# What learning math sometimes looks like – when a problem suddenly makes sense

I really like this problem from the 1985 AJHSME:

Problem #20 from the 1985 AJHSME

Here’s the problem:

In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January $1$ fall that year?

This problem has always given my son a little trouble. Talking through it tonight he struggles a bit to get going, but slowly gets to the point where he’s drawn a calendar that represents the situation. It doesn’t look like a normal calendar, though.

Around 4:30 in the video he’s talking about why the picture looks wrong, but then comes a really cool idea to fix that problem – re-draw the calendar! From this new point of view the solution to the problem just falls off the page.

What my son went through talking this problem tonight is what learning math often looks like. From one point of view a problem can be nearly intractable and frustrating, but from another everything seems to make perfect sense.

# My favorite math class

Saw a twitter hash tag #myfavoritemathclass going around a few days ago. I actually struggled to come up with just one to call my favorite, but it was really fun thinking through all of the classes. There were roughly four categories for me:

Junior High

Mrs. Whitney’s geometry class.

Up to that point in my math education everything was all about calculation. For me that was fine – I’m much more of a calculator by nature than anything else – but the new ideas about proof and constructions in geometry class were really interesting. The constructions, especially, I suppose and some of the problems that were not really accessible to me then stayed with me for a long time. I remember, for example, learning that $\cos(72^{o})$ = $(\sqrt{5} - 1) /4$ in high school and immediately heading off to construct a pentagon.

High School

This one is the only no-brainer – Mr. Waterman’s Enrichment Math class. This class was mainly for the math team at my high school and I took the class all three years I was in high school. Mr. Waterman also taught calculus, linear algebra, and differential equations, but it was enrichment math that really shaped how I think about math.

We learned all kinds of different math – basically whatever Mr. Waterman felt like talking about that day. Nothing was off limits – advanced geometry, number theory, fun algebra problems from old contests. Also super fun was having former students come back to give talks. After graduating I came back to give a talk to the class every year that Mr. Waterman was at Central.

College

There’s almost no way for me to pin it down to one class here – nearly every class had something amazing. The overall theme for me in college was learning that you could actually study math for math’s sake!

My introduction to complex analysis from freshman year convinced me to major in math. I remember nearly falling out of my chair seeing this integral on a homework assignment:

My analysis class out of Rudin was one of the biggest struggles that I’ve ever had in math. The professor even told me at the end of the semester that I didn’t have what it took to be a mathematician. Whatever, dude – I was just glad to have that class behind me ☺

The next analysis class was incredible – I don’t have the book anymore, but I think it was called “Measure Theory and Integration.” Who knows why, but Parseval’s theorem was really fascinating to me. I found a research job programming the data transmissions for a satellite telescope using ideas from Fourier analysis the summer after taking that class.

The most absurdly hard class for me in college was topology, which was by professor Munkres out of the topology book he wrote. At the start of the class he told us that he’d been teaching the class since the 60s and graded on a curve based on all of the students that he ever had. Given the size of the class, he estimated that there would be 2 A’s an 1 A-. There were 2 IMO gold medalists and 1 silver medalist in the class – ha ha. I didn’t really understand topology even remotely until grad school.

The class that was the most fun was Mike Artin’s Algebra course out of the now published Algebra book he was writing at the time. He had such a fun teaching style and was probably 2nd only to Mr. Waterman in his ability to communicate his love for the subject through his teaching. I was really happy to run into him at MIT’s graduation last spring and show him a fun project that we’d done based on a picture in his book.

A 3d project for kids and adults inspired by Kip Thorne

The class that just blew me away was Richard Stanley’s combinatorics class. The ideas that he showed us about generating functions were incredible – if there’s one thing that I wish I understood better now, it is the connection between algebraic expressions and counting. He had so many amazing examples – counting is really cool.