KENNESAW STATE UNIVERSITY
GRADUATE COURSE PROPOSAL OR REVISION,
Cover Sheet (10/02/2002)
Course Number/Program Name MATH 7395
Department Mathematics and Statistics
Degree Title (if applicable) M.A.T., M.Ed., and Ed.D.
Proposed Effective Date Spring, 2014
Check one or more of the following and complete the appropriate sections:
X New Course Proposal
Course Title Change
Course Number Change
Course Credit Change
Course Prerequisite Change
Course Description Change
Sections to be Completed
II, III, IV, V, VII
I, II, III
I, II, III
I, II, III
I, II, III
I, II, III
Notes:
If proposed changes to an existing course are substantial (credit hours, title, and description), a new course with a
new number should be proposed.
A new Course Proposal (Sections II, III, IV, V, VII) is required for each new course proposed as part of a new
program. Current catalog information (Section I) is required for each existing course incorporated into the
program.
Minor changes to a course can use the simplified E-Z Course Change Form.
Submitted by:
Approved
Charles Koppelman
Faculty Member
4/9/13
Date
Not Approved
Department Curriculum Committee Date
Approved
Approved
Approved
Approved
Approved
Approved
Not Approved
Department Chair
Date
College Curriculum Committee
Date
College Dean
Date
GPCC Chair
Date
Dean, Graduate College
Date
Not Approved
Not Approved
Not Approved
Not Approved
Not Approved
Vice President for Academic Affairs Date
Approved
Not Approved
President
Date
KENNESAW STATE UNIVERSITY
GRADUATE COURSE/CONCENTRATION/PROGRAM CHANGE
I.
Current Information (Fill in for changes)
Page Number in Current Catalog
Course Prefix and Number
Course Title
Class Hours
____Laboratory Hours_______Credit Hours_____
Prerequisites
Description (or Current Degree Requirements)
II.
Proposed Information (Fill in for changes and new courses)
Course Prefix and Number MATH 7395_________________________
Course Title ___ Non-Euclidean Geometry_____________
Class Hours
3___Laboratory Hours___0___CreditHours____3___
Prerequisites Math 3395 or Math 7714 or consent of instructor
Description (or Proposed Degree Requirements)
This course examines the development of the axiomatic basis for non-Euclidean
geometry and its relationship to Euclidean geometry, and analyzes proofs of important
theorems in hyperbolic geometry. Topics will include Hilbert’s axioms, finite and
infinite affine and projective planes, neutral geometry, Hilbert planes, Euclidean planes,
and hyperbolic planes. Special emphasis will be given to the nature of geometric proof
and historical attempts to prove the Euclidean parallel postulate.
III.
Justification
Currently there is only one graduate level geometry course (Math 7714) offered at Kennesaw
State University. The main focus of Math 7714 is Euclidean Geometry and so there is
considerable overlap between its content and the content of Math 3395, which is the only
geometry course available to undergraduate secondary math education majors. Graduate
students in mathematics education need a more thorough grounding in the history and
development of modern geometric concepts and geometric proof than is currently available
to them in Math 7714. The proposed course, Non-Euclidean Geometry, fills this need.
In spring 2012, the Department of Mathematics and Statistics offered Introduction to NonEuclidean Geometry as a special topics course. A combination of graduate and
undergraduate students enrolled in the course, which was double-numbered Math 4490/7900.
The students who were enrolled in the course began with marked weaknesses in their ability
to develop and present a cogent, well-developed, and logical proof. An emphasis in
Introduction to Non-Euclidean Geometry was an examination of the nature of proof in the
context of the historical development of Non-Euclidean Geometry and students are required
to develop both direct and indirect proofs based on Hilbert’s axioms. In their final
evaluations, one student wrote “This course is extremely important in the math curriculum.
It teaches a critical thinking skill not available to other courses. Intro to Non-Euclidean
Geometry teaches careful examination of axioms and their consequences. The course helps
develop logic, logical equivalence, and implication in a challenging and engaging way.”
In Spring, 2013, the Mathematics Education Committee agreed to include Non-Euclidean
Geometry as an elective course in the Graduate Math/MAED Course Rotation schedule, to be
first offered in spring 2014. The course would be open to math M.A.T students, M.Ed
students, and Ed.D. students. The course would be double-numbered so that undergraduates
could enroll in MATH 4490 (Special Topics in Mathematics).
