or, How to Call Out Mistakes without Trampling the Mistaken
It was the end of our first day on limits—a deep and slippery concept, the engine of calculus—when Melanie exclaimed, “Wait. Shouldn’t that limit be 4, not 6?”
Nope—it was 6. Melanie’s error suggested that she’d missed the lesson’s most basic truth, an idea that the class had spent the day paraphrasing, analyzing, and shouting in chorus. Talking one-on-one, I could have coached her through the misconception. But hers was a public declaration, in front of the whole room.
Even before the words had left Melanie’s mouth, I could hear the groan welling up among the students, murmured ridicule and the slapping of foreheads soon to follow. They all knew it. She didn’t. From Melanie’s blushing, you could read her self-esteem falling like a mercury thermometer.
And so I found myself confronting one of the teacher’s daily puzzles: what do you say when a student is wrong?
In the classroom—and beyond it—dealing with mistakes is delicate and crucial territory. At one extreme, we can deride errors, ridicule them, chase them out of town with a pitchfork—to the dismay of the poor kids who were trying their best.
At the other extreme, we risk coddling wrong statements, embracing them as “perspectives,” waiting in vain for the mistaken to see the light, and in the meantime allowing confusion to reign.
It’s no easy balance to strike, exposing falsehoods without steamrolling egos. To solve the predicament means tackling a deeper issue: what kind of dialogue, exactly, are we having? Which is primary: ideas, or the people voicing them?
Most of our daily conversations are centered on people. Consider the meandering path of dinner table chatter. It drifts from topic to topic, following tangents, pursuing loose connections. Questions are posed and never answered. Thoughts dissipate, half-articulated. A conversation, after all, is about sharing each other’s company. It’s a meal, not an interrogation.
Academic discussions differ. We’re hashing out ideas, each person pitching in to advance the conversation, not necessarily tabulating whose insight is whose. I say A; you say B; and we both think of C simultaneously. It’d be counterproductive to insist that your ideas deserve special status simply because they’re yours. The truth is indifferent to its speakers. Reason doesn’t notice who gives it voice.
In an academic dialogue, ideas ought to matter more than egos.
The challenge for teachers is that we’re having both conversations at once. Our students matter as people—we care about their thoughts, their feelings, their ambitions and fears. But ideas matter, too—we want to spread truth, to help our students master bodies of fact that past generations have painstakingly uncovered (and that textbook authors have clumsily compiled).
So how do we address wrong ideas, without trampling the people who espouse them? Here are four time-tested techniques that I’ve stolen from teachers far nimbler than I.
“That’s exactly the mistake I made when I first learned this material.” Or: “That’s a tricky point. In last year’s class, more than half of them made that error on the quiz.” Or the simplest: “That’s a really natural mistake to make.”
“It sounds like you’re generalizing from the pattern we saw earlier. But I’m not so sure those same results will apply here.” Even the most egregious mistakes have their roots in some identifiable thought process, and tracing that process helps students recognize that their ideas are not them. A train of thought can derail, and it doesn’t make the thinker any less worthy or intelligent.
“Hmm… go ahead and check that calculation again, then get back to me.” Private mistakes sting far less than public failures.
“Remember, mistakes are the essence of learning. When someone commits an error, we respectfully help them understand it, so that everybody can learn from it.” At its best, conversation is a beautiful thing—collaborative thinking, a shared brain. But students need to be taught the rules of the game.
With proper preparation, a class will rarely respond with the hostility they showed Melanie that day. But sometimes even a friendly pack can turn on one of its own.
“I’m glad you brought that up, Melanie,” I shouted over the groans. “It goes to show how tricky this transition is. You’ve all spent the last three years approaching graphs one way. This new perspective is the conceptual core of calculus, and the mental shift will take some time.”
The groans faded. Melanie nodded.
This is excellent! We have a lot of discussion in math and this comes up somewhat frequently. I’ve found myself doing a mixture of a few of these things.
Thanks for reading! Let me know if you’ve got other strategies. (Originally I’d included ‘Cut the tension with humor’ on the list, before I realized that this is one of the few moments in class/life when I go out of my way to AVOID humor, since it’s liable to come out at the expense of the kid who’s made a mistake.)
I love your writings Ben 🙂
Thanks for reading!
I really look forward to your posts!
I had ended up loathing math by high school, and now you’re making me fall in love again.
Glad to hear it! There’s lots of great math stuff out there on the internet these days – Vi Hart and James Grime in video, Steven Strogatz and Keith Devlin in prose, many others…
Awesome Blog post! Given time pressure we face in an over crowded Mathematics curriculum it is important to remember that we need take the time to make mistakes a comfortable part of the learning process. If students aren’t making mistakes, they aren’t attempting challenging enough material but if mistakes aren’t handled with care they can lead to a lifelong affliction of Math Anxiety. We want students to ask and attempt without fearing failure.
