items within a body of information for the assessment of knowledge. Often it is useful to partition such a body of information into sub-collections. As we are primarily interested in psychological applications, we refer to these sub-collections as tests. We extend the concept of surmise relations between items within tests to surmise relations between tests. Three di erent kinds of surmise relations between tests are investigated with respect to their properties. Furthermore, the corresponding knowledge spaces for tests and their bases are introduced. The relationship of this set theoretical approach to a Boolean matrix representation is discussed. Finally, we give a short overview about the further research regarding this mathematical model. It will be the foundation for a software system that will be used for analyzing test data. Other applications in ÿelds like curriculum development and structuring hyper-texts can easily be imagined.

Knowledge space theory was already applied for building adaptive testing and training systems which contain test and training problems including their prerequisite structure, and which offer explanations and help for solving these problems. Normally, however, lessons prepare students for solving problems. In this paper, we present a method for systematically structuring an adaptive eLearning course containing test problems as well as lessons. This method is based on knowledge space theory extended by component-wise representation of problems and on applying demand analysis. A course developed with such a method can be fed into an adaptive tutoring system, realized, e.g., within the adaptive tutoring system RATH.

The assessment of knowledge and learning progress in the context of game-based learning requires novel, noninvasive, and embedded approaches. In the present paper, we introduce a mathematical framework which relates the (problem solution) behavior of a learner in the game context to the learner's available and lacking competencies. We argue that a problem situation and its status at a certain point in time can be described by a set of game props and their current properties or states. In the course of the game, the learner can perform different actions to modify the props and, consequently, change the problem situation. Each action is evaluated with respect to its correctness or appropriateness for accomplishing a given task which, in turn, enables conclusions about the competence state of the learner. This assessment procedure serves as the basis for adaptive interventions, for instance, by providing the learner with guidance or feedback.

Elaborations of Competence-based Knowledge Space Theory (CbKST) incorporate skills that refer to the conceptual information of the domain as well as to the activities learners are expected to perform in this context. Thus, they are suggested as a formal knowledge representation framework that is able to take into account current activity-oriented pedagogical trends in designing effective Units of Learning (UoL). The broad array of required behaviour to be achieved by learners demands a search for instruments like taxonomies that allow for conceptualising activities, and consequently, skills and learning objectives. It is shown that the availability of such a taxonomy-based framework may be utilised in order to enhance the access and interface functionalities of learning systems. In particular, the selection of proper learning units and the delivery of effective feedback mechanisms on the teaching and learning progress are facilitated.

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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 journals. The application of the probabilistic theory to the high school mathematics test is original to this article. Consider the task of assessing an individual's knowledge state with regard to a specified collection of problems or questions. For concreteness, a small sample of such problems is shown below, taken from the standard high school mathematics curriculum. This example will be used throughout the article.

Knowledge space theory provides a formal model for representing students' knowledge and describing the structure of a domain of knowledge. A similar formal structure can be used to described the structure of hypertexts. The combination of knowledge space theory and the formal hypertext model leads to a framework for intelligent tutoring systems which provides individualized learning paths to a student. Using powerful procedures from relational database systems and from knowledge space theory, w e get e. g. an ecient selection of appropriate teaching documents.

A flexible approach for structuring and merging distributed learning object is presented. At the basis of this approach there is a formal representation of a learning object, called attribute structure. Attribute structures are labeled directed graphs representing structured information on the learning objects. When two or more learning objects are merged, the corresponding attribute structures are unified, and the unified structure is attached to the resulting learning object.

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.

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.

