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.”
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.”