Skip to main content

Cognitive Neuroscience and Algebra: Challenging Some Traditional Beliefs

  • Chapter
  • First Online:
And the Rest is Just Algebra

Abstract

Recent studies using neuroimaging technology with tasks touching on various areas of mathematics are raising a great deal of excitement with their findings. This chapter presents some key work related to higher level mathematical reasoning and a few insights arising from these studies with respect to our current understanding of algebra learning. After a general introduction on cognitive neuroscience and its recent advances relevant to mathematics education, the chapter focuses on two studies in particular, one on the algebraic solving method and the other on representing functions. The chapter concludes with a discussion of the ways in which these results from the newly emerging field, which is at times referred to as mathematics educational neuroscience, offer the potential of casting a quite different light on how we think about students’ processing of algebra-related material.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274.

    Article  Google Scholar 

  • Blanton, M., Brizuela, B. M., Gardiner, A. M., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in six-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46, 511–558.

    Article  Google Scholar 

  • Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J.-S. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46, 39–87.

    Article  Google Scholar 

  • Bloedy-Vinner, H. (1994). The analgebraic mode of thinking: The case of parameter. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 88–95). Lisbon, Portugal: PME.

    Google Scholar 

  • Bloedy-Vinner, H. (2001). Beyond unknowns and variables—Parameters and dummy variables in high school algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 177–189). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Bruer, J. T. (1997). Education and the brain: A bridge too far. Educational Researcher, 26(8), 4–16.

    Article  Google Scholar 

  • Cai, J., Lew, H. C., Morris, A., Moyer, J. C., Ng, S. F., & Schmittau, J. (2005). The development of students’ algebraic thinking in earlier grades: A cross-cultural comparative perspective. Zentralblatt für Didaktik der Mathematik, 37, 5–15.

    Article  Google Scholar 

  • Campbell, S. R. (2010). Embodied minds and dancing brains: New opportunities for research in mathematics education. In B. Sriraman & L. English (Eds.), Theories of mathematics education (pp. 309–331). Berlin: Springer. doi:10.1007/978-3-642-00742-2_31.

    Chapter  Google Scholar 

  • De Smedt, B., Ansari, D., Grabner, R. H., Hannula-Sormunen, M., Schneider, M., & Verschaffel, L. (2011). Cognitive neuroscience meets mathematics education: It takes two to tango. Educational Research Review, 6, 232–237.

    Article  Google Scholar 

  • De Smedt, B., & Verschaffel, L. (2010). Traveling down the road: From cognitive neuroscience to mathematics education … and back. ZDM: The International Journal on Mathematics Education, 42, 649–654. doi:10.1007/s11858-010-0282-5.

    Article  Google Scholar 

  • Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20(3–6), 487–506.

    Article  Google Scholar 

  • Dresler, T., Obersteiner, A., Schecklmann, M., Vogel, A. C. M., Ehlis, A.-C., Richter, M. M., et al. (2009). Arithmetic tasks in different formats and their influence on behavior and brain oxygenation as assessed with near-infrared spectroscopy (NIRS): A study involving primary and secondary school children. Journal of Neural Transmission, 12(16), 1689–1700.

    Article  Google Scholar 

  • Fischer, K. W. (2009). Mind, brain, and education: Building a scientific groundwork for learning and teaching. Mind, Brain and Education, 3(1), 3–16. doi:10.1111/j.1751-228X.2008.01048.x.

    Article  Google Scholar 

  • Hernandez-García, L., Wager, T., & Jonides, J. (2002). Functional brain imaging. In H. Pashler & J. Wixted (Eds.), Stevens’ handbook of experimental psychology (Methodology in experimental psychology 3rd ed., Vol. 4, pp. 175–221). New York: Wiley. http://onlinelibrary.wiley.com/doi/10.1002/0471214426.pas0405/full.

    Google Scholar 

  • Hoch, M., & Dreyfus, T. (2004). Structure sense in high school algebra: The effects of brackets. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 49–56). Bergen, Norway: PME.

    Google Scholar 

  • Khng, K. H., & Lee, K. (2009). Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving. Learning and Individual Differences, 19, 262–268.

    Article  Google Scholar 

  • Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York: Macmillan.

    Google Scholar 

  • Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age.

    Google Scholar 

  • Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13, 129–164.

