From Biological Brain to Mathematical Mind: The Long-Term Evolution of Mathematical Thinking

  • David TallEmail author
Part of the Mathematics in Mind book series (MATHMIN)


In this chapter we consider how research into the operation of the brain can give practical advice to teachers and learners to assist them in their long-term development of mathematical thinking. At one level, there is extensive research in neurophysiology that gives some insights into the structure and operation of the brain; for example, magnetic resonance imagery (MRI) gives a three-dimensional picture of brain structure and fMRI (functional MRI) reveals changes in neural activity by measuring blood flow to reveal which parts of the brain are more active over a period of time. But this blood flow can only be measured to a resolution of 1 or 2 seconds and does not reveal the full subtlety of the underlying electrochemical activity involved in human thinking which operates over much shorter periods.


  1. Amalric, M. and Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians, Proceedings of the National Academy of Science of the USA. 113 (18) 4909–4917. Scholar
  2. Chomsky, N. (2006). Language and mind. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  3. Dehaene, S., Spelke E., Pinel, P., Stanescu, R., and Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science 284 (5416): 970–974.CrossRefGoogle Scholar
  4. Filloy, E., and Rojano, T. (1989). Solving equations, the transition from arithmetic to algebra, For the Learning of Mathematics 9 (2): 19–25.Google Scholar
  5. Gravemeijer, K., Bruin-Muurling G., Kraemer J. M., van Stiphout, I. (2016). Shortcomings of mathematics education reform in The Netherlands: A Paradigm Case? Mathematical Thinking and Learning 18 (1): 25–44, DOI: Scholar
  6. Gray, E. M. and Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education 26 (2): 115–141.Google Scholar
  7. Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton: Princeton University Press (Dover edition, New York 1954).Google Scholar
  8. Hodds, M., Alcock, L., and Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education 45: 62–101.CrossRefGoogle Scholar
  9. Inglis, M., and Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43, 358–390.CrossRefGoogle Scholar
  10. Isoda, M. and Tall, D. O. (eds.) (2018). Mathematics: Junior high school 1, 2, 3, English Edition. Tokyo: Gakkotosyo.Google Scholar
  11. Katz, M. and Tall, D. O. (2012). The tension between intuitive infinitesimals and formal analysis. In: Bharath Sriraman, (ed.), Crossroads in the history of mathematics and mathematics education, (The Montana Mathematics Enthusiast Monographs in Mathematics Education 12), pp. 71–90.Google Scholar
  12. Kozielecki, J. (1987). Koncepcja transgresyjna człowieka [Transgressive concept of a man]. Warszawa: PWN.Google Scholar
  13. Lave, J. and Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. ISBN 0-521-42374-0.CrossRefGoogle Scholar
  14. Maruyama, M., Pallier, C., Jobert, A., Sigman, M., Dehaene, S. (2012). The cortical representation of simple mathematical expressions. Neuroimage 61(4):1444–1460.CrossRefGoogle Scholar
  15. McGowen, M. and Tall, D. O. (2010). Metaphor or met-before? The effects of previous experience on the practice and theory of learning mathematics. Journal of Mathematical Behavior 29: 169–179.CrossRefGoogle Scholar
  16. Monti, M. M., Parsons, L. M., Osherson D. N. (2012) Thought beyond language: Neural dissociation of algebra and natural language. Psychological Science 23(8): 914–922.CrossRefGoogle Scholar
  17. Nakai, T., Sakai, K. L. (2014). Neural mechanisms underlying the computation of hierarchical tree structures in mathematics. PLoS One 9(11): e111439.CrossRefGoogle Scholar
  18. Piaget, J. (1952). The child’s conception of number. New York: Norton.Google Scholar
  19. Pieronkiewicz, B. (2014). Report of mathematical transgressions conference, Annals of the Polish Mathematical Society, 5th Series: Didactica Mathematicae 36: 163–168.Google Scholar
  20. PISA (2015). Programme for International Student Assessment. Retrieved from, November 30, 2018.Google Scholar
  21. Shum, J., Hermes, D., Foster, B. L., Dastjerdi, M., Rangarajan,V., Winawer, J., Miller, K., Parvizi, J. (2013). A brain area for visual numerals. Journal of Neuroscience 1: 6709–6715.CrossRefGoogle Scholar
  22. Stewart, I. N. & Tall, D. O. (2014). Foundations of Mathematics, Second Edition. Oxford: OUP. ISBN: 9780198531654Google Scholar
  23. Stewart, I. N. & Tall, D. O. (2015). Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition. Boca Raton: CRC. ISBN 9781498738392Google Scholar
  24. Stewart, I. N. and Tall, D. O. (2018). Complex analysis, Second edition. Cambridge: Cambridge University Press. ISBN: 978-1108436793.CrossRefGoogle Scholar
  25. Tall, D. O, Thomas, M. O. J., Davis, G. E., Gray, E. M., and Simpson, A. P. (2000). What is the object of the encapsulation of a process? Journal of Mathematical Behavior 18 (2): 1–19.Google Scholar
  26. Tall, D. O. (2013). How humans learn to think mathematically. New York: Cambridge University Press. ISBN: 9781139565202.CrossRefGoogle Scholar
  27. Tall, D. O. (2019). Complementing supportive and problematic aspects of mathematics to resolve transgressions in long-term sense making. The Fourth Interdisciplinary Scientific Conference on Mathematical Transgressions, Krakow (to appear).Google Scholar
  28. Tall, D. O. and Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics 86 (1): 97–124.CrossRefGoogle Scholar
  29. Tall, D. O., Tall, N. D. and Tall, S. J. (2017). Problem posing in the long-term conceptual development of a gifted child. In: Martin Stein (ed.) A life’s time for mathematics education and problem solving. On the occasion of Andràs Ambrus75th Birthday, pp. 445–457. WTM-Verlag: Münster.Google Scholar
  30. Tall, D., de Lima R. and Healy, L. (2014). Evolving a three-world framework for solving algebraic equations in the light of what a student has met before. Journal of Mathematical Behavior 34: 1–13.CrossRefGoogle Scholar
  31. Van den Heuvel-Panhuizen M. and Drijvers P. (2014). Realistic mathematics education. In: S. Lerman (ed.), Encyclopedia of Mathematics Education,, Springer: Dordrecht. Scholar
  32. TIMSS (2015). TIMSS 2015 results. Retrieved from, November 30, 2018.
  33. Verhoef, N. C., and Tall, D. O. (2011). Teacher’s professional development through lesson study: effects on mathematical knowledge for teaching. Proceedings of the 35e Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 297–304). Ankara, Turkey.Google Scholar
  34. Verhoef, N. C., Tall, D. O., Coenders, F. and Smaalen, D. (2014). The complexities of a lesson study in a Dutch situation: Mathematics Teacher Learning International Journal of Science and Mathematics Education 12: 859–881. doi: Scholar
  35. Limbic System (n.d.) Wikipedia: Limbic System. Retrieved from November 30, 2018.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of WarwickCoventryUK

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