On Friday I realized—yet again—that my too-clever-for-their-own-good students were finding ways to answer questions without understanding the ideas.
Rather than reckon with the concept of slope, they were memorizing a complex rule:
That’s all true, so far as it goes, but it’s as opaque and sinister as the tax code.
“Math is supposed to make sense!” I told them, and in my flailing to explain why, I found myself reaching for my favorite rhetorical tool: the overly-detailed analogy.
So, to see what math class is like for memorization-driven students, imagine that you’re a household robot.
Your job is to clean the dishes.
Now, you don’t know anything about food, meals, or human culinary practices—you’re a robot, after all. All you know is what you’re told. And your only wish in life is to follow your instructions, and to carry out the right procedures.
You’re thrilled when a nice family takes you home, teaches you how to recognize dishes as “dirty,” “clean,” and “dry,” and then lays out your simple rules to obey:
Of course, you don’t know what dishes are for (decoration? sport?). You don’t know how they get dirty (carelessness?). You don’t know the point of cleaning them (don’t they keep getting dirty?). And you don’t know why they belong in the cupboards afterwards (they’re hidden when clean and displayed when dirty?). You just know that these are the steps.
And you follow steps. You’re a good robot.
On your first day, you notice some dry dishes out on the table. You start stacking them in the cupboard, just like you’ve been told, when your owner starts screaming. “What are you doing, you dumb machine? I just set the table a minute ago!”
You’re confused. Weren’t you just following the rule that says ‘Stack the dry dishes’?
Soon, the rules are amended:
Okay. Weird, and contradictory, but whatever. One exception isn’t so hard to learn.
A little later, you notice the plates getting dirty. There’s all kinds of lumpy organic stuff piling up on them. So you grab them and start cleaning.
“NO!” you hear. “STUPID ROBOT! That was our lasagna!”
What’s all this “lasagna” talk? Isn’t this organic dirt on the dishes? Aren’t dirty dishes supposed to be cleaned? Isn’t that, like, your entire job?
Soon, your first rule is amended:
“Wait until the large dirt particles are gone?” How paradoxical. You’re supposed to focus your cleaning on the less dirty dishes, waiting for the really dirty dishes to magically clean themselves halfway?
Well, no matter. Strange and inconsistent rules are still rules, and—being a good robot—you follow rules.
But the next day, they’re yelling at you again. “Why didn’t you clean that plate? You left it out overnight!”
You look at it. It’s still got lots of dirt on it. Didn’t they say not to clean it if it’s got such big particles left?
A moment later, you’re programmed with a new first rule:
You’re starting to get anxious now. What once seemed a simple scheme of rules has grown rather complicated. You could really use some encouragement for your efforts, and so you leap at the first chance to put this newly modified rule into action. The removal of particles on one dish has stopped, so you begin to clean it.
But no, only more shouting—“I was just pausing to take a sip of water! I wasn’t done!”—and a new, even more complicated rule:
It all seems cruelly arbitrary. This once-straightforward task (“clean the dirty dishes”) has become a nightmare cobweb of exceptions and contradictions, in which you must calibrate your behaviors precisely based on bizarre conditions and inscrutable requirements.
Ladies and gentlemen, welcome to math class.
What’s missing, of course, is the crucial understanding of what it’s all for. It turns out that you can’t succeed as a dishwasher until you understand why dishes get dirty in the first place.
“The point of dishes,” the robot needs to be told, “is to hold humans’ food while they eat it. When they’re done eating, that’s when the dishes need to be cleaned.”
Every rule – even the craziest, most arbitrary mandate – has a reason rooted in this essential purpose. (Why leave the dishes with big particles? Because the person is still eating!) And so it is in math class. If you understand slope not as “that list of steps I’m supposed to follow” but as “a rate of change,” things start making more sense. (Why is it the ratio of the coefficients? Because, look what happens when x increases by 1!)
You get to work a lot less, and think a lot more.
Now, conceptual understanding alone isn’t enough, any more than procedures alone are enough. You must connect the two, tracing how the rules emerge from the concepts. Only then can you learn to apply procedures flexibly, and to anticipate exceptions. Only then will you get the pat on the back that every robot craves.
With my students on Friday, I garbled the whole analogy. I tend to do that.
But there’s a simple takeaway. Even if you don’t care about understanding for its own sake; even if you’re indifferent to the beauty and deeper logic of mathematics; even if you care only about test results and right answers; even then, you should remember that the “how” is rooted in the “why,” and you’re unlikely to master methods if you disregard their reasons.