Educational Studies in Mathematics

, Volume 71, Issue 3, pp 263–278 | Cite as

Intuitive vs analytical thinking: four perspectives

  • Uri Leron
  • Orit Hazzan


This article is an attempt to place mathematical thinking in the context of more general theories of human cognition. We describe and compare four perspectives—mathematics, mathematics education, cognitive psychology, and evolutionary psychology—each offering a different view on mathematical thinking and learning and, in particular, on the source of mathematical errors and on ways of dealing with them in the classroom. The four perspectives represent four levels of explanation, and we see them not as competing but as complementing each other. In the classroom or in research data, all four perspectives may be observed. They may differentially account for the behavior of different students on the same task, the same student in different stages of development, or even the same student in different stages of working on a complex task. We first introduce each of the perspectives by reviewing its basic ideas and research base. We then show each perspective at work, by applying it to the analysis of typical mathematical misconceptions. Our illustrations are based on two tasks: one from statistics (taken from the psychological research literature) and one from abstract algebra (based on our own research).


Mathematics education Cognitive psychology Evolutionary psychology Mathematical errors Rationality debate Dual process theory Medical diagnosis problem Bayesian thinking Base-rate neglect Group theory Lagrange's theorem 



The first author gratefully acknowledges a Bellagio Residency grant from the Rockefeller Foundation, where some of the ideas presented in this paper were developed.


