A Short Comment on Hints and Comments

I often find myself at a desk, agonizing over whether or not it’s worth it to give comments on student work. I have a similar worry that arises when kids are working in class. Should I say something? What should I say? Will it help?

Clearly, sometimes comments don’t work. Sometimes hints help. Sometimes they don’t. True, there’s always going to be some guesswork involved in teaching. Nothing works, at least not all the time. But come on! There has got to be some trends, some patterns that I can lean on.

A few weeks ago, I was talking to a kid about whether 3x^2 or (3x)^2 is equal to 9x^2. She wasn’t sure, and I found myself in that agonizing moment: what do I say? should I give a hint? let her flail?

I decided to speak up, and I found myself offering the following hints:

  • “For this sort of problem, you’re probably looking to either use number testing or an area model.”
  • “Why don’t you try number testing?”
  • “Can you find a shortcut using the area model?”

These hints worked for this kid, and I’m left wondering why?

I wasn’t trying to do a whole lot of conceptual development during this one-on-one discussion. These were comments that were just asking this kid to connect strategies she had thought about in a whole-group setting to her personal work. I was using my hints/comments to draw a connection, not to develop an idea.

I was hoping that this idea would help me make decisions about when to give feedback in other situations. Today, I was trying to figure out whether to give comments on the work of my 4th Graders, who are working on multiplication. As I stared at the pile, I felt my gut say “AVOID!” Were my instincts right, or was I just being lazy?

I realized that I had no special language with which to give comments to my 4th Graders. I wanted to do for them what I did for my algebra student: refer them back to models or strategies that they had some understanding of already. Otherwise comment-writing gets insanely laborious and tricky. (How do you teach a kid to use a new strategy in two lines?) I wanted to leave them comments like “try doubling!” or “what if you broke apart this number by 10s?” or “put this multiplication in a word problem.”

I couldn’t do any of this, though, because we hadn’t focused on these strategies and we hadn’t named them, yet. So I didn’t write comments.

Instead, I ran a opening mental-math lesson. I asked this series of questions:

  • 8 x 3
  • 20 x 3
  • 28 x 3

We named the strategy that many of them used: it’s splitting up the numbers, or breaking then numbers apart. Then I asked them to practice it with a new problem:

  • 13 x 12

And we collected a few different ways of doing this, and talked about why it works.

And then, when they were working alone on practice with multiplication, I went up to A. She was using laborious doubling methods for her multiplication. I said to her, “Could you try breaking these numbers apart?” She said, “I prefer to use doubling and adding.” I said, “Well, that’s good too, but if you were going to break them apart, how would you?”

And she said 37 x 4 is the same as 30 x 4 and 7 x 4, and she solved it, and now we’re in business.

So, in short, one thing that works for me is giving hints that refer back to strategies we’ve already talked about.

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7 thoughts on “A Short Comment on Hints and Comments

  1. I also like referring to things we have done before to solve problems, but what kept this student from trying those? It seems she knew what to do- so why did she avoid it? What kept her working on a strategy that was failing or hard instead of using something else?
    These are the things I am trying to figure out in my Ss, why they make the choices they do- it’s never random.

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    1. I also like referring to things we have done before to solve problems, but what kept this student from trying those? It seems she knew what to do- so why did she avoid it?

      I think it’s just that — there’s a difference between “getting” something and being able to use it in a problem.

      Haven’t you ever felt that? It’s what happens when I read a math textbook, understand what I’ve read, and then find myself unable to answer any of the problems. There’s a kind of understanding that we can have without being quite able to put it into practice, without help.

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  2. Sometimes, I think that when we tell kids there are lots of ways to solve a problem, and no single right way, it can backfire. For some students, the first way that they think of seems like a reasonable strategy to them, and even when it starts to get difficult, they sometimes dig their heels in and stubbornly forge ahead. It can be hard for them to admit that another way might be easier, because eventually their first method worked. And of course, when we stress that the amount of time that they take shouldn’t be a factor, that can backfire too. It can be difficult to emphasize the importance of simple and efficient solutions to problems without reverting to stressing the speed in which they solve problems. How can we find that balance in what we want students to focus on (efficiency, not speed) when thinking about math?

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    1. How can we find that balance in what we want students to focus on (efficiency, not speed) when thinking about math?

      Yeah! I know what you’re talking about.

      If I think through how my post relates to your comment, I come up with a potential answer: leave the kids alone, if they’re stuck on their way of thinking. Teach them, outside of the context of this problem, some new techniques and strategies. Then, if they get stuck again, point them towards this new, more efficient strategy. Repeat until it sticks.

      Like I said, that’s what I get when I try to use this post to answer your question. But maybe that answer doesn’t hold up?

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      1. Yup – leaving them alone is so often the right answer, and yet sometimes so hard. I work with a lot of kids with Asperger’s too, and for many, their rigid thinking can become a barrier to new approaches. However, their struggles with generalizing can make it easier to present a new way to solve a similar problem (that they don’t necessarily see as similar), and eventually wind back to the original problem.

        I guess it sometimes depends on how important a particular problem/technique is, and how much time you’re willing to spend as a teacher, either in the class our outside of it. I always try to see the end game – in the end, what will help the student the most. Of course, I think I get that wrong a lot of the time, too.

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  3. Great post.

    I feel I need to clarify my “Nothing Works” slogan.

    It’s a response to those who are done thinking about how to teach, because they have the answer. You see this on all sides. Traditionalists who think that teacher explanation is the end-all and be-all. Progressives who object to teacher explanations ever. More generally, people who promote one-size-fits-all strategies.

    None of this applies to you.

    Good teaching is not about rejecting strategies because they are not universal. It’s about knowing as many strategies as possible, so as to be able to improvise effectively with a mix of them. The positive way to say “nothing works” is “know as many things as possible”.

    What’s great about your blog is that you analyze concrete teaching challenges, and figure out what did and did not work. With that approach, you will continually refine your practice, and you are in no danger of turning off your teacher brain.

    You are eager to generalize from your experiences. I do advise caution there, and I agree with Ethan that the goal is often to find balance. But if you didn’t have that drive, you wouldn’t write, and we’d all be the poorer for it. The main thing is to stay alert, so that if a situation arises when a previous generalization does not apply, you are not trapped. Deal with it, then write about it!

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    1. Thanks for the clarification, Henri. I especially appreciate the idea that what matters is making sure t keep the thinking going.

      You are eager to generalize from your experiences. I do advise caution there, and I agree with Ethan that the goal is often to find balance.

      More and more, I realize that the object of my study isn’t teaching, but my teaching. That’s something that I’m trying to do a better job representing in my writing, too. Thanks for the reminder.

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