The Real Math

There’s a phrase my college students often use that used to drive me crazy. The phrase is “do the math.” When my students use this phrase they always, without fail, mean, “do the algorithm,” or “follow the procedure,” as in, “I solved the problem with a picture, and then I did the math to make sure I was right.” The implication, of course, is that other solution methods are not doing math, that you’re only actually doing math if you’re doing what you learned in school.

A couple days ago I had an interesting interaction with my 7-year-old. Bear with me for a moment. These are related thoughts. He had taken it upon himself to count up the number of dots on the step stool in our bathroom.

Please ignore the dust. We haven’t washed it in awhile and none of the camera filters I used could hide it.

He proudly showed me how he counted the number of dots in a row (13) and the number of rows (9), and then broke the 9 apart into 5 and 4, used skip-counting to multiply 13 by 5, and then by 4, and then added everything together to get 117. It was wonderful. He came up with it all himself, and it made my math teacher heart happy. And to top it off, once he finished showing me his method he wondered if he could think about it differently, and he broke apart the 13 into 10 and 3 and solved it anew and got the same answer. I was just melting with delight.

But that’s not what I want to write about. What I want to write about is the thing he said next, which was: “Now I’m going to figure it out for real.”

Wait, what? He just did figure it out for real! Twice!

My son ran into the bathroom and grabbed the step stool and brought it in and set it on the chair (not the table, thank goodness) and proceeded to count all the dots one by one by one by one. 117 is a lot of dots to count, with a lot of room for error, but it wasn’t until nearly the end that he did indeed make an error. I watched as he counted the dots in the 11th column and then brought his finger back down to the 11th column and counted them all over again, and as a result his final count was 126, not 117.

“Huh,” he said. “I guess there are 126 dots.” He was completely unfazed by this. Never mind that he had already solved the problem and gotten 117, and then solved it in a different way and gotten 117 again. To him, the count was obviously the more trustworthy method.

Of course I intervened. I pointed out that he got 117 twice, and shouldn’t his count match up with his earlier solutions? (He didn’t buy it.) I suggested that he count again, and I helped him keep track of the dots this time, and he found it very exciting when they turned up 117. In fact, he ran to tell his dad (who had already seen his first solution):

“Daddy! I found the proof that there were 117 dots!”

“Oh, really?” his dad asked, with interest. “How did you find the proof?”

“I counted!”

Of course I immediately wrote this all down.

Okay, so now to connect the dots (ha ha). This experience with my son reminded me of an experience with a college student just a few weeks ago, who came to me with her solution to a fraction division problem, one that she had solved beautifully with picture. “But then I did the math,” she told me (there it is, did the math), “and I got a different answer. I just don’t understand what I did wrong in the picture.”

As it turned out, she hadn’t done anything wrong in the picture. She’d just made a simplification error in the algorithm, when she did “the math”, and she never caught it because when the answers turned out to be different, she immediately assumed her picture was wrong, and that was what she checked, over and over again. It never occurred to her to double check her algorithm.

It was just like my son, who didn’t even consider that counting could lead him astray. The bug in relying on the old reliable method is that it’s much harder to see when it fails.

Like I said, the phrase “doing the math” used to drive me crazy. And it used to drive me crazy that when my students solve a problem in a new, much more conceptually meaningful way, they always want to check themselves by “doing the math”. But it doesn’t drive me crazy anymore (or at least not as much as it used to). Because for my students, “doing the math” has meant doing a procedure or an algorithm for years and years and years, and as long as they’re following the steps it’s been pretty reliable. New methods, even if they make sense, even if they make more sense than the algorithm ever did, are still new, and there’s still a level of distrust that they’ll actually work. Or maybe not distrust so much as surprise that it really does work. When they check themselves with “the math” it helps them build their trust in the new way. It’s confirmation.

That’s exactly what my son was doing, too. He knows how to count, but multiplying? That’s strange, uncharged, exciting new territory for him. Multiplication is still a little bit like magic, and counting is real, but it’s the counting that helps him pull back the curtain on multiplication magic and come to see that his reasoning, if not yet completely trustworthy, is just as real.

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