, Volume 45, Issue 2, pp 253–265 | Cite as

Illumination: an affective experience?

  • Peter LiljedahlEmail author
Original Article


What is the nature of illumination in mathematics? That is, what is it that sets illumination apart from other mathematical experiences? In this article the answer to this question is pursued through a qualitative study that seeks to compare and contrast the AHA! experiences of preservice teachers with those of prominent research mathematicians. Using a methodology of analytic induction in conjunction with historical and contemporary theories of discovery, creativity, and invention along with theories of affect the anecdotal reflections of participants from these two populations are analysed. Results indicate that, although manifested differently in the two populations, what sets illumination apart from other mathematical experiences are the affective aspects of the experience.


Preservice Teacher Positive Emotion Creative Process Mathematical Idea Research Mathematician 
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.


  1. Ajzen, I. (1988). Attitudes, personality, and behaviour. Milton Keynes: Open University Press.Google Scholar
  2. Ashcraft, M. (1989). Human memory and cognition. Glenview: Scott, Foresman and Company.Google Scholar
  3. Bailin, S. (1994). Achieving extraordinary ends: An essay on creativity. Norwood: Ablex Publishing Corporation.Google Scholar
  4. Barnes, M. (2000). ‘Magical’ moments in mathematics: Insights into the process of coming to know. For the Learning of Mathematics, 20(1), 33–43.Google Scholar
  5. Bruner, J. (1964). Bruner on knowing. Cambridge: Harvard University Press.Google Scholar
  6. Burton, L. (1999). The practices of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37(2), 121–143.CrossRefGoogle Scholar
  7. Csikszentmihalyi, C. (1996). Creativity: Flow and the psychology of discovery and invention. New York: HarperCollins Publishers.Google Scholar
  8. Davis, P., & Hersch, R. (1980). The mathematical experience. Boston: Birkhauser.Google Scholar
  9. DeBellis, V., & Goldin, G. (2006). Affect and meta-affect in mathematical problem solving: A representational perspective. Educational Studies in Mathematics, 63(2), 131–147.CrossRefGoogle Scholar
  10. Dewey, J. (1933). How we think. Boston: D.C. Heath and Company.Google Scholar
  11. Feynman, R. (1999). The pleasure of finding things out. Cambridge: Perseus Publishing.Google Scholar
  12. Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht: Kluwer Academic Publishers Group.Google Scholar
  13. Ghiselin, B. (1952). The creative process: Reflections on invention in the arts and sciences. Berkeley: University of California Press.Google Scholar
  14. Glaser, B. & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago: Aldine Publishing Co.Google Scholar
  15. Hadamard, J. (1945). The psychology of invention in the mathematical field. New York: Dover Publications.Google Scholar
  16. Kneller, G. (1965). The art and science of creativity. New York: Holt, Reinhart, and Winstone, Inc.Google Scholar
  17. Koestler, A. (1964). The act of creation. New York: The Macmillan Company.Google Scholar
  18. Liljedahl, P. (2009). In the words of the creators. In R. Leikin, A. Berman, & B. Koichu (Eds.), Mathematical creativity and the education of gifted children (pp. 51–70). Rotterdam: Sense Publishers.Google Scholar
  19. Liljedahl, P., & Allen, D. (in press). Mathematical discovery. In E. G. Carayannis (Ed.), Encyclopedia of creativity, invention, innovation, and entrepreneurship. New York: Springer.Google Scholar
  20. Liljedahl, P., & Sriraman, B. (2006). Musings on mathematical creativity. For The Learning of Mathematics, 26(1), 20–23.Google Scholar
  21. McLeod, D. (1992). Research on the affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575–596). New York: Macmillan.Google Scholar
  22. Nova (1993). The proof. Aired on PBS on October 28, 1997. Accessed 11 December 2003.
  23. O’pt Eynde, P., De Corte, E., & Verschaffel, L. (2001). Problem solving in the mathematics classroom: A socio-constructivist account of the role of students’ emotions. Proceedings of 25th Annual Conference for the Psychology of Mathematics Education, 4, 25–32.Google Scholar
  24. Patton, M. Q. (2002). Qualitative research and evaluation methods. Thousand Oaks: Sage.Google Scholar
  25. Perkins, D. (2000). Archimedes’ bathtub: The art of breakthrough thinking. New York: W.W. Norton & Company.Google Scholar
  26. Poincaré, H. (1952). Science and method. New York: Dover Publications, Inc.Google Scholar
  27. Pólya, G. (1965/1981). Mathematical discovery: On understanding, learning and teaching problem solving (vol. 2). New York: Wiley.Google Scholar
  28. Rota, G. (1997). Indiscrete thoughts. Boston: Birkhauser.Google Scholar
  29. Sfard, A. (2004). Personal Communication.Google Scholar
  30. Sinclair, N. (2002). The kissing triangles: The aesthetics of mathematical discovery. International Journal of Computers for Mathematics Learning, 7(1), 45–63.CrossRefGoogle Scholar
  31. Wallas, G. (1926). The art of thought. New York: Harcourt Brace.Google Scholar
  32. Whittlesea, B. (1993). Illusions of familiarity. Journal of Experimental Psychology: Learning, Memory, and Cognition, 19(7), 1235–1253.CrossRefGoogle Scholar
  33. Whittlesea, B., & Williams, L. (2001). The discrepancy-attribution hypothesis: The heuristic basis of feelings of familiarity. Journal for Experimental Psychology: Learning, Memory, and Cognition, 27(1), 3–13.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2012

Authors and Affiliations

  1. 1.Simon Fraser UniversityBurnabyCanada

Personalised recommendations