Educational Studies in Mathematics

, Volume 22, Issue 1, pp 1–36 | Cite as

On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin

  • Anna Sfard


This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: structurally-as objects, and operationally-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions.

On the grounds of historical examples and in the light of cognitive schema theory we conjecture that the operational conception is, for most people, the first step in the acquisition of new mathematical notions. Thorough analysis of the stages in concept formation leads us to the conclusion that transition from computational operations to abstract objects is a long and inherently difficult process, accomplished in three steps: interiorization, condensation, and reification. In this paper, special attention is given to the complex phenomenon of reification, which seems inherently so difficult that at certain levels it may remain practically out of reach for certain students.


Schema Theory Complex Phenomenon Abstract Object Mathematical Conception Concept Formation 
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. Anderson, J. R.: 1976, Language, Memory, and Thought, Erlbaum, Hillsdale, N.J.Google Scholar
  2. Behr, M., Erlwanger, S., and Nichols, E.: 1976, How children view equality sentences (PMDC Technical Report No. 3), Florida State University (ERIC Document Reproduction Service No. ED144802).Google Scholar
  3. Bishop, A. J.: 1988, ‘A review of research on visualisation in mathematics education’, in Proceedings of The Twelfth International Conference for the Psychology of Mathematics Education, Hungary, Vol. 1, pp. 170–176.Google Scholar
  4. Brownell, W. A.: 1935, ‘Psychological considerations in learning and teaching arithmetic’, in The Teaching of Arithmetic: Tenth Yearbook of the NCTM, Columbia University Press, New York.Google Scholar
  5. Cajori, F. A.: 1985, History of Mathematics, Fourth Edition, Chelsea Publishing Company, New York.Google Scholar
  6. Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., and Reys, R.: 1980, ‘Results of the second NAEP mathematics assessment: Secondary school’, The Mathematics Teacher 73(5), 329–338.Google Scholar
  7. Clements, K.: 1981, ‘Visual imagery and school mathematics’, For the Learning of Mathematics 2(2), 33–38.Google Scholar
  8. Courant, R. and John, F.: 1965, Introduction to Calculus and Analysis, Vol. I, Interscience Publishers, New York.Google Scholar
  9. Davis, R. B.: 1975, ‘Cognitive processes involved in solving simple algebraic equations’, Journal of Children's Mathematical Behavior 1(3), 7–35.Google Scholar
  10. Davis, P. J. and Hersh, R.: 1983, The Mathematical Experience, Penguin Books, London.Google Scholar
  11. Dörfler, W.: 1987, ‘Empirical investigation of the construction of cognitive schemata from actions’, in Proceedings of the Eleventh International Conference of PME, Vol. III, pp. 3–9.Google Scholar
  12. Dörfler, W.: 1989, ‘Protocols of actions as a cognitive tool for knowledge construction’, in Proceedings of the Thirteenth International Conference of PME, Paris, Vol. 1, pp. 212–9.Google Scholar
  13. Dubinsky, E. and Lewin, P.: 1986, ‘Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness’, Journal of Mathematical Behavior 5, 55–92.Google Scholar
  14. Eisenberg, T. and Dreyfus, T. (eds.): 1989, Visualization in the mathematics curriculum, Special issue of Focus on Learning Problems in Mathematics 11(1&2).Google Scholar
  15. Frege, G.: 1970, ‘What is function’, in Geach, P. and Black, M. (eds.), Translations from the Philosophical Writings of Gottlob Frege, Blackwell, Oxford.Google Scholar
  16. Hadamard, J. S.: 1949, The Psychology of Invention in the Mathematics Field, Princeton University Press, NJ.Google Scholar
  17. Halmos, P. R.: 1985, ‘Pure thought is better yet...’, The College Mathematics Journal 16, 14–16.Google Scholar
  18. Halmos, P. R.: 1985a, I Want to be a Mathematician, An Autobiography, Springer, New York.Google Scholar
  19. Henrici, P.: 1974, ‘The influence of computing on mathematical research and education’, in Proceedings of Symposia in Applied Mathematics, Vol. 20, American Mathematical Society, Providence.Google Scholar