This study compares Hungarian 7 th to 11 th graders' and American 9 th to 11 th graders' thinking patterns in describing an atom. A new evaluation method, the combination of phenomenography and knowledge space theory was used to explore students' reasoning and to follow the change in students' cognitive structures. According to the phenomenographic analysis of the responses, three main categories, 'units of matter', 'constituents of atoms' and 'model of atoms', were identified. Connections between these categories were determined by adapting Knowledge Space Theory to the hierarchy of categories. Results showed that during the instruction, the initial uniform model for representation of students' knowledge structure became more diffuse but at the end of the instruction the organisation of the categories in students' minds could be represented again by a single model. In the initial model, the 'units of matter' category was independent of the 'constituents of atoms' and 'model of atoms' categories, and the 'model of atoms' category was built on the category 'constituents of atoms'. Significant change in connections among categories could be detected only in the case of Hungarian students. In the reasoning of Hungarian 11 th graders, the hierarchy between 'constituents of atoms' and 'model of atoms' was reversed. [Chem.

There are various methods in knowledge space theory for building knowledge structures or surmise relations from data. Few of them have been thoroughly analyzed, making difficult to decide which of these methods provide good results and when to apply each of the methods. In this paper, we investigate the method inductive item tree analysis and discuss the advantages and disadvantages of this algorithm. In particular, we introduce some corrections and improvements to it, resulting in two newly proposed algorithms. These algorithms and the original inductive item tree analysis procedure are compared in a simulation study and with empirical data.

The paper describes two approaches towards knowledge assessment which can be applied within any CALL application. The first approach uses test items to define a universe of empirical knowledge states. We show that the relations between items ('are about equal', 'is harder than') and subjects ('are about equal', 'is better than') can be easily computed within a CALL system, and that the empirical results give insight into the structure of problems and their solutions. Given a standardized test procedure, a CALL system can use this method to describe the knowledge state of a tested subject either using a reference population, a knowledge structure generated by experts (see below), or by computing a knowledge structure using the given sample. The second approach is based on querying four experts about the skills minimally necessary to solve problems of the item set of the first approach. The results of these queries can be transformed into theoretical knowledge structures, and the empirical data can be used to test the prediction of the experts. The results show that ¡ The experts disagree remarkably, ¡ Most of the experts did not express even simple relationships which are observable by the empirical data analysis, ¡ Situational factors such as the appearance of learning tasks in the same lecture are observable in the empirical data as well as in the opinions of the experts.

Let Q be a finite set of problems. A knowledge state is the set of problems a person is capable of solving. A family K of knowledge states is called a knowledge structure if the empty set and Q are elements of K. When K is closed under union, the structure is called a knowledge space. There exist effective algorithms to generate a knowledge space by querying an expert using an entail relation. In this paper we show that more general knowledge structures can be generated using modified query procedures. The general theory of the Galois connections between entail relations and knowledge structures is explored. Finally, we present query procedures which can be applied to generate any knowledge structure.

Traditional approaches of assessment in the medical domain are insufficient for evaluating trainees' technical skills. Currently, many European medical training bodies are attempting to introduce competence-based training programmes for technical skills as well as other domains (e.g., communication, professional behaviour, clinical cognition). These efforts are limited due to the absence of appropriate assessment tools. Based on Competence-based Knowledge Space Theory (CbKST), a collaborative project MedCAP intends to develop a valid and reliable competence assessment procedure for one important medical skill, spinal anaesthesia. The paper briefly overviews the current states of training and assessment for medical procedural skills, describes the core ideas of CbKST, and introduces the ongoing project that will transfer the innovative approach of CbKST in personalized learning and competence assessment to the medical domain.

One of the methods of historical knowledge assessment is the application of knowledge space theory. One of its most important elements is the construction of knowledge structure. This presupposes a prerequisite relation system between knowledge items, which can mostly be proved in a "logical way" in the case of exact sciences, however we cannot disregard the empirical justification of these relations in the case of history. The present study puts the aforementioned as its aim in one topic of Hungarian history, namely the life of Lajos Kossuth. The profundity of the questions equals the knowledge level expected at school-leaving exams. We made a questionnaire, which was filled in by 169 high-school and university students. We selected two groups of its questions, which can be related to more restricted topics. We examined the relations between them in each groups that we supposed to exist and based on this, we set up two prerequisite relation models.

