Foundations and Structure of Mathematics 1 - Fall 2004


TIME: 5:00 p.m. - 8:00 p.m. Tuesday

PLACE: Executive Auditorium of the Higher Education Center at Virginia Highlands Community College

INSTRUCTOR: Dr. Robert Gardner

OFFICE: Room 308G of Gilbreath Hall at ETSU

E-MAIL: or


OFFICE HOURS: 1:00-2:00 T, 10:25-11:20 W

PHONE: (423)439-6977 (Math Office (423)439-4349)

TEXT: The text for this course is The Heart of Mathematics: An Invitation to Effective Thinking, 2nd Edition, by E. B. Burger and M. Starbird, Key College Publishing, 2004. The first edition is also acceptable (the text in the sections we will cover is virtually identical from the first to the second edition, but some of the pictures are different). You can order a copy of the first edition from for around $49 postage paid (see the Amazon site for this book for details). I will refer to the "manipulatives kit" but you don't need to buy it. I will have 3-D glasses available for each of us when we cover the sections which require them. I will also try to find some of the manipulatives (which involve cut-outs) online for you to download.

ABOUT THE COURSE: This course (as well as the others in the ETSU program for elementary and middle school teachers in southwest Virginia) will be consistent with NCATE/NCTM standards and be guided by the "PUFM" philosophy ("Profound Understanding of Fundamental Mathematics"). Performance based assessment will be used to modify instruction as needed.

CLASS WEBPAGE: A webpage for this class is available at:
Although we will use Blackboard in this class, the majority of material available to you online will be accessible through this website.

BLACKBOARD: A Blackboard site is available for this class. We will use it primarily for "Discussions." I will use this site to communicate with you as a group, so it is important that you have the e-mail address you normally use as part of your "profile" in Blackboard.

GRADING: Grades will be based on student performance on written homework assignments (HW), a written report combined with in-class oral presentation (RO), and class participation (CP, which will include in-class discussion and individual student presentation of homework problems) as follows:

AVERAGE = (HW + RO + CP)/3.
Grades will be assigned on a 10 point scale with "plus" and "minus" grades being assigned as appropriate (up or down 3 points, as is consistent with the grade points assigned to plus and minus grades by ETSU).

WRITTEN HOMEWORK ASSIGNMENTS: Two or three written problems will be assigned each class period and collected during the next class. These problems will be graded for accuracy and returned during the following class meeting.

IN-CLASS HOMEWORK PRESENTATIONS: A student will be selected to present each written homework problem in class (with access to their written solution). Each student is expected to present at least one problem during the term. If necessary, second chances will be given.

CLASS PARTICIPATION: I will lecture on each topic or present a video. We will then, as a group, discuss the topic at hand. Each student is expected to participate in some of the discussions. We will also carry on discussions through Blackboard. Each student is expected to post comments on Blackboard at least three times.

WRITTEN REPORT: A written report of at least five pages in length will be turned in some time near the end of the term. A number of suggested topics will be given on the class webpage (see below) and you should get your topic approved by the instructor before getting too involved in your project. It is expected that each report will include mathematical content (for example, if one were to do a report on Rene Descartes, then the report should include a discussion of his contributions to mathematics and their influence on future mathematical development). The written report will be due on November 30!!!

ORAL PRESENTATION: Each student will give an in-class presentation of roughly 10 minutes in length on his/her written report. The last three class days will be devoted to student presentations. Your instructor will be quite impressed with multimedia presentations (assuming we have access to the appropriate equipment). Alternatives to an in-class presentation will be considered!!! Make a proposal to the instructor. Presentations will be given during the last two class meetings (November 30 and December 7).

