# For the Teacher: 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].
Some justifications for a study of non-Euclidean geometry are as follows:
• 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.
NonEuclid creates an interactive environment for learning about and exploring non-Euclidean geometry on the high school or undergraduate level. The software package includes explanations, activities, and strategies for incorporating non-Euclidean geometry into high school curriculum.

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.
In Euclidean geometry, we can use this definition to prove the theorem that "parallel lines are equidistant along their length". When students are asked to prove this theorem, they often complain "I can SEE that they are equidistant - what are you asking me to do?" This is because most of us first learned about parallel lines when we were very young. We where shown pictures, and told "these are parallel lines". We use mental images of parallel lines, squares and circles as our definitions. This, in geometry, is completely wrong!!! In common language, we begin with an object or idea. A definition is no more then an attempt to describe in words the preexisting object or idea. For example, a Dog is something that exists in the real world. When we look up the word "dog" in the dictionary, we find a bunch of words that try to discribed as consicesly and accurately as posible what a dog is. A dog (and all objects in common language), is a priori to its definition. In geometry, the definition is primary. Geometry begins with definitions of abstract, unvisualized objects. The properties of an abstract object follow as consequences of the definition and are called "theorems". For example, parallel lines DO NOT exist in the world. Parallel lines are nothing more and nothing less then "infinite lines in the same plane that do not intersect". This distinction is very difficult to understand and is the source of much confusion about geometric proofs.

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]

NonEuclid Home
Next Topic - References & Further Reading