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1. Isn’t this an interesting multiplication mistake?

2. I used to ask “how could we help this student?” or “why do you think this student made this mistake?” I still think that these questions are valuable to ask when looking deeply at student thinking. But, when teaching, the better question seems to be not “what mistake did this student make?” but instead “what could this student know that might help her?”

In this case, I’d say that this student could use more versatile ways of breaking numbers apart more than any sort of reflection on the errors of her ways.

Every once in a while people get in touch with me because they don’t like that this site is focused on mistakes. I think this is probably what they’re getting at.

  • Howard Phillips

    It took me quite a while to figure out what was wrong with these. Same mistake it seems.
    She breaks up one of the 2 digit numbers in a bizarre way. I look at 12 x 14
    12 = 10 and 2 …..ok, then
    10 = 2 x 5 …..ok, then getting 140
    but there is still a 2 to deal with, and she multiples 140 by 2 instead of multiplying the 14 by 2 and adding to the 140
    So in effect she has done 12 x 14 = 5 x 2 x 2 x 14, but she doesn’t see it that way.

    In the second one she has done the same but to the second factor.

    maybe “Get out the blocks” is where to go back to.

  • Max Ray

    When I skimmed this post in my email, I read it as, “what DOES this student know that might help her?” which I became totally fascinated by as a question. I’m really interested in what this mistake reveals about how the student currently thinks multiplication works, and what that helps us know about how to help her.

    I notice that she begins with a strategy that works well for, say, 10s (also powers of two): Break a number into its factors and multiply the other number by all factors, as in 14 * 10 = (14 * 5) * 2.

    She over-applies that strategy to partitions, rather than factors: 14 * 12 =/= (14 * 10) * 2, but 14 * 12 = 14 * (10 + 2) = 14 * 10 + 14 * 2.

    I wonder if she has seen peers use both “breaking apart” strategies and is treating them interchangeably, unaware of the difference between them.

    I wonder if she would make the same mistake on a smaller-numbers problem that she could connect to a direct modeling strategy.

    I wonder how her peers who aren’t making this mistake keep separate when they need to multiply with their broken-apart numbers, and when they need to add with them.

    I wonder what will help her fix her thinking so that she doesn’t make this mistake in the future, when number become even bigger or more tempting.

    I wonder what she would do with a problem like 14 * 102 or 14 * 1002.

    Possible interventions:
    – have her check her work with pictures or counting
    – help her notice that her partial result for 14 * 10 is right, but her final answer for 14 * 12 is not, and see if she can notice the difference between 14 * (5 * 2) = 14 * 10, and 14 * (10 + 2) =/= 14 * 12.
    OR
    – try to figure out from her why she thinks she’s getting it wrong and what is confusing about the strategies she’s aware of
    – see if she’s willing to learn a new strategy (either a multiplying by factors strategy or an accurately using-your-excellent-knowledge-of-finding-tens strategy)

    • Michael Pershan

      A lovely analysis, Max.

      I need to think more about the gap between your question and mine. I think that in order to figure out what this student *could* know we’d want to know what in her thinking she *does* know that might help her.

      So why not just ask “what does this student know that could help her?” Maybe these aren’t good reasons, but here’s what I have on my mind now:

      1. What happens when we (or some other teacher) struggle to find something in a student’s work that seems promising? We might fail to set an appropriate goal for this student. (A comment I sometimes hear on this site is “we need to go back to the basics” which, to my mind, isn’t super duper helpful.)
      2. What if the student has resources in her work that could be used to successfully solve a problem, even though a more appropriate model or strategy is sitting on the shelf, unused.

      In this case, my concern is that we’ll overly focus on helping her develop this strategy instead of helping her take her performance to the next level by helping her develop a more powerful method. A fine balance is needed here, but I don’t want to be limited by the morsels a student has given us in her work.

      • Max Ray

        Good questions — they seem to cover the gamut of fears people have about giving written feedback to students who have made mistakes.

        The easy answer is “What DOES this student know…?” is not the right question to ask about every math mistake or piece of work. Sometimes what the student clearly doesn’t know is way more important. Example: a student who writes:
        2^3 = 6
        2^4 = 8
        3^1 = 3
        What’s salient here is that clearly the student doesn’t know exponential notation (and, specifically, they have decided to answer questions with this notation using multiplication). And I’m not even convinced that knowing that they do know these multiplication facts is particularly high leverage.

