Designing Learning Environments for Developing Understanding of Geometry and Space

By Richard Lehrer; Daniel Chazan | Go to book overview

Part I
Why Teach Geometry?

Over the past half-century in the United States, geometry has taken less and less of a central role in mathematics curricula. Many have even questioned its relevance in this high-tech, computer-dependent world. Among mathematics educators, researchers, mathematicians, and scientists, however, criticism has focused on the lack of emphasis in mathematics education on reasoning about geometry and space, on the traditional (Euclidean) emphasis on proof, and on the approaches taken to teaching geometry and space. These criticisms and the mathematics reform movement associated with the NCTM Standards ( 1989, 1991, 1995) form the backdrop to the first section of this book. Why, indeed, should students study geometry? How and when do we begin to teach geometry? What should the study of geometry entail? How do we generalize geometry to children's overall education?

Without linking their discussion to particular grade levels or courses, Goldenberg, Cuoco, and Mark make a strong plea for regular attention to geometry throughout the curriculum. Their examples--from the dissection of shapes to the use of area models for multiplication, from proofs without words to dynamically linked graphs--suggest a wide range of activities possible in a geometry curriculum. Throughout their chapter, they examine the central role of visualization in mathematics-- noting along the way the importance of developing habits of mind that create a fertile environment for reflection, argument, and thought.

Gravemeijer, in the following chapter, takes a close look at how we teach geometry, suggesting (from 20 years of experience in The Netherlands with Realistic Mathematics Education) that geometry instruction build on children's informal knowledge of their environment. Drawing on Dutch curricular materials and examples taken from an American reform curriculum, he suggests ways of supporting children as they build their own models of geometric concepts and of guiding them through the reinvention (and abstraction) of the intended mathematics.

In the context of secondary mathematics reform, Chazan and Yerushalmy suggest that it is more important to focus on students' classroom activity than on the particular type of geometry taught. Although they appreciate the argument for introducing students to new mathematical advances, they argue that there still are benefits to be derived from courses in Euclidean geometry, provided that the way such courses are taught changes. Noting Schoenfeld ( 1988) scathing critique of well-taught traditional geometry courses, they argue that dynamic-geometry programs support the creation of radically different Euclidean-geometry courses. In such courses, students' conjectures play a central role, and deductive rea

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