Can a Kite Be a Triangle? Aesthetics and Creative Discourse in an Interactive Geometric Environment

Abstract

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.