Crying, Eureka

Last night, I took the final for my Discrete Math class. It's been a wonderful, fun, and somewhat amazing class (amazing in that I had no idea what discrete math was when I started, and only registered for the class because the one I wanted to take was cancelled and it was in a convenient time slot).

It has also been a stressful class. Although I want to study math forever, I really have a love/hate relationship with it. In short, I love math that I already understand, and sort of hate the struggle to learn new math. But once I start to understand the new math I will love it as well. (I have a very similar experience with music; I almost always hate listening to anything new, but as soon as the newness wears off - usually around the 2nd or 3rd exposure - I'm all over it.) I have a low threshold for frustration, so if I don't understand new math right away I tend to get disproportionately upset.

Occasionally, I get to learn new math that I immediately understand. This tends to be things like definitions around functions (onto, one-to-one, etc.), propositional logic, set theory, probability, and so on. These are really exciting and wonderful experiences for me - all the joy of a new way of thinking, with none of the suffering.

Anyway, returning to my Discrete class. Some of it has been the easy, delicious stuff. Other parts were really hard. The hardest part for me was recursively defined relations. These are sequences like the Fibonacci numbers, where the most natural definition is how to get the "next" number. In class, we "solved" these by coming up with the closed-form rule: the rule that tells you how to get, for instance, the 1000th Fibonacci number without having to figure out the preceding 999. We used tricks. We used generating functions.

On the final exam, we were given a series that didn't respond to any of these methods. We were to solve it "by inspection", which means "look at it and figure it out." Looking back, this one seems simple to me, but I had a very hard time with it. Here it is:

a[1] = 1
a[k+1] = k^2 * a[k]

so the terms are

a[1] = 1 (given)
a[2] = 1^2 * a[1] = 1
a[3] = 2^2 * a[2] = 4
a[4] = 3^2 * a[3] = 36
a[5] = 4^2 * a[4] = 576
and so on

I won't give the solution here in case anyone wants to figure it out for themselves, but taking the test, I stared at this. I tried to reason through it. I drew a picture. I wrote out the first six terms.

And then I cried. I knew I needed to get an A on the test in order to have a good chance of an A in the class, and this question was 1/6 of the points. And solving recursively defined series by inspection is something I can "never" do. So I actually cried. And then I dried my eyes and tried to calm down. I started writing out the terms in terms of how they were calculated, and this got me to the solution (which I at first didn't recognize, until I noticed that it really did correspond to each term). Then I had to do a (very easy) proof by induction to show that my solution was correct, and I was done.

I've noticed that I often get a difficult math thing immediately after crying. I think crying is one way of frustration "breaking", in the way that sweating is the breaking of a fever. Once the frustration breaks, you're calm again, and ready to proceed a bit diligently (if hopelessly), and insight can make its way through your brain again.

I am getting much better all the time at figuring out what to try next, in math. It used to be that I'd get frustrated and give up, and then the next day it would occur to me what else I could have tried, and then I wouldn't bother. Now I can often think of the next thing to try right away, and just do it. I'm sure my tolerance for frustration is increasing too. And I am succeeding at my explicit goal of learning that I can get better at math through effort.

It's been a good semester.