Teaching Geometry through Geography: A Computer Approach

by Peter F. Ash, PhD

Introduction

There is general agreement that mathematics education in the United States is in crisis. (See, e.g., Everybody Counts: A Report to the Nation on the Future of Mathematics Education by the National Research Council, 1989.) The number of college students majoring in mathematics has declined over the past two decades as has the mathematical proficiency of college graduates. Our better students are becoming no match for the better students in Japan and Europe, causing concern about the future global competitiveness of the American economy.

The decline in mathematical competence is tied to students' lack of interest in mathematics. The typical freshman college mathematics course for science majors is a calculus course that has not changed much in the last thirty years. But the world has changed and students know it. Mathematics that does not fully reflect the changes brought about by the computer seems irrelevant to many students.

To regain the interest and enthusiasm of students we need a mathematics curriculum that uses real-world applications, both to motivate students and to teach them mathematics relevant in the computer age. Computers also can be used to individualize instruction, allowing students to proceed at their own pace.

Geometry is ideally suited to computerization because it can rely on computer graphics to illuminate concepts and excite students. My series of software packages will teach geometry through a specific application area: geography. I have chosen geography for several reasons:

• Everyone who has ever looked at a map has used geography.
  

• Modern Geographic Information System (GIS) software has revolutionized the processing of geographic data. Most students will be unfamiliar with this application. Its versatility and power will astound them.
  

• As the common root of the words geography and geometry indicates, the ancient Greeks developed geometry to solve problems of land measurement. Geography is the first application of geometry.

Audience

The software is designed as a series of packages, each covering a specific topic in geometry. Because of the modular nature of the software and because its hypertext-based nature allows the learner to explore each topic at different depths, the software will appeal to a broad market, including:

• high school geometry courses
  

• college mathematics classes at the freshman or sophomore level
  

• self-study (the home computer market).

Coverage

Currently-planned packages are:

1. Ways of mapping the globe to a flat surface: projections and their properties.
2. Understanding contour maps: an intuitive introduction to surface geometry concepts such as gradients and line integrals.
3. The traveling salesperson problem and routing: an introduction to graph theory.
4. The four-color map problem: an introduction to topology.
5. Finding the shortest distance between two points on the globe: great circles and geodesics.
6. Surveying: plane trigonometry, spherical trigonometry, and non-Euclidean geometry.
7. Environmental analysis: computations of area and statistical analysis using GIS software.
8. Spatial analysis: Voronoi diagrams; their properties and applications.