An approach to automated learning path generation inside a domain ontology supporting a web tutoring system is presented. Even if a terminological ontology definition is needed in real systems to enable reasoning and/or planning techniques, and to take into account the modern learning theories, the task to apply a planner to such an ontology is very hard because the definition of actions along with their preconditions and effects has to take into account the semantics of the relations among concepts, and it results in building an ontology of learning. The proposed methodology is inspired to the Knowledge Space Theory, and proposes some heuristics to transform the original ontology in a weighted graph where the A* algorithm is used to find the path. The proposed approach is applied to the implementation of a web based tutoring system about the Java programming language.

Computerized assessment of knowledge is one of the most promising applications of knowledge space theory. The first requirement is to map a picture as accurate as possible of the organization of the knowledge. So far, all the procedures designed for this purpose rely exclusively on the query of an expert. Several experiments have shown the limitations of that approach in realistic conditions. One source of difficulty is the very high sensitivity of the querying algorithms to an expert's mistakes. Another source of difficulty concerns the validity of the expert: his or her expertise can exhibit large discrepancies with the knowledge structure of the actual population. To solve these difficulties, the present paper proposes and simulates a two-step procedure. The first step implements a modification of an existing querying procedure. The modification lowers the incidence of wrong answers on the knowledge structure. The second step consists in a refinement mechanism which relies on the knowledge assessment of many subjects to refine the very structure used by these assessments. For each step, it is shown that the underlying knowledge space can be recovered.

In the teaching of introductory statistics to first year students economics and business, the Maastricht University uses a blended learning environment that allows students to design a individualized environment by attuning available learning tools to personal preferences. The blended learning environment consists of tutorials based on the problem-based learning principle, lectures, independent learning and an electronic learning environment based upon knowledge space theory: ALEKS. Usage of only the first component is required; the usage of other components can be set according to individual preferences. In this study, we will focus on the intensity of the use of the electronic learning environment ALEKS and investigate the relationship between this and student learning dispositions in a cluster analytic study. Data of about 4650 students taking this course are used. This study aims to contribute on the topic ‘development of tools to improve students’learning of statistics’.

This sequence can be coded as a "word" xjf..z and a complete set of such words forms a "descriptive language" for the family $. This class of languages is defined precisely, and some connections between such languages and families of sets are investigated. It is shown in particular that when + is closed under intersections and unions, and satisfies the topological condition known as T,, then I) can be recovered exactly from any of its descriptive languages. These results have an application in the assessment of knowledge. In this framework, the set X is a set of questions, and any set SE I) represents a possible knowledge state, containing all the questions that some individual is capable of solving. A subclass of the descriptive languages are then the "assessment languages." Any such language defines a nonredundant algorithm for determining the knowledge state of any individual.

The authors describe a new approach towards assessing skills of medical trainees. Based on experiences from previous projects with (i) applying virtual environments for medical training and (ii) competence assessment and personalisation in technology enhanced learning environments, a system for personalised medical training with virtual environments is built. Thus, the practical training of motor skills is connected with the user-oriented view of personalised computer-based testing and training. The results of this integration will be tested using a haptic device for training spinal anaesthesia.

The authors present an approach for implementing a system for the assessment of medical competences using a haptic simulation device. Based on Competence based Knowledge Space Theory (CbKST), information on the learners' competences is gathered from different sources (test questions, data from the simulator, and supervising experts' assessments). The envisaged architecture consists of three core modules, an LMS (Moodle) containing user model and content objects and realising the interface between system and user, a simulator interface as an own service connecting the LMS to the (external) simulator system, and a CbKST service offering the assessment logic and visualisations of the assessment result for learner and teacher.

