Course
Motivation:
One of the goals in teaching mathematics is to try to get students to
understand the ”proposition–proof” method of deciding whether or not given
statements are true, as well as determining what types of applications such
statements will give. Usually, the first formal exposure students have to this
method is during a high school geometry course, where we have to be satisfied
with “statement–reason” fill-in-the-blanks types of “proofs.” Since there
is no further exposure to theorems and proofs until students take their first
year of calculus, they will have forgotten almost everything about that part of
high school geometry, usually retaining only the mechanical material covered on
the ACT.
Moreover,
during the first year of calculus, students and instructors are working so hard
on rules of derivatives that formal proofs of important theorems, while perhaps
covered in class, are not typically expected of students on exams. We try
to rectify this with a second-year course involving direct and indirect methods
of proof, but again by the end of the semester this goal has been partially
lost because there is new material in the course content which also must be
learned. (At the U of M we used to do this with the linear algebra course; now
we are trying it with discrete mathematics.)
Thus,
College Geometry is a course created for third-year college students as an
in-depth study of how the propositional calculus is used to obtain results in a
given area of mathematics, and how this leads to unanswered questions and
possible ways to find answers to these questions. Classic Greek scholars
created this logical method of synthesis and analysis, with the first
applications being those in the areas of geometry and number theory (another
third-year course).
Course Description:
This course is roughly divided into three parts, with Part I being Euclidean
Geometry, picking up where classical (high school) geometry leaves off and
proceeding to more modern times.
Part II is an introduction to Projective Geometry, where the Euclidean
postulates are stripped down to their bare essentials, investigating what
results may thus be obtained, leading to the modern development of Euclidean
geometry.
Part III is typically referred to as Noneuclidean Geometry. We begin with the Erlanger Program, which states that a geometry is defined in terms of its invariants under certain transformations. After demonstrating this with projective and affine geometry, we discuss how the set of isometries (rigid motions) provides another way to study Euclidean geometry. Finally, in studying the history of the parallel postulate, we learn how other types of geometric descriptions of physical space (hyperbolic, elliptic, spherical) have been developed and how they are used.
Note: In previous incarnations of this course the topic of Analytic Geometry was also included. The present course offering de-emphasizes this topic to some extent, but for students with a strong calculus background there will be opportunities to earn Extra Credit by completing selected assignments in analytic geometry, such as equations for conic sections, inversion, evolutes and involutes, and the geometric applications of concepts covered in Calculus III. Extra credit may also be available for future teachers in presenting how certain topics might be taught in high school geometry.
Course Prerequisites:
MATH 2701 or 2702, plus at least one year of calculus.
Text:
Fundamental Concepts of Geometry, Bruce E. Meserve
(
Originally published by Addison-Wesley in 1955, reprinted in
1983, currently offered by Dover Publications (www.doverpublications.com).
Class Website:
http://www.msci.memphis.edu/faculty/dwiggins/3581/default.htm
Class Meetings:
MWF
Instructor:
D. P. Dwiggins, PhD (send e-mail: ddwiggns@memphis.edu)
Office:
Room 368, Dunn Hall (third floor)
Hours:
Tuesdays and Thursdays, 9:00-noon
Afternoon hours available upon request.
Telephone:
678-4174
Course Evaluation:
Homework will be assigned and collected on a biweekly basis, leading to
a daily average worth 100 points. There
will also be two mid-term exams,
worth 100 points each, along with the final (comprehensive) exam, also
worth 100 points.
Grade Calculation:
The homework and exams give a total of 400 points, with the final exam (if
higher)
replacing the lowest of the other three 100-point scores. Dividing by 400 gives
the semester average (%), with grades assigned according to the Grade Scale
.
Attendance Policy:
As needed for purposes of reporting to the University.