Designing Learning Environments for Developing Understanding of Geometry and Space

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FIG. 16.24 The graph of ln(x + 2) / x constructed in Sketchpad 3.0.

In this chapter I have attempted to illustrate, by example, the opportunities for exploring and integrating many ideas and topics in mathematics with the Geometer's Sketchpad. Examples were presented from elementary-, middle-, and high-school classrooms where teachers have been using Sketchpad in a variety of ways for diverse purposes. Although most of these examples illustrate the exciting potential of Sketchpad for students' creative explorations, some examples also illustrate potential problems. Some of the problems that emerged from these different implementations of Sketchpad were discussed, such as the difference between "drawing" and "constructing" (see Finzer & Bennett, 1995, for a more in-depth discussion on this topic), demonstration versus proof (see de Villiers, chap. 15, this volume, for a thoughtful discussion of proof in a drag-mode geometry environment), and pedagogical problems associated with a teacher's lack of experience with Sketchpad and the perceived constraints of the "traditional" curriculum.

Examples of resources and ideas for exploring topics outside the normal geometry curriculum were briefly described. These topics included conics, trigonometry, coordinate geometry, and algebraic functions. Goldenberg et al. (chap. 1, this volume) provide other examples. Finally, the potential of Sketchpad for helping teachers and students integrate their developing knowledge of algebra and geometry was illustrated through an exploration of the new locus and graphing features of version 3.0. According to Goldenberg et al. (chap. 1, this volume), such an integration will help students connect with mathematics and build a "habits of mind" perspective that generates mathematical power.

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