  1. Barbey, A. K., & Sloman, S. A. (2007). Base-rate respect: From ecological rationality to dual processes. The Behavioral and Brain Sciences, 30(3), 241–254.Google Scholar
  2. Barkow, J., Cosmides, L., & Tooby, J. (Eds.) (1992). In The adapted mind: Evolutionary psychology and the generation of culture. Oxford University Press: Oxford.Google Scholar
  3. Bjorklund, D. F., & Pellegrini, A. D. (2002). The origins of human nature: Evolutionary developmental psychology. Washington, D.C.: American Psychological Association Press.CrossRefGoogle Scholar
  4. Buss, D. M. (2005). The handbook of evolutionary psychology. Hoboken: Wiley.Google Scholar
  5. Casscells, W., Schoenberger, A., & Grayboys, T. (1978). Interpretation by physicians of clinical laboratory results. New England Journal of Medicine, 299, 999–1000.CrossRefGoogle Scholar
  6. Cosmides, L., & Tooby, J. (1992). Cognitive adaptations for social exchange. In J.Barkow, L.Cosmides, & J.Tooby (Eds.), The adapted mind: Evolutionary psychology and the generation of culture (pp. 163–228). Oxford: Oxford University Press.Google Scholar
  7. Cosmides, L., & Tooby, J. (1996). Are humans good intuitive statisticians after all? Rethinking some conclusions from the literature on judgment under uncertainty. Cognition, 58, 1–73. doi: 10.1016/0010-0277(95)00664-8.CrossRefGoogle Scholar
  8. Cosmides, L., & Tooby, J. (1997). Evolutionary psychology: A primer, Center for Evolutionary Psychology, University of California, Santa Barbara. Accessed 26 September 2008.
  9. Davis, R. B., Maher, C. A. & Noddings, N. (Eds.) (1990). Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education, Monograph no. 4.Google Scholar
  10. Evans, J. S. B. T. (2003). In two minds: dual-process accounts of reasoning. Trends in Cognitive Sciences, 7(10), 454–459. doi: 10.1016/j.tics.2003.08.012.CrossRefGoogle Scholar
  11. Evans, J. S. B. T. (2006). The heuristic–analytic theory of reasoning: extension and evaluation. Psychonomic Bulletin & Review, 13, 378–395.Google Scholar
  12. Evans, J. S. B. T. (2008). Dual-processing accounts of reasoning, judgment, and social cognition. Annual Review of Psychology, 59, 61–624.CrossRefGoogle Scholar
  13. Gallian, J. A. (1990). Contemporary abstract algebra (2nd ed.). Lexington: D. C. Heath.Google Scholar
  14. Geary, D. (2002). Principles of evolutionary educational psychology. Learning and Individual Differences, 12, 317–345. doi: 10.1016/S1041-6080(02)00046-8.CrossRefGoogle Scholar
  15. Gigerenzer, G. (2005). I think, therefore I err. Social Research, 72(1), 1–24. doi: 10.1007/s11205-004-4512-5.CrossRefGoogle Scholar
  16. Gigerenzer, G., Todd, P. M., & ABC research group (1999). Simple heuristics that make us smart. New York: Oxford University Press.Google Scholar
  17. Gilovich, T., Griffin, D., & Kahneman, D. (Eds.) (2002). In Heuristics and biases: The psychology of intuitive judgment. Cambridge: Cambridge University Press.Google Scholar
  18. Hazzan, O., & Leron, U. (1996). Students’ use and misuse of mathematical theorems: the case of Lagrange’s theorem. For the Learning of Mathematics, 16, 23–26.Google Scholar
  19. Kahneman, D. (Nobel Prize Lecture, December 8, 2002). Maps of bounded rationality: A perspective on intuitive judgment and choice. In T. Frangsmyr (Ed.), Les Prix Nobel (pp. 416–499). Accessed 26 September 2008.
  20. Kahneman, D., & Frederick, S. (2005). A model of heuristic judgment. In K. Holyoak, & R. G. Morrison (Eds.), The Cambridge handbook of thinking and reasoning pp. 267–294. Cambridge: Cambridge University Press.Google Scholar
  21. Leron, U. (2003). Origins of mathematical thinking: A synthesis, Proceedings CERME3, Bellaria, Italy, March, 2003. Accessed 26 September 2008.
  22. Leron, U., & Hazzan, O. (1997). The world according to Johnny: a coping perspective in mathematics education. Educational Studies in Mathematics, 32, 265–292. doi: 10.1023/A:1002908608251.CrossRefGoogle Scholar
  23. Leron, U., & Hazzan, O. (2006). The rationality debate: application of cognitive psychology to mathematics education. Educational Studies in Mathematics, 62(2), 105–126. doi: 10.1007/s10649-006-4833-1.CrossRefGoogle Scholar
  24. Looren de Jong, H. (2002). Levels of explanation in biological psychology. Philosophical Psychology, 15(4), 441–462. doi: 10.1080/0951508021000042003.CrossRefGoogle Scholar
  25. Nesher, P., & Teubal, E. (1975). Verbal cues as an interfering factor in verbal problem solving. Educational Studies in Mathematics, 6, 41–51. doi: 10.1007/BF00590023.CrossRefGoogle Scholar
  26. Nisan, N., & Schocken, S. (2005). The elements of computing systems: Building a modern computer from first principles. Cambridge: MIT.Google Scholar
  27. Paz, T., & Leron, U. (2008) The slippery road from actions on objects to functions and variables. Journal for Research in Mathematics Education, in press.Google Scholar
  28. Pinker, S. (1997). How the mind works. New York: Norton.Google Scholar
  29. Pinker, S. (2002). The blank slate: The modern denial of human nature. Nyborg: Viking.Google Scholar
  30. Ridley, M. (2003). Nature via nurture: Genes, experience, and what makes us human. London: Harper Collins.Google Scholar
  31. Samuels, R., Stitch, S., & Tremoulet, P. (1999). Rethinking rationality: From bleak implications to Darwinian modules. In E. LePore, & Z. Pylyshyn (Eds.), What is cognitive science? (pp. 74–120). Oxford: Blackwell.Google Scholar
  32. Stanovich, K. (2008). Distinguishing the reflective, algorithmic, and autonomous minds: Is it time for a tri-process theory? In J. Evans, & K. Frankish (Eds.), In two minds: Dual processes and beyond. Oxford: Oxford University Press.Google Scholar
  33. Stanovich, K. E., & West, R. F. (2000). Individual differences in reasoning: Implications for the rationality debate. The Behavioral and Brain Sciences, 23, 645–726. doi: 10.1017/S0140525X00003435.CrossRefGoogle Scholar
  34. Stanovich, K. E., & West, R. F. (2003). Evolutionary versus instrumental goals: How evolutionary psychology misconceives human rationality. In D. E. Over (Ed.), Evolution and the psychology of thinking: The debate (pp. 171–230). Hove: Psychology Press.Google Scholar
  35. Stein, R. (1996). Without good reason: The rationality debate in philosophy and cognitive science. Oxford: Oxford University Press.Google Scholar
  36. Tooby, J., & Cosmides, L. (2005). Conceptual foundations of evolutionary psychology. In D. M. Buss (Ed.), The handbook of evolutionary psychology (pp. 5–67). New York: Wiley.Google Scholar
  37. Wason, P., & Johnson-Laird, P. (1972). The psychology of reasoning: Structure and content. Cambridge: Harvard University Press.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Education in Technology and ScienceTechnion–Israel Institute of TechnologyHaifaIsrael

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