Spotlight: University of Central Florida Teaching Axioms
Implemented in College Geometry


Mathematics Education majors at the University of Central
Florida are required to take College Geometry, a course which
emphasizes Euclidean Geometry. The course design was based on
three "axioms" for good teaching. 

Axiom 1: You must know what you are going to teach.
Axiom 2: You must know how to teach what you will teach.
Axiom 3: You should know more than what you will teach.

Axiom 1:  You must know what you are gong to teach.  Although
the instructors did not assume that students enrolling in this
course would know Euclidean geometry, there was no intent to
convert a college geometry course into a high school geometry
course. It seemed , however, that a thorough review of high
school Euclidean Geometry would be needed.

Much has been written lately about the influence of technology
on teaching mathematics. The faculty decided to use IBM's
Geometry Two software which contained tutorials, a sequenced
list of axioms and theorems, and a drawing tool covering
construction, transformation, and vector geometry. The software
includes most of the things typically found in a formal high
school geometry course. There are proofs of theorems where
statements are given and the student must supply the reasons;
reasons are given and statements must be supplied; some of the
steps in the proof must be supplied; and proofs that must be
developed entirely.

During the first two weeks of class, students were introduced
to the software and given some instruction on the use of
personal computers linked to a server that contains the
Geometry Two software. The students were shown how to use the
drawing tool and some activities in the software. Then a
theorem was completed to show how the software and its set of
integrated rules is used to develop a proof. One important
aspect of this development was to show that the proof may be
completed in different ways, limited only by the sets of
theorems, axioms and rules that have been developed prior to
the particular problem in consideration.

These students were required to work through the software on
their own. Two quizzes were given on this software in which
each student was randomly assigned a proof that had to be
completed on the computer. A screen print of the completed
proof was submitted for evaluation.

Axiom 2: You must know how to teach what you will teach.  If
the only teaching methodology students observe in a university
mathematics class is "lecture" then when they teach
mathematics, "lecture" is what they are most likely to do.
While "lecture" has a place in the classrooms, the research
literature shows that effective teaching requires the use of
many techniques that are chosen for the particular subject. A
college geometry class is an ideal setting for demonstrating
the effectiveness of guided learning and cooperation groups.

In this course, students were placed in small groups and given
assignments. The teaching staff played many roles. On occasion
it was necessary to lecture in order to provide some basic
information, but this was kept to a minimum. Their primary role
during group work was to encourage and provide hints. They
tried not to provide the solution when a group was stuck.
Groups must have the freedom to create solutions. Geometry is
particularly good for this since the method for solving a
problem is seldom unique. There were typically four or five
different approaches to a problem from eight groups. In the
process, they learned geometry and got ideas about how to teach
the subject.

Axiom 3: You should know more than what you will teach.  The
standard high school geometry course is basic Euclidean
Geometry. Prospective high school teachers should know more
than this. In addition to Geometry Two we used Howard EvesU
Fundamentals of Modern Elementary Geometry. Most of the text
part of the course focused on the Menlaus and Ceva Theorems,
Pascal's Mystic Hexagram Theorem for a Circle, Desargues'
Two-Triangle Theorem, a number of theorems about orthogonal
circles, and some Euclidean constructions.

On the first day of class the students were typically
overwhelmed. By the end of the semester, they were amazed at
how much geometry they covered. Student reaction to the course
has been extremely positive. Some sample statements include:

"I now know some of the thrills, aggravations, excitements, and
frustrations of cooperative groups."

"I did not know there was such a variety of approaches in
geometry-and we just scratched the surface."

Teaching a course in this manner requires much more work than
teaching a lecture course. There is more preparation required,
and evaluation is more time-consuming since every solution must
be read very carefully with extensive and precise comments
provided to the students. But it is a lot of fun too.


For more information, contact:
Douglas K Brumbaugh
Professor of Mathematics, or
Joby Milo Anthony
Associate Professor of Mathematics 
University of Central Florida
Orlando, FL 32816
Phone: (407) 823P2045
Fax:  (407) 823-5135
E-mail: brumbad@pegasus.cc.ucf.edu
.