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June 1st, 2010, 01:51 AM  #1 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  What you wish you knew  designing a course
I have recently been asked to help design a new introductory course for entering freshmen math majors. In general, we expect almost all of the freshmen to have taken a calculus course in high school, but about half will be repeating Calc I at the university. This is a small course, only one hour per week. But I think that with one hour per week, many great things can be accomplished. So I wondered what do you wish that you knew early on? How do you explain what math really is? For example, when I learnt Galois Theory, I was actually studying in Budapest for a semester. And it turns out that everyone around me knew things about symmetric polynomials and complex numbers that I simply wasn't familiar with. In fact, I thought that I should have been familiar with them. But I think that symmetric polynomials, complex numbers, and many other ideas from classical algebra are easily accessible to an entering freshmen math major, and so they merit consideration for this course. I also want to give a sense that math can be more than just arithmetic, and perhaps even start looking into the idea of proofs (although there is a stronger proofs/vector spaces course further on). Math education is something that interests me greatly, and I'd be thrilled to hear your ideas and discuss what we think. I also know that it's difficult to say what you would have wanted to know, as this school has its own background, and people will come in with a variety of skills as it is. But nonetheless, this is a start. What do you think? 
June 1st, 2010, 11:55 AM  #2  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: What you wish you knew  designing a course
That sounds like fun! Another idea that I think is worth studying is cyclotomic polynomials. There are some rather accessible approaches to the subject, and they come back quite a bit. I really think you should at least show limits from a very rigorous perspective, and describe (at least briefly) how easy it is to extend this idea. I very likely would have been more open to calculus (and analysis) if it had been better discussed when I was first learning... of course, maybe it was discussed well, and I was too naive and blase to notice. There are other things, certainly, but I cannot think of them offhand. I'll definitely give this some thought. Quote:
 
June 2nd, 2010, 07:10 AM  #3  
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: What you wish you knew  designing a course Quote:
 
June 2nd, 2010, 07:54 AM  #4 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: What you wish you knew  designing a course
Last semester, as in spring or fall? I was there last fall. I didn't actually take Galois theory... as a subject, it scares me for some reason. I know Csaba because I spent a lot of time hanging out in Anna's office, and Csaba would always give me shit about drinking the coffee he made... So eventually I started making it if I showed up early. At the risk of derailing the thread, what classes did you take? Back on subject: I think the "crash course" that they have at BSM has a lot of the ideas that you should be trying to show Most of them I hadn't seen until more advanced courses, but they are low level, approachable things which really show the deeper structural relationships that get mathematematicians excitied. 
June 3rd, 2010, 08:51 AM  #5 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: What you wish you knew  designing a course
And I must admit, the classical algebra course that Cliff taught in the beginning was great, and he introduced me to the book Polynomials, by Barbeau. I suspect this will play a big role in the course.


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