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:
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 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)
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]
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Copyright©: Joel Castellanos, 1994-2007