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The characteristics of mathematical creativity

Abstract

Mathematical creativity ensures the growth of mathematics as a whole. However the source of this growth, the creativity of the mathematician is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics; a qualitative study involving five creative mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview transcripts and to verify the theory driven hypotheses. The results indicate that in general, the mathematicians’ creative process followed the four-stage Gestalt model of preparation–incubation–illumination–verification. It was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity. In addition contemporary models of creativity from psychology were reviewed and used to interpret the characteristics of mathematical creativity.

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Notes

  1. 1.

    In all vignettes I = interviewer; A, B, C, D, E = mathematicians.

References

  1. Amabile, T. M. (1983). Social psychology of creativity: A componential conceptualization. Journal of Personality and Social Psychology, 45, 357–376.

  2. Arnheim, R. (1962). Picasso’s Guernica. Berkeley: University of California Press.

  3. Birkhoff, G. (1969). Mathematics and psychology. SIAM Review, 11, 429–469.

  4. Corbin, J., & Strauss, A. (1998). Basics of qualitative research. Thousand Oaks: Sage Publications.

  5. Csikszentmihalyi, M. (1988). Society, culture, and person: A systems view of creativity. In R. J. Sternberg (Ed.), The nature of creativity: Contemporary psychological perspectives (pp. 325–339). Cambridge: Cambridge University Press.

  6. Csikszentmihalyi, M. (2000). Implications of a systems perspective for the study of creativity. In R. J. Sternberg (Ed.), Handbook of creativity (pp. 313–338). Cambridge: Cambridge University Press.

  7. Davis, P. J., & Hersh, R. (1981). The mathematical experience. New York: Houghton Mifflin.

  8. English, L. D. (1991). Young children’s combinatoric strategies. Educational Studies in Mathematics, 22, 451–474.

  9. English, L. D. (1993). Children’s strategies in solving two- and three-dimensional combinatorial problems. Journal for Research in Mathematics Education, 24(3), 255–273.

  10. Ernest, P. (1991). The Philosophy of mathematics education. Briston: The Falmer Press.

  11. Ernest, P. (1994). Conversation as a metaphor for mathematics and learning. Proceedings of the British society for research into learning mathematics day conference, Manchester Metropolitan University (pp. 58–63). Nottingham: BSRLM.

  12. Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42–53). Dordrecht: Kluwer Academic Publishers.

  13. Frensch, P., & Sternberg, R. (1992). Complex problem solving: Principles and mechanisms. Mahwah: Lawrence Erlbaum and Associates.

  14. Gallian, J. A. (1994). Contemporary abstract algebra. Lexington: D.C. Heath and Co.

  15. Gardner, H. (1993). Frames of mind. New York: Basic Books.

  16. Gardner, H. (1997). Extraordinary minds. New York: Basic Books.

  17. Gruber, H. E. (1981). Darwin on man. Chicago: University of Chicago Press.

  18. Gruber, H. E., & Wallace, D. B. (2000). The case study method and evolving systems approach for understanding unique creative people at work. In R. J. Sternberg (Ed.), Handbook of creativity (pp. 93–115). Cambridge: Cambridge University Press.

  19. Hadamard, J. W. (1945). Essay on the psychology of invention in the mathematical field. Princeton: Princeton University Press. (page references are to Dover edition, New York 1954).

  20. Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking (pp. 54–60). Dordrecht: Kluwer Academic Publishers.

  21. Hung, D. (2000). Some insights into the generalizations of mathematical meanings. Journal of Mathematical Behavior, 19, 63–82.

  22. Krutetskii, V. A. (1976). In: J. Kilpatrick, I. Wirszup (Eds.) & J. Teller (Trans.), The psychology of mathematical abilities in school children. Chicago: University of Chicago Press.

  23. L’Enseigement Mathematique (1902), 4, 208–211, and (1904), 6, 376.

  24. Lester, F. K. (1985). Methodological considerations in research on mathematical problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving. Multiple research perspectives (pp. 41–70). Hillsdale: Lawrence Erlbaum and Associates.

  25. Maher, C. A., & Kiczek, R. D. (2000). Long term building of mathematical ideas related to proof making. Contributions to Paolo Boero, G. Harel, C. Maher, M. Miyasaki. (organisers) Proof and proving in mathematics education. ICME9-TSG 12. Tokyo/Makuhari, Japan.

  26. Maher, C. A., & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.

  27. Maher, C. A., & Speiser, M. (1997). How far can you go with block towers? Stephanie’s intellectual development. Journal of Mathematical Behavior, 16(2), 125–132.

  28. Manin, Y. I. (1977). A course in mathematical logic. New York: Springer.

  29. Minsky, M. (1985). The society of mind. New York: Simon & Schuster Inc.

  30. Muir, A. (1988). The psychology of mathematical creativity. Mathematical Intelligencer, 10(1), 33–37.

  31. Nicolle, C. (1932). Biologie de l’invention. Paris: Alcan.

  32. Patton, M. Q. (2002). Qualitative research and evaluation methods. Thousand Oaks: Sage Publications.

  33. Policastro, E., & Gardner, H. (2000). From case studies to robust generalizations: An approach to the study of creativity. In R. J. Sternberg (Ed.), Handbook of creativity (pp. 213–225). Cambridge: Cambridge University Press.

