Patrick Stevens is an undergraduate mathematics student at the University of Cambridge, and I’ve really been enjoying his blog recently. He’s been doing a series of posts about discovering proofs of standard real analysis theorems. He writes that the series is “mostly intended so that I start finding the results intuitive – having once found a proof myself, I hope to be able to reproduce it without too much effort in the exam.” When I teach analysis, one of my main goals is for students to start developing their mathematical intuition, to learn how to “follow their noses.”
Hector the dog is following his nose, perhaps sniffing out a proof of the Heine-Borel theorem. Image: SaudS, via Wikimedia Commons.
It’s fun for me to watch Stevens follow his nose and figure out these proofs, especially because I’ve done most of them with my students recently. In addition to figuring out the proofs, Stevens also writes about figuring out statements of theorems themselves.
“A little while ago I set myself the exercise of stating and proving the Contraction Mapping Theorem. It turned out that I mis-stated it in three different aspects (“contraction”, “non-empty” and “complete”), but I was able to correct the statement because there were several points in the proof where it was very natural to do a certain thing (and where that thing turned out to rely on a correct statement of the theorem).”
I’m trying to figure out how to incorporate these posts into my analysis class the next time I teach it. My instinct is to make them recommended reading for my students, but I’m not sure the best way to make that an active learning moment for them, rather than just another time for them to watch someone else do math. Perhaps a writing assignment where they walk through the details like Stevens does would be better. Or I could suggest that they work along with Stevens and try to figure out what the next step will be. If they come up with different steps, it would be good for them to figure out how they are different and whether they are both valid ways to continue the proof.
In a post about making topology simpler, Stevens tackles the eternal confusion that the words “open” and “closed” create. I talked about that confusion on my other blog last September, and there is a pretty great video about it called Hitler Learns Topology. (I don’t usually like Hitler parody videos, but this one cracks me up.)
I’ve been reading Stevens’ blog for a while, and I would be remiss if I did not highlight my favorite of his posts so far, Slightly silly Sylow pseudo-sonnets. Yes, they are poems about the Sylow theorems. Here’s the first one:
“Suppose we have a finite group called G.
This group has size m times a power of p.
We choose m to have coprimality:
the power of p‘s the biggest it can be.
Then One: a subgroup of that size do we
assert exists. Two: conjugate are Sy-
low p-subgroups. And m‘s nought mod np
And np=1(modp); that’s Three.”
I am second/third year mathematics major at Texas A&M-Central Texas. I am currently going through the calculus classes. I have come to the point where mathematics is more about intuition than knowledge of formulae. How does a “newbie” begin training themself to think more intuitively? What resources are out there? How about practing proofs? Where can I begin?
I’d second Taylor’s comment that “tinkering with problems I didn’t know how to solve was very helpful to me.” Just rehearsing proofs is really bad for intuition – ideally, you need to be in a position where you could find most of the proofs out for yourself. In fact, I wouldn’t be surprised if many of our lecturers were working out the proofs as they went along (I know for a fact one of mine was).
Having said that, there are a couple of very common themes in analysis. Tim Gowers has an excellent worked example (http://gowers.wordpress.com/2014/02/03/how-to-work-out-proofs-in-analysis-i/) in which he goes through a large swathe of the available tactics. The trick is really in getting a feel for which tactic is going to be most useful. I think of it as “The question gives me this kind of knowledge; what kind of knowledge am I aiming to achieve, and which of my tactics will get me that kind of knowledge?” Hmm, I’m having to stop myself writing a tract here – might have to decant that into a separate post.
During my undergrad I struggled in an Intro to Analysis class more than I have ever struggled in any class in my life. What I found helped was what helps with any creative endeavor. Keep doing it and not caring when your proof is bad. I memorized all the definitions and then literally tried every question in our analysis text. I studied the example proofs and then I just plugged away. I was very bad at first but as I tried more and more I developed a sense of different things to try. You can train your brain to think in new ways and develop a sense for what works just like you can train it to apply an algorithm, its just that the former takes more time. (In my experience.) Tinkering with problems I didn’t know how to solve was very helpful to me.