    Article  Google Scholar 

  • Lagrange, J.-B. (2000). L’intégration d’instruments informatiques dans l’enseignement : une approche par les techniques [The integration of computer tools into teaching: An approach according to techniques]. Educational Studies in Mathematics, 43, 1–30.

    Article  Google Scholar 

  • Lee, K., Lim, Z. Y., Yeong, S. H. M., Ng, S. F., Venkatraman, V., & Chee, M. W. L. (2007). Strategic differences in algebraic problem solving: Neuroanatomical correlates. Brain Research, 1155, 163–171.

    Article  Google Scholar 

  • Lee, K., Yeong, S. H. M., Ng, S. F., Venkatraman, V., Graham, S., & Chee, M. W. L. (2010). Computing solutions to algebraic problems using a symbolic versus a schematic strategy. ZDM: The International Journal on Mathematics Education, 42, 591–605. doi:10.1007/s11858-010-0265-6.

    Article  Google Scholar 

  • Leikin, R., Waisman, I., Shaul, S., & Leikin, M. (2012). An ERP study with gifted and excelling male adolescents: Solving short insight-based problems. In T. Y. Tso (Ed.), Proceedings of the 36th International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 83–90). Taiwan, Taipei: PME.

    Google Scholar 

  • Luna, B., Garver, K. E., Urban, T. A., Lazar, N. A., & Sweeney, J. A. (2004). Maturation of cognitive processes from late childhood to adulthood. Child Development, 75, 1357–1372.

    Article  Google Scholar 

  • Menon, V. (2010). Developmental cognitive neuroscience of arithmetic: Implications for learning and education. ZDM Mathematics Education, 42, 515–525. doi:10.1007/s11858-010-0242-0.

    Article  Google Scholar 

  • Nathan, M. J., & Koedinger, K. R. (2000). Teachers’ and researchers’ beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31, 168–190.

    Article  Google Scholar 

  • Nathan, M. J., & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Educational Research Journal, 40, 905–928.

    Article  Google Scholar 

  • Newman, S. D., Willoughby, G., & Pruce, B. (2011). The effect of problem structure on problem-solving: An fMRI study of word versus number problems. Brain Research, 1410, 77–88.

    Article  Google Scholar 

  • Ng, S. F. (2004). Developing algebraic thinking in early grades: Case study of the Singapore primary mathematics curriculum. The Mathematics Educator, 8(1), 39–59.

    Google Scholar 

  • Obersteiner, A., Dresler, T., Reiss, K., Vogel, A. C. M., Pekrun, R., & Fallgatter, A. J. (2010). Bringing brain imaging to the school to assess arithmetic problem solving: Chances and limitations in combining educational and neuroscientific research. ZDM—The International Journal on Mathematics Education, 42, 541–554. doi:10.1007/s11858-010-0256-7.

    Article  Google Scholar 

  • Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2012). Algebra in elementary school. Enseignement de l’algèbre élémentaire (Special Issue of Recherches en Didactique des Mathématiques) (pp. 107–122).

    Google Scholar 

  • Susac, A., Bubic, A., Kaponja, J., Planinic, M., & Palmovic, M. (2014). Eye movements reveal students’ strategies in simple equation solving. International Journal of Science and Mathematics Education, 12, 555–577.

    Article  Google Scholar 

  • Thomas, M. J., Wilson, A. J., Corballis, M. C., Lim, V. K., & Yoon, C. (2010). Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function. ZDM—The International Journal on Mathematics Education, 42, 607–619. doi:10.1007/s11858-010-0272-7.

    Article  Google Scholar 

  • Turner, D. A. (2011). Which part of ‘two way street’ did you not understand? Redressing the balance of neuroscience and education. Educational Research Review, 6, 224–232.

    Article  Google Scholar 

  • Waisman, I., Leikin, M., Shaul, S., & Leikin, R. (2014). Brain activity associated with translation between graphical and symbolic representations of functions in generally gifted and excelling in mathematics adolescents. International Journal of Science and Mathematics Education, 12, 669–696.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carolyn Kieran .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kieran, C. (2017). Cognitive Neuroscience and Algebra: Challenging Some Traditional Beliefs. In: Stewart, S. (eds) And the Rest is Just Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-45053-7_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45053-7_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45052-0

  • Online ISBN: 978-3-319-45053-7

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics