# Designing Learning Environments for Developing Understanding of Geometry and Space

# Designing Learning Environments for Developing Understanding of Geometry and Space

## Synopsis

Rather than looking to high school geometry as the locus--and all too often, the apex--of geometric reasoning, the contributors to this volume suggest that reasoning about space can and should be successfully integrated with other forms of mathematics, starting at the elementary level and continuing through high school. Reintegrating spatial reasoning into the mathematical mainstream--indeed, placing it at the core of K-12 mathematics environments that promote learning with understanding--will mean increased attention to problems in modeling, structure, and design and reinvigoration of traditional topics such as measure, dimension, and form. Further, the editors' position is that the teaching of geometry and spatial visualization in school should not be compressed into a characterization of Greek geometry, but should include attention to contributions to the mathematics of space that developed subsequent to those of the Greeks.

This volume is essential reading for those involved in mathematics education at all levels, including university faculty, researchers, and graduate students.

## Excerpt

Despite a long intellectual history dating back to the origins of civilization, and a recent resurgence as cutting-edge mathematics, geometry and spatial visualization in school are often compressed into a caricature of Greek geometry, generally reserved for the second year of high school. The resulting impoverished view of the mathematics of space rebounds throughout schooling generally to diminish student (and adult) understanding of mathematics.

Among mathematicians and mathematics educators, there is increasing consensus, however, that geometry and spatial visualization deserve a more prominent role in school mathematics. Formalist views of mathematics as a "game" in which abstract symbols are manipulated (views ascendant in the second half of the 19th and the early part of the 20th century) are now challenged by views emphasizing the role of "empirical" methods in mathematics. Contemporary mathematicians studying chaos, fractals, and nonlinear dynamics rely on computer-generated visual representations to perform and display the results of experiments. Moreover, not only do new computer technologies make mathematical experimentation possible and plausible, but these technologies have been widely adopted in a range of cultural practices. At the same time, the mathematics education reform movement accords a central role to mathematical exploration and sense-making and supports the use of technology and visual representations. This shift implies the need to reexamine the nature of the school mathematics curriculum, the goals and aims of teaching, and the design of instruction.

Rather than looking to high-school geometry as the locus (and all too often, the apex) of geometric reasoning, the authors of this volume, many of whom were active in the National Center for Research in Mathematical Sciences Education (NCRMSE), suggest that reasoning about space can and should be successfully integrated with other forms of mathematics, starting at the elementary level and continuing through high school. Reintegrating spatial reasoning into the mathematical mainstream (indeed, placing it at the core of K-12 mathematics environments that promote learning with understanding) will mean increased attention to problems in modeling, structure, and design and reinvigoration of traditional topics like measure, dimension, and form: Geometry education should include contributions to the mathematics of space that were developed after those of the Greeks.

This volume reflects our appreciation of the interactive roles of subject matter, teachers, students, and technologies in designing classrooms that promote understanding of geometry and space. Although these elments of geometry education are mutually constituted, the volume is organized to highlight our vision of a general geometry education, the de-