Math 450: GEOMETRY
COURSE: Math 450, Geometry, 3 credits
INSTRUCTOR: Dr. Larry Krajewski
Office: MC 526
Phone: (608) 796-3658 (office)
(608) 782-1648 (home) [no calls after 10 p.m.]
Office hours: 3M, 11W, 12F & by appointment
PREREQUISITES: C or better in Math 260
TEXTBOOK: Geometry by Discovery by David Gay, John Wiley & Sons, 1998
Finite Geometry Notes
FINAL EXAM: Wednesday, 10 May 2000, 12:50-2:50 p.m.
-become acquainted with historical developments in geometry
-explore the many applications of geometry in various areas of mathematics
-provide a variety of geometric concepts and tools for use in other branches of mathematics
-present Euclidean geometry as a mathematical system and as one of several geometries
-present geometry as a rich source of mathematical models
-provide informal expository developments of school geometry
-challenge the prospective teacher to consider what high school geometry could be
1. Axiomatic Systems
A. Sets of Axioms
B. Finite Geometries
2. Problem Solving Strategies
5. Shortest Path Problems
7. Best Shapes
Problem Sets 60%
There will be no tests. I am asking you to keep a journal and there will be weekly problem sets. There will
also be daily exercises generated from class work or the textbook. A final project is due on final exam day.
Now a few words about the journals. I have been reading lately about writing, learning, and mathematics
(see bibliography). Many people have reported the advantages of using writing, especially journals, in
their classes. Among the benefits cited were improved ability in problem solving, higher scores on tests,
and improved communication. I truly believe that one can learn better if one has to describe in writing what
one is thinking. It forces one to clarify ideas and identify areas of difficulty. I will read them and write my
comments or answers to your questions. The following scale will be used:
+ : exceptionally good entry
û : satisfactory
_ : less than what I expect
Each student is required to prepare a term project to be presented as a science_fair type exhibit.
Purposes of the project are:
1. to gain experience in researching a mathematics topic;
2. to become familiar with reference sources in mathematics;
3. to become an expert in some topic in geometry;
4. to gain experience in presenting information effectively to others, and
5. to have fun!
Projects will be judged on both mathematical content and quality of presentation. Be prepared to answer
questions when you present your project. Consult your text, pages 383-399, for project ideas. You must
clear your choice of topics with me and have a topic selected by spring break.
NOTES TO THE STUDENTS
I will try to challenge you with interesting and relevant problems. The British philosopher and
mathematician Bertrand Russell once said, "Most people would rather die than think_and most do!"
Thinking is a dynamic process. To be a critical thinker you need to be truthful, open_minded, empathetic,
questioning, active, autonomous, rational, self_critical, and flexible. You should be able to analyze and
solve problems effectively; generate, test, and organize ideas; form, relate and apply concepts; construct
and evaluate arguments; explore issues from multiple perspectives; reason analytically with concepts,
relationships, and abstract properties; develop evidence and reason to support views; exchange ideas
with others in a systematic fashion; apply knowledge to new situations; and become aware of your own
thinking processes in order to monitor and direct it.
Some of you may have had mathematics courses that were based on the transmission, or absorption, view
of teaching and learning. In this view, students passively "absorb" mathematical structures invented by
others and recorded in texts or known by authoritative adults. Teaching consists of transmitting sets of
established facts, skills, and concepts to students. I do not accept this view. I am a constructivist.
Constructivists believe that knowledge is actively created or invented by the person, not passively
received from the environment. No one true reality exists, only individual interpretations of the world.
These interpretations are shaped by experience and social interactions. Thus, learning mathematics
should be thought of as a process, of adapting to and organizing one's quantitative world, not discovering
preexisting ideas imposed by others.
Consequently, I have three goals when I teach. The first is to help you develop mathematical structures
that are more complex, abstract, and powerful than the ones you currently possess so that you will be
capable of solving a wide variety of meaningful problems. The second is to help you become autonomous
and self_motivated in your mathematical activities. You will not "get" mathematics from me but from your
own explorations, thinking, reflecting, and participation in discussions. As independent students you will
see your responsibility is to make sense of, and communicate about, mathematics. Hopefully you will see
mathematics as an open-ended, creative activity and not a rigid collection of recipes. And the last is to help
you become a skeptical student who looks for evidence, example, counterexample and proof, not simply
because school exercises demand it, but because of an internalized compulsion to know and to
This is not to imply that you should work alone. Research has shown that students learn better if they work
cooperatively in small groups to solve problems and learn to argue convincingly for their approach among
conflicting ideas and methods. You will have several opportunities to work together in class on problems.
Another bit of research points to the value of working together on homework. In the 1970s, Uri Treisman,
a mathematician at the University of California at Berkeley began an extensive study to determine why
students did poorly in calculus. Currently about 40_50% of those who start calculus do not finish.
Professor Treisman found that students who did homework in groups were far more likely to do well than
students who worked alone. He observed students who worked in groups of three or more. One student
would get an answer that was wrong; a second student would find the error and correct it. The process
was repeated continually with the result that virtually all the students in the group understood how to
approach the problem under discussion correctly. For students who worked alone, misconceptions went
unchallenged. This led to a downward spiral of frustration and self_doubt. So I encourage you to work
together on homework.
AMERICANS WITH DISABILITIES ACT
If you are a person with a disability and require any auxiliary aids, services or other accommodations for this
class, please see me and Wayne Wojciechowski, The Americans With Disabilities Act Coordinator (MC 320
_ 796_3085) within ten days to discuss your accommodation needs.
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Abelson, Harold and Andrea diSessa, Turtle Geometry, MIT Press, 1980.
Albers, Donald, "Geometry is Alive and Well and Living in Paris Under an Assumed Name," Two Year
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Barnsley, Michael, Fractals Everywhere, Academic Press, 1988.
Beck, Anatole et.al., Excursions into Mathematics, Worth, 1969.
"Beyond Euclid_Turning the Incredible into the Obvious," QUANTUM, Sept/Oct 1992, 19_23.
Chakerian, G.D. et.al., Geometry_A Guided Inquiry, Houghton-Mifflin, 1972.
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