Preface To the Student
How this class works
Every semester I collect written evaluations from my classes and the students who have taken this course have been overwhelmingly positive in their comments. While challenged by the material and the method, they enjoyed the class. If you'd like, I'll read random comments from the last semester I taught it. I'll print them out, let you randomly select a number and I'll read that comment.
The reason I open with a discussion of my evaluations, is because this class will be taught in a way that is (most likely) different from any mathematics classes you have encountered in the past. Most of the class will be devoted to students working problems at the board and half of your grade will be determined by the amount of mathematics that you produce in this class. I use the word produce because it is my belief that the best way to learn mathematics is by doing mathematics. Therefore, just as I learned to ride a bike by getting on and falling off, I expect that you will learn mathematics by attempting it and (occasionally) falling off! You will have a set of notes (you are reading them now) that you will turn into a book by working through the problems. I urge you to seriously consider the value of becoming an independent thinker who tackles doing mathematics (and everything else in life) on your own, rather than waiting for someone else to show you how to do things.
A common pitfall
There are two ways in which students often approach my classes. The first is to say, “I'll wait and see how this works and then see if I like it and put some problems up later in the semester after I catch on.” Think of the course as a forty yard dash. Do you really want to wait and see how fast the other runners are before you start running? If you try to do the problems every night, then either you will get a problem (YEA!) and be able to put it on the board or you will struggle with the problem and learn a lot in your struggle. If you have worked hard on a problem, you may go to the board with your work and show us what you tried. Doing so will almost surely give you further ideas on how to solve it. If someone else puts the problem on the board then you will be able to ask questions and help yourself and others understand it, which also counts toward your presentation grade. And you will be able to say to yourself, “Ahhhh, now I see where I went wrong and now I can do this one and a few more for next class.” If you do not try problems each night, then you will watch the student put the problem on the board, but perhaps will not quite catch all the details and when you study for the tests or try the next problems you will have only a loose idea of how to tackle such problems. Basically, you have seen it only once in this case. The first student saw it once when s/he tackled it on his own, again when either s/he put it on the board or another student presented it, and a third time when s/he studied for the next test or quiz. Hence the difference between these two approaches is the difference between participating and watching a movie. I hope that each of you will tackle this course with the attitude that you will learn this material and thus will both enjoy and benefit from the class.
Because the board work constitutes half of your grade, let's put your mind at ease regarding this part of the class. First, by coming to class everyday you will earn a 60% on board work. Every problem you present pushes that grade higher. You may come see me anytime for an indication of what grade your current level of participation will earn you at the end of the semester. And I give one presentation credit for an office visit within the first two weeks of the semester. Here are some rules and guidelines associated with the board work.
Your only sources are to be me and these notes.
You are not allowed external materials (web, books, etc.).
You are not allowed to work with or ask questions of other individuals (friends, classmates, teachers, etc.).
You may not present a problem that you have seen in another class.
If there is a problem that you have spent a lot of time on and you don't want to see a solution to it yet, you may step out of the room while it is presented and turn it in for some credit.
I will call for volunteers every day and will pick the person with the least presentations to present a given problem. You may inform me that you have a problem in advance (which I appreciate), but the problem still goes to the person with the least presentations on the day I call for solutions. Ties are broken either randomly (at the beginning) or by test grades (lower test grades taking priority). A student who has not gone to the board on a given day will be given precedence over a student who has gone to the board that day. To “present” a problem at the board means to have written the problem statement up, to have written a correct solution using complete mathematical sentences, and to have answered all students' questions regarding the problem. When you have worked hard on a problem, even if you aren't sure your work is correct, go to the board. Going to the board can only help your grade. Since you will be communicating with other students in class on a regular basis, here are several guidelines that will help you. Remember that the whole class is on your side because everyone wants to see the problem presented correctly and to understand it.
When you speak, don't use the words “obvious,” “stupid,” or “trivial.” Don't attack anyone personally or try to intimidate anyone. Don't get mad or upset at anyone, even yourself. And if you do, try to get over it quickly. Don't be upset when you make a mistake - brush it off and learn from it. Don't let anything go on the board that you don't fully understand. Don't say to yourself, “I'll figure this out at home.” Don't use concepts we have not defined. Don't try to put up a problem you have not written up.
Do prepare arguments in advance. Do be polite and respectful. Do learn from your mistakes. Do ask questions such as, “Can you tell me how you got the third line?” Do let people answer when they are asked a question. Do refer to earlier results and definitions by number when possible. For example, “by Definition 29” we know that...
How to study
Read over your notes from class every day.
Make a list of questions to ask at the beginning of the next class.
Review the recent problems.
Work on several new problems.
Write up as many solutions as you can so that you can simply copy your solutions onto the board the next day.
Some problems are hard. If you don't get one, don't give up on it. Move on to another problem and come back to that problem later. The problems worth solving in life are not solved in five minutes.
What is discrete mathematics?
The word “discrete” means finite in this setting. Thus, discrete mathematics means the study of the topics in mathematics that can be approached without using infinite sets. Even so, you'll see a few infinite sets in the course.
W. Ted Mahavier