Should I see my son’s misconceptions?


Yosef turns three tomorrow — happy birthday, kid! My sister got him some new puzzles for his birthday, and that’s how we spent a big chunk of the afternoon.

This is his first foray into “big kid” puzzles. We had no idea he was ready for them, and he can do a lot of it on his own, though he always asks for help. (Like every three year-old, he likes attention from grown-ups.)

While he’s been playing, I’ve been watching and trying to make sense of how he’s thinking. As far as I can tell, his main strategy is to match the pictures of pieces: fish goes with fish, yellow with yellow, etc. He also has an eye for missing pieces — meaning, he matches holes with pieces that are congruent to the holes.

One thing that’s fascinated me: he doesn’t really notice the difference between edge pieces and interior pieces. Here he is, trying to stick an edge piece into the middle of the puzzle.

IMG_0760 (1)

I got curious, so I started asking him about potential fits. Could this piece go here?


He never mentions the shape of the piece, or the way that it would partly stick out. When I asked him about this piece he only mentioned the color. When I swapped out the yellow piece would another green-sea colored one, he would try to smoosh it into the hole. Only then would he tell me why it wouldn’t work — “It doesn’t go in the hole.”

Now, I honestly don’t care how well or poorly he solves puzzles. But learning stuff is fun, and I was curious whether I could help him see the difference between edge and inside pieces.

“Look Yosef,” I said. Just like in that picture, one of those inside pieces was along the top row of pieces, protruding out of the frame. I put my finger at the top of the puzzle on the top left side of the puzzle, and I slowly dragged my finger to the right. “My finger can just keep going, going, going…until it bumps into this. Bonk! This piece doesn’t belong!”

Yosef laughed. “Bonk!” he said. “Bonk!”

“But look Daddy. If my finger goes like this” — he loops down into the interior of the puzzle, far below the top row of pieces, slowly meanders up until it reaches the false piece, right under where my finger had bumped — “if it goes like this, then it doesn’t bump.”

Which was true! Had to cop to that.

He returned to the puzzle. He matched pictures — dolphin into dolphin, clownfish to clownfish — and every so often mystified me by quickly intuiting where a piece went. He also continued to shove edge pieces all along the inside of the puzzle.

I wasn’t lying when I said that I didn’t care how he plays with these puzzles…but doesn’t it just kill you to watch someone painstakingly — delicately with care — try like seventeen different ways of putting an inside piece into the side of a puzzle?

I mostly keep my mouth shut and let him have fun. He’s clearly not seeing edge pieces yet, which is interesting, but obviously fine.

Every once in a while though, I nudge at his understanding. “Pass me an edge piece,” I’ll say, hoping that he starts thinking of edge pieces as a distinctive category. If he asks me to fill in part of the puzzle I’ll talk aloud about my thinking: “This piece couldn’t go here because it doesn’t have a straight side.”

I have no idea if this stuff is connecting with him. Learning to see shapes in different ways is messy and slow. My little two-second nudges won’t make much of a difference to him — right up until he becomes ready for them, and then they might.


It’s pretty clear to me that there are things about shapes and puzzle pieces that Yosef doesn’t yet understand. He’s three. Of course there are. But how should I think about his understanding? In some quarters of the math education world, the answer is contentious.

Brian Lawler is someone who has been incredibly patient with me on Twitter, as we’ve gone back and forth discussing his positions on the nature of mathematical knowledge, teaching and learning. He passionately believes that any talk of misconception is not only wrong-headed, but also the act of labeling someone as holding a misconception is harmful to that person. Ditto for a smattering of other terms that imply that the other person’s thinking is worse than your’s, or on the way to some better understanding — this includes talk of alternate conceptions, early conceptions, preconceptions.

Rochelle Gutierrez likewise asks us to refuse to talk of misconceptions.

