Learning Spaces: Interdisciplinary Applied Mathematics
Springer Science & Business Media, Sep 10, 2010 - Mathematics - 417 pages
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.
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2 Knowledge Structures and Learning Spaces
3 Knowledge Spaces
4 WellGraded Knowledge Structures
5 Surmise Systems
6 Skill Maps Labels and Filters
7 Entailments and the Maximal Mesh
8 Galois Connections
12 Stochastic Learning Paths
A Continuous Markov Procedure
14 A Markov Chain Procedure
15 Building a Knowledge Space
16 Building a Learning space
17 Analyzing the Validity of an Assessment
18 Open Problems