  20. Hiebert, J. and Lefevre, P.: 1986, ‘Conceptual and procedural knowledge in mathematics: An introductory analysis’, in Hiebert, J. (ed.), Conceptual and Procedural Knowledge: The Case of Mathematics, Erlbaum, Hillsdale, NJ.Google Scholar
  21. Jourdain, P. E. B.: 1956, ‘The nature of mathematics’, in Newman, J. R. (ed.), The World of Mathematics, Simon and Schuster, New York.Google Scholar
  22. Kaput, J. J.: 1979, ‘Mathematics and learning: Roots of epistemological status’, in Lochhead, J. and Clement, J. (eds.), Cognitive Process Instruction, Franklin Institute Press.Google Scholar
  23. Kieran, C.: 1981, ‘Concepts associated with the equality symbol’, Educational Studies in Mathematics 12(3), 317–326.Google Scholar
  24. Kilpatrick, J.: 1988, ‘Editorial’, Journal for Research in Mathematics Education 19(4).Google Scholar
  25. Kleiner, I.: 1988, ‘Evolution of the function concept: A brief survey’, College Mathematics Journal 20(4), 282–300.Google Scholar
  26. Lesh, R. and Landau, M. (eds.): 1983, Acquisition of Mathematics Concepts and Processes, Academic Press, New York.Google Scholar
  27. Markovits, Z., Eylon, B., and Bruckheimer, M.: 1986, ‘Functions today and yesterday’, For the Learning of Mathematics 6(2), 18–24.Google Scholar
  28. Maurer, S. B.: 1985, ‘The algorithmic way of life is best’, College Mathematics Journal 16, 2–5.Google Scholar
  29. Miller, G. A.: 1956, ‘The magic number seven plus minus two: Some limits on our capacity for processing information’, Psychological Review 63, 81–96.Google Scholar
  30. Otte, M.: 1984, ‘Komplementarität’, Dialektik 8, 60–75.Google Scholar
  31. Paivio, A.: 1971, Imagery and Verbal Processes, Holt, Rinehart, and Winston, New York.Google Scholar
  32. Penrose, R.: 1989, The Emperor's New Mind, Oxford University Press, Oxford.Google Scholar
  33. Piaget, J.: 1952, The Child's Conception of Number, Routledge and Kegan, London.Google Scholar
  34. Piaget, J.: 1970, Genetic Epistemology, W. W. Norton, New York.Google Scholar
  35. Poincaré, H.: 1952, Science and Method, Dover Publications, New York.Google Scholar
  36. Sfard, A.: 1987, ‘Two conceptions of mathematical notions: operational and structural’, in Proceedings of the Eleventh International Conference of PME, Montreal, Vol. 3, pp. 162–9.Google Scholar
  37. Sfard, A.: 1988, ‘Operational vs structural method of teaching mathematics: A case study’, in Proceedings of the Twelfth International Conference of PME, Hungary, pp. 560–7.Google Scholar
  38. Sfard, A.: 1989, ‘Transition from operational to structural conception: the notion of function revisited’, in Proceedings of the Thirteenth International Conference of PME, Paris, Vol. 3, pp. 151–8.Google Scholar
  39. Sinclair, H. and Sinclair, A.: 1986, ‘Children's mastery of written numerals and the construction of basic number concepts’, in Hiebert, J. (ed.), Conceptual and Procedural Knowledge: The Case of Mathematics, Erlbaum, Hillsdale, N.J.Google Scholar
  40. Skemp, R. R.: 1971, The Psychology of Learning Mathematics, Penguin Books, Harmondsworth, England.Google Scholar
  41. Skemp, R. R.: 1976, ‘Relational understanding and instrumental understanding’, Mathematics Teacher 77, 20–26.Google Scholar
  42. Stein, S. K.: 1988, ‘Gersham's law: Algorithm drives out thought’, Journal of Mathematical Behavior 7, 79–84.Google Scholar
  43. Steiner, H.-G.: 1985, ‘Theory of mathematics education: An introduction’, For the Learning of Mathematics 5(2), 11–17.Google Scholar
  44. Tahta, D. (ed.): 1972, A Boolean Anthology, Derby, ATM.Google Scholar
  45. Thompson, P. W.: 1985, ‘Experience, problem solving, and learning mathematics: considerations in developing mathematical curricula’, in E. A.Silver (ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, Erlbaum, Hillsdale, N.J.Google Scholar
  46. Vinner, S. and Dreyfus, T.: 1989, ‘Images and definitions for the concept of function’, Journal for Research in Mathematics Education 20(5), 356–66.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Anna Sfard
    • 1
  1. 1.The Science Teaching CentreThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations