Inventing (in) early geometry, or How creativity inheres in the doing of mathematics

Authors

  • Jean-Francois Maheux UQAM
  • Wolf-Michael Roth University of Victoria

https://doi.org/10.4471/redimat.2015.57

Downloads

Abstract

Inventing is fundamental to mathematical activity, should one be a professional mathematician or a primary school student. Research on mathematical creativity generally is organized along three axes according to its focus on the final product, the overall process, or the individual person. Through these conceptualizations, however, research rarely considers how mathematical actions themselves are fundamentally creative. In an action-oriented perspective, every single act is recognized as creative, whereas discovery and invention emerge as the result of the incoming of the unexpected qua unexpected—which can take place at any moment in the most mundane, everyday action. In this article, we conceptualize mathematical actions as inherently creative of the activity within which professional mathematicians and primary school students experience (some) mathematics for a first time. To make our case, we develop the microanalysis of an exemplary episode of third-grade geometry (age 8-9 years) in which two children and an adult work with a tangram set. Our analysis characterizes inventing (in) geometry as a serendipitous, open-ended experience of working with traces in the receiving and the offering of something novel. In concluding, we propose considering that inventing in early geometry is also inventing geometry itself: an inventing-in-the-act which also result in being invented as a (professional or school) geometer

Downloads

Download data is not yet available.

Author Biographies

Jean-Francois Maheux, UQAM

Dept. Mathematiques, section didactique

Wolf-Michael Roth, University of Victoria

Lansdowne Professor

Applied Cognitive Science

References

Anglin, W. S. & Lambek, J. (1995). The Heritage of Thales. Springer

Google Scholar Crossref

Balka, D. S. (1974). The development of an instrument to measure creative ability in mathematics. Dissertation Abstracts International, 36(01), 98.

Google Scholar Crossref

Baltag, A. (1999, August). A logic of epistemic actions. In Electronic Proceedings of the FACAS workshop, held at ESSLLI.

Google Scholar Crossref

Châtelet, G. (1993). Les enjeux du mobile. Paris: Éditions du Seuil.

Google Scholar Crossref

Châtelet, G. (2010). L’enchantement du virtuel. Paris: Presses de l’École normale supérieure.

Google Scholar Crossref

Csikszentmihalyi C. (1996). Creativity: flow and the psychology of discovery and invention. New York: Harper Collins

Google Scholar Crossref

de Freitas, E., & Sinclair, N. (2013). New materialist ontologies in mathematics education. Educational Studies in Mathematics, 83(3), 453-470.

Google Scholar Crossref

Derrida, J. (1962). Introduction. In Husserl, E. L'origine de la géométrie. Paris: Presses universitaires de France.

Google Scholar Crossref

Derrida, J. (1993). Khôra. Paris: Galilée.

Google Scholar Crossref

Derrida, J. (2007). Psyche: Inventions of the Other. Paris: Galilée.

Google Scholar Crossref

Dreyfus, T., & Eisenberg, T. (1996). On different facets of mathematical thinking. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 253 - 284). Mahwah, NJ: Lawrence Erlbaum Associates.

Google Scholar Crossref

Dufrenne, M. (1953). Phénoménologie de l'expérience esthétique. Paris: Presse Universitaires de France.

Google Scholar Crossref

Evans, E.W. (1964). Measuring the ability of students to respond in creative mathematical situations at the late elementary and early junior high school level. Dissertation Abstracts, 25 (12), 7107.

Google Scholar Crossref

Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational studies in mathematics, 39(1-3), 111-129.

Google Scholar Crossref

Hadamard J. (1945) The psychology of invention in the mathematical field. New York: Dover.

Google Scholar Crossref

Haylock, D. W. (1984). Aspects of mathematical creativity in children aged 11-12. Doctoral dissertation, Chelsea College, University of London.

Google Scholar Crossref

Henry, M. (2000). Incarnation. Paris: Éditions du Seuil.

Google Scholar Crossref

Husserl, E. (1976). Husserliana Band VI: Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie. Eine Einleitung in die phänomenologische Philosophie. The Hague, The Netherlands: Martinus Nijhoff.

Google Scholar Crossref

Kandinsky, W. (1913). Kandinsky 1901–1913. Berlin: Der Sturm.

