Learning Spaces: Interdisciplinary Applied Mathematics

Front Cover
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.

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

1 Overview and Basic Mathematical Concepts
1
2 Knowledge Structures and Learning Spaces
23
3 Knowledge Spaces
42
4 WellGraded Knowledge Structures
61
5 Surmise Systems
81
6 Skill Maps Labels and Filters
102
7 Entailments and the Maximal Mesh
119
8 Galois Connections
132
12 Stochastic Learning Paths
215
A Continuous Markov Procedure
241
14 A Markov Chain Procedure
273
15 Building a Knowledge Space
297
16 Building a Learning space
334
17 Analyzing the Validity of an Assessment
359
18 Open Problems
375
Glossary
378

9 Descriptive and Assessment Languages
151
10 Learning Spaces and Media
163
11 Probabilistic Knowledge Structures
187
Bibliography
397
Index
409
Copyright

Other editions - View all

Common terms and phrases

About the author (2010)

Jean-Paul Doignon is a professor at the mathematics department of the Université Libre de Bruxelles, Belgium. His research covers various aspects of discrete mathematics (graphs, ordered sets, convex polytopes, etc.) and applications to behavioral sciences (preference modeling, choice representation, knowledge assessment, etc.). Jean-Claude Falmagne is emeritus professor of cognitive sciences at the University of California, Irvine. His research interests span various areas, focusing on the application of mathematics to educational technology, psychophysics, choice theory, and the philosophy of science, in particular measurement theory.