| I haven't been a mathematician for decades, and in particular I don't recall the details in the examples I'm about to give. :) But I'd say yes up to a point. In particular, one matches problems to generalized patterns, which in math often has the feel of transforming a problem into another form. 1. If you're going to take a test for school, you can recognize patterns pretty easily. For example, Harvard has a 3-day Qualifying Exam as a PhD requirement, given twice a year. Once, after looking at the exam pages for the first two days, I said "Hmm, there hasn't been a Schwarz Reflection Principle question yet." The next evening, I got a nice hug from Lisa Mantini. :) 2. If you're doing actual research math, it's of course much harder. For example, the theorem in my thesis was scooped by a few months by Neyman and Mertens. What's more, they did it as a special case of a fixed-point theorem, while I just proved it by brute force. And now some simpler anecdotes from my undergraduate days. 3. In my E&M course, we were supposed to work out fields for a cylindrical wire with an off-center cylindrical hole. This was a LOT easier in curvilinear coordinates than Euclidean. When I showed that to the professor in his office, he literally stood up and applauded. 4. In an abstract algebra course, there was a typo in the book for one of the exercises as to whether a certain polynomial was reducible. I solved the hard form by transforming the problem into one about a polynomial with matrix variables. I took that story with me when I went to graduate school. Ron Livne's eyes lit up, and he transformed it again into something about -- well, into something about transformations on the polynomial's complex roots. |