This post is an argument that there isn’t much of a difference between questions and non-questions when it comes to hints.
Well, wait, hold on, that’s dumb. There are lots of important differences between questions and non-questions. I should focus up my claim.
Claim: Whether you ask a question, make a request or just say something to a student during a problem-solving session, you’re doing mathematical work for that student.
- Locus of Attention – Feedback can draw the student’s attention towards something that matters for their learning.
- Motivation – You can ask a student to do something that they otherwise wouldn’t do.
I think that this is true of teaching more broadly. I think that everything that a teacher does for learning can probably be dropped into either the canister marked “redirect attention” or the one marked “motivation.”
What we pay attention to matters so much. There is a mathematical way of seeing and noticing. Our ability to focus our attention in mathematically productive ways is a sign of problem-solving expertise (Schoenfeld 1992).
When a teacher asks a question of a student, they are (inevitably) redirecting that student’s attention to something that they aren’t currently thinking about. To take an example from Max, if you ask “What do you notice about this?” you are redirecting a student’s attention back towards the scenario, asking them to see things with fresh eyes. A more experienced problem solver probably wouldn’t need this question. Per Schoenfeld, that student would know that it’s time to go back and redirect her attention to noticing things about the scenario.
Annie wants the student to do all the mathematical work, but she also wants to redirect their attention in productive ways:
As a sometimes coach in a wide range of schools/districts/populations, I don’t think that I work with students that are so unusual, and I find that when a kid asks me a question or tells me they’re stuck, probably 19 times out of 20 (or maybe more like 49 out of 50, though my colleagues would probably guess at 99 out of 100), what I say is, “Tell me something about the problem.” That’s it. There are small modifications – if they asked about question 3, I’ll say, “Tell me something about question 3.” Or, if I’m feeling radical and they’ve actually done some math, “Tell me about what you’ve tried so far.”
I added that emphasis there, because this seems like a great example of a bit of important mathematical work that experts often do but that our students often don’t. Going back to Schoenfeld, this isn’t just a confidence or a motivational issue. It’s a marker of a certain kind of mathematical expertise, perhaps teachable (though perhaps not so easily at all).
The world of math education is full of strong distinctions and lines in the sand. There are GOODWORDS and there are BADWORDS, and also GOODTHINGS and BADTHINGS. A GOODTHING is a question that asks students to think about their own thinking. A BADTHING is a hint, even one that amounts to asking kids to think about their own thinking. (e.g. “It’s often helpful in moments like these to just ask yourself what you notice about the scenario. Make a list, even.”)
I take from Dewey (and the other pragmatists) a “rejection of sharp dichotomies.” Everything that we say to a kid alters the locus of their attention, hopefully in a good way, and we need to own that if we are to understand it and wield that power productively. Minimialistic hints, if they help learning, work for the same reason that any sort of hint could conceivably work: you’re doing mathematical work for the student that sets them up to think about something valuable.
And this is pretty much all we can do.