with the aid of a computer-assisted design tool--a KidCAD. Here the technology facilitates a transition between experience and geometric representation. Further highlighting the diverse uses of technology, Zech and her colleagues at the Cognition and Technology Group at Vanderbilt University discuss and illustrate their work in developing visual (computer- based) toolkits to help middle-school students carry out real-world activities such as wayfinding and the designing of playgrounds. In the last chapter, Renninger, Weimar, and Klotz indicate that if we recast the form, substance, and tools of a geometry education, then it also becomes imperative to reconsider teachers' professional development. They discuss the development of a national electronic forum for geometry teachers and their students that encourages learning, discussion, and problem solving, as well as providing a resource for the field.
As Hershkowitz suggests in the epilogue, the contributors to this volume advance several related agendas for mathematics education. First, the authors help us better understand the wide range and influence of spatial reasoning and geometry in mathematics. The research presented here suggests that instead of the current arrangement of years of arithmetic with occasional small helpings of geometry, geometry and spatial reasoning can and should be incorporated as a central feature of a general mathematics education: geometry for all. Second, the contributors emphasize the diversity and range of student thinking encompassed by spatial reasoning and geometry. Not only are existing theories called into question, but several fruitful avenues for new theoretical development in mathematics education are suggested. Third, contributors explore how the development of spatial thinking is tied to tools, ranging from modest (but powerful) ones like PolydronsTM to mechanical curve-drawing devices and the new notational forms made possible by computer-based technologies. Taken together, the research suggests renewed curricular focus on geometry and space: Geometry is not only central to reform in mathematics curricula and the instructional focus on learning with understanding, but with its inherent (and in these studies, enhanced) emphasis on conjecture, argumentation, deductive proof, and reflection, is also central to a solid general education and, as many in this volume note, to good habits of mind.
Preparation of this volume was supported by the National Center for Research in Mathematical Sciences Education (NCRMSE), which in turn was supported by a grant from the Office of Educational Research and Improvement, U.S. Department of Education (grant No. R117G1002) and by the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison. Any opinions, findings, or conclusions are those of the authors and do not necessarily reflect the views of the supporting agencies.