Saw this neat tweet from Steven Strogatz last night:
The short summary of his analysis course made me wonder how much an intro analysis course has changed in ~25 years. So, on an early trip into Boston today, I grabbed my old copy of Rudin and flipped through it. To me it looks like Strogatz’s course and Dan Strook’s back in 1991 were remarkably similar. The one topic on Strogatz’s list that wasn’t really part of Strook’s course is Fourier Series, but that was actually the next class in my undergrad analysis sequence so I’m not too surprised it wasn’t covered.
Anyway, with about 30 min to write something, here are some of my random thoughts I had about intro to analysis and intro to proof 25 years after I took that class . . . .
The reason that Strogatz’s post caught me attention is that I’ve been thinking a lot about my own math education over the last couple of years, and about that intro analysis course in particular. Some of Cathy O’Neil’s ideas have been really helpful to me – see here, for example:
She also had some great thoughts on the topic of an “intro to proof” class during Harvard’s Gender Equity in Mathematics discussion.
Two other pieces that got me thinking about ‘intro to proof” ideas are (i) Numberphile’s famous 1 + 2 + 3 + . . . = -1/12 video:
(plus the important Follow up video from Ed Frenkel )
and (ii) Jordan Ellenberg’s How not to be Wrong. In the discussion of series in that book (and I’m going from memory here, because I didn’t think to grab this book on the way in) he uses Grandi’s series as an example of defining a seemingly-divergent series to have a value. The idea that struck me (again, from memory so not quite a direct quote) went something like this:
“We can say the series 1 – 1 + 1 – 1 + . . . . diverges, or, if we want to define the series to have a value, the only value that makes sense is 1/2.”
Now, I understand that one of the main points of an introductory analysis class is to learn basic techniques of analysis and I’m not even remotely questioning that. Looking back, though, I wish I would have seen ideas similar to what Ellenberg talks about with Grandi’s series or that Numberphile presented with the -1/12 series (though probably more like how the ideas were presented by Frenkel) in that course.
That there is a way to make sense of 1 + 2 + 3 + . . . = -1/12 has to be at least as interesting as the idea that 1 – 1/2 + 1/3 – 1/4 + . . . = ln(2). In fact, after seeing the Numberphile video I went straight to the library to grab a copy of Hardy’s Divergent Series book to see what the hell was going on – I’d never seen that idea before!
Anyway, I think students would find it extremely interesting to see that there’s a bit more to series than meets the eye. Even in areas that seem as clear cut as convergent and divergent series, mathematicians have found some amazingly interesting things to say!
Now, back to the general idea of an “intro to proof” class. I know that I had to work super hard to understand proofs as an undergrad. In fact, it is a nothing short of a miracle that I got a B in that class. In terms of helping math students learn about proofs, I love the idea behind the two classes that O’Neil got going at Berkeley and Barnard. One point from the post I liked above that was completely missing from my intro analysis class is this one:
“But of course the most important thing was that I clearly stated at the beginning of each class in the first two weeks that proving things in math was a skill like any other that you get good at through practice.”
I think you can go even further than that, though, and give students a peek at the process that mathematicians going through in their work. This Numberphile interview with Ken Ribet is something I wish I would have seen when I was younger (and I know the topic isn’t analysis, but I don’t think that matters all that much):
To me, seeing the ideas about collaboration, working and revising, failing multiple times, and even not quite understanding that you’ve understood something (!) would be eye-opening for kids learning about proof. It would, I think, plant the idea that when you are struggling to understand this theorem of Abel (which, I think is the one from “Abel’s test” in Strogatz’s note):
If the series converge to and , then
you’d know that Abel had to do a heck of a lot of hard work to understand it himself!