  34. Poincaré, H. (1948). Science and method. New York: Dover.

  35. Polya, G. (1945). How to solve it. Princeton: Princeton University Press.

  36. Polya, G. (1954). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. II). Princeton: Princeton University Press.

  37. Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic.

  38. Skemp, R. (1986). The psychology of learning mathematics. London: Penguin Books.

  39. Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations. The Journal of Secondary Gifted Education, XIV(3), 151–165.

  40. Sriraman, B. (2004a). The influence of Platonism on mathematics research and theological beliefs. Theology and Science, 2(1), 131–147.

  41. Sriraman, B. (2004b). Discovering a mathematical principle: The case of Matt. Mathematics in School, 33(2), 25–31.

  42. Sternberg, R. J. (1979). Human intelligence: Perspectives on its theory and measurement. Norwood: Ablex Publishing Co.

  43. Sternberg, R. J. (1985). Human abilities: An information processing approach. New York: W. H. Freeman.

  44. Sternberg, R. J. (2000). Handbook of creativity. Cambridge: Cambridge University Press.

  45. Sternberg, R. J., & Lubart, T. I. (1996). Investing in creativity. American Psychologist, 51, 677–688.

  46. Sternberg, R. J., & Lubart, T. I. (2000). The concept of creativity: Prospects and paradigms. In R. J. Sternberg (Ed.), Handbook of creativity (pp. 93–115). Cambridge: Cambridge University Press.

  47. Taylor, S. J., & Bogdan, R. (1984). Introduction to qualitative research methods: The search for meanings. New York: Wiley.

  48. Torrance, E. P. (1974). Torrance tests of creative thinking: Norms-technical manual. Lexington: Ginn.

  49. Ulam, S. (1976). Adventures of a mathematician. New York: Scribner’s.

  50. Usiskin, Z. P. (1987). Resolving the continuing dilemmas in school geometry. In M. M. Lindquist & A. P. Shulte (Eds.), Learning and teaching geometry, K–12: 1987 Yearbook (pp. 17–31). Reston: National Council of Teachers of Mathematics.

  51. Wallas, G. (1926). The art of thought. New York: Harcourt, Brace & Jovanovich.

  52. Weisberg, R. W. (1993). Creativity: Beyond the myth of genius. New York: Freeman.

  53. Wertheimer, M. (1945). Productive thinking. New York: Harper.

  54. Wittgenstein, L. (1978). Remarks on the foundations of mathematics (Rev. ed.). Cambridge: Massachusetts Institute of Technology Press.

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Author information

Correspondence to Bharath Sriraman.

Additional information

Reprint of Sriraman, B. (2004). The characteristics of mathematical creativity. The Mathematics Educator. 14 (1), 19–34. Reprinted with permission from Mathematics Education Student Association at The University of Georgia. ©2004 Bharath Sriraman.

Appendix: Interview protocol

Appendix: Interview protocol

(The interview instrument was developed by modifying questions from questionnaires in L’Enseigement Mathematique (1902) and Muir (1988))

  1. 1.

    Describe your place of work and your role within it?

  2. 2.

    Are you free to choose the mathematical problems you tackle or are they determined by your work place?

  3. 3.

    Do you work and publish mainly as an individual or as part of a group?

  4. 4.

    Is supervision of Research a positive or negative factor in your work?

  5. 5.

    Do you structure your time for mathematics?

  6. 6.

    What are your favorite leisure activities apart from mathematics?

  7. 7.

    Do you recall any immediate family influences, teachers, colleagues or texts, of primary importance in your mathematical development?

  8. 8.

    In which areas were you initially self-educated? In which areas do you work now? If different, what have been the reasons for changing?

  9. 9.

    Do you strive to obtain a broad overview of mathematics, not of immediate relevance to your area of research?

  10. 10.

    Do you make a distinction between thought processes in learning and research?

  11. 11.

    When you are about to begin a new topic, do you prefer to assimilate what is known first or do you try your own approach?

  12. 12.

    Do you concentrate on one problem for a protracted period of time or on several problems at the same time?

  13. 13.

    Have your best ideas been the result of prolonged deliberate effort, or have they occurred when you were engaged in other unrelated tasks?

  14. 14.

    How do you form an intuition about the truth of a proposition?

  15. 15.

    Do computers play a role in your creative work (mathematical thinking)?

  16. 16.

    What types of mental imagery do you use when thinking about mathematical objects?

Questions regarding foundational and theological issues have been omitted in this protocol. The discussion resulting from these questions are reported in Sriraman, B. (2004). The influence of Platonism on mathematics research and theological beliefs. Theology and Science, 2 (1), 131–147.

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Sriraman, B. The characteristics of mathematical creativity. ZDM Mathematics Education 41, 13 (2009) doi:10.1007/s11858-008-0114-z

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Keywords

  • Domain specific creativity
  • Gestalt psychology
  • Jacques Hadamard
  • Systems views of creativity
  • Theories of creativity
  • Mathematical creativity