Its good to add your tools and advice to my arsenal of techniques to encourage a growth mindset amongst my students.
Thanks Ben!
Hey, thanks for reading, Hayley! Like you say, it’s so important to be comfortable with mistakes (which is why I make so many, of course – just building up my tolerance).
Haha Of course. You’re modelling productive mistake making!
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I always think that for each member of an audience that voices a confusion there are several more with the same thought sitting in silence.
I suppose there are 2 lessons in that moment – the subject matter and the way to approach learning. If you speak up, even when you’re wrong, you will learn. So it is vital that the response is respectful rather than dismissive. And as you’ve said before, often learning maths involves feeling stupid, and that has to be accepted rather than rebuked.
I definitely agree. I’ve had several conversations with my 6th-8th graders this week about helping each other with mistakes, and not seeing error as damning.
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That’s what I thinking about.
This is great, Ben. I spent September focusing on growth mindset this year, helping students understand that we make mistakes when we challenge ourselves and jump out of our comfort zone.The best learning happens there. It is easy to always answer questions that we know everything about, but it’s harder to answer questions where we challenge out thinking. We are at the point now where we “celebrate” and share mistakes – showing incorrect thinking actually helps to solidify the correct thinking (the old “what it isn’t/non-example” on the Frayer model). There are always other students who are at the same place that Melanie is, they just aren’t as vocal about it! Making the classroom a safe space and lessening the fear of failure is an integral part of the learning environment. Thanks for sharing – I always enjoy your posts.
Thanks for reading and commenting!
I think you’re right that the best learning happens just outside of the comfort zone – far enough to give you a little adrenaline rush, but not so far that you feel lost. Definitely important to help kids find that space.
I have enjoyed your posts for the humor, but this one is really terrific and gets to the heart of teaching and parenting. People who become fearful of making mistakes or looking foolish stop learning, so the key is always to make mistakes an accepted, normal part of any learning process. I had a teacher deride and mock me in front of my peers in high school and it took years for me to believe that I was capable in that particular subject. Very thoughtful post.
Thanks for reading. I’m sorry you had that experience; I actually wrote a related essay for The Atlantic (http://www.theatlantic.com/education/archive/2014/06/how-i-became-an-unfair-teacher/371908/), in case it’s any interest. Anyway, I’m glad to hear you did overcome that memory eventually.
If they’re not making mistakes, they’re not learning. I try to identify what was right in the answer and ask questions to help them guide themselves to the right track.
Yeah, well said. Highlighting what’s right is as important as identifying what’s wrong.
Very wise!
Thanks, Peggy!
In class, we spend a lot of time fixing mistakes. I will often share work that I’ve created that is incorrect. As a class, we find the mistakes and fix it.
Then, when I see a mistake in a student’s work, I’ll often put it up on the doc cam and celebrate scholars who examine the work and find the mistake. Quite frequently the student who made the mistake will find it. It’s a positive experience and helps students critically examine work.
That sounds great – what a positive class culture!
I do less formal, usually oral versions of teaching through mistakes (“So I can just multiply these two numbers, right? Wait, why not?”) but I like your systematic approach.
This is a great topic. For the past year, one new strategy that I’ve used is calling a salvageable mistake “a bit of a fixer upper,” quoting the trolls from Frozen. This hopefully keeps the mood light when a mistake happens, and continuing the song lyrics hopefully encourages students that they can fix the mistake (with a little bit of love): <a href="http://meangreenmath.com/2014/06/14/a-bit-of-a-fixer-upper/"http://meangreenmath.com/2014/06/14/a-bit-of-a-fixer-upper/
Oh man. Now you’ve got me listening to that song. All wisdom comes from Frozen.
This is great! I also try to avoid saying things like “This is easy!” or letting students get away with saying it – it’s easy for me, and it’s easy when you see it, but it’s not easy for everyone.
Well put! With some big year-group tests, we’ve been talking a lot about how easy is a subjective and not-always-useful concept.
This is pretty good advice for all of us to remember when dealing with other adults, too. 😀
True! I’d say we adults make at least as many mistakes as kids do…
i am from pakistan . very interesting postshatsoff to you may god bless you. keep it up (y)
Thanks for reading!
I often *thank* a student for making a mistake and say something like, “I’m so glad you said that,” because it gives us an opportunity to explore a misconception that many might have (and just weren’t brave enough to voice) or even a related tangent. I usually make it seem like this was something I forgot to mention and their role was to remind me to discuss this specific issue.
Mmm, I like that rhetorical device. I think you’re right – it’s quite reasonable to thank a kid for being gutsy enough to speak up.
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Your drawings kill me. I can’t handle it. Too. Much. Awesome. Seriously, I laugh ’til I cry. Thanks!
If you are my Chinese-born Engineering Calculus professor, you inadvertently call people stupid.