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I work with a fifth grader鈥攍et鈥檚 call her Sasha鈥攚ho struggles with math. She鈥檚 anxious about it, she鈥檚 told me. Sasha doesn鈥檛 like how it feels when it seems like the other kids are getting it and she鈥檚 not. She asks me to give her a practice problem like the ones she鈥檚 working on in class.

I think for a moment, then type onto our Zoom chalkboard: 鈥淚 recently bought an 8-kilogram bag of Kitty Kibble. Assuming that I 诲辞苍鈥檛 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?鈥�

Onscreen, I watch her read. First, she smiles鈥攕he likes it when my cat makes it into word problems鈥攁nd then her expression darkens. She blinks, then swallows. Blinks again.聽Then she looks up. 鈥淔ifteen,鈥� she says matter-of-factly.

鈥淚nteresting,鈥� I say. 鈥淗ow did you get that answer?鈥� When she doesn鈥檛 reply, I suggest we read the problem again, clarifying for ourselves what information is provided and what we鈥檙e trying to find out. Sasha doesn鈥檛 like to do this, particularly鈥攕he鈥檚 told me that she finds rereading boring鈥攂ut it often helps her clarify her thinking.

I begin. 鈥淚 recently bought鈥斺��

Sasha interrupts. 鈥淎ctually, there鈥檚 something I need to tell you.鈥�

鈥淲hat鈥檚 that?鈥�

鈥淭his weekend, I actually petted a very cute dog.鈥� She sighs wistfully. 鈥淚t had curly fur.鈥�

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鈥檚 more than that: Sasha鈥檚 mind, like many people鈥檚, often goes elsewhere, especially when she鈥檚 faced with a task she finds daunting. What Sasha struggles with is getting it back on track, especially when she doesn鈥檛 quite know where it鈥檚 going. She doesn鈥檛 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鈥檛 make sense. What about this? Or this? Or this? And the reason Sasha doesn鈥檛 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鈥攖he answer, the sentence鈥攁nd 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 诲辞苍鈥檛 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鈥檝e seen it in practice, it always struck me as cursory and incomplete. When I鈥檓 truly engaged in reading, I 诲辞苍鈥檛 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鈥檛 know this鈥攚hat it鈥檚 like for me, what it might look like for them鈥攚ithout 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鈥攖he way she might look at it one way and then another, brushing aside a thought or two about what to have for dinner鈥攐r 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鈥檛 seem to lead as well to this paragraph you鈥檙e reading now. And the sentence before that went like this, in the first draft: 鈥淲e should talk about what happens when you solve a problem, or write a sentence: what鈥檚 happening inside your head when the pencil scribbles, pauses, hovers, erases.鈥� 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鈥檓 still not sure that it totally belongs, that it鈥檚 not a little too something鈥擨 诲辞苍鈥檛 know, flowery?鈥攆or the rest of this piece.

You catch my drift: The finished product always reveals less than it obscures鈥攁nd 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鈥檛 be as served by reading the paragraph you read first鈥攖he final draft鈥攁s much as following the slippery, sometimes chaotic actions I took to get there.

I鈥檝e 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. 鈥淚 think it would help me,鈥� I say after a moment, 鈥渢o read the problem again.鈥� So I do.

鈥淚 recently bought an 8-kilogram bag of Kitty Kibble. Assuming that I 诲辞苍鈥檛 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?鈥�

鈥淭his is actually a pretty tricky problem, now that I read it,鈥� I say. 鈥淚鈥檓 not sure how to start.鈥� I take a long pause. Then: 鈥淢ind if I think out loud for a sec?鈥�

She nods.

鈥淪o, OK. Eight is how much food there is at the start, right? And Tabitha eats 50 every day. That鈥檚 taking away, so I think that鈥檚 subtraction. OK.鈥� I nod to myself. 鈥淎ctually, I鈥檓 feeling a bit hungry. I鈥檒l get a snack after we do this problem. OK, so, 8 minus 50鈥︹��

I pause. 鈥淲ait a sec.鈥�

Sasha is following along鈥攕keptically at first, it seems, but then with genuine interest.

I shake my head. 鈥淪omething鈥檚 wrong here. With something like cat food, you can鈥檛 take away 50 from 8.鈥�

Sasha鈥檚 brow is furrowed, reading the problem. Then, after a minute, her eyes widen.

鈥淓mily, you made a little mistake,鈥� she says, her voice reassuring. 鈥淵ou forgot to think about the units!鈥�

鈥淲hat do you mean?鈥�

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鈥檚 proud of herself. I can see it through the screen.

And, I think, she can see my smile, too.聽