Skip to main content
SpringerLink
Log in

Overview and Basic Mathematical Concepts

  • Chapter
  • First Online:
Learning Spaces

  • 1410 Accesses

Abstract

A student is facing a teacher, who is probing his1 knowledge of high school mathematics. The student, a new recruit, is freshly arrived from a foreign country, and important questions must be answered. To which grade should the student be assigned? What are his strengths and weaknesses? Should the student take a remedial course in some subject? Which topics is he ready to learn? The teacher will ask a question and listen to the student’s response. Other questions will then be asked. After a few questions, a picture of the student’s state of knowledge will emerge, which will become increasingly sharper in the course of the examination.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • F.M. Lord. Individualized testing and item characteristic curve theory. In D.H. Krantz, R.C. Atkinson, R.D. Luce, and P. Suppes, editors, Contemporary Developments in Mathematical Psychology, Vol. II. Measurement, Psychophysics and Neural Information Processing. Freeman, San Francisco, CA, 1974.

    Google Scholar 

  • K.S. Fu. Syntactic Methods in Pattern Recognition. Academic Pres, New York, 1974.

    MATH  Google Scholar 

  • D.J. Weiss. New Horizons in Testing: Latent Trait Theory and Computerized Testing Theory. Academic Press, New York, 1983.

    Google Scholar 

  • G. Pilato, R. Pirrone, and R. Rizzo. A KST–based system for student tutoring. Applied Artificial Intelligence, 22:283–308, 2008.

    Article  Google Scholar 

  • M.C. Desmarais and X. Pu. A Bayesian student model without hidden nodes and its comparison with item response theory. International Journal of Artifficial Intelligence in Education, 15:291–323, 2005.

    Google Scholar 

  • R.O. Duda and P.E. Hart. Pattern Classification and Scene Analysis. John Wiley, New York, 1973.

    MATH  Google Scholar 

  • M. Koppen and J.-P. Doignon. How to build a knowledge space by querying an expert. Journal of Mathematical Psychology, 34:311–331, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  • D. Albert, A. Nussbaumer, and C. Steiner. Using visual guidance and feedback based on competence structures for personalising e-learning experience. In ICCE 2008 — The 16th International Conference on Computers in Education, pages 3–10. Asia–Pacific Society for Computers in Education, 2008.

    Google Scholar 

  • J.-P. Doignon and J.-Cl. Falmagne. Spaces for the Assessment of Knowledge. International Journal of Man-Machine Studies, 23:175–196, 1985.

    Article  MATH  Google Scholar 

  • J. Dugundji. Topology. Allyn and Bacon, Boston, 1966.

    MATH  Google Scholar 

  • J.-P. Doignon and J.-Cl. Falmagne. Parametrization of knowledge structures. Discrete Applied Mathematics, 21:87–100, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Durnin and J.M. Scandura. An algorithmic approach to assessing behavioral potential: comparison with item forms and hierarchical technologies. Journal of Educational, 65:262–272, 1973.

    Google Scholar 

  • K. Van Lehn. Student modeling. In M.C. Polson and J.J. Richardson, editors, Foundations of Intelligent Tutoring Systems. Erlbaum, Hillsdale, N.J., 1988.

    Google Scholar 

  • C.E. Dowling, C. Hockemeyer, and A.H. Ludwig. Adaptive assessment and training using the neighbourhood of knowledge spaces. In C. Frasson, G. Gauthier, and A. Lesgold, editors, Intelligent Tutoring Systems, pages 578–586. Springer–Verlag, Berlin, 1996.

    Google Scholar 

  • J. Heller. A formal framework for characterizing querying algorithms. Journal of Mathematical Psychology, 48(1):1–8, 2004.

    Article  MathSciNet  Google Scholar 

  • J.-Cl. Falmagne. Stochastic token theory. Journal of Mathematical Psychology, 41(2):129–143, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • O. Conlan, C. Hockemeyer, V. Wade, and D. Albert. Metadata driven approaches to facilitate adaptivity in personalized eLearning systems. The Journal of Information and Systems in Education, 1:38–44, 2002.

