Text: Mathematical Experimentation: A Laboratory Bridge Course available from the Mathematics Department at cost.
This is a course about the process of doing mathematics, rather than a course about some topic in particular. That makes it a little hard to describe, because what happens in the course will be, to some extent, up to all of us. There will be definite starting points -- four of them -- found in a text, and it would be possible merely to do what the text suggests and be satisfied with that. But it would be more in the spirit of the course to launch out unpredictably and see what happens.
The four units that are our starting points are listed below. Each of them connects, or is a "bridge," to some higher mathematics that you can experiment with and come to know by experience before you ever study it formally. Or, if you have met it before, you may find that running into it in an experimental way gives you a new angle on it and new understanding. The first unit, on iteration of linear maps, is a little of both. Practical experience with the phenomenon of convergence should be useful in a later course on analysis, while in a way it is nothing new -- you have already run into the notion of "limit" in a calculus course.
Grading
If you do all 4 written papers satisfactorily (this means writing a paper that is correct and in good style) and if you also give a good talk, you will get at least a B+. For any kind of A the content of your papers and talk should also be interesting. The most interesting thing you can do is to be inventive yourself in raising and investigating your own questions.
Style
Good mathematical style is clear, not too stiff, and not verbose. Above all, it should be logical and correct. Here is where you will almost certainly find yourself rewriting, and striving to say things more precisely and more succinctly -- it is not so easy! Plan to use LATEX for these papers. On a PC the program WINEDT is an excellent text editor and toolkit for doing this. On a Mac use Texshop. There is an extensive template and more LATEX hints under "resources" on the ELLA site. I think you'll be pleased to see how professional your mathematical writing looks when you use these tools. I will help you with this, of course, and we will also have a TA for the course, Jialu Chen '12 (chen22j). She will be available for all kinds of technical help, and also for help with writing, proofs -- everything!
As you think about mathematical writing, have a look at the MAA publications in the Mathematics Department Office, just to the left as you go in. Journals like the Monthly, College Mathematics, and Mathematics Magazine have articles that are quite readable. Even if you can't follow their arguments, look at HOW they say what they say: snappy introduction, clear articulation of the ideas. You might even find a topic here that you could follow up for Unit 5.
A possibility to keep in mind is presenting your own work at the Hudson River Conference, this year at Skidmore College, Saratoga Springs, NY, on April 16. This is an undergraduate mathematics conference held every spring somewhere in New York or New England and hundreds of people attend. See the Hudson River Conference website for info.