Below, I discuss
Mathematics and Politics: Strategy, Voting, Power and Proof, by Alan Taylor.
A calculator will be helpful.
The text, and a calculator if you have one, should be brought to class every day.
The more you look around you, the more you will discover that math is everywhere. This semester, we'll focus on politics as a (timely) forum for exploring mathematics.
We will use the concept of escalation to discuss game-tree analyses, and to explore the limitations that we bump into even when using state-of-the-art computers. We will also explore a couple of the most common ``games'', and how they can be used to get a feel for what might have happened during the arms race and the Cuban Missile Crisis. We will learn about some of the intricacies of yes-no voting, and consider the power individuals have, when they can have more than one vote. Considering the recent mess in Florida, you might find it interesting to discover what mathematics has to say about how much power Florida has in the electoral college! We will also consider various different voting methods, and discuss ways of determining whether a method is ``fair'' or not.
I hope you will come to appreciate both the potential and the limitations of mathematical analysis in political science, becoming a sophisticated and skeptical consumer of mathematical models and of the "rationality assumption." Throughout the course you will see some very beautiful mathematics, and raise some important questions in political science.
This class requires no specific mathematical background (beyond arithmetic and some algebra), nor does it assume any background in political science (I myself have no political science background). Calculation is emphasized less, probably, than in most of your previous courses, and deductive reasoning is emphasized more.
I come to this class knowing that the class will consist of students with a variety of backgrounds. Some of you are starting the course with a very positive attitude towards mathematics, some may not have had the best mathematical experience, while still others of you have had just plain bad mathematical experiences. I try to establish an atmosphere that is challenging and engaging for all of you, but supportive and reassuring for those who need it. I want you all to succeed, and most of you will succeed if you follow the right strategy (see ``How can you succeed?'' and ``Is this the right class for you'' below). I think this material is really interesting, and hopefully you will too! For those who have had negative experiences in the past, or who are ``only taking this class because you have to'', keep an open mind!
As is true with all courses, how much you actually learn in Math Thought is entirely up to you. As you read through how the course is structured, you will see that a lot is expected of you. Your responsibilities for this class will consist of reading the textbook daily, doing a few problems individually daily, working weekly or semi-weekly on group homework, three exams, and a comprehensive final.
Is This the Right Course for You?:
Being interested, or open to the possibility of being interested, in how math can be used to unravel some of the murky world of politics, and being willing to work hard are the primary qualifications for this course.
As I mentioned above, I do not assume very much in the way of mathematical background; all I ask is that if you have previously had negative mathematical experiences you put them aside and be open to the possibility that you might enjoy this class.
There are some people, however, for whom this is not the right class. Students whose schedule makes more than one math class undesirable should think about their needs: those who might major or minor in psychology, sociology, biology, education, political science, or economics may be required (or recommended) to take other math courses. People who want to learn techniques they'll use often (as opposed to learning some of the ways math relates to the world around you) should consider taking Statistics or Universal Machines.
While an active interest in politics is not necessary, if you are one of those people who refuse to listen to any discussion of the intricacies of politics, you may want to wait until a class comes along that you are interested in -- it's very hard to work in a class that's completely uninteresting to you. To determine whether you're interested, ask yourself: were you interested in the recent election brouhaha? Would you be interested to figure out how much power the president, or Florida, has? Does learning about different ways of voting, and their benefits and problems sound interesting? How about learning a little ``game-theory''? You could also look through the chapter headings in the book and read through some exercises (just don't mistake intimidation because you haven't yet learned some of the terms for a lack of interest).
Also, if you plan on not putting a lot of effort into this class, you should wait until you can put the effort into it: you will get more out of it (as well as being more likely to pass it), and it's more fair to the rest of the class.
How can you succeed in this course?:
As I mentioned above, being interested (or willing to be interested) and being willing to work hard are the two critical factors for success in this course.
Plan to spend an average of 9 hours a week outside of class working on this course. As usual, some weeks you will spend more time on this class, while others will be less frenetic (relatively speaking). Also, of course, be aware that some people can get away with spending less time on the course than others can. Just because your friend or neighbor isn't working hard doesn't mean that you won't have to.
Another key to success: admit it when a concept or connection is eluding you. Come see me in my office, and/or consult with friends.
And finally: think about forming study groups. As I discuss below, working alone has its value, but so does working in groups.
Please come visit me! Chat about mathematical thoughts, or bring questions. If you do have questions, come right away, don't stew over them and let them evolve into a serious situation!
If you come during my office hours, you of course do not need to make an appointment beforehand -- just stop by! If you can't make my office hours, make an appointment with me for another time.
It may come up that I need to e-mail the class about something: a change to the problem set, a clarification of a problem. Please make sure I have the e-mail address you use regularly, and please do check your e-mail regularly. You can also, of course, always e-mail me at firstname.lastname@example.org.
Reading technical material is an extremely valuable skill, and is becoming more necessary in all areas of our lives all the time. One of the goals of this class is that you become more comfortable reading mathematical prose.
Before each class meeting, I expect you to have read the material that we will be discussing that day. At first it may seem difficult, but stick with it -- as you get used to Taylor's writing style, and as you practice, you'll find it getting easier. The key to reading mathematics is (as you've probably heard before) to read it with pencil in hand. Whenever you come to an example, try to work it out yourself; fill in the missing details in the book. Really read the diagrams and try to figure out where they come from and what they have to do with the text.