Elaborations of Competence-based Knowledge Space Theory (CbKST) incorporate skills that refer to the conceptual information of the domain as well as to the activities learners are expected to perform in this context. Thus, they are suggested as a formal knowledge representation framework that is able to take into account current activity-oriented pedagogical trends in designing effective Units of Learning (UoL). The broad array of required behaviour to be achieved by learners demands a search for instruments like taxonomies that allow for conceptualising activities, and consequently, skills and learning objectives. It is shown that the availability of such a taxonomy-based framework may be utilised in order to enhance the access and interface functionalities of learning systems. In particular, the selection of proper learning units and the delivery of effective feedback mechanisms on the teaching and learning progress are facilitated.

This study compares Hungarian 7 th to 11 th graders' and American 9 th to 11 th graders' thinking patterns in describing an atom. A new evaluation method, the combination of phenomenography and knowledge space theory was used to explore students' reasoning and to follow the change in students' cognitive structures. According to the phenomenographic analysis of the responses, three main categories, 'units of matter', 'constituents of atoms' and 'model of atoms', were identified. Connections between these categories were determined by adapting Knowledge Space Theory to the hierarchy of categories. Results showed that during the instruction, the initial uniform model for representation of students' knowledge structure became more diffuse but at the end of the instruction the organisation of the categories in students' minds could be represented again by a single model. In the initial model, the 'units of matter' category was independent of the 'constituents of atoms' and 'model of atoms' categories, and the 'model of atoms' category was built on the category 'constituents of atoms'. Significant change in connections among categories could be detected only in the case of Hungarian students. In the reasoning of Hungarian 11 th graders, the hierarchy between 'constituents of atoms' and 'model of atoms' was reversed. [Chem.

Partial orders can be represented by subsets of a given set ordered by inclusion. Special kinds of such set representations are investigated because they facilitate the exploration of properties of knowledge spaces. A new type of order relation is deÿned which is closely related to interval orders. These orders and the interval orders are investigated with respect to their parsimonious set representations. The theorems are applied to knowledge space theory.

Elaborations of Competence-based Knowledge Space Theory (CbKST) incorporate skills that refer to the conceptual information of the domain as well as to the activities learners are expected to perform in this context. Thus, they are suggested as a formal knowledge representation framework that is able to take into account current activity-oriented pedagogical trends in designing effective Units of Learning (UoL). The broad array of required behaviour to be achieved by learners demands a search for instruments like taxonomies that allow for conceptualising activities, and consequently, skills and learning objectives. It is shown that the availability of such a taxonomy-based framework may be utilised in order to enhance the access and interface functionalities of learning systems. In particular, the selection of proper learning units and the delivery of effective feedback mechanisms on the teaching and learning progress are facilitated.

Quasi-orders, that is, reflexive and transitive binary relations, have numerous applications. In educational theories, the dependencies of mastery among the problems of a test can be modeled by quasi-orders. Methods such as item tree or Boolean analysis that mine for quasi-orders in empirical data are sensitive to the underlying quasi-order structure. These data mining techniques have to be compared based on extensive simulation studies, with unbiased samples of randomly generated quasi-orders at their basis. In this paper, we develop techniques that can provide the required quasi-order samples. We introduce a discrete doubly inductive procedure for incrementally constructing the set of all quasi-orders on a finite item set. A randomization of this deterministic procedure allows us to generate representative samples of random quasi-orders. With an outer level inductive algorithm, we consider the uniform random extensions of the trace quasi-orders to higher dimension. This is combined with an inner level inductive algorithm to correct the extensions that violate the transitivity property. The inner level correction step entails sampling biases. We propose three algorithms for bias correction and investigate them in simulation. It is evident that, on even up to 50 items, the new algorithms create close to representative quasi-order samples within acceptable computing time. Hence, the principled approach is a significant improvement to existing methods that are used to draw quasi-orders uniformly at random but cannot cope with reasonably large item sets.

