“I love what you did with x right here,” Peter Hirzel says to his 9th-grade algebra student. “I think I can hear the wheels spinning there,” he adds with a smile.

 
He’s looking over a student’s shoulder, seeing what she’s done to balance an equation and get her x standing alone on one side of the equal sign. She’d had a number of choices – divide by this, multiply by that – and Mr. Hirzel is looking carefully at this student’s work to see not only which path she’s taking, but how she chooses, and what kinds of paths are most natural or attractive to her when she has more than one to choose from. He’s watching her work and listening to her think out loud – his work is the watching and listening – but not so much to ensure that she’s doing what he wants, or following his instructions. Instead, he’s trying to figure out her own instructions, how she’s guiding herself through a challenging problem, in part by applying principles he has given her, but in part by opening the toolbox of her mind, interpreting his instructions, discovering insights, and making her way as an original thinker.

 
In another classroom, in another city, a music teacher stops a brass band after they make their way through a challenging phrase in a short musical score and he says to his saxophone player, “OK, OK, I can hear what you’re doing. Where you just blew ba-pa-BAH? Try it with a softer touch on that opening note. Go softer on that -ba- right when you open there. I think I just heard what’s holding you back, now. It’s that little ba. Just hit it a little softer, and you’re right there.” The student picks up his saxophone, plays the phrase, and his teacher smiles a big smile. “That’s it! You got it! You were just jumping on that one note a little too much, a little too eager. That’s all you do now, just soften that up, and everything else you’ve got going on, it’s all just right. You’re already there.” (The teacher is Poncho Williams, one of the truly great Jazz horn players alive today).

 
That phrase sticks in my mind: “You’re already there.” We tend to see a student do something wrong, take a false step, and assume that we need to start over, go back and retrace the steps along the path that got our student lost. But so often, a student isn’t lost. Instead, the student might be hitting one note on a slant, might be thinking about x in a slightly different way, or even seeing something in text that I, as a teacher, failed to see. We erase too much that’s too good but starting all over.

 
Here’s how this plays out in an English class for me every now and then: I’ll be reading a student essay about Homer’s Iliad, as an example, and I’ll see a line saying something like, “Achilles, the ultimate hero in the Iliad, is the greatest fighter in the first half of the book, and he wins many victories.” There may be a page or so to follow, but unless I’m careful, I’ll give in to the urge to stop reading for a moment and write a note in the margin like this:
You must have gotten Achilles mixed up with another character – maybe Agamemnon? – who fights and wins battles in those early chapter of the book. Do you remember our class discussions about how important it was that Achilles removed himself from battle and refused to fight? That’s perhaps the most important event in the beginning of The Iliad. Please work harder to make sure you’re observing and absorbing these key details.

 

But I would be wrong if this student continues with something like this: “While it’s true that Achilles does not fight, he is actually a greater hero, and a greater warrior, for staying out of the literal battle. He overcomes expectations and provides a model of a hero who says ‘no’ to the idea that any man has a duty to fight and kill other men.” Now, I might argue about whether this interpretation is true, but it surely is much more sophisticated that I’d have guessed from the first sentence.

 

As the argument continues on, I say to myself, “Oh, there’s a lot more to this student’s thoughts than I’d imagined at first. What looked like a factual mistake was actually a deeper way of thinking about and writing about the story.” And then I have to ask myself how I might have reacted to this idea if it were presented in class. Even more important – what if the student were a careful thinker, but not a brave talker? What if the student had the whole idea in mind, but only expressed the first part – “Achilles was a great fighter in the beginning of the book”? Would I have said “No, he wasn’t – you got it wrong”? I hope not. I hope I’d respond with a genuine question, something like, “Really? How so?” Or maybe even something as bold as, “Wow, I don’t think I see that – can you show me?”
Instead of pushing back the student’s seemingly wrong answer, the far better instinct would be to lean closer, to try to hear more and act on the assumption that students often, even when I least expect it, have more going on below the surface than most of us imagine.