Google Scholar Crossref

Klee, P. (1953). Pedagogical sketchbook. New York: Praeger.

Google Scholar Crossref

Liljedahl, P. & Allan, D. (2013). Mathematical Discovery. In E.G. Carayannis (Ed.), Encyclopedia of Creativity, Invention, Innovation, and Entrepreneurship (pp. 1228-1233). Springer.

Google Scholar Crossref

Mann, E. L. (2006). Mathematical creativity and school mathematics: Indicators of mathematical creativity in middle school students. Doctoral dissertation, University of Connecticut.

Google Scholar Crossref

Marx, K./Engels, F. (1983). Werke Band 42. Berlin: Dietz.

Google Scholar Crossref

Maturana, H. (2009). The Origins of humanness in the biology of love. Exeter: Imprint Academic.

Google Scholar Crossref

Merleau-Ponty, M. (1964). Le visible et l’invisible. Paris: Gallimard.

Google Scholar Crossref

Poincaré, H. (1952). Mathematical creation. In Ghistin, B. (Ed.) The creative process (pp. 22-31).

Google Scholar Crossref

Rorty, R. (1989). Contingency, irony, and solidarity. Cambridge: Cambridge University Press.

Google Scholar Crossref

Roth, W. M. (1995). From 'wiggly structures' to 'unshaky towers': Problem framing, solution finding, and negotiation of courses of actions during a civil engineering unit for elementary students. Research in Science Education, 25, 365–381.

Google Scholar Crossref

Roth, W. M. (2011). Geometry as objective science in elementary classrooms: Mathematics in the flesh. New York: Routledge.

Google Scholar Crossref

Roth, W. M. (2013). To Event: Toward a post-Constructivist of theorizing and researching the living curriculum as Event*-in-the-Making. Curriculum Inquiry, 43(3), 388-417.

Google Scholar Crossref

Roth, W. M. (2014a). Curriculum*-in-the-making: A post-constructivist perspective. New York: Peter Lang.

Google Scholar Crossref

Roth, W. M. (2014b). Learning in the discovery sciences: The history of a "radical" conceptual change or The scientific revolution that was not. Journal of the Learning Sciences, 1-39.

Google Scholar Crossref

Roth, W. M., & Hwang, S. (2006). Does mathematical learning occur in going from concrete to abstract or in going from abstract to concrete? The Journal of Mathematical Behavior, 25(4), 334-344.

Google Scholar Crossref

Runco, M. A. (1993). Creativity as an educational objective for disadvantaged students. Storrs, CT: The National Research Center on the Gifted and Talented.

Google Scholar Crossref

Runco, M. A. (2004). Creativity. In S. T. Fiske, D. L. Schacter, & C. Zahn-Waxler (Eds.), Annual Review of Psychology (pp. 657 – 687). Palo Alto: Annual Reviews.

Google Scholar Crossref

Sinclair, N., de Freitas, E., & Ferrara, F. (2013). Virtual encounters: The murky and furtive world of mathematical inventiveness. ZDM, 45(2), 239-252.

Google Scholar Crossref

Sriraman, B (2010). Thinking to mathematician’s mathematics creativity. Studies in Dialectics of Nature, 7, 85-88.

Google Scholar Crossref

Sriraman, B. & Dahl, B. (2009). On bringing interdisciplinary ideas to gifted education. In L.V. Shavinina (Ed). The International Handbook of Giftedness (pp. 1235-1256). Springer.

Google Scholar Crossref

Treffinger, D. J., Young, G. C., Selby, E. C., & Shepardson, C. (2002). Assessing creativity: A guide for educators. Storrs, CT: The National Research Center on the Gifted and Talented.

Google Scholar Crossref

Watson, J. D. (2012). The double helix: A personal account of the discovery of the structure of DNA. New York: Simon & Schuster.

Google Scholar Crossref

Zhang, X. G. (2013). Thinking analysis to the process of mathematical creativity of mathematicians. Philosophy of Mathematics Education Journal, 27. http://people.exeter.ac.uk/PErnest/pome27/

Google Scholar Crossref

Downloads

Published

2015-02-24

Almetric

Dimensions

Issue

Section

Articles