Can a Kite Be a Triangle? Aesthetics and Creative Discourse in an Interactive Geometric Environment
In this chapter we will use the lens of aesthetics (Sinclair in Mathematics and beauty: aesthetic approaches to teaching children. Teachers College Press, New York, NY, 2006) to explore mathematical creativity in an interactive geometric environment from three different perspectives: inquiry, teaching, and mathematical resolution. We will be illustrating the mathematical creativity with an episode where academically talented middle school students are working with Shape Makers (Battista in Shapemakers. Key Curriculum Press, Emeryville, CA, 2003) in Geometer’s Sketchpad. At one point in the lesson, a student makes a triangle looking shape with the Kite Maker Tool and asks, “Can a triangle be a kite?” We see creativity reflected in three ways: in the generation of ideas, in the teacher’s instructional choices, and in the resolution of the mathematical discussion by the students. The creative and aesthetic qualities of open inquiry, the Geometers’ Sketchpad, and teacher moves created a setting where students and the teacher made aesthetically motivational, generative, and evaluative choices to build understanding of geometric properties of kites and triangles as well as the limitations of sets of geometric properties in classifying geometric shapes.
KeywordsInteractive geometry Aesthetics Student discourse
- Appel, K., & Haken, W. (1977). Every planar map is four colorable. Part I: Discharging. Illinois Journal of Mathematics, 21(3), 429–490.Google Scholar
- Ball, D. (2000). Working on the inside: Using one’s own practice as a site for studying teaching and learning. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 365–402). Mahwah, NJ: Lawrence Erlbaum Associates. https://eric.ed.gov/id=ED478318.
- Baron, L. M. (2010). Helping teachers generate better homework: MAA makes time for WeBWorK. MAA Focus, 30(5), 18–19.Google Scholar
- Battista, M. (2003). Shapemakers. Emeryville, CA: Key Curriculum Press.Google Scholar
- Burbules, N. C. (2006) Rethinking the virtual. In J. Weiss et al. (Eds.), The International handbook of virtual learning environment (pp. 37–58). Dorderecht, The Netherlands: Springer.Google Scholar
- Jackiw, N. (1991). Geometer’s sketchpad [computer software]. Emeryville, CA: Key Curriculum Press.Google Scholar
- Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In Proceedings of the 31st International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 161–168). Seoul, South Korea: The Korea Society of Educational Studies in Mathematics.Google Scholar
- Murphy, R., Gallagher, L., Krumm, A., Mislevy, J., & Hafter, A. (2014). Research on the use of Khan Academy in schools. Menlo Park, CA: SRI Education.Google Scholar
- National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common core state standards for mathematics: Standards for Mathematical Practice. Washington, DC: Authors.Google Scholar
- National Research Council. (2001). In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
- Richardson, L., & St. Pierre, E. A. (2005). Writing: A method of inquiry. In N. K. Denzin & Y. S. Lincoln (Eds.), The Sage handbook of qualitative research (pp. 959–978). Thousand Island, CA: Sage Publications.Google Scholar
- Sinclair, N. (2006). Mathematics and beauty: Aesthetic approaches to teaching children. New York, NY: Teachers College Press.Google Scholar
- Sinclair, N. (2008). Notes on the aesthetic dimension of mathematics education. In Proceedings of the 100th International Commission of Mathematics Instruction Symposium (https://www.unige.ch/math/EnsMath/Rome2008/WG5/Papers/SINCL.pdf). Rome, Italy: The International Commission of Mathematics Instruction.
- Sriraman, B., Yaftian, N., & Lee, K. H. (2011). Mathematical creativity and mathematics education. In The elements of creativity and giftedness in mathematics (pp. 119–130). Rotterdam, The Netherlands: Sense Publishers.Google Scholar
- Tent, M. B. W. (2006). The prince of mathematics: Carl Friedrich Gauss. Wellesley, MA: A K Peters.Google Scholar
- Wilson, R. J. (2002). Four colors suffice: How the map problem was solved. Princeton, NJ: Princeton University Press.Google Scholar