Designing Learning Environments for Developing Understanding of Geometry and Space

By Richard Lehrer; Daniel Chazan | Go to book overview

10
Sixth-Grade Students' Conceptions of Stability in Engineering Contexts

James A. Middleton

Arizona State University

Robert Corbett

Kahler Slater Architects

Geometry, as a curricular subject for primary- and middle-school students, provides a descriptive medium by which real-world actions and objects can be described in terms of their physical properties. Geometric models can be applied to real-world problems to simplify complex problem situations, and many algebraic and numeric ideas can be fostered by looking at them through a geometric perspective ( National Council of Teachers of Mathematics [NCTM], 1989). The complex spatial patterns of the real world can be simplified into component relationships such as points, lines, angles, transformations, similarity, and dimensionality, and these physically simpler (but cognitively more abstract) ideas can be operated on mentally--changed, recombined, transformed--whereas the physical objects themselves may not be. In other words, the field of geometry can be applied to realistic situations, and conversely, realistic situations give rise to geometric thinking:

The Euclidean space with all its objects is a rich structure, although it is poor if compared with all I perceive around, its colours, polished and rough surfaces, sounds, smells, movements. But thanks to the impoverishment it furnishes a certain context, which for some reasons suits us extremely well. . . . This Euclidean space has never been an aim in itself, but rather it has been the mental and mathematically conceptual substratum for what is done in it. ( Freudenthal, 1983, p. 224)

As Freudenthal went on to describe, the development of students' knowledge of geometry reflects a tension between experience in an irregular world and this notational/representational system (or conceptual

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