Inventing (in) early geometry, or How creativity inheres in the doing of mathematics
https://doi.org/10.4471/redimat.2015.57
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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) geometerDownloads
References
Anglin, W. S. & Lambek, J. (1995). The Heritage of Thales. Springer
Google Scholar CrossrefBalka, D. S. (1974). The development of an instrument to measure creative ability in mathematics. Dissertation Abstracts International, 36(01), 98.
Google Scholar CrossrefBaltag, A. (1999, August). A logic of epistemic actions. In Electronic Proceedings of the FACAS workshop, held at ESSLLI.
Google Scholar CrossrefChâtelet, G. (1993). Les enjeux du mobile. Paris: Éditions du Seuil.
Google Scholar CrossrefChâtelet, G. (2010). L’enchantement du virtuel. Paris: Presses de l’École normale supérieure.
Google Scholar CrossrefCsikszentmihalyi C. (1996). Creativity: flow and the psychology of discovery and invention. New York: Harper Collins
Google Scholar Crossrefde Freitas, E., & Sinclair, N. (2013). New materialist ontologies in mathematics education. Educational Studies in Mathematics, 83(3), 453-470.
Google Scholar CrossrefDerrida, J. (1962). Introduction. In Husserl, E. L'origine de la géométrie. Paris: Presses universitaires de France.
Google Scholar CrossrefDerrida, J. (1993). Khôra. Paris: Galilée.
Google Scholar CrossrefDerrida, J. (2007). Psyche: Inventions of the Other. Paris: Galilée.
Google Scholar CrossrefDreyfus, 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 CrossrefDufrenne, M. (1953). Phénoménologie de l'expérience esthétique. Paris: Presse Universitaires de France.
Google Scholar CrossrefEvans, 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 CrossrefGravemeijer, 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 CrossrefHadamard J. (1945) The psychology of invention in the mathematical field. New York: Dover.
Google Scholar CrossrefHaylock, D. W. (1984). Aspects of mathematical creativity in children aged 11-12. Doctoral dissertation, Chelsea College, University of London.
Google Scholar CrossrefHenry, M. (2000). Incarnation. Paris: Éditions du Seuil.
Google Scholar CrossrefHusserl, 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 CrossrefKandinsky, W. (1913). Kandinsky 1901–1913. Berlin: Der Sturm.
Google Scholar CrossrefKlee, P. (1953). Pedagogical sketchbook. New York: Praeger.
Google Scholar CrossrefLiljedahl, P. & Allan, D. (2013). Mathematical Discovery. In E.G. Carayannis (Ed.), Encyclopedia of Creativity, Invention, Innovation, and Entrepreneurship (pp. 1228-1233). Springer.
Google Scholar CrossrefMann, E. L. (2006). Mathematical creativity and school mathematics: Indicators of mathematical creativity in middle school students. Doctoral dissertation, University of Connecticut.
Google Scholar CrossrefMarx, K./Engels, F. (1983). Werke Band 42. Berlin: Dietz.
Google Scholar CrossrefMaturana, H. (2009). The Origins of humanness in the biology of love. Exeter: Imprint Academic.
Google Scholar CrossrefMerleau-Ponty, M. (1964). Le visible et l’invisible. Paris: Gallimard.
Google Scholar CrossrefPoincaré, H. (1952). Mathematical creation. In Ghistin, B. (Ed.) The creative process (pp. 22-31).
Google Scholar CrossrefRorty, R. (1989). Contingency, irony, and solidarity. Cambridge: Cambridge University Press.
Google Scholar CrossrefRoth, 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 CrossrefRoth, W. M. (2011). Geometry as objective science in elementary classrooms: Mathematics in the flesh. New York: Routledge.
Google Scholar CrossrefRoth, 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 CrossrefRoth, W. M. (2014a). Curriculum*-in-the-making: A post-constructivist perspective. New York: Peter Lang.
Google Scholar CrossrefRoth, 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 CrossrefRoth, 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 CrossrefRunco, M. A. (1993). Creativity as an educational objective for disadvantaged students. Storrs, CT: The National Research Center on the Gifted and Talented.
Google Scholar CrossrefRunco, 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 CrossrefSinclair, N., de Freitas, E., & Ferrara, F. (2013). Virtual encounters: The murky and furtive world of mathematical inventiveness. ZDM, 45(2), 239-252.
Google Scholar CrossrefSriraman, B (2010). Thinking to mathematician’s mathematics creativity. Studies in Dialectics of Nature, 7, 85-88.
Google Scholar CrossrefSriraman, 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 CrossrefTreffinger, 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 CrossrefWatson, J. D. (2012). The double helix: A personal account of the discovery of the structure of DNA. New York: Simon & Schuster.
Google Scholar CrossrefZhang, 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/
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