Educational Studies in Mathematics

, Volume 39, Issue 1–3, pp 131–147

Using Intuition From Everyday Life in 'Filling' the gap in Children's Extension of Their Number Concept to Include the Negative Numbers

  • Liora Linchevski
  • Julian Williams


We report here an instructional method designed to address the cognitive gaps in children's mathematical development where operational conceptions give rise to structural conceptions (such as when the subtraction process leads to the negative number concept). The method involves the linking of process and object conceptions through semiotic activity with models which first record processes in situations outside mathematics and subsequently mediate activity with the signs of mathematics. We describe two experiments in teaching integers, an interesting case in which previous literature has focused on the dichotomy between the algebraic approach and the modelling approach to instruction. We conceptualise modelling as the transformation of outside-school knowledge into school mathematics, and discuss the opportunities and difficulties involved.


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  1. Barton, B.: 1996, 'Making sense of ethnomathematics: ethnomathematics is making sense', Educational Studies in Mathematics 31, 201–233.CrossRefGoogle Scholar
  2. Behr, M., Lesh, R., Post, T. and Silver, E.: 1983, 'Rational number concepts', in R. Lesh and M. Landau (eds.), Acquisition of Mathematical Concepts and Processes, Academic Press, New York.Google Scholar
  3. Brown, J. S., Collins, A. and Duguid, P.: 1989, 'Situated cognition and the culture of learning', Educational Researcher 18(1), 32–42.CrossRefGoogle Scholar
  4. Chaiklin, S. and Lave, J.: 1993, Understanding Practice, Cambridge University Press, Cambridge, U.K.Google Scholar
  5. Dirks, M. K.: 1984, 'The integer abacus', Arithmetic Teacher 31(7), 50–54.Google Scholar
  6. Engestrom, Y.: 1991, 'Non scolae sed vitae discimus: Toward overcoming the encapsulation of school learning', Learning and Instruction 1(3), 243–259.CrossRefGoogle Scholar
  7. Fischbein, E.: 1987, Intuition in Science and Mathematics, D. Reidel, Dordrecht.Google Scholar
  8. Freudenthal, H.: 1983, Didactical Phenomenology of Mathematical Structures, Kluwer Academic Publishers, Dordrecht.Google Scholar
  9. Glaeser, G.: 1981, 'Epistemologie des nombres relatifs', Recherches en Didactiques des Mathematiques 2(3), 303–346.Google Scholar
  10. Gravemeijer, K.: 1994, 'Educational development and developmental research in mathematics education', Journal for Research in Mathematics Education 25(5), 443–471.CrossRefGoogle Scholar
  11. Gray, E. M. and Tall, D. O.: 1994, 'Duality, ambiguity, and flexibility: a 'proceptual' view of simple arithmetic', Journal for Research in Mathematics Education 25(2), 116–140.CrossRefGoogle Scholar
  12. Heckman, P. E. and Weissglass, J.: 1994, 'Contextualised mathematics instruction: moving beyond recent proposals', For the Learning of Mathematics 14(1), 29–33.Google Scholar
  13. Hefendehl-Hebeker, L.: 1991, 'Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs', For The Learning of Mathematics 11(1), 26–32.Google Scholar
  14. Lave, J.: 1988, Cognition in Practice: Mind, Mathematics and Culture in Everyday Life, Cambridge University Press, New York.Google Scholar
  15. Lave, J.: 1996, 'Teaching, as learning, in practice', Mind, Culture, and Activity 3(3), 149–164.CrossRefGoogle Scholar
  16. Lave, J. and Wenger, E.: 1991, Situated Learning: Legitimate Peripheral Participation, Cambridge University Press, Cambridge, UK.Google Scholar
  17. Leont'ev, A. N.: 1981, 'The problem of activity in soviet psychology', in J. V. Wertsch (trans. and ed.) The Concept of Activity in Soviet Psychology, M. E. Sharpe, Armonk, N.Y., pp. 37–71.Google Scholar
  18. Liebeck, P.: 1990, 'Scores and forfeits — an intuitive model for integer arithmetic', Educational Studies in Mathematics 21(3), 221–239.CrossRefGoogle Scholar
  19. Linchevski, L. and Williams, J. S.: 1996, 'Situated intuition, concrete manipulations and mathematical concepts: the case of integers', Proceedings of the Twentieth International Group for the Psychology of Mathematics Education Vol. 3, University of Valencia, Valencia, pp. 265–272.Google Scholar
  20. Lytle, P. A.: 1994, 'Investigation of a model based on neutralization of opposites to teach integers', Proceedings of the Nineteenth International Group for the Psychology of Mathematics Education, Universidade Federal de Pernambuco, Recife, Brazil, pp. 192–199.Google Scholar
  21. Semadeni, Z.: 1984, 'A principle of concretization permanence for the formation of arithmetical concepts', Educational Studies in Mathematics 15(4), 379–395.CrossRefGoogle Scholar
  22. Sfard, A.: 1991, 'The dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin', Educational Studies in Mathematics 22, 1–36.CrossRefGoogle Scholar
  23. Sfard, A. and Linchevski, L.: 1994, 'The gains and pitfalls of reification: the case of algebra', Educational Studies in Mathematics 26, 87–124.CrossRefGoogle Scholar
  24. Treffers, A.: 1987, Three Dimensions: A model of Goal and Theory Description in Mathematics Instruction — the Wiskobas Project, Kluwer Academic Publishers, Dordrecht.Google Scholar
  25. Walkerdine, V.: 1988, The Mastery of Reason: Cognitive Development and the Production of Rationality, Routledge, London.Google Scholar
  26. Wertsch, J. V.: 1991, Voices of the Mind: a Sociocultural Approach to Mediated Action, Harvester, London.Google Scholar
  27. Wertsch, J. V.: 1996, 'The primacy of mediated action in sociocultural theory', Mind, Culture, and Activity 1(4), 202–208.Google Scholar
  28. Williams, J. S. and Linchevski, L.: 1997, Situated Intuitions, Concrete Manipulations and the Construction of the Integers: Comparing two Experiments, paper presented to the annual conference of the American Educational Research Association, Chicago, March 1997.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Liora Linchevski
    • 1
  • Julian Williams
    • 2
  1. 1.Liora Linchevski, School of EducationHebrew University of JerusalemMount ScopusIsrael
  2. 2.Faculty of EducationUniversity of ManchesterU.K.

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