Innovative computer software has transformed geometry instruction over the last two decades. Many veteran teachers remember the mesmerizing effect of tangram-like games from the earliest days of the personal computer. Today, the same compelling graphics are appearing in the geometry classroom.

The infusion of technology into the teaching of geometry started in the early 1980s as pioneering schools adopted the Logo programming language. Students and teachers have benefited from this pairing of subject content and medium. From early elementary grades through high school, Logo software enhances instruction; provides motivation; and enables students to use modern tools in the study of shapes, angles, and theorems.

The wealth and depth of resources available via the Web alone are a treasure for teachers at every level. Many students are naturally drawn to technology. The use of carefully selected software can really drive home a lesson, unit, or entire syllabus in secondary courses in geometry, CAD (computer-aided design), and computer graphics.

Search and Ye Shall Find

A little time spent investigating websites can lead to worthwhile online activities to enhance lessons. While a simple Google search on tessellations may yield far too many sites to evaluate effectively, narrowing the search by adding a qualifier such as "tutorials" or "lesson plans" can streamline a search for relevant websites. Topics ranging from keystones to quilting, from trusses to tetrahedrons, from arches to Archimedes can be enhanced with carefully selected websites and activities. Examples abound: Carbon nanotubes made up of strings of "Buckyballs" or the recent reinterpretation of an ancient manuscript by Archimedes still invites exploration in a field previously associated in students' minds with Euclid and ancient Greece.

Elementary Grades

The judicious use of Internet resources and software packages that allow students to explore basic spatial relationships complement the use of manipulatives such as pattern blocks, tangrams, geo-boards, and origami. Even in a one- or two-computer classroom, small groups of students can complete age-appropriate and educationally sound online activities.

Technology can be thought of as another tool which, along with graph paper, rulers, construction paper, building blocks, and field trips, can make learning rich and varied. Technology can serve effectively in a complementary and supportive role in the early elementary classroom. Tactile and sensory explorations of math manipulatives paired with interactive activities stimulate a child's curiosity and foster deeper awareness of meaning in snowflakes, chessboards, soccer balls, quilts, and tangrams.

National Library of Virtual Manipulatives for Interactive Mathematics

(matti.usu.edu/nlvm/nav/ vlibrary.html)

This resource offers Web-based activities for many age groups. After pairing traditional manipulatives with their virtual counterparts, students can discuss the uses and advantages of both types of activity. Help students place these activities in context by citing real-world examples such as surgeons trying new techniques on virtual patients, scientists using simulation to test the latest Mars probe, or auto-safety experts testing for the best designs of airbags and reinforced side-panels.

Software Titles

Web-based resources offer excellent advantages in classrooms with broadband connectivity: no additional cost for extra software installation, maintenance, or upgrades with the concomitant support staff. However, even for classrooms without fast and reliable Web access, many software packages which do not rely on fast Internet connections are time- and teacher-tested.

Students in third and fourth grades can explore properties of shapes, reflection and rotation, symmetry, polygons, and more in two- and three-dimensional environments using ShapeUp, available on Macintosh and PC platforms from Sunburst Software. Other titles such as Exploring Shapes Through Patterns are suitable for kindergarten and first grade.

Middle Grades

In grades five through eight, many schools have gone beyond basic instruction in keyboarding and word processing with the introduction of advanced animation and graphics software that affords even deeper and richer opportunities for exploring geometry. The widespread use of powerful Shockwave and Flash animation on websites has created a plethora of games, activities, and interactive demonstrations of every geometric concept from Euclid's theorems to hypercubes, tessellations, and M. C. Escher.

LEGO Logo Mindstorms and Micro Worlds Pro

Since its inception in the late 1960s at MIT, the Logo programming language has been a suecess in elementary schools. In as early as third grade, students use MicroWorlds Pro (one of the latest incarnations in the long history of Logo) to explore basic "Turtle Geometry" and more advanced geometric constructions. In the last decade, the introduction of Mindstorms to the Logo line has allowed for the exploration of robotics and object-oriented programming as part of many elementary curricula, especially in after-school programs. LEGO Logo Mindstorms and MicroWorlds Pro are especially suited for extracurricular activities because the flexibility and openended nature of project-based learning is an integral component of the constructivist pedagogy on which Logo was founded.

The connection between Logo and popular LEGO construction kits made possible with Logo Mindstorms can provide further investigations in robotics, programming, and inquiry in schools that have embraced the use of Logo in the earlier grades. Mindstorms could serve as either a stand-alone activity or in a sequence that builds on early work in MicroWorlds.

The Geometer's Sketchpad

The Geometer's Sketchpad (GSP) has earned numerous accolades such as "Best Mathematics Education Software of all Time," as it was named by the Stevens Institute of Technology Mentor Teachers Survey.

Originally developed at Swarthmore College in the 1980s for the teaching of the standard high school geometry curriculum accompanying a textbook, this program is being used in many school as early as the fifth grade in introductory investigations of the properties of squares, parallelograms, and rhombi; construction of traditional quilt blocks; and the exploration of ratio and proportion. One school reports the study of cardioids on Valentine's Day with The Geometer's Sketchpad. We've come a long way from cut-and-fold construction paper hearts!

Adapting Commercial Software for Classroom Use

In my collaborations with math teachers in middle schools, I have used several industry-standard vector-based computer graphics software programs including Adobe Illustrator, Macromedia Freehand, and Macromedia Flash to create projects on topics ranging from tessellations, regular polygons, Islamic patterning, and the Platonic and Archimedian solids. The use of the software allows students to achieve results that would be beyond their capabilities to construct by hand alone. We start with the simple premise that any concept that can be taught with traditional tools such as compasses, protractors, and templates can be carried into new media with analogous directives and outcomes. One reason for the success of these endeavors is that students are able to learn the most sophisticated software readily and rapidly (I like to say that students can learn complex software faster than I can teach it). Many students are attracted to the complexity and power of the software and revel in using the same software tools that professionals employ.

High School

The Geometer's Sketchpad

At the high school level, the GSP has been recognized as a model of seamless integration of software, curricula, and pedagogy possibly unequaled in any other area of K-12 education (with the exception of ubiquitous word-processing). As a result of the intense and ongoing support of college and university programs in the development of this software, the GSP remains the best approach to the application of computer software in an educational setting. I think of this program as the mothership of all software titles. So much has been written about this application that little can be added here. The constant update of resources and continued upgrades ensure that the GSP is likely to keep its position as a leader for years.

Other Related Courseware

In addition to mathematics instruction in general and geometry topics in particular, several other high school-level courses employ state-of-the-art software in syllabi ranging from computer-aided design to computer graphics and even set and stage design.

AutoCAD and Vectorworks