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I was wondering if anyone had any experience with an undergraduate course that emphasized the building of mathematical theories or if they'd ever heard of this being done? How did the class work (did the professor list axioms and definitions each week and see what the students were able to prove, did you discuss a specific mathematical theory and the various problems that arose during its creation?)

Does this sort of thing work well for undergraduates? I'd be very interested in taking a class like this because I've never really had any experience with that aspect of mathematics. So if there are any classes that are commonly taught in this sort of style, I'd be interested to know that as well.

Thanks.

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Could you say a little more about what you mean by "theory building"? Tom Stephens' answer below interprets this to mean an approach in which the students assume primary responsibility for proving the theorems (keyword: "Moore Method"). Is that what you mean? –  Pete L. Clark Aug 12 '10 at 3:40
    
I'm not sure what you mean by this, but if you have only taken general freshman-level mathematics at a US school (calculus, the usual diff eq class, etc), maybe you are simply asking about classes where students actually write proofs and where the exercises they do are not restricted to calculations they have been previously shown exactly how to do. In that case, yes, most courses intended for math majors are like that. –  Mike Benfield Aug 12 '10 at 11:37

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I have benefited from more than one professor who taught more or less according to the Moore Method. These professors happened to come out of SUNY Binghamton in the early 1980's.

The courses required lots of input from the students. One professor in particular would ask the class to make a conjecture about some construction he would put on the board and then name the conjecture after the student - we were then personally responsible for success or failure of our ideas. After the first few humiliations, this became wonderfully empowering and fun.

The biggest impact I have noticed from taking these courses has been on my independent studying in other classes. I notice that I am able to replicate the types of attacks on my own reasoning that my professor used to - hence leading me closer toward the boundaries of my own understanding.

I don't really know how to respond to your idea of theory building... it seems to me that mathematics is not created upward from luckily chosen axioms, but in a highly nonlinear and back-and-forth fashion. The only example I know of to illustrate this clearly is the development of topology, starting in the late 1800's with Cantor and more or less culminating in the early 1900's with Hausdorff (excellent discussions with references appear in Engelking, General Topology).

I agree that it would be a great benefit to you to find courses which at least approach what you have in mind. Good luck to you.

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Moore method was basically what I was looking for, I think. I was interested in taking a class where the professor doesn't give you thinks to prove but allows you to explore and come up with things to prove. By theory building, I meant something that investigates how modern areas of mathematics arose historically and the sorts of problems that arise when a new area of math is born. I've already found some resources that have answered this part of the question as well. –  WWright Aug 13 '10 at 1:36

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