One of my goals in this class is to increase your awareness of mathematical culture, and of how math is used in the world around us. To this end, I would like to have you write a paper about a topic that goes beyond the scope of this course (either by applying what we've learned, or by introducing some mathematics that isn't included in M305G).
The paper is "optional" in the sense that you can earn an adequate participation grade without writing it (and a fortiori, the participation grade itself is optional). If you do want to earn the maximum possible participation grade of 15%, you must write a paper. If you do not write a paper, the highest participation grade you can earn is 12%. The paper itself is worth up to 5%.
If, in light of all this, you wish to write a paper (and I sincerely hope that you will choose to do so), please keep in mind the following instructions.
When writing the paper, your goal should be to tell someone who is not a mathematician (say, another M305G student) about a mathematical idea or application you have learned outside of class. While the writing should be suitable for someone who is not a mathematician, that does not mean that the paper should be devoid of technical content. No matter what you write about, there should be some mathematics going on; this can consist of solving an example problem, outlining the proof of a theorem, carefully explaining the logic behind a certain idea, et cetera.
Since the purpose of your paper is to relate some interesting mathematics to someone who may not already be interested in mathematics, I encourage you to have fun with it, and not feel the need to be extremely formal or scholarly in tone. However, I do expect your paper to be written in proper English, with a reasonable effort made to avoid mistakes in spelling, punctuation, and grammar.
The MPG Illusion: In light of our nation's recent energy crisis, Richard P. Larrick and Jack B. Soll have written an excellent article for Science Magazine, titled "The MPG Illusion," discussing why the use of miles per gallon as a measure of fuel efficiency can lead consumers to make less-than-optimal decisions when buying (or not buying) cars. Based on what I've seen, the mathematics in the article is quite simple and can be understood by anyone who understands high school math.
The mathematics of democracy. The hotly contested 2000 presidential election drew attention to the idea that in most American elections for public office, a third-party contender can affect a close contest between two candidates X and Y by drawing votes away from Y and enabling X to win, even if a majority of voters prefer candidate Y over candidate X. (NB: I'm not taking a position on whether such a majority actually existed, but the perception is there.) People have proposed several alternative voting methods in an effort to remedy this problem; however, there is a theorem called Arrow's paradox which states that it is impossible to construct a voting system that has all of the properties we would like for it to have. Arrow's paradox sets forth several criteria that a voting system "should" have; discuss these criteria, and take a look at some alternative voting systems and see whether these systems have the desired criteria.
The slide rule. Back in ancient times (before the 1980's or so), people used slide rules to perform the calculations that we use calculators and computers to do today. Learn how to use a slide rule (there are some "virtual slide rules" available on the web, so you don't have to go and hunt for one) and explain the mathematics behind how the slide rule works.
Winning Ways. Winning Ways for Your Mathematical Plays, by Berlekamp, Conway, and Guy is a terrific book (written by three first-rate mathematicians) about strategies for various two-player games. Find a copy of this book, pick out a game (or several), and discuss the strategy and the mathematics behind it. The mathematics in this book will probably be at various levels, some of which will seem alien to you. But you should be able to find something that you can work with.
The Monty Hall paradox. The Monty Hall paradox is a deceptively simple problem in probability that most people - including some mathematicians - get wrong when they hear it for the first time. Explain the problem, the solution, and why the solution is called a "paradox" even though it really isn't. This topic would also be great as part of a paper on "paradoxical" problems in probability in general.
Mathematical induction. Find out what mathematical induction is, and discuss some theorems that can be proven by mathematical induction. (Better yet, give some examples of proof by mathematical induction.)
As mentioned above, your paper should be written in proper English. Writing mistakes are a form of sloppiness, and I reserve the right to deduct points for sloppy work (though I won't do this unless the sloppiness is really noticeable).
Any sources you use in your paper should be properly cited. I'm not a huge stickler for any particular format (though some of your professors will be); the main thing here is to avoid giving the impression that you are responsible for an idea, phrase, or sentence that is due to somebody else.
I don't have a particular length in mind for this paper, but I think it should take at least three full double-spaced pages to discuss an interesting topic in the depth and detail I want to see for this assignment. Don't worry about turning in a paper that's too long (as long as you're not just restating things over and over); I don't mind reading something long if it's quality work.
The due date for the paper is Saturday, August 16, the day of the final, though you can turn in your paper anytime before that. You must give me a hard copy of the paper; an electronic copy won't cut it (I want to be able to sit down and read and enjoy it). I won't have time to comment extensively on your paper, but I will try to let you know at some point how many points (on a 0- to 5-point scale) I awarded for it.
I have an unwavering belief in the integrity of my students, but based on some unfortunate past experiences I have to state the following: I have a very keen eye for papers that are not the original work of my students. Copying text en masse is not an acceptable strategy for any paper you'll write in school (even if you do state the source); it's simply unethical, and it's wrong. A paper that just restates what an article says, without any original ideas or work on the part of the student, is just as bad. The fact that this is "just" an extra credit paper doesn't mean that I will be more forgiving of plagiarism in the unlikely case that it does happen. An academic integrity blemish on your university record can slam shut a lot of doors for you... so please don't put me in a position where I have to do that to you.