[Mailbag] What Do You Do with the Ideas You Used to Call “Mistakes”

Guillaume Paré, in the very interesting comments of my last post where I urged us to reconsider mistakes:

I do agree with what is written, but I am still wondering what I’m supposed to do with that information and the student’s copy.

T: Oh this is so interesting! You’ve actually answered a different question correctly. Check this out:

T: How does that help you come up with an answer to the original question? Talk about it. I’ll be back.

That’s a script right there. It works for any incorrect answer. The script is all-purpose and all-weather but it has two challenging requirements:

  1. You have to actually believe that student ideas are interesting, especially ones that don’t correctly answer the question you were trying to ask.
  2. You have to identify the question the student answered correctly.

This is why I want to learn more math and more math and more math.

The more math I know, the more power I have not just to show off at parties but also to appreciate student ideas and to identify the different interesting questions they’re answering correctly.

Barbara Pearl, via email:

Can you write about it briefly again in a simpler way so I can try and understand it? When students make a mistake or answer something incorrectly, you want to …

I want to teach in a way that honors the specific student and also the general ways people learn.

So in any interaction with students, I need to a) understand the sense they’re making of mathematics, b) celebrate that sense, saying loudly “I see you making sense!”, and then c) help them develop that sense, connecting the question they answered correctly to a question they haven’t yet answered correctly.

Rachel, in the comments:

So, if we don’t call it a mistake, then what do we call it?

I don’t have any problem saying a student’s answer is incorrect, that they didn’t correctly answer the question I was trying to ask. But my favorite mathematical questions defy categories like “correct” and “incorrect” entirely:

  • So how would you describe the pattern?
  • What do you think will happen next?
  • Would a table, equation, or graph be more useful to you here?
  • How are you thinking about the question right now?
  • What extra information do you think would be helpful?

How can you call any answer to those questions a mistake or incorrect? What would that even mean? Those descriptions feel inadequate next to the complexity of the mathematical ideas contained in those answers, which I interpret as a signal that I’m asking questions that matter.

Featured Comment

Denise Gaskins quoting WW Sawyer:

Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.

Daniel Peter:

“So, if we don’t call it a mistake, then what do we call it?”

THINKING

cheesemonkeysf:

I find that I have to keep insisting that they restate the question in their own words. The culture of “right answer” is filled with shame and shaming, and students will try repeatedly to just give me the “correct” answer to the original question. But this is a missed opportunity for developing understanding, in my view.

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

19 Comments

  1. Reply

    Funny coincidence: after reading your post on respecting how students think by asking your five questions, I happened to read this post on the 1972 Lordstown strike where workers wanted more autonomy over their workplace (link below). If all a teacher (or “teaching software”) does is say “Right” or “Wrong” that’s a bit like a foreman capriciously ordering workers “do this, don’t do that,” not recognizing they could have their own sense about quality.

    http://www.lawyersgunsmoneyblog.com/2018/11/lordstown-work-america

  2. Reply

    Featured Comment

    These two posts have reminded me of my all-time favorite W.W. Sawyer quote from Vision in Elementary Mathematics”
    “Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.”
    • I think that’s really interesting. I was teaching a middle school class last week and I asked “How are you thinking about the question right now?” instead of a related, known-answer question.

      This was less for the individual student’s sake and more for my sake and also for the class’s sake. When a student gives an answer that I can judge as incorrect or correct, I find it really hard to meet those two requirements above. I need to spend some time thinking about a student’s answer to figure out what question they’re answering. I get sweaty with the entire class watching me think through it. Plus, the rest of the class hears the answer and some of them are even worse equipped than I am to draw a connection between that answer. and theirs.

      Perhaps it’s better for whole-group discussion to ask questions that are more general, questions that the class will find easier to integrate into their own ideas, questions that don’t have an answer that is easily evaluate as correct or incorrect.