These scholars aren’t talking about me doing math with my kid — they’re talking about the ways math teaching can beat down kids in a lot of school situations. Still, their arguments are that thinking in terms of misconceptions or even not-there-yet conceptions is harmful — even violent — to a math learner. Their framework should apply to me doing a puzzle with my son too, I think.

Now, it doesn’t seem entirely accurate to me to say that Yosef has misconceptions about puzzles or shapes. It’s not like he actively thinks that edges don’t matter — he just doesn’t see the difference between edge and interior pieces yet. Yet he has so many amazing things in his little-kid brain that help him put pieces together. He absolutely has a conception of shape that is letting him have a blast with puzzles, and he loves doing them with me. I love playing puzzles with him. I love him.

Is it still harmful for me to think and talk about the things he doesn’t yet see?

I try to be a good father to my children. This is not always easy with a three-year old, but I really do try. I think I’m getting more patient — hopefully in time for the baby’s turn at toddlerhood — and I try hard to give Yosef room to play with toys the way he wants to play with them. I certainly don’t want to visit “intellectual violence” (as the phrase goes) on him by getting all up in his face about the right way to do a freaking 48-piece puzzle. I’d rather save our conflict for when he’s dropping a block on his baby sister’s head.

At the same time, part of our play is learning. The kid wants to put the puzzle pieces in on his own. He wants me to help. He likes learning new things — he’s a kid, he’s deeply curious about the world. The world includes mathematical language. Every time we put on his shoes we end up doing a whole routine about left/right: No, that’s not left. It’s right. No, not right, right. Right. Left. That’s right. His conception of left/right is relatively weak — it’ll get stronger.

Kids love improving their conceptions of the world, if they really get a chance to feel like it’s their own conceptions that are improving. Humans are curious creatures, and we like improving. There are a million ways for this to go wrong and to turn into abuse — in a lot of schools, this is happening.

In a lot of schools and homes, though, it isn’t. I don’t think it’s inherently abusive to see your child’s misconceptions or to help them see things in a new and richer way. It can be, of course, and that abuse needs to be detailed and discussed.


Some people might disagree with the above, but not many. The real question is a linguistic, or even a strategic one:

Does refusing to talk of ‘misconceptions’ cut down on the abuse?

Pretty much everyone I talk to online seems to think that this is a good way to chip away at the problem of abusive teaching practices. The first way this might chip away at the problem: the word “misconception” itself connotes the bad teaching practices. You can hardly use the word without being further nudged towards abuse — or you might nudge a colleague to abuse. If you eliminate the word, you eliminate the subconscious priming of yourself and of colleagues.

The second way: Changing your pedagogy is hard, and it’s easy to forget your principles. The refusal to talk of “misconceptions” is a relatively easy change to make, but it’s like a red string around your finger. It will remind you of your commitment to the proper pedagogy, and you’ll continuously improve as a result.

I actually think there really is something to that second thing, but I also think it’s incredibly risky for the cause of good pedagogy to tie it to refusing to use the word “misconception.”

It’s because my kid’s conception of shape really does have room to improve.

We see misconceptions in children because it really is true that there’s stuff that they don’t yet know. Noticing this doesn’t have to be an act of violence — in fact, I don’t think that it usually is. Usually it’s like me playing with my son and noticing there’s stuff he doesn’t yet know how to do, even as my mind is blown because oh my god my son is into puzzles! When did our baby turn into a kid?

Is it good pedagogy to ask people who don’t already see their pedagogy as abusive to forswear from using words that they use all the time? Isn’t this exactly the sort of “intellectual violence” that we’re being urged to refrain from? Shouldn’t we start with the way people actually see the world, rather than asking them to use language that is not their own?

Seriously: imagine what a teacher whose well-meaning administrator announces that they don’t want to hear any talk of misconceptions any longer, that this is now school policy. Is that good pedagogy?

