I find that lots of students are really good at how.
Like, how do you factorize a quadratic? How to you differentiate a cubic? How do you solve a system of simultaneous linear equations? How do you poach an egg?
(Apparently you need a gentle whirlpool to get the egg moving. Whirlpools: the unsung hero of the breakfast table.)
Why are they so skilled at how? It’s because students like procedures. They like certainty, clarity, the feeling that you know exactly what to do at every moment.
But they struggle with why. And – even more basically – they struggle with what.
I find that questions like this elicit one of two responses from students. Either this:
These aren’t questions students are accustomed to answering in math class. In history, perhaps, where they have to write IDs of historical figures and events; or even in science, where they have to understand each component’s role in a theory.
But not in math. We math teachers tend to ask lots of how questions, and not so many what questions.
If you ask me, that’s sort of sad. They’re experts in how, and they can’t even tell you what the how is for.
And in this case, it turns out, there’s a pretty satisfying answer.
First, note that quadratics are much more complicated and interesting than their simple flat-brained cousins, the linears:
And then, note that quadratics are much simpler than their roller-coaster contortionist siblings, the cubics, quartics, and other high-degree polynomials:
To me, this is the appeal of quadratics. As degree-2 polynomials, they occupy a sweet spot between the dull degree-1’s, and the intimidating, intractable degree-3’s.
Just as Goldilocks sought the perfect bed (not too hard, not too soft) and the perfect porridge (not too hot, not too cold), so the mathematician seeks the perfect polynomial. Not too hard, not too easy. Not too complex, not too simple.
Just about right.
Of course, this line of reasoning is open to an obvious attack. Okay, a disgruntled student might say, you’ve convinced me that, if I’m going to study polynomials, I ought to focus on quadratics first.
But why should I study polynomials to begin with?
The answer to that is trickier, I think. You might as well ask this:
This question has as many different answers as mathematics has teachers. Some like to focus on the applications of math. Some argue it’s all bout the beauty. Some just say, “Because,” and then sigh, because it’s been a long day.
But for me, it’s about thinking.
And the role that the quadratic plays in polynomials… well, that’s exactly the role that mathematics plays human thought.
In every walk of life, humans need to reason.So of course, they can learn these intellectual skills in other places. You don’t need math. But gosh, does math make it easier!
You can learn to taxonomize in biology, by considering the classification of organisms. But your taxonomies will never be perfect, because life doesn’t fit into neat little boxes. (I’m looking at you, protists.)
Life doesn’t… but math does.
Or you can learn to dissect arguments in civics. But emotions will flare. It’ll be tough to agree on premises. And even if you do, words like “justice,” “freedom,” and “common good” are subject to fuzzy interpretations and subtle misunderstandings. All words are like that: a little vague, tricky to pin down.
Except in math.
Logic shows up everywhere. But in math, it’s the whole game. Math isolates the operations of logic and reason so that we can master them.
In short: math is the playground of reason.
This post is hastily adapted from a talk I gave yesterday at University of Birmingham, titled Death to the Quadratic Formula (or, Long Live the Quadratic Formula). Thanks to Dave Smith and the IMA for the invitation!