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    I like the idea of showing the student the question that they answered correctly. I’ve done that myself before on occasion, but I never formalized it the way you have.
    My default has been, “can you expain your thinking?” for BOTH correct and incorrect answers, but sometimes I feel that doesn’t quite do it. (Ex: the student says, “I just added 17 each time,” and doesn’t catch the error.) It’s good to have another tool that keeps the responsibility for thinking in the student’s court.

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    I find that Dan’s script above can be made MUCH more valuable for students if I emphasize critical reading skills. Students need to be able to identify what that different question *IS* to which they found an answer.

    By asking students to restate the original question/problem in their own words (i.e., What is this question *actually* asking for?), I feel like we can better capture the learning potential within this particular mathematical experience.

    Featured Comment

    I find that I have to keep insisting that they restate the question in their own words. The culture of “right answer” is filled with shame and shaming, and students will try repeatedly to just give me the “correct” answer to the original question. But this is a missed opportunity for developing understanding, in my view.

    I find that at least 99% of adult life consists of badly worded instructions to unclear tasks. I believe helping students to develop their own clarity and critical reading skills is a valuable way of helping them to navigate the somewhat messed-up world they are going to inherit.

    Thanks for flagging an interesting discussion!

    – Elizabeth (@cheesemonkeysf)

    • I appreciate this contribution to the thread Elizabeth! I find your response resonates with my classroom experience using CPM with English Learners. However, I was unable to develop a culture in my class where students would re-read prompts or critically analyze text without me demanding it… and I couldn’t be the language support each of my students needed every day.
      What might you suggest for a teacher struggling to ask “students to restate the original question/problem in their own words (i.e., What is this question *actually* asking for?)” Is this something you ask of all your students on every prompt? How does it become part of the class culture?

    • Thanks Cheesemonkeysf. I’m with you about the “right answer” mindset. Such thinking is a precursor to focusing on grades, not learning. Something that always leaves a bad taste in my mouth. The guy who teaches science two doors and I independently stopped putting scores on assignments this year and have started to write far more comments trying to direct students to the part of their own written thinking where unclarity is present. The task for the student is to revisit their own thinking.

    • @Evan – I actually ask them to analyze the text of the prompt/question, as in asking them What is the verb in the instruction/prompt? Circle it in red. What does that verb mean?

      I ask them about prefixes and suffixes and roots (they are often SHOCKED that math teachers might know this strange wizardry) and I will ask them (as Dan often does) to write a guess in their notebook.

      I will also do a whole-class group noticing (thank you, Math Forum) if I need to, asking each person to say something they noticed and writing it on the board. I have actually been known to fill up more than one whiteboard with these noticings — but this practice is important to communicated to them that I am not going to fold. THEY ARE GOING TO HAVE TO GO BACK TO THE TEXT AND THEY ARE GOING TO HAVE TO ANSWER MY Q BEFORE WE MOVE ON!

      I have even assigned this as a graded HW assignment when I’ve had to.

      A lot of being the language support as you put it (I like that framing) involves out-stubborning them. This communicates to them that this is IMPORTANT, that they are responsible for their learning, and that I am not going to let them off the hook for this. Does that help at all?

      – Elizabeth (@cheesemonkeysf)

    • @Stephen – I envy you your ability to write significant comments to your students in a timely fashion (or perhaps your class size… or perhaps your energy level. :) ). I have >185 students, so it’s just not practical for me to write & turn around comments like this in a timely way. Perhaps that’s why I’ve had to expand my repertoire of in-class techniques like this. It’s a little painful because until my current school, I had always had small enough classes to be able to establish powerful relationships with students through writing comments.

      So I say good for you and keep going!!!

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    I am firmly in “state your unpopular opinion” territory, but here is why I’m uncomfortable with this and the last post.

    1. It still seems to me that the meaning of “mistake” we’re dealing with is new and designed to conform to the distinction. We might as well have invented a new word.

    2. There is a small but important movement to entirely remove language that applies any sort of evaluation on student thinking. So no misconception, no mistake, no incorrect, no wrong. These last two posts seem to me uneasily balanced on the edge of this movement. The arguments of these posts are really arguments for removing much more than “mistake” from teacher talk, and likewise “So, if we don’t call it a mistake, then what do we call it? THINKING.”