    Google Scholar 

  • C. Hockemeyer. Tools and utilities for knowledge spaces. Unpublished technical report, Institut für Psychologie, Karl–Franzens–Universität Graz, Austria, 2001.

    Google Scholar 

  • G. Birkhoff. Rings of sets. Duke Mathematical Journal, 3:443–454, 1937.

    Article  MathSciNet  Google Scholar 

  • M.D. Kickmeier-Rust, B. Marte, S.B. Linek, T. Lalonde, and D. Albert. Learning with computer games: Micro level feedback and interventions. In M. E. Auer, editor, Proceedings of the International Conference on Interactive Computer Aided Learning (ICL), Kassel, 2008. Kassel University Press. CD–ROM publication.

    Google Scholar 

  • P. Suppes. Axiomatic Set Theory. Van Nostrand, Princeton, N.J., 1960.

    MATH  Google Scholar 

  • D. Albert, editor. Knowledge Structures. Springer-Verlag, Berlin–Heidelberg, 1994.

    Google Scholar 

  • F.S. Roberts. Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences. Addison-Wesley, Reading, Mass., 1979.

    MATH  Google Scholar 

  • K. Jameson. Empirical methods for generative semiotics models: an application to the roman majuscules. In W.C. Watt, editor, Writing Systems and Cognition, Neuropsychology and Cognition Series. Kluwer Academic Publishers, Dordrecht, 1992.

    Google Scholar 

  • D. Eppstein, J.-Cl. Falmagne, and H.B. Uzun. On verifying and engineering the wellgradedness of a union-closed family. Journal of Mathematical Psychology, 53(1):34–39, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  • J.-P. Doignon and J.-Cl. Falmagne. Knowledge Spaces. Springer-Verlag, Berlin, Heidelberg, and New York, 1999.

    MATH  Google Scholar 

  • M. Koppen. Extracting human expertise for constructing knowledge spaces: An algorithm. Journal of Mathematical Psychology, 37:1–20, 1993.

    Article  MATH  Google Scholar 

  • J. Nunnally and I. Bernstein. Psychometric Theory. MacGraw-Hill, New York, 1994.

    Google Scholar 

  • J.-Cl. Falmagne, M. Koppen, M. Villano, J.-P. Doignon, and L. Johannesen. Introduction to knowledge spaces: how to build, test and search them. Psychological Review, 97:204–224, 1990.

    Article  Google Scholar 

  • E. Cosyn and H.B. Uzun. Note on two sufficient axioms for a well-graded knowledge space. Journal of Mathematical Psychology, 53(1):40–42, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  • D. Albert and J. Lukas, editors. Knowledge Spaces: Theories, Empirical Research, Applications. Lawrence Erlbaum Associates, Mahwah, NJ, 1999.

    Google Scholar 

  • D. Albert and Th. Held. Establishing knowledge spaces by systematical problem construction. In D. Albert, editor, Knowledge Structures, volume 1, pages 78–112. Springer–Verlag, Berlin–Heidelberg, 1994.

    Google Scholar 

  • H. Wainer and S. Messick. Principles of Modern Psychological Development: A Festschrift for Frederic Lord. Lawrence Erlbaum Associates, Hillsdale, N.J., 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Cognitive Sciences, Institute of Mathematical Behavioral Sciences, University of California, Irvine, 92697-5100, Irvine, California, USA

    Prof. Dr. Jean-Claude Falmagne

  2. Département de Mathématique, Université Libre de Bruxelles, c.p. 216 Bd du Triomphe, 1050, Bruxelles, Belgium

    Prof. Dr. Jean-Paul Doignon

Authors
  1. Prof. Dr. Jean-Claude Falmagne
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Prof. Dr. Jean-Paul Doignon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Claude Falmagne .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Falmagne, JC., Doignon, JP. (2011). Overview and Basic Mathematical Concepts. In: Learning Spaces. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01039-2_1

Download citation

Publish with us

Policies and ethics

Search

Navigation