Why is it Important for Students to Study Hyperbolic Geometry?

The National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics outlines K-12 school mathematics goals and reform through the next decade. This document calls for all college-intending high school students to:

- "Develop an understanding of an axiomatic system through investigating
and comparing Euclidean and Non-Euclidean geometries". [NCTM-89].

- The word "definition" has a very precise meaning in geometry that is quite different from its meaning in common language. Confusion on this concept is the source of many difficulties in understanding the processes of geometric proofs. The strangeness and counter-intuitiveness of non-Euclidean geometry helps students to directly and starkly perceive the differences between Definitions and Theorems as they are used in geometry.
- Non-Euclidean geometry is becoming increasingly important in its role in modern science and technology.
- A study of non-Euclidean geometry make clear that geometry is not something that was completed 3,000 years ago in Greece. It is a current and active field of research.

The following is an example of how studying hyperbolic geometry, helps students understand Euclidean geometry:

The definition of parallel lines (in both Euclidean and hyperbolic geometry) is:

- Parallel lines are infinite lines in the same plane that do not intersect.

A study of hyperbolic geometry helps us to break away from our pictorial definitions by offering us a world in which the pictures are all changed - yet the exact meaning of the words used in each definition remain unchanged. hyperbolic geometry helps us focus on the importance of words.

Several interactive software tools have been developed which allow students to explore Euclidean geometry. For example, The Geometric Supposer (available from Sunburst) and The Geometry Sketch Pad (available from Key Curriculum Press). These software tools have been fairly popular in schools. Because of its graphics capability, the computer offers a high degree of visualization by quickly drawing and measuring geometric figures with a precision that otherwise would require complex drawing instruments, technical skills, and time. These graphics capabilities allow students to explore geometric patterns and theorems not in the usual curriculum. Using these geometry programs, high school students have actually discovered several completely new theorems. [Kedder-85]

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