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  • Book
  • © 2011

Learning Spaces

Interdisciplinary Applied Mathematics

Authors:

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  • Introduces learning space as a special case of knowledge space.

  • Exposes theory and several applications of learning spaces and ancillary assessment procedures.

  • Presents ALEKS as a practical application of learning spaces for an efficient web based learning environment.

  • Includes supplementary material: sn.pub/extras

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eBook
USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-01039-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 139.99
Price excludes VAT (USA)
Hardcover Book
USD 199.99
Price excludes VAT (USA)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xv
  2. Overview and Basic Mathematical Concepts

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 1-21
  3. Knowledge Structures and Learning Spaces

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 23-41
  4. Knowledge Spaces

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 43-60
  5. Well-Graded Knowledge Structures

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 61-79
  6. Surmise Systems

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 81-101
  7. Skill Maps, Labels and Filters

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 103-117
  8. Entailments and the Maximal Mesh

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 119-131
  9. Galois Connections*

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 133-150
  10. Descriptive and Assessment Languages*

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 151-162
  11. Learning Spaces and Media

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 163-185
  12. Probabilistic Knowledge Structures

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 187-214
  13. Stochastic Learning Paths*

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 215-240
  14. Uncovering the Latent State: A Continuous Markov Procedure

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 241-272
  15. A Markov Chain Procedure

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 273-296
  16. Building a Knowledge Space

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 297-333
  17. Building a Learning space

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 335-357
  18. Analyzing the Validity of an Assessment

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 359-374
  19. Open Problems

    • Jean-Claude Falmagne, Jean-Paul Doignon
    Pages 375-377
  20. Back Matter

    Pages 379-417

About this book

Learning spaces offer a rigorous mathematical foundation for various practical systems of knowledge assessment. An example is offered by the ALEKS system (Assessment and LEarning in Knowledge Spaces), a software for the assessment of mathematical knowledge. From a mathematical standpoint, learning spaces as well as knowledge spaces (which made the title of the first edition) generalize partially ordered sets. They are investigated both from a combinatorial and a stochastic viewpoint. The results are applied to real and simulated data. The book gives a systematic presentation of research and extends the results to new situations. It is of interest to mathematically oriented readers in education, computer science and combinatorics at research and graduate levels. The text contains numerous examples and exercises, and an extensive bibliography.

Keywords

  • ALEKS
  • Markov processes
  • antimatroid
  • knowledge
  • knowledge assessment
  • knowledge space
  • learning space
  • mathematical psychology
  • online learning
  • psychometric methods
  • skill map

Reviews

From the reviews:

“The book deals with the construction of knowledge spaces and learning spaces … . Thus, the creative mathematician will find material capable of entertaining him or her for some time. The practitioner may be interested in applications. … there is no doubt that reading and working with this book will be rewarding for the mathematician and useful for scientists from very different areas. In many aspects it has the potential to serve as a guideline to a new and theoretically better founded form of psychometry.” (Reinhard Suck, SIAM Review, Vol. 54 (2), 2012)

“This book is an enlarged second edition of the 1999 ‘Knowledge Spaces’ by the same authors. … The authors cover both deterministic and probabilistic models, justify their findings and give good examples and applications, such as pattern recognition and medical diagnosis. … We recommend it to doctoral and postdoctoral studies.” (George Stoica, Zentralblatt MATH, Vol. 1205, 2011)

Authors and Affiliations

  • School of Social Sciences, Dept. Cognitive Sciences, University of California, Irvine, Irvine, USA

    Jean-Claude Falmagne

  • Dépt. Mathematique, Université Libre de Bruxelles, Bruxelles, Belgium

    Jean-Paul Doignon

About the authors

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.

Bibliographic Information

Buying options

eBook
USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-01039-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 139.99
Price excludes VAT (USA)
Hardcover Book
USD 199.99
Price excludes VAT (USA)