IV.
Additional Information (for New Courses only)
Instructor: Charles Koppelman
Text: Euclidean and Non-Euclidean Geometries, Fourth Edition - Author:
Marvin J. Greenberg (published by W.H. Freeman, 2008)
Prerequisites: Math 3395 or Math 7714 or consent of instructor
Objectives:
Students will be able to:
o apply Euclid’s postulates and identify the gaps that led to the development of nonEuclidean geometry
o apply the rules of logic and its notation
o apply the incidence axioms and prove the propositions that derive from them.
o identify finite models of incidence geometry and establish isomorphisms between
models
o determine whether a finite model is an affine plane, projective plane, or neither,
and prove their conclusion
o distinguish between consistent and inconsistent axiomatic systems
o apply the principle of duality for projective planes
o form the projective completion of an affine plane
o apply the betweenness axioms
o prove the propositions that derive from the betweenness axioms
o apply the propositions to prove Pasch’s theorem and the crossbar theorem
o apply the congruence axioms
o prove selected propositions and theorems that derive from the congruence axioms
o apply Archimedes axiom, Aristotle’s axiom, Dedekind’s axiom and the other
continuity axioms and their corollaries
o identify the relationship between Hilbert planes, neutral geometry, and Euclidean
planes
o apply and prove parts of the measurement theorem based on Aristotle’s axiom and
Dedekind’s axiom
o prove the equivalence of Hilbert’s Euclidean parallel postulate, Euclid V, and the
angle sum theorem
o prove the equivalence of selected theorems in Euclidean geometry and Hilbert’s
Euclidean parallel postulate
o identify Saccheri and Lambert quadrilaterals and prove related theorems including
the uniformity theorem, Saccheri’s angle theorem, and the Saccheri-Legendre
theorem
o identify the logical flaws in historical attempts to prove the Euclidean parallel
postulate, including proofs by Proclus, J. Bolyai, Wallis, and Legendre
o define the defect of a triangle and prove the additive property of the defect
o prove that rectangles do not exist in a Hilbert plane satisfying the acute angle
hypothesis
o prove the universal non-Euclidean theorem
o prove that non-congruent, similar triangles do not exist in a Hilbert plane
satisfying the acute angle hypothesis
o prove theorems associated with parallel lines that admit a common perpendicular
o supply key steps and reasons in the proofs of theorems relating to parallel lines
containing limiting parallel rays
o prove that Hilbert’s Hyperbolic Axiom of parallels holds in a non-Euclidean plane
satisfying Dedekind’s axiom (real hyperbolic plane)
o prove that parallel lines containing limiting parallel rays do not admit a common
perpendicular or congruent alternate interior angles.
Instructional Method
In addition to teacher-directed presentations, students will work in groups to share
ideas and present solutions to key exercises to the class.
Method of Evaluation
There will be two (2) group problem sets each worth 10% of the final grade. Satisfactory
completion of assigned end-of-chapter exercises is worth 10% of the final grade. There
will be a midterm and a final exam, each worth 30% of the final grade. Students will also
work in groups to complete at least one independent project worth 10% of the final grade.
V.
Resources and Funding Required (New Courses only)
Resource
Amount
Faculty
Other Personnel
Equipment
Supplies
Travel
New Books
New Journals
Other (Specify)
1,050/year
0
0
0
0
0
0
TOTAL
1,050/year
Funding Required Beyond
Normal Departmental Growth
0
Explanation of costs: This
proposal would require the
hiring of one part-time instructor
to teach one course for one
semester every other year.
VI. COURSE MASTER FORM
This form will be completed by the requesting department and will be sent to the Office of the
Registrar once the course has been approved by the Office of the President.
The form is required for all new courses.
DISCIPLINE
COURSE NUMBER
COURSE TITLE FOR LABEL
(Note: Limit 30 spaces)
CLASS-LAB-CREDIT HOURS
Approval, Effective Term
Grades Allowed (Regular or S/U)
If course used to satisfy CPC, what areas?
Learning Support Programs courses which are
required as prerequisites
Mathematics
MATH 7395
Non-Euclidean Geometry
3–0–3
Spring 2014
Regular
APPROVED:
________________________________________________
Vice President for Academic Affairs or Designee __
VII Attach Syllabus