Table of Contents

Preface; Contents; 1 Overview and Basic Mathematical Concepts; 2 Knowledge Structures and Learning Spaces; 3 Knowledge Spaces; 4 Well-Graded Knowledge Structures; 5 Surmise Systems; 6 Skill Maps, Labels and Filters; 7 Entailments and the Maximal Mesh; 8 Galois Connections*; 9 Descriptive and Assessment Languages*; 10 Learning Spaces and Media; 11 Probabilistic Knowledge Structures; 12 Stochastic Learning Paths*; 13 Uncovering the Latent State: A Continuous Markov Procedure; 14 A Markov Chain Procedure; 15 Building a Knowledge Space; 16 Building a Learning space. Learning spaces offer a rigorous mathematical foundation for practical systems of educational technology. Learning spaces generalize partially ordered sets and are special cases of knowledge spaces. The various structures are investigated from the standpoints of combinatorial properties and stochastic processes. Leaning spaces have become the essential structures to be used in assessing students' competence of various topics. A practical example is offered by ALEKS, a Web-based, artificially intelligent assessment and learning system in mathematics and other scholarly fields. At the heart of A.