The notes you take in class will probably serve as the main text for the course, with TaylorÍs book forming the backbone. The book will give you an introduction to what I'll be doing, while your notes will have alternative explanations and extra examples.
It is essential to take clear and complete notes. If you miss a class, borrow someone elseÍs notes, quickly photocopy them so you can return the originals, and then recopy the notes in your own notebook. (What is the point, do you think, of recopying rather than just adding the photocopies to your notebook?) Keep all your notes in the same notebook, rather than sprinkling them randomly among your various notebooks.
Individual Problem Sets:
Learning math is best accomplished through a combination of group and individual efforts. Your daily homework, which will encourage you to practice the skills recently learned from the reading and from class, should be done individually. You may, of course, consult with your friends and neighbors, but the final result must reflect your own understanding, word choice, and work!
The individual homework will generally consist of 1-5 ``drill'' problems. I will only grade these on a cursory basis: the entire problem set will be graded out of 5 points, reflecting only whether you seemed to have the right idea and whether you did all the problems. I will not actually correct the problem sets (i.e. I may not even comment as to whether your answers are right or wrong); instead I will post solutions outside my door for a week or so. It is your responsibility to check how you did, and to ask me questions if you don't understand something.
While these problems do not require the level of polish the group problem sets discussed below do, they must be neat enough for me to read; if I can not read any portion of a problem set, I will give up and the entire problem set will get a 1. (I just don't have the time to decipher any more). For those of you with messy handwriting -- do your best, but also don't panic; I do have lots of experience reading student papers.
For your own benefit, I do suggest recopying or summarizing each problem. While time-consuming, it also makes the returned homework much more useful -- you can study from it when exam times roll around.
You may not divide the problems up between the two of you! Each of you must work on each problem. After you have jointly worked out solutions to each problem, you should jointly rough out how you're going to write up the solution.
The final draft, however, for each group problem set should be written by just one member of the group, the ``primary author''.
Points on the group homework will be based on each person's honest assessment of the effort and contribution made by each member. Groups also must make note of who was the primary author for each problem set, and the primary author must alternate.
Here are some guidelines for how the group problem sets should be
I will set aside 5-10 minutes at the beginning of each class to answer questions on the homework due the next class period (which, of course, means you need to be getting an early start). Problem sets are due at the beginning of class each day.
Consult the Guidelines for Homework Presentation for information on how your problem sets should look.
|Late problem sets will not be accepted! No exceptions!|
I plan to drop the lowest three non-zero problem set scores at the end of the semester. If it is warranted, I may be persuaded to drop one zero instead.
On an exam, you can expect to see some combination of problems similar to homework problems (both individual and group), problems that apply the concepts we've studied to an unfamiliar situation, explanations of concepts, and occasionally proofs.
While last-minute cramming may be the study technique you've used in the past, or the technique you think most appropriate to a gen ed class, it probably will not serve you well in this class. I suggest re-reading the notes, summarizing the important concepts and techniques while you go; redoing as many problems as you possibly can ( not merely reading your solutions), and practicing writing proofs and explanations of concepts. By this practice, I do not mean memorization -- the words may be different every time. It takes practice explaining something yourself to really be able to explain a concept. So read through an explanation and work on it until you really feel you understand it. Take a break to give yourself time to forget the specific words. Then take a blank piece of paper and try to write out the explanation or proof yourself. Compare to the original and repeat until you've got it down pat!
Obviously, from the above discussion, studying for an exam in this class is not a small thing. You'll have to spread it out over several days, so be organized.
I will give three exams, to make sure throughout the semester that you are learning the material. Each of these will be given on Monday evenings, and I will let you spend up to 2 hours on them (I will write them intending for it to take the ``average'' student about 1 hour, so that should give you plenty of time.)
These exams are tentatively scheduled for:
Notify me in advance if you will be missing an exam, either by phone or by e-mail. If your reason for missing is acceptable, we will arrange that you take the exam early. If you miss an exam without notifying me in advance, I reserve the right not to give you a make-up exam. I will not give any individual more than one make-up exam during the semester.
Clearly, missing class is not a wise idea. If you do miss class, it is of course your responsibility to find out any assignments, and to get a copy of the notes and of any hand-outs.
I expect to use the weights below, although I reserve the right to change my mind if the semester does not go as expected.
|Individual Problem Sets||17%|
|Group Problem Sets||18%|
I expect you to abide by the Honor Code. If I have any reason to suspect that perhaps a violation has occured, I will ask the Judicial Board to investigate the matter. Below are some guidelines on what constitutes violations of the honor code in this class.
Problem Sets: You may work with anybody you want. You may use any references that help you figure out how to do the problem on your own; you may not use any references (people, old projects, books, the web, for instance) which tell you how to solve it or lead you to the solution. You must understand how to do every problem, and you must cite references if you've received assistance from any source. When doing group projects or group problem sets, you may not divide it into different parts--you must do them all together, and you must make sure every member of your group understands every part. Exams: You may not use any notes, books, or colleagues as reference during the midterms and final, except for your ``cheat sheet'', which must conform to my stated rules. You may not use a calculator unless I specify that you may, and you may not use a graphing calculator.
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278