    …but then why are we comfortable using the language of correct or incorrect? And why are we comfortable saying kids are bad at expressing their ideas but uncomfortable talking about whether they’ve made a mistake or not?

    3. The premise of the last bit of the post — that correct/incorrect can’t apply to complex mathematical questions — doesn’t match with my experience of mathematics. Proofs can be correct or incorrect…though of course there can be amazing, beautiful ideas contained in flawed proofs! And isn’t that the point, ultimately? That we should be able to train our ears to hear what’s valuable and correct in a student’s thinking, that we should be able to communicate that, but while also telling our students how to improve? Beyond this, what are we talking about? (I agree: we shouldn’t kick off the conversation by telling kids “You’ve made a mistake!”)

    4. These last two posts are implicitly a critique of growth mindset culture in math edu. Hear hear! Too many math teachers are in the business of shutting down students’ ideas but putting a cherry on top. (Stronger you make do mistakes, young Jedi.) But that critique hasn’t quite been made explicit. Isn’t that what we’re talking about here?

    5. The math that most benefits from frank talk of correctness is the math of skills. This is a lot of what we do in the classroom! The end of this post says that these types of questions — the ones that can be judged correct/incorrect easily — don’t matter. (Or don’t very much matter?) I think that’s wrong, and it’s signing away a lot of wonderful math. I’ve seen kids come alive with beautifully complex questions and with simple questions too. (Can’t Three Act Problems be simply judged to be correct or not? Isn’t that the point of a video — it’s the answer key?)

    6. As usual in these sorts of things, as we get more specific I find more agreement. I use something like the script you offer a couple times a day, at least. (Compliment — Restatement — Correction — Try Again).

    But the advice to always find a question that the student answered correctly is not always useful. The point isn’t to randomly find a question that the kid correctly answered — the point is to value the students’ thinking. Knowing more math (so that we can match kids’ thinking with mathematical questions) seems to me of dubious value. Kids want to know that there is something valuable about their thinking for THIS question, not some other one.

    7. I strongly suspect that this is a cultural thing, and that different groups of students (/teachers) in this large country and larger world have different relationships to all this language. In the yeshivas that I attended, we always talked about this and that great sage and the mistakes they made or their incorrect assumptions. In the schools that I’ve taught in with the math that I’ve taught, I strongly sense that my students value forthright, explicit information about when they’ve made a mistake (so they can improve). It’s even sort of funny when people are shy or implicit about when a mistake has been made.

    I suspect that these last two posts are optimized to resonate with a particular culture that I don’t have access to, and that this partly explains my tin ear.

    • Thanks for the thoughtful comment here, Michael. Your perspective from studying in yeshivas is a particularly helpful lens on your last comments. At this point I don’t mean to say much more than this:

      That we should be able to train our ears to hear what’s valuable and correct in a student’s thinking, that we should be able to communicate that, but while also telling our students how to improve?

      I tend to think the implicit definition of “mistake” in math class works against that specialized hearing, and that the growth mindsetization of mistakes amounts to doing the wrong thing in the right way.

  6. Reply

    I have a nice friend in Hong Kong who came to our house for a dinner a few years ago. In that occasion i remember myself asking him what had changed in Hong Kong after becoming part of the People’s Republic of China. He answered that not much had changed, but he had noticed that leaders had started facing problems of the town differently: they seemed to wonder more about what the higher in land hierarchies would have wanted them to choose, rather than what the best solution to the problem would be. I wonder if sometimes our students do the same thing and try to guess what we want them to say rather than try to understand what the question is about. If this is true then naming mistakes differently might help them relax and focus on the real object. Thank you so much for the idea

  7. Reply

    I love this post. It’s amazing how many teachers would call what the student did a “mistake.” As a future teacher, I strive to make sure that I don’t ever say that the student did something wrong, but rather I try to figure out the student’s thought process and what they tried to do. Dan, thank you for your great responses to the questions from your last post because they have given me great ideas on how to approach student “mistakes.”

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