The correlational agreement coefficient CA(V, D) [van Leeuwe, J.F.J., 1974. Item tree analysis. Nederlands Tijdschrift voor de Psychologie 29, 475-484.] is a descriptive measure for the fit of a quasi-order V on an item set to a binary data set D. The coefficient is based on the comparison between the empirical correlations of the items to their assumed theoretical correlations. These theoretical correlations are derived from the assumption that the quasi-order is a correct representation of the data. In a recent paper Ü nlü and Albert [Ü nlü , A., Albert, D., 2004. The correlational agreement coefficient CA(V, D)-A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.] presented a detailed mathematical investigation of CA(V, D). They describe a number of problems of this coefficient which show in their opinion that its use to compare quasi-orders is questionable. We do not agree with some of the statements in Ü nlü and Albert [Ü nlü , A., Albert, D., 2004. The correlational agreement coefficient CA(V, D)-A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.]. Especially we try to show that some of the problems of CA(V, D) mentioned in Ü nlü and Albert [Ü nlü , A., Albert, D., 2004. The correlational agreement coefficient CA(V, D)-A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.] are in fact properties which a good measure of fit for a quasi-order should have.

Item Tree Analysis (ITA) is an explorative method of data analysis which can be used to establish a hierarchical structure on a set of dichotomous items from a questionnaire or test. There are currently two different algorithms available to perform an ITA. We describe a computer program called ITA 2.0 which implements both of these algorithms. In addition we show with a concrete data set how the program can be used for the analysis of questionnaire data.

The correlational agreement coefficient CA(V, D) [van Leeuwe, J.F.J., 1974. Item tree analysis. Nederlands Tijdschrift voor de Psychologie 29, 475-484.] is a descriptive measure for the fit of a quasi-order V on an item set to a binary data set D. The coefficient is based on the comparison between the empirical correlations of the items to their assumed theoretical correlations. These theoretical correlations are derived from the assumption that the quasi-order is a correct representation of the data. In a recent paper Ü nlü and Albert [Ü nlü , A., Albert, D., 2004. The correlational agreement coefficient CA(V, D)-A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.] presented a detailed mathematical investigation of CA(V, D). They describe a number of problems of this coefficient which show in their opinion that its use to compare quasi-orders is questionable. We do not agree with some of the statements in Ü nlü and Albert [Ü nlü , A., Albert, D., 2004. The correlational agreement coefficient CA(V, D)-A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.]. Especially we try to show that some of the problems of CA(V, D) mentioned in Ü nlü and Albert [Ü nlü , A., Albert, D., 2004. The correlational agreement coefficient CA(V, D)-A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.] are in fact properties which a good measure of fit for a quasi-order should have.

I am grateful to professor J. C. Falmagne and four anonymous referees for their suggestions concerning the mathematical argumentation and to Diana Bursy for improving the English style. Correspondence and reprint requests should be sent to Martin Schrepp,

A knowledge domain is represented in knowledge space theory by a finite set Q of problems. The knowledge of a subject in the domain is described by the subset of problems from Q that the subject is able to solve. The set of all possible knowledge states of subjects is called the knowledge structure on Q. A crucial problem is how a hypothetical knowledge structure, obtained for example by querying an expert, can be evaluated empirically. We describe a method for comparing two knowledge structures concerning their ability to describe a set of empirically observed response patterns. Our comparison method is tested in two simulation studies. The results of these studies show that the method is highly accurate in its evaluation of knowledge structures.

The connection between the theory of knowledge spaces and formal models of cognitive problem solving processes is examined. This connection is described for a model of the cognitive processes of subjects in solving letter series completion problems, which is based on ideas from Simon and Kotovsky (1963). The model is formulated as an algorithm, depending on its ability to solve letter series completion problems on two parameters. Different abilities of subjects in solving such problems can be represented within the model by different choices for these parameters. It is shown that the model determines surmise relations on sets of letter series completion problems. These surmise relations, respectively the corresponding quasi-ordinal knowledge spaces, are used in two experimental investigations to test the underlying process model empirically. The results of these experiments show that the surmise relations derived from the process model are able to predict the difficulty of letter series completion problems in a satisfactory manner.

