I work with a fifth grader—let’s call her Sasha—who struggles with math. She’s anxious about it, she’s told me. Sasha doesn’t like how it feels when it seems like the other kids are getting it and she’s not. She asks me to give her a practice problem like the ones she’s working on in class.
I think for a moment, then type onto our Zoom chalkboard: “I recently bought an 8-kilogram bag of Kitty Kibble. Assuming that I don’t go to the store again and Tabitha eats 50 grams of food a day, after how many days will she be completely out of food?”
Onscreen, I watch her read. First, she smiles—she likes it when my cat makes it into word problems—and then her expression darkens. She blinks, then swallows. Blinks again. Then she looks up. “Fifteen,” she says matter-of-factly.
“Interesting,” I say. “How did you get that answer?” When she doesn’t reply, I suggest we read the problem again, clarifying for ourselves what information is provided and what we’re trying to find out. Sasha doesn’t like to do this, particularly—she’s told me that she finds rereading boring—but it often helps her clarify her thinking.
I begin. “I recently bought—”
Sasha interrupts. “Actually, there’s something I need to tell you.”
“This weekend, I actually petted a very cute dog.” She sighs wistfully. “It had curly fur.”
At first, I thought this was a stalling tactic. Sasha has a bunch of those: check out this Zoom filter, look at my stuffed animal, just give me a sec to go to the bathroom. But after working with her for some time, I realize it’s more than that: Sasha’s mind, like many people’s, often goes elsewhere, especially when she’s faced with a task she finds daunting. What Sasha struggles with is getting it back on track, especially when she doesn’t quite know where it’s going. She doesn’t know how to sit with a hard problem and poke and prod until she finds a path worth journeying down. Maybe it could be... Wait, no, that doesn’t make sense. What about this? Or this? Or this? And the reason Sasha doesn’t know how to do this, it seems to me, is because, as is the case for nearly all of us, she was never really taught.
At its core, learning is a change in the content, the patterns, and the movement of thought. In the physics of the intellectual universe, thoughts are the atoms out of which everything is made, bouncing around to form the molecules, elements, and matter of cognition. But thoughts, like atoms, are invisible: Even in the realm of education, we most often talk about finished products—the answer, the sentence—and not the messy, iterative, highly personal processes that built them.
And even when we do talk about process, we tend to do so in superficial terms: one or two steps we took, perhaps, but not everything we considered, tried, ruled out. We don’t talk about what thought looks like, what it sounds like, how it feels: the tension and excitement of holding on to multiple options at once, the anxiety of forging ahead and drawing a blank. The dead ends.
In my elementary teaching program, they taught us to model certain elements of thinking: explaining the steps of solving a problem or stopping at the end of a chapter to make a prediction out loud. But then, and when I’ve seen it in practice, it always struck me as cursory and incomplete. When I’m truly engaged in reading, I don’t stop every so often to make a prediction: My mind is whirring and jumping around as I read, making connections and predictions, and my heart is often along for the ride. The process of reading, in other words, is complex, creative, discursive, and very distinctly mine. My students wouldn’t know this—what it’s like for me, what it might look like for them—without explicit modeling.
When we teach students the components of thought, we should get granular. We should talk about what happens when a mathematician solves a problem—the way she might look at it one way and then another, brushing aside a thought or two about what to have for dinner—or when a writer composes a sentence: the pencil scribbling, pausing, hovering, erasing. We should teach students that creation is always a process, and the process is as complex as it is variable.
Take that last paragraph. The version you just read looks nothing like the original. The last sentence only appeared a few minutes ago, replacing a much more declarative statement that, on reflection, didn’t seem to lead as well to this paragraph you’re reading now. And the sentence before that went like this, in the first draft: “We should talk about what happens when you solve a problem, or write a sentence: what’s happening inside your head when the pencil scribbles, pauses, hovers, erases.” In fact, you should know that last phrase, the one with the pencil, was the genesis of the entire section: I scribbled it down in my notes section as I was writing the section that came before that. To tell you the truth, though, I’m still not sure that it totally belongs, that it’s not a little too something—I don’t know, flowery?—for the rest of this piece.
You catch my drift: The finished product always reveals less than it obscures—and it obscures, well, pretty much everything. If, for some reason, you wanted to learn exactly how I develop and describe metaphors about thought, you wouldn’t be as served by reading the paragraph you read first—the final draft—as much as following the slippery, sometimes chaotic actions I took to get there.
I’ve started using this process with my students: distilling and narrating my thought processes in detail, not only laying bare how my own mind works but making clear that their thoughts, as messy and unique as they may be, are welcome to the party too.
With Sasha, I take a long pause. “I think it would help me,” I say after a moment, “to read the problem again.” So I do.
“I recently bought an 8-kilogram bag of Kitty Kibble. Assuming that I don’t go to the store again and Tabitha eats 50 grams of food a day, after how many days will she be completely out of food?”
“This is actually a pretty tricky problem, now that I read it,” I say. “I’m not sure how to start.” I take a long pause. Then: “Mind if I think out loud for a sec?”
“So, OK. Eight is how much food there is at the start, right? And Tabitha eats 50 every day. That’s taking away, so I think that’s subtraction. OK.” I nod to myself. “Actually, I’m feeling a bit hungry. I’ll get a snack after we do this problem. OK, so, 8 minus 50…”
I pause. “Wait a sec.”
Sasha is following along—skeptically at first, it seems, but then with genuine interest.
I shake my head. “Something’s wrong here. With something like cat food, you can’t take away 50 from 8.”
Sasha’s brow is furrowed, reading the problem. Then, after a minute, her eyes widen.
“Emily, you made a little mistake,” she says, her voice reassuring. “You forgot to think about the units!”
“What do you mean?”
And Sasha explains. Then, thinking out loud, she walks us through the problem. She makes a few mistakes, as all mathematicians do, and then arrives at the answer. She checks her work by doing the problem backward.
She’s proud of herself. I can see it through the screen.
And, I think, she can see my smile, too.