Reflection #1 - Due Monday, August 27
We have touched on the question "What is mathematics?" this week and will continue to do so
over the course of this class. Talk about your own personal experience of mathematics.
Please be up-front and honest!
Some questions you may wish to talk about: Have you always loved math? Have you always hated math?
Did you use to love math, but then had a bad experience at some point? What are some of your earliest
memories about math (either good or bad!)? Are you excited about this class? What do you wish to get
out of this class?
Reflection #2 - Due Monday, September 10
In class, we talked about Fibonacci numbers and various places where they show up in nature. We
also used these numbers to generate the golden ratio, and we talked about golden rectangles and their
possible connection to art and architecture. My main question is, simply, "Do you believe any of this?"
Some questions you may wish to address: How can we explain the appearance of Fibonacci numbers in pineapples,
sunflowers, pinecones, etc.? Does nature understand mathematics? Is God a mathematician? Do Fibonacci numbers
really show up in nature more than other numbers? Can you find more examples of Fibonacci numbers in nature
that we didn't get to talk about in class? Are golden rectangles more "ideal" than other rectangles?
Do famous works of art and architecture really contain instances of golden rectangles?
Reflection #3 - Due Wednesday, September 19
We have seen two real mathematical proofs: a proof that there are infinitely many primes and a proof that
the square root of two is irrational. We also watched the Nova special called
"The Proof" about Sir Andrew Wiles'
proof of Fermat's Last Theorem. This reflection deals with your feelings about mathematical proof and/or your thoughts
about the movie we watched.
Some questions you may wish to address: Could you follow the proofs we did in class? Should we care at all about
"pure math" (math with no apparent connection to anything in real life)? Were you surprised at how emotional Wiles got
when he talked about his proof? Did the mathematicians in the movie fit your (or society's) stereotypes about
mathematicians?
Reflection #4 - Due Wednesday, October 3
Chapter 3 really stretches the mind. By thinking deeply about what it means to count, we learned how to
compare the sizes of infinite sets. We showed that there are different sizes of infinity, and we learned how to
create a larger set from any set. This means there are actually infinitely many sizes of infinity! We also
talked about the Continuum Hypothesis, which is neither true nor false! This reflection deals with your feelings
about all this craziness.
Some questions you may wish to address: Did this chapter interest you at all? Did you enjoy thinking about deep
philosophical/mathematical questions even though they have no direct bearing on your life? Were you surprised
that certain people personally attacked Cantor because of his ideas? Were you shocked to learn that there are
infinitely many sizes of infinity? Isn't it wild that certain mathematical questions are neither true nor false?
Reflection #5 - Due Wednesday, October 24
Chapter 4 is all about geometry. We first talked about and proved (three different ways!) the famous
Pythagorean Theorem, then we studied the Platonic solids. Our view of geometry was greatly expanded when we
threw out the Parallel Postulate and developed non-Euclidean geometry. This reflection is basically just your
personal thoughts/experiences pertaining to geometry.
Some questions you may wish to address: What were some of your experiences with geometry in high school? Have your
thoughts about geometry changed at all? Did you find it interesting to prove the Pythagorean Theorem? Was it
overkill to prove this theorem three different ways? Are the Platonic solids really all that "ideal"? Does
non-Euclidean geometry interest you?
Reflection #6 - Due Wednesday, November 7
Chapter 5 really stretches (pun intended!) the idea of geometry. Instead of the rigid geometry of Chapter 4, we
looked at "rubber sheet geometry," where two objects are considered equivalent if one can be molded/shaped into the
other without cutting or gluing. This may seem like a silly exercise, at first, but this idea actually opens up a
strange new world where spheres are cubes, donuts are coffee cups, and shapes that appear to be linked are
not actually linked. How did you feel about this chapter?
Some specific questions you may wish to address: Is the idea of rubber geometry fun to think about? Was it hard
to visualize/describe how to mold certain shapes into others? Were you surprised when I cut the Moebius strip in
half and it stayed in one piece? Are you fascinated (or at least interested) in things like the Klein bottle,
which is a one-sided surface without an edge? Is any of this really mathematics?
Reflection #7 - Due Monday, November 19
In Chapter 6, we talked about how a seemingly silly question about walking across bridges became
an entirely new branch of mathematics: graph theory. "Childish" activities involving coloring maps and tracing
pictures turned into deep and important mathematical questions. Of all the topics we've covered so far, this
one really illustrates the importance of "play" in mathematics. How do you feel about graph theory?
Some specific questions you may wish to address: Does knowing the trick to tracing out pictures ruin all the fun? Are
you surprised that there is no known way to quickly see whether or not a graph is Hamiltonian? Did you find the four-
color theorem interesting and/or surprising? Are mathematicians going too far when they turn child's play into
complex theories with esoteric terminology (planar graph, bipartite graph on n,m vertices, etc.)? Shouldn't we just
leave well enough alone?!
Reflection #8 - Due Friday, November 30
We really learned a lot this semester. We covered a little bit of number theory, set theory, geometry,
topology, graph theory, and abstract algebra. This last reflection is sort of a free-for-all discussion about
the entire course.
Some specific questions you may wish to address: What was your favorite thing you learned about in this class? What
was your least favorite thing? Did your feelings about mathematics change throughout the course of the semester?
Do you feel like you have a better understanding of what mathematicians really study? Do you want to learn more?