        Of course that leaves us with the question, “how do you know when to ask, ‘what does this students already know?'”

        I think your questions 1) and 2) help me answer start to answer that:
        1) To me it’s really not as much about finding nuggets to work with and build from, but about knowing as much as we can about the student’s understanding and why their work so far seems reasonable to them. The reason being, if we can’t unpack what’s reasonable, our interventions may be seen as unreasonable and taken up in ways we didn’t intend. So I seek to find out what the student does know in the sense that I seek to find out the student’s current working model of the task, and I try to identify the last place the student’s model intersected with reasonableness. Sometimes, that intersection with reasonableness was so long ago that I can’t find it, or I am reasonably suspicious that the student doesn’t have a current working model beyond “put the numbers in the number blender and hope,” or it’s too likely that what the student does understand about the situation will actually get in the way of new learning — for example a student is trying to learn about area, and keeps reverting back to an understanding of perimeter. If I try to work on updating her perimeter concept and demonstrating that perimeter and area are different, that will further cement her confusion around perimeter and area. My goal in that case would be to get her to re-think area as a totally new kind of problem, about totally different questions… not a good place to “start from current understanding” if current understanding is about something that’s actually conceptually quite different.

        2) Like, for example, a student who solves 11 + 2 and 12 + 3 by counting on from the larger number, but who falls apart at 2 + 11 and 3 + 12 because they run out of fingers, and so they start laboriously constructing for themselves a breaking-apart strategy that their working memory is not quite up to — so they will be okay with saying, “2 + 11 is 12 and one more is 13” but stumble a little with “3 + 12 is 13 and… three more, 13 and 3 is sixteen… no, wait…”. Mathematically, it would be so beautiful if they could see 2 + 11 and 11 + 2 as being the same and solve 2 + 11 by counting on from 11. Do we spend the time with them getting good at their breaking apart strategy instead? Here are the factors I would consider:

        a) Which strategies are more long-term useful? In this case, the answer would have to be “both”

        b) Which strategies are the students more likely to be able to carry out? It seems that this student doesn’t quite have the working memory chops to get fluent at her breaking-apart strategy, so she may not get comfortable with 2 + 11 or 3 + 12 using her current strategy. If I thought she could get good at her own strategy quickly I’d help her use her own strategy as a lens through which to confirm and appreciate the commutative property strategy.

        c) Which strategies will support the student to feel confident and feel like they “own” the mathematics? I think in the imaginary case of the student working on 2 + 11, it will be important to eventually support the student to feel ownership of both approaches. Perhaps by having her show the class her thinking about solving 2 + 11, and letting other students who are just better at remembering partial sums take up her method and admire her thinking on it. And then also by praising her work and thinking on 11 + 2 and using blocks to show her how she can user her same smart thinking on 2 + 11, and having her practice several of these with models with the task, “show how to count these to take advantage of your smart ‘counting-on’ thinking”

        It’s a lot to consider, especially if you’re teaching 30+ of these learners at a time, but if you could give each kid’s approach* to each problem the thoughtful response it deserves, it would probably involve this kind of thinking.

        *And the truth is, in a class of 30+, you’ll have a handful of kids whose approaches always show that they understand the math and are fluent in carrying it out, a bunch who make the exact same mistake because of having the same underlying mental model, another group who make a second, less-popular mistake, a few kids who get it but get it in a really unique way, and a few who don’t get it in an equally unique way. It’s okay to give the same feedback to a whole cluster of kids.

  • Mark P

    Isn’t the issue largely one of layout?

    If she wrote: (i’m assuming she because of the pink pen)

    12 x 14

    10 x 14 + 2 x 14

    2 x 5 x 14 + 28

    2 x 70 + 28

    140 + 28 = 168

    which is what was intended, she would be right.

    I see it as a person trying to follow a taught pattern, but instead of completing each step, she half completes the first step but then carries on and does 10 x 14 before returning to the 2 — by which time she has forgotten what she meant to do with it.

    Getting her to write the split into tens and singles times the target first, before moving on, might eliminate the problem.

    (It’s amusing that people should worry that a Maths site is about mistakes. If students didn’t make mistakes the job would be easy. There should be more sites about mistakes and erroneous thinking.)