MTH 301 -- Combinatorics

MWF 1 -- Bradley Hall 225

Objectives

This course has three goals of roughly equal importance. First is the ability to think mathematically, which involves solving problems, producing and analyzing arguments, and so on. Second is the ability to communicate mathematics, both orally and in writing. Third is familiarity with some of the topics and techniques in combinatorics.

Class Format

This course will probably be quite unlike any math course you have had before. We will use a method called guided discovery. This method fosters creativity and independent thinking. It is also fun. I will pose problems of various types. You will solve the problems. You will then present your results to the class. These presentations are a major part of the course.

Try to solve the problems and write up your results before we discuss them in class. Do not consult books, the internet or other sources. You may collaborate with each other, with a couple of conditions: you should make a serious independent attempt at solving a problem before discussing it with another student, and the discussion should consist of ideas and hints rather than complete solutions. Of course, if you are stuck on a problem, you should not hesitate to ask me for help outside of class. Work far enough ahead of the classroom presentations that there is time for this consultation.

Each day, I will select people to present their results in class. Be prepared! When you are presenting, strive to be clear and organized. When you are observing a presentation, it is your responsibility to follow the logic and verify that it is correct for yourself. If you cannot follow the reasoning, you should ask a question of the presenter. I may ask you to re-explain someone else's argument.

We use this format because it is directly aimed at our objectives, particularly the first and second. It gets you involved in doing and creating mathematics, rather than merely watching and imitating. This may be among the most challenging math courses you take, and I think you'll find the experience worthwhile. Also, the material is neat, so I hope you have fun with it.

Evaluation

Your grade is based on a combination of homework and participation (30%), two in-class exams (20% each) and a final exam (30%).

Letter grades have essentially the following meaning:
D = You are clearly putting in effort but not reliably assimilating material.
C = You can reliably reproduce and apply what you've seen in class.
B = You are consistently producing mathematics.
A = The mathematics you produce is high quality and answers difficult questions.

Homework

Your standing homework assignment is to write up solutions to all the problems. Ideally, you will have at least a draft of a problem's solution before that problem's classroom treatment. Among other things, that helps improve your presentations. At the latest, you should have each solution written up before the class after the one in which it is presented. Writing up solutions is an essential part of learning mathematics. Keep a notebook containing all of your write-ups. You should be ready to submit this notebook for grading at any time. Please make every effort to keep your notebook neat and clean, and leave space for comments. Any comments you do receive should be incorporated into revised solutions. I highly recommend keeping your notebook in electronic form, preferably using the mathematical typesetting system LaTeX.

Participation

This includes your presentations and your questions on others' presentations. Quantity is important, but so is quality, including clarity and insight. Be willing to make mistakes, but also be willing to learn from them. Do not let false or unclear statements pass, but do address them courteously.

Exams

The in-class exams are scheduled for 21 February and 4 April. The final is on 14 May at 9:00. All exams are cumulative.