Introduction. What is Math?
The NOVA episode A Mathematical Mystery Tour. Fermat's Last Theorem and the Simpsons.
(1) The Simpsons and Fermat.
(2) Outline for Mathematical Myster Tour.
Read Section 4.1.
4.1 Pythagoras and His Hypotenuse.
The Pythagorean Theorem and proofs, an extensive discussion of manipulatives, and the Parallel Postulate. The Ascent of Man episode "The Music of the Spheres."
(1) "The Pythagorean Theorem" Handout.
(2) Outline for "The Music of the Spheres".
Homework Set 1.
Pythagoras (continued), widescreen TV's
Presentations of the homework assignments were given. The Parallel Postulate and the Pythagorean Theorem were further explored. Applications involving distances in 3-dimensions and widescreen TV's were given.
(1) "The Pythagorean Theorem" Handout.
(2) Virtual Manipulative - The Pythagorean Theorem
Read Section 4.6.
4.6 The Shape of Reality
Noneuclidean geometries and alternatives to the Parallel Postulate were explored. Spherical geometry was illustrated using a globe and string. 3-D goggles were handed out and 3-D pictures of the saddle surface were viewed. The taxicab metric and the "Spider and Bug" problem were discussed.
(1) "Shape of Reality" Handout.
(2) My "Shape of Space" webpage.
(1) Read photocopied pages from The Shape of Space.
(2) Homework Set 2.
The Shape of Space
We played tic-tac-toe on a 2-torus, watched the video The Shape of Space, discussed different possible (topological) shapes of our universe, and mentioned recent research by NASA into the subject.
(1) My "Shape of Space" webpage.
(2) Curriculum materials for The Shape of Space
Read Section 4.7.
The Fourth Dimension
Dimension, "fundamental directions," coordinates, hypercubes and distance, what would we see in 4-dimensional space?, vector spaces and n-dimensional space. We watched part of a Cosmos episode with Carl Sagan, and an Innovations episode.
(1) Outline for "To the Nth Dimension... and Back."
(2) "The Fourth Dimension (and More!)" Handout. (This is in PDF format and requires a PDF reader to view).
Homework Set 3.
The Fourth Dimension (continued)
Coordinates, hypercubes, distance, what would we see in 4-dimensional space?, readings from Hyperspace, watched "There's Math in Deviled Eggs - Strategies for Teaching Young Children."
(1) "The Fourth Dimension (and More!)" Handout. (This is in PDF format and requires a PDF reader to view).
(2) 3-D hypercube image from the text.
(3) "There's Math in Deviled Eggs" handout.
(4) Eric Saltsman's hypercube webpage.
(5) Drew Olbrich's rotating cube and hypercube webpage
(6) A rotating hypercube "movie."
(1) Homework Set 4.
(2) Read Sections 4.5 and 5.3.
(3) Make each of the 5 Platonic solids out of paper (go here for templates).
4.5 The Platonic Solids Turn Amorous, 5.3 Feeling Edgy?
(1) Templates for Platonic Solids.
(2) "Platonic Solids" handout.
Homework Set 5.
Read 7.1, 7.2, and 7.4 (due 11/9).
The Mathematics of The Simpsons
A special presentation by Dr. Sara Greenwald of Appalachian State University. Supper will be arranged as well.
(1) Dr. Greenwald's SimpsonMath webpage.
(2) Dr. Andrew Hestler's SimpsonMath webpage.
(3) Dr. Greenwald's "Mathematics of Futurama" webpage.
7.1 Chance Surprises
7.2 Predicting the Future in an Uncertain World
7.4 Down for the Count
(1) An Analytic soultion to the "Self Duals" problem.
(2) Probability handout.
Homework Set 6.
Read 7.5 and 7.6.
7.5 Great Expectations
7.6 What the Average American Has
Expectation and Statistics handout
Homework Set 7 (due December 7).
Written Reports due next class.
Student Presentations
Written reports due!!!
Student Presentations
7.5 Great Expectations (continued)
7.6 What the Average American Has (continued)
"Round Table" discussion
Expectation and Statistics handout

Homework Set 1.
Do two of the following, of an appropriate level:
Homework Set 2.
Do three of the following, of an appropriate level:
Homework Set 3.
Do three of the following:
Homework Set 4.
Do one of the following:
Homework Set 5.
Do three of the following:
Homework Set 6.
Do three of the following:

Homework Set 7.
Do three of the following:


You should have your topic approved by the instructor by Tuesday October 12! The report should include at least three references, one of which is not a website. Soem possible topics include (but are not limited to):