Bringing Dynamic Geometry to Three Dimensions: The Use of SketchUp in Mathematics Education

Background

In the past fifteen years, the available technologies for use in mathematics classrooms have expanded far beyond the handheld calculator. Many contemporary technologies have made an impact on the teaching and learning of mathematics. Interactive applets and virtual manipulatives in Web 2.0 and Smartboard tools can help elementary students understand the base ten number system and help secondary students visualize the algebraic process of completing the square (Moyer, Bolyard & Spikell, 2002); Dynamic geometry software such as Geometers Sketchpad (GSP) and GeoGebra can allow middle and secondary students to construct shapes and explore their properties with thousands of examples by dragging vertices, while also offering a concrete connection between varying quantities of shapes and algebraic functions (Jones, 2000; Wasserman & Arkan, 2011); Statistics packages such as Fathom and Excel spreadsheets can help middle school students visualize data and secondary students better understand sampling distributions and inferential statistics (Lock, 2002; Meletiou-Mavrotheris, 2003); And the incorporation of Computer Algebra Systems (CAS) into graphing calculators can make symbolic manipulation less burdensome and free secondary students to focus on deeper conceptual understandings and applications (Heid & Edwards, 2001). Each of these technologies have uniquely added to and altered the way K-12 mathematics content can be taught and accessed in school classrooms.

Many of these technologies allow students to interact dynamically with software, which produces real-time changes and modifications based on direct input from students. On GSP and GeoGebra platforms, students can construct, for example, a rectangle, and, by dragging one corner, produce thousands of different rectangles, providing a live interaction between the student and the two-dimensional object. The interface allows students to see, among others, what properties hold for these thousands of cases; in this situation, for example, students may notice that the lengths of the two diagonals of the rectangle are always congruent. The interactive features allow students to use inductive reasoning to help form and justify conclusions about two-dimensional figures. (However, this aspect may also become a limitation that causes students not to see a need for further evidence, confusing inductive reasoning with deductive proof (deVilliers, 2006).) The use of dynamic, two-dimensional geometry software, has proven to be an effective classroom tool; among many extant studies, for example, Saha, Ayub, & Tarmizi (2010) showed a statistically significant gain in students’ performance on a visual-spatial reasoning test about coordinate geometry that used the GeoGebra platform as opposed to a traditional classroom.

This chapter will provide examples for how the use of one contemporary technology, SketchUp^{TM 1} (which will be referred to as GSU), can effectively support implementation of the Common Core State Standards for Mathematics (CCSSM, 2010) as a dynamic three-dimensional software. GSU is free software frequently used for design-related careers such as architecture and interior design. (SketchUp Pro is also available for purchase.) The National Council of Teachers of Mathematics (2000) advocates that, “students should develop visualization skills through hands-on experiences with a variety of geometric objects and through the use of technology that allows them to turn, shrink, and deform two- and three-dimensional objects” (p. 43). While the GSU tools used to create three-dimensional objects are not based solely on Euclidean elements such as points, lines, and planes (i.e., users cannot create an individual “point” in GSU, or a line or plane, in the strictly mathematical sense), CCSSM standards that address visual and spatial reasoning for three-dimensional objects in grades K-12, in particular, can be supported effectively by GSU. The primary purpose of this chapter is to discuss the potential role that GSU can have in mathematics education in the era of CCSSM, by offering effective ways to integrate its’ unique and dynamic features into K-12 mathematics classrooms. This chapter will focus on the technological and dynamic aspects of GSU that make its’ contribution distinct from physical models and other manipulatives used for teaching three-dimensional geometry. Similar to what GSP and GeoGebra have done for two-dimensional geometry in mathematics education, three-dimensional dynamic technologies such as GSU have the potential to do for three-dimensional geometry.