The Assessment of Knowledge, in Theory and in Practice

  • Jean-Claude Falmagne
  • Eric Cosyn
  • Jean-Paul Doignon
  • Nicolas Thiéry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3874)


This paper is adapted from a book and many scholarly articles. It reviews the main ideas of a theory for the assessment of a student’s knowledge in a topic and gives details on a practical implementation in the form of a software. The basic concept of the theory is the ‘knowledge state,’ which is the complete set of problems that an individual is capable of solving in a particular topic, such as Arithmetic or Elementary Algebra. The task of the assessor—which is always a computer—consists in uncovering the particular state of the student being assessed, among all the feasible states. Even though the number of knowledge states for a topic may exceed several hundred thousand, these large numbers are well within the capacity of current home or school computers. The result of an assessment consists in two short lists of problems which may be labelled: ‘What the student can do’ and ‘What the student is ready to learn.’ In the most important applications of the theory, these two lists specify the exact knowledge state of the individual being assessed. Moreover, the family of feasible states is specified by two combinatorial axioms which are pedagogically sound from the standpoint of learning. The resulting mathematical structure is related to closure spaces and thus also to concept lattices. This work is presented against the contrasting background of common methods of assessing human competence through standardized tests providing numerical scores. The philosophy of these methods, and their scientific origin in nineteenth century physics, are briefly examined.


Word Problem Knowledge Structure Precedence Relation Knowledge State Mathematical Psychology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Albert, D. (ed.): Knowledge Structures. Springer, New York (1994)Google Scholar
  2. Albert, D., Lukas, J. (eds.): Knowledge Spaces: Theories, Empirical Research, Applications. Lawrence Erlbaum Associates, Mahwah (1999)Google Scholar
  3. Birkhoff, G.: Rings of sets. Duke Mathematical Journal 3, 443–454 (1937)CrossRefMathSciNetGoogle Scholar
  4. Cosyn, E., Uzun, H.B.: Axioms for Learning Spaces. Journal of Mathematical Psychology (2005) (to be submitted)Google Scholar
  5. Doignon, J.-P., Falmagne, J.-C.: Spaces for the assessment of knowledge. International Journal of Man-Machine Studies 23, 175–196 (1985)zbMATHCrossRefGoogle Scholar
  6. Doignon, J.-P., Falmagne, J.-C.: Knowledge Spaces. Springer, Berlin (1999)zbMATHGoogle Scholar
  7. Dowling, C.E.: Applying the basis of a knowledge space for controlling the questioning of an expert. Journal of Mathematical Psychology 37, 21–48 (1993a)zbMATHCrossRefMathSciNetGoogle Scholar
  8. Dowling, C.E., Hockemeyer, C.: Computing the intersection of knowledge spaces using only their basis. In: Dowling, C.E., Roberts, F.S., Theuns, P. (eds.) Recent Progress in Mathematical Psychology, pp. 133–141. Lawrence Erlbaum Associates Ltd., Mahwah (1998)Google Scholar
  9. Dowling, C.E.: On the irredundant construction of knowledge spaces. Journal of Mathematical Psychology 37, 49–62 (1993b)zbMATHCrossRefMathSciNetGoogle Scholar
  10. Falmagne, J.-C., Doignon, J.-P.: A class of stochastic procedures for the assessment of knowledge. British Journal of Mathematical and Statistical Psychology 41, 1–23 (1988a)zbMATHMathSciNetGoogle Scholar
  11. Falmagne, J.-C., Doignon, J.-P.: A Markovian procedure for assessing the state of a system. Journal of Mathematical Psychology 32, 232–258 (1988b)zbMATHCrossRefMathSciNetGoogle Scholar
  12. Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Berlin (1999); Mathematical foundations, Translated from the 1996 German original by C. FranzkeGoogle Scholar
  13. Heller, J.: A formal framework for characterizing querying algorithms. Journal of Mathematical Psychology 48, 1–8 (2004)CrossRefMathSciNetGoogle Scholar
  14. Kelvin, W.T.: Popular Lectures and Addresses, vol. 1-3. MacMillan, London (1889); Electrical Units of Measurement. In: Constitution of Matter, vol. 1Google Scholar
  15. Koppen, M.: Extracting human expertise for constructing knowledge spaces: An algorithm. Journal of Mathematical Psychology 37, 1–20 (1993)zbMATHCrossRefGoogle Scholar
  16. Koppen, M.: The construction of knowledge spaces by querying experts. In: Fischer, G.H., Laming, D. (eds.) Contributions to Mathematical Psychology, Psychometrics, and Methodology, pp. 137–147. Springer, New York (1994)Google Scholar
  17. Pearson, K.: The Life, Letters and Labours of Francis Galton, vol. 2. Cambridge University Press, London (1924) Researches of Middle LifeGoogle Scholar
  18. Roberts, F.S.: Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  19. Rusch, A., Wille, R.: Knowledge spaces and formal concept analysis. In: Bock, H.-H., Polasek, W. (eds.) Data analysis and information systems, pp. 427–436. Springer, Heidelberg (1996)Google Scholar
  20. Suck, R.: The basis of a knowledge space and a generalized interval order. In: Abstract of a Talk presented at the OSDA 1998, Amherst, MA, September 1998. Electronic Notes in Discrete Mathematics, vol. 2 (1999a)Google Scholar
  21. Suck, R.: A dimension–related metric on the lattice of knowledge spaces. Journal of Mathematical Psychology 43, 394–409 (1999b)zbMATHCrossRefMathSciNetGoogle Scholar
  22. Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore (1992)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jean-Claude Falmagne
    • 1
  • Eric Cosyn
    • 2
  • Jean-Paul Doignon
    • 3
  • Nicolas Thiéry
    • 2
  1. 1.Dept. of Cognitive SciencesUniversity of CaliforniaIrvineUSA
  2. 2.ALEKS Corporation 
  3. 3.Free University of Brussels 

Personalised recommendations