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.

**Argument:**

Back on my old blog I wrote a post about the mechanism through which feedback can impact learning. I took my cue from Kluger and DeNisi and identified two mechanisms:

- 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.

I’m reading Lesh & Zawojewski’s 2007 piece on problem solving (do you have a stable link to that?), and it’s given me some new ideas. I’ve got a new framework for hints that I want to try out.

I’m going to argue that hints can do two things. One is to promote mathematical thinking. That is, a student is no longer thinking mathematically because of roadblocks in the problem solving process — lack of knowledge, insufficient strategy, etc. Here, I think I buy your argument about hints, especially:

“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.”

I think we can spend a great deal of very productive time sorting that out. But I want to differentiate it from my second point:

There is significant evidence in the literature that student metacognition positively influences student effectiveness in problem solving. From a description of Schoenfeld in the Lesh, Zawojewski article:

“He described how when small groups of students were engaged in challenging mathematical problems, he (as their teacher) circulated through the classroom asking specific questions: What (exactly) are you doing? (Can you describe it precisely?) Why are you doing it? (How does it fit into the solution?) How does it help you? (What will you do with the outcome when you obtain it?).”

This was an example of a teacher instructing with the goal of enhancing metacognition, and the article presents evidence (Kramarski, Mevarech, and Arami (2002)) that metacognition helps student performance, on both standard tasks and “authentic” tasks.

One critical piece of the problem solving process that I believe this metacognitive activity promotes is a problem solver’s ability to recognize when a strategy is proving unproductive, and to reverse course with a new approach. This principle, and similar principles that involve progress monitoring and a willingness to try alternative approaches, is critical to solving novel problems successfully, and it seems these metacognitive strategies may be helpful in doing so.

So I want to argue that a hint can serve to purposes. A hint can promote mathematical thinking, so that students are thinking more deeply after the hint than before it. I think you make really good points here. But I would argue that hints serve a second purpose, which is to promote metacognition with respect to students’ problem solving processes, and that this metacognition looks different than deep mathematical thinking, while also promoting successful problem solving.

What do you think? Is this distinction worth making? What am I missing?

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I need to reread Lesh and Zawojewski (my personal stable link is Lesh & Zawojewski (2007)). My impression from their chapter is that their read of the literature problematizes the distinction between learning metacognition and learning other aspects of math.

I take it that metacognition is a real component of mathematical knowledge, but that it usually develops alongside learned procedures, concepts, strategies, etc. In other words, I see metacognition as an important element of mathematical knowledge.

And just as we can’t just learn set theory and thereby learn every area of math that can be derived from it, I don’t think we can learn domain-agnostic metacognition and then apply it to every area of math. We need (and I think this is supported by Lesh?) that we need to think of metacognition as something that looks a bit different for every lesson, problem, unit, course.

One of my ideas for a project to work on is to articulate this vision unit by unit for a HS geometry course. But I need to figure out a way to split it up so that I’m not accidentally committing to some huge project that turns out to be a waste of time and useless and wrong.

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