Since the beginning of research in artificial intelligence several attempts have been made to construct intelligent tutorial systems (ITS). Such an ITS consists in general of a representation of the knowledge in a special domain, a diagnostic procedure to determine the knowledge of a student working with the system, teaching material, and a procedure for adaptive teaching. This chapter demonstrates how an extension of the theory of knowledge spaces can be used for the design of a domain-independent ITS. The main components of this ITS are a representation of the skills necessary in the knowledge domain and their dependencies as a surmise system, a set of questions related through a skill assignment to the skill states used for knowledge diagnosis, and a rule that relates skill states to teaching operations. The ITS is adaptive with respect to the consideration of the prior knowledge a student possess and with respect to the learning speed of a student. The strict formalized description of the systems components and their interactions during the teaching process guarantees that an ITS with the described properties can be implemented easily.

Let Q be a finite set of problems. A knowledge state is the set of problems a person is capable of solving. A family K of knowledge states is called a knowledge structure if the empty set and Q are elements of K. When K is closed under union, the structure is called a knowledge space. There exist effective algorithms to generate a knowledge space by querying an expert using an entail relation. In this paper we show that more general knowledge structures can be generated using modified query procedures. The general theory of the Galois connections between entail relations and knowledge structures is explored. Finally, we present query procedures which can be applied to generate any knowledge structure.

The authors present an approach for implementing a system for the assessment of medical competences using a haptic simulation device. Based on Competence based Knowledge Space Theory (CbKST), information on the learners' competences is gathered from different sources (test questions, data from the simulator, and supervising experts' assessments). The envisaged architecture consists of three core modules, an LMS (Moodle) containing user model and content objects and realising the interface between system and user, a simulator interface as an own service connecting the LMS to the (external) simulator system, and a CbKST service offering the assessment logic and visualisations of the assessment result for learner and teacher.

Traditional approaches of assessment in the medical domain are insufficient for evaluating trainees' technical skills. Currently, many European medical training bodies are attempting to introduce competence-based training programmes for technical skills as well as other domains (e.g., communication, professional behaviour, clinical cognition). These efforts are limited due to the absence of appropriate assessment tools. Based on Competence-based Knowledge Space Theory (CbKST), a collaborative project MedCAP intends to develop a valid and reliable competence assessment procedure for one important medical skill, spinal anaesthesia. The paper briefly overviews the current states of training and assessment for medical procedural skills, describes the core ideas of CbKST, and introduces the ongoing project that will transfer the innovative approach of CbKST in personalized learning and competence assessment to the medical domain.

In the past, formal models of cognitive psychology successfully contributed to the development of computerized adaptive tutorial systems. Emotional and motivational aspects, however, were rarely considered, although a variety of studies demonstrated their significant influence on learning and performance. The aim of the current pilot study was to investigate whether the sound and empirically valid knowledge space theory is able to cover learning and performance in two different motivational states, which were hope for success and fear of failure. Moreover, within a factorial design these motivational states were combined with two different learning conditions. Prestructured learning sessions within an adaptive tutorial system were contrasted with rather free text-based learning. The data collected with 104 high school students in the domain of elementary probability theory indicate that knowledge space theory is able to represent the responses obtained in a post-test for both motivational states, as well as for both learning conditions. These results lay out a promising route to integrating cognitive and emotional/motivational aspects into a comprehensive psychological model for adaptive tutorial systems.

In this paper two levels of content aggregation are considered: (i) a 'conceptual level' of the content of the learning objects, and (2) a 'didactic preference level' of the specific preferences expressed by a taecher or educational institution. We then present a knowledge structure model in which the learning objects are organized according to constraints imposed at these two distinct levels. The structure on the learning objects is then used to restrict the number of learning paths available to a learner during her/his navigation of the learning environment. In particular, only paths that are consistent with both the content constraints and the didactic constraints are available.