Three Minus One

On the first day back from summer, the math teachers were eating breakfast together at school and talking about kids. It was my first year teaching eighth grade, so I slide my roster across the table to Mr. B, who taught a group of seventh graders that drove him nuts last year.

“Watch out for S,” Mr. B said. “He lacks the capacity to sit still and listen.”

I wouldn’t go that far, but I know what he means. And S was having a hard time keeping up with some of the material from class, so I’ve been meeting with him once a week, during one of my planning periods.

S has a homework assignment about negative numbers. The kids of B-period had come in with a huge mess of ideas about negative numbers. I’ve made some progress with a bunch of them over the past few months, but not really with S, whose thinking still shows the magnetic tug of ways of thinking that have so far resisted my efforts to expose. Maybe his ideas aren’t stable enough to build on?

During our last meeting, S was working on a homework assignment I had given his class. (Was I surprised that it was a bit hard for them? No.) Here was the work:


S: I had a hard time with these.

Me: Awesome. I think it’s often helpful to be more specific when you’re asking for help.

S: Like, I just didn’t get these, exactly. Wasn’t sure how to think about them.

Me: OK good, but can you be more particular? Which question? Did you know what the question was asking?

S: It was just all sort of hard.

And so on. Eventually, he asks about the first question.

S: So, for this one, is it asking to like pick which one of these is x?

No, no, that’s not what it means. Here’s what it means.

S: Ah, OK, got it that makes a lot of sense. So if x is 1, then this is 3 -1. And that’s -2, right?

I drag out a blank pause. I’m hoping he sorts this out on his own, because I am without ideas if he can’t.

S: Is it -2? Is it 2?

Here I jump in. I worry that he’s holding the problem in his head and trying to think about it too, so I write it down for him on his page: 3 – 1.

S: Ah, OK, it’s -2.

I pause again. In that pause, I feel a sort of sadness. Though it’s hard to identify in the moment, later I come to think that it’s sadness for the sort of chaos that S is showing in his world at this moment. What does it feel like to call into doubt something as basic to thinking as three minus one? It reminds me of the way reality slips away in so much of Phillip K. Dick’s writing:

“He felt all at once like an ineffectual moth, fluttering at the windowpane of reality, dimly seeing it from outside.”

S continued to flutter around three minus one. If three minus one was up for reconsideration, what else wasn’t, in that moment? I wonder if S had something in common with Descartes’ skeptic:

“So serious are the doubts into which I have been thrown as a result of yesterday’s meditation that I can neither put them out of my mind nor see any way of resolving them. It feels as if I have fallen unexpectedly in a deep whirlpool which tumbles me around so that I can neither stand on the bottom nor swim up to the top.”

I remember, in this moment, wondering whether I should talk or listen as S worked his way around 3-1. Would the experience of falling back to the correct answer be helpful for him, a sort of anchor to help him escape whirlpools to come? Or would the experience of hearing my instruction be more grounding? Would it be less embarrassing? Was he embarrassed?

I don’t know any of this, and I don’t think I ever will.

S: Oh wait it’s 2.

I asked him what helped him decide, but he was inarticulate. I offered the suggestion that thinking about other subtraction that he knows might be helpful when in this sort of doubt, but I’m sure he did this while trying to find his way. We moved on — there was more to talk about, and he led the way forward.

It is not hard for me to imagine being unmoored in the way S was. What do I know, after all, about learning and teaching? What did I know in that moment? All my beliefs were subject to revision, I had no confidence about my perceptions of S’s reality. This might be the key for better understanding what it’s like to be unsure of something as fundamental as three minus one. Because if I can’t be sure of how a child I see daily was thinking about three minus one, then what do I really know at all?


3 thoughts on “Three Minus One

  1. What indeed do we really know at all. I had a similar moment of speechless confusion when a student of mine was unable to multiply 5 x 10. He guessed 20. I was lost in that deep whirlpool – it was like in that moment that I simply forgot everything I know about teaching. The whole-class discussion of place value just ended and I went over and sat next to him. I’m so heartened to know that this is not my experience alone – it was deeply unsettling.

    From the comfort of the sidelines I would have asked your guy to draw a number line, but I’m guessing that in the moment I would have fallen into whatever the opposite of a Zen-like realization is.


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