Defining A New Semantics of Space: Computers, Software, and the Electronic World
Computer tools (software, computer-based video toolkits, electronic discussion, and news groups) have the potential not only to define new approaches to geometry education, but also to define a new semantics of space, a potential explored by the contributors to this final section of the volume.
Goldenberg and Cuoco start off by surveying issues raised by dynamic geometry, a geometry that, although derived from Euclid, has a number of distinctive features with no parallels in Euclidean geometry. The authors examine the effects on teaching and learning of one key feature of dynamic geometry--dragging--and look in particular at the way students perceive figures (and geometry) because of the defaults built into the "drag" mode of operation of these electronic tools. They look at the notion of continuum as applied to two-dimensional figures, at the possibility of two-dimensional "monsters" (e.g., self-intersecting bow-tie quadrilaterals), and examine the implications of "stretchy" line segments, which preserve ratios between (non-Euclidean) "movable" points rather than the particular, measured distances between (Euclidean) fixed points.
De Villiers, in chapter 15, also notes the effect of the dynamic geometry on the teaching and learning of geometry. Raising the issue of the traditional geometric ritual of proof, he examines student conceptions of proof, suggesting that dynamic geometry pushes proof away from verification toward discovery and deductive explanation. This theme is revisited by Olive (in chap. 16), who also focuses on yet another aspect of dynamic geometry raised by Goldenberg and Cuoco, the potential fusion of geometry and algebra. Olive describes a broad spectrum of student work with dynamic geometry, ranging from transformations and symmetry to functions and conic sections.
The next two chapters explore a different dynamic, that of geometry as applied mathematics. Watt describes a prototype fifth-grade geometry unit and child-appropriate computer-aided design (CAD) program. Fifth- grade students used KidCAD to model furniture rearrangment in their classroom and to design a new school space. Watt characterizes this activity as negotiation between the model space and the actual space of the world. Noting the simplicity of this software in comparison to dynamic