This September, I gave my 7th-graders an elegant little problem about a 12-step staircase. You’re climbing from the bottom to the top, using combinations of single and double steps. The question is, how many ways can you do this?
I was stunned when some of my students offered answers almost immediately. “145!” one screamed, as if he had just gotten bingo. “Am I right?”
“Whoa, that was fast!” I said. “Why 145?”
“12 times 12, plus 1!” he announced. “Am I right?”
“But…” I hesitated. “But why 12 times 12? Why plus 1? Are we just doing random computations that sound like fun?”
He listened to my questioning with the same patience you’d give a friend’s mediocre guitar solo. Then he launched right back into his chorus: “So,” he said, “am I right?”
To him, at that moment, “doing math” meant “making a number smoothie.” You take the numbers in front of you, throw them all into the blender, and mash the “pulse” button until you get something.
The funny thing is, in our classes, this often works. You see a thick block of text; you pick out the numbers; you run them through the formula; and voila, you’ve got a solution, no thinking required!
It’s like I’m trying to teach you to make smoothies, and I always start by laying out exactly the ingredients you need. “Here’s a banana, a cup of strawberries, a cup of milk, and a cup of vanilla yogurt. Can you make a smoothie?”
Of course you can!
But what happens when I throw extra ingredients in front of you? “Here’s a banana, an onion, a cup of strawberries, paprika, dry pasta, cooked pasta, milk, vegetable oil, a cup of vanilla yogurt, and toothpaste. Can you make a smoothie?”
Maybe you can. Or maybe you’ll conjure up a foul paprika pasta paste.
The number smoothie is a classic way to avoid thinking. It is the blind mashing of a button. Whereas real mathematics is thoughtful, selective, and carefully considered, the number smoothie is precisely the opposite: indiscriminate, wanton, thoughtless.
It doesn’t mean you should never use formulas, any more than you should swear off smoothies. I’ve written before about the need to understand where formulas come from, but even after you’ve done that, a formula needn’t be a lifeless tool.
Learning a formula doesn’t need to be the terminus of thought.
Instead, try something like this:
For the blue, pink, and purple triangles, you’ve got too much information. It’s the mathematical equivalent of needing to sort out the onions and toothpaste from the real ingredients. For the green, you don’t have enough information—the mathematical equivalent of needing to root through the fridge to find the missing strawberries.
Or you could ask a question like this:
Now, there are no ingredients at all! You’ve got to go grocery shopping all by yourself. It’s a newfound freedom (and, for many students, a new level of cognitive challenge).
Or, you could ask questions like this:
Whereas before you just blended yourself a puree and called it quits, these questions demand a more sophisticated kind of cooking. Now, using the formula is just one small step along the path.
Or how about this:
You’ve been given an ingredient that looks like a strawberry, but isn’t. You’d better know your stuff if you want to recognize this poisonous imposter-berry before it’s too late.
Occasionally, we teachers grow frustrated with our formula-thirsty students. (Okay, more like “often” or “weekly.”) Sometimes, we even denounce formulas altogether, deriding them as “brainless plug-and-chug” or “not real math.”
Of course, that’s going too far. The intelligent use of formulas is an important part of mathematics. But we’re right about one thing: there’s a lot more to formulas than just throwing numbers into a blender.