Books on Knowledge Spaces and Structures

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Falmagne, J.-C., Albert D., Doble, D., Eppstein D., Hu X. (2013)
Knowledge Spaces: Applications in Education. Berlin: Springer-Verlag.

Picture showing cover of Knowledge Spaces: Applications in Education

The book describes up-to-date applications and relevant theoretical results. These applications come from various places, but the most important one, numerically speaking, is the internet based educational system ALEKS. The ALEKS system is bilingual English-Spanish and covers all of mathematics, from third grade to the end of high school, and chemistry. It is also widely used in higher education because US students are often poorly prepared when they reach the university level. The chapter by Taagepera and Arasasingham deals with the application of knowledge spaces, independent of ALEKS, to the teaching of college chemistry. The four chapters by Albert and his collaborators strive to give cognitive interpretations to the combinatoric structures obtained and used by the ALEKS system. The contribution by Eppstein is technical and develops means of searching the knowledge structure efficiently.
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Falmagne J.-C., Doignon, J.-P. (2011)
Learning Spaces: Interdisciplinary Applied Mathematics. Berlin: Springer-Verlag.

Picture showing cover of Learning Spaces: Interdisciplinary Applied Mathematics

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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Albert, D., & Lukas, J. (Eds.) (1999). Knowledge Spaces: Theories, Empirical Research, Applications. Mahwah, NJ: Lawrence Erlbaum Associates.
Picture showing cover of Knowledge Spaces: Theories, Empirical Research, Applications


  • Knowledge Structures: What they are and how they can be used in Cognitive Psychology, Test Theory, and the Design of Learning Environments (Joseph Lukas & Dietrich Albert)

Theoretical Developments and Empirical Investigations

  • Component-Based Knowledge Spaces in Problem Solving and Inductive Reasoning (Dietrich Albert & Theo Held)
  • Component-Based Construction of Surmise Relations for Chess Problems (Martin Schrepp, Theo Held, & Dietrich Albert)
  • An Integrated Approach for Constructing, Coding, and Structuring a Body of Word Problems (Theo Held)
    Modeling Knowledge as Competence and Performance (Klaus Korossy)
  • An Empirical Test of a Process Model for Letter Series Completion Problems (Martin Schrepp)


  • Organizing and Controlling Learning Processes within Competence-Performance Structures (Klaus Korossy)
  • Structure and Design of an INtelligent Tutorial System based on Skill Assessment (Dietrich Albert & Martin Schrepp)
  • Application of Doignon and Falmagne’s Theory of Knowledge Structures to the Assessment of Motor Learning Processes (Susanne Narciss)
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Doignon, J.-P., & Falmagne, J.-C. (1999)
Knowledge Spaces. Berlin: Springer-Verlag.

Picture showing cover of Knowledge Spaces

  • Overview and Mathematical Glossary
  • Knowledge Structure and Spaces
  • Well-Graded Knowledge Structures
  • Surmise Systems
  • Skill Maps, Labels and Filters
  • Entailments and the Maximal Mesh
  • Galois Connections
  • Probabilistic Knowledge Structures
  • Stochastic Learning Paths
  • Descriptive and Assessment Languages
  • Uncovering the State of an Individual: A Continuous Markov Procedure
  • Uncovering the State of an Individual: A Markov Chain Procedure
  • Building the Knowledge Structure in Practise
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