Learning Spaces: Interdisciplinary Applied Mathematics [Hardcover]

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 ALEKS is an artificial intelligence engine that assesses each student individually and continously.

The book is of interest to mathematically oriented readers in education, computer science, engineering, and combinatorics at research and graduate levels. Numerous examples and exercises are included, together with an extensive bibliography.

Learning spaces offer a rigorous mathematical foundation for practical systems of educational technology. Describing the underlying theory and applications involved with ancillary assessment procedures, Learning Spaces provides an overview for mathematically oriented readers in education, computer science, and additional disciplines.

Overview and Mathematical Glossary.- Knowledge Structures and Learning Spaces.- Knowledge Spaces.- Well-Graded Families.- Surmise Systems.- Skill Maps, Labels and Filters.- Entailments and the Maximal Mesh.- Galois Connections.- Descriptive and Assessment Languages.- Greedoids,  Learning Spaces, and Antimatroids.- Learning Spaces and Media.- Probabilistic Knowledge Structures.- Stochastic Learning Paths.- A Continuous Markov Procedure.- A Markov Chain Procedure.- Building a Knowledge Structure.- Building a Learning Space.- Applications.- Open Problems

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