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

#### Abstract

*qua*unexpectedâ€”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

#### Full Text:

PDF#### References

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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.

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

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

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.

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

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.

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.

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

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.

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

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

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.

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.

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

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/

DOI: http://dx.doi.org/10.4471/redimat.2015.57

#### Article Metrics

_{Metrics powered by PLOS ALM}

**REDIMAT - Journal of Research in Mathematics Education** | ISSN: 2014-3621

Legal Deposit: B.34289-2012 | https://redimat.hipatiapress.com | redimat@hipatiapress.com