Eight Important Things I Learned From Math

Since I’m going to be out of the country and possibly without Internet for a couple weeks, I decided to post early. Here’s the follow-up to last week. I hope to get something posted next week sometime, but, if not, see you Sunday, June 28!

For a lot of people, the question “what did you learn from math?” is obvious: you learned how to do your taxes, calculate a tip, integrate a function. Few people seem to see math as imparting more general lessons that can be applicable to life in general.

I guess I’m one of those few.

1. How to lay out a problem.

The number-one skill I acquired in math class is the ability to figure out what the problem actually is. Whether you’re given a word problem or a couple equations, the first thing you need to do is decide exactly what it is you’re trying to accomplish. Are you maximizing a particular variable? Solving for x? Shooting for a proof? What are the base equations you have to work with?

It’s surprising how often this comes up in real life. Not necessarily the math-flavoured part (“Let x represent…” – although that, too, is useful far more often than you’d think), but the need to define a problem and organize the information you already have.

2. How to turn the world upside-down.

Some math problems seem impossible the way they’re first phrased. Like the infamous birthday problem: what are the odds that at least two people in a class of (insert number here) have the same birthday?

When you first look at this question, it seems like you’re being asked to do the – well, if not impossible, the highly improbable: figure out all the different combinations of two or more people having the same birthday, calculate the probabilities for those , and add them all up besides. And you could do this problem that way. But it’s much easier to stand on your head and ask instead: what’s the probability of no one in the class having the same birthday? Subtract that from one, and you’ll have your answer.

Sometimes, math teaches you, solving a problem is all in the way you look at it.

3. How not to be a hero.

In the first few math classes I took, I did my homework like this: write down appropriate questions. Hole up in my room. Finish them all, and if I don’t get one, work really hard until I do.

It took me until third year to realize that math professors don’t mean for you to solve every problem on your own. First, if you’re stuck, they expect you to ask them about it – and most of them will all but walk you through the solution step by step. But second and more importantly, they expect you not to think you’re stuck until you’ve worked through the problem with your friends. As long as you include the names of the people with whom you collaborated, you’re golden.

In other words, the point isn’t to do it alone. The point is to get ‘er done.

4. How to focus on the process, not the answer.

You can have the right final answers on every single question on a math exam and still get the whole thing wrong. Why? Because the point of math class isn’t to get the right answer. The point of math class is to do things the right way.

Philosophers argue over whether math is empirical  – whether mathematical knowledge is derived from experience or reason alone. And historically, it can go either way: on one hand, ideas like imaginary numbers were discovered precisely because theory said we needed them, even if there was no real-life justification anyone could point to; contrariwise, it took years after the invention of calculus for someone to define the concept of limit on which the whole field depends.

If you’re a mathematical creative genius like Leibniz or Newton, then you can leave the details to your Weierstrauss. But if you’re a student in Calculus 101, just guessing the answer or following your intuition isn’t enough. You have to understand and convey the logical process leading from problem to solution, or else when you wade in deeper or try to invent something new, you’re going to be lost.

After all, Leibniz and Newton were able to come up with the concept of calculus in the first place only because they knew how to take ordinary intuitions and systematically extend them to their logical limits.

5. How to present my thoughts for other people.

When I was in elementary and high school, I was one of those awful kids for whom math just made sense. Like a puzzle or something, where it all clicked together in my head and, presto change-o, out popped an answer onto the page. And, okay, this sometimes happened for me in undergrad, too.

This was all well and good when all I had to do was fill in the correct answers on worksheets and tests, but it didn’t help at all when I became a TA, tried to talk about problems with my friends, or even just plain messed up a step while working through a question. “Because it just does!” isn’t a helpful way to answer a classmate or a student who wants to know why the equation you just wrote down simplifies to a certain form; “but it was just an arithmetical error!” doesn’t change the teacher’s mind when all you actually wrote down is the wrong answer.

I’d go so far as to say that when I can’t explain what I’m doing, I don”t really know what I’m doing. And that, I’m discovering, goes for everything else along with math.

6. How to sweat the details.

I’m knee-deep in a calculus question. I’m integrating as fast as I can – aha! An answer! I flip to the back of the book, expecting to have my brilliant solution confirmed, and… I’m wrong. Why? At one point in my integration, the denominator of the fraction I was using became zero. A fraction with a zero denominator doesn’t exist. And because the integration process is explicitly defined only to work over certain values, it’s inapplicable as-is under these circumstances.

One single DNE among a tonne – well, an uncountable infinite number of perfectly good values seems like it shouldn’t matter. But it does. And math will screw you up if you don’t pay attention to details like that. Certain methods are applicable only under strict circumstances, and if you don’t know what those are (or, worse, decide not to care), you may get away with it for a while. But sooner or later it’ll come back to bite you on the butt. As my third-grade teacher used to remind us, if something’s worth doing, it’s worth doing well – and thinking through…

7. How not to sweat the details.

… at least, it’s worth thinking through if you need a precise answer. On the other hand, there’s a time and a place for quick ‘n’ dirty math. It all depends on what exactly you’re trying to figure out. The tip at a restaurant? Just round it to easier numbers and guesstimate. Really difficult multivariate calculus problem? Figure out if there are any coefficients or constants you can safely assume are small enough to be negligible.

Like I said above, math is all about figuring out what problem you’re trying to solve. You’ll often have extra information that you don’t really need to take into account. The knack is in figuring out which info you can safely ignore and which matters to your answer.

8. How to appreciate more than words

Language was and is my first love: in kindergarten, they had to physically remove me from the reading corner (I went to a Montessori school before first grade) in order to convince me to try out other subjects. But there are times when you just can’t say in words and qualities what you can in numbers and quantities.

It’s like Milo learns in The Phantom Tollbooth – language and math are the warp and weft of knowledge. A “mere” number isn’t any more or less creative, vital, or interesting than a “mere” word. There are ideas you can’t express in numbers, and ideas you can’t express in words. That doesn’t mean one is intrinsically better than the other.

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