There is real injustice and unkindness in this world, and I have no clue how to fix it. I think a focus on which words are allowed or not is a tactical mistake. Take any word that you associate with fear, abuse, pain; there are others out there who associate those same words with love, play and growth. To bridge those gaps we need to talk with each other and find a common language. That can only happen if we agree to use each others’ words.


7 thoughts on “Should I see my son’s misconceptions?

  1. I dunno. I find the concern about “misconception” really hard to wrap my head around.

    An analogy: mistakes. We all seem to agree that a statement can be incorrect, that errors happen. We don’t defend their factual value; instead, we defend their emotional and experiential value. We aim (rightly, I think) to de-stigmatize mistakes, to ensure no one is ashamed or afraid to make them, to embrace them as steps on the path to mastery.

    But we don’t deny that mistakes occur, or that truths are generally better than falsehoods. We don’t try rebrand mistakes as “takes.”

    Is there a reason this same logic doesn’t apply to “misconceptions”?

    (Also: happy birthday Yosef! I think he’s lucky to have a dad who takes such joy and care in sharing puzzles with him.)

    Liked by 3 people

  2. I love this whole post. I think I agree with all of it, and I adore the description of you being there with him, gently sharing something you’re doing, but letting him explore on his own too. “[D]oesn’t it just kill you to watch someone painstakingly — delicately with care — try like seventeen different ways of putting an inside piece into the side of a puzzle?” — oh yes!! but at the same time I’m laughing at myself the way you seem to be at yourself.

    I don’t have a problem with using the word “misconception,” especially since I think it’s mostly used talking about students, not to students. However, when I think of how I cringe at “rigor” or “grit,” both of which are probably used by some absolutely wonderful teachers and administrators, I can sympathize with misconception-haters.

    I also think that as teachers we sometimes overanalyze our own impact. We can worry a lot about whether to call a kid’s idea a “conception” or a “misconception,” but frankly, the student next to them isn’t going to use either of those words; they’ll just say it’s “wrong.” As Ben indicated, setting a tone for how the class community reacts to mistakes or misconceptions seems more important than what word we use.

    Liked by 1 person

  3. Your post brought up several thoughts for me. First a personal thought, a reminder of a puzzle we had for years when the grandchildren were younger. It was much less complicated that the one that Yosef was playing with. It was one of those old wooden ones with half a dozen pieces to fit into matching spaces on a board – the duck in the duck shape, the sailboat in the sailboat shape, the doggie with the doggie, etc. Our granddaughter was a puzzle whiz at age 2 and enjoyed snapping the pieces into place over and over again. That said, Charlotte hadn’t spoken a word yet, which made her mother a bit nervous. When she’d say to Charlotte, “Which is the duck? Where does the sailboat go?” Charlotte would happily comply, just without talking. (When Charlotte did start talking, it was in complete sentences.)

    One time a friend visited with his son, also age 2. Zeke was a total bust with the puzzle, trying over and over again to fit a piece, any piece, into a space, any space. He seemed completely unaware of matching shapes or turning them this way or that. He played with the puzzle for a while, not stressed, just picking pieces up, trying, and putting them down again. The entire time, he described, in complete sentences, what he was doing. “This is a duck. Maybe it goes here. This is a sailboat. I can’t get the piece to fit. I’ll try the doggie.” He was happy to have help and seemed to be pleased when all of the pieces were in the puzzle. But then he noticed a broom in the corner of the room and moved on to sweeping.

    OK, what does this have to do with misconceptions in the classroom? I’m not sure, except that the key word that popped out at me in your blog was “yet.” More important than what Charlotte and Zeke each hadn’t yet figured out was what each of them did understand. Charlotte’s visual and motor ability was as natural to her as was Zeke’s impressive relating to the world through language.

    I think of my work in the classroom as becoming aware of students’ assets and building on them. Sometimes a misconception is an asset, as it can be springboard to learning. But a misconception isn’t something to be fixed, but rather something that can guide what I might suggest as an exploration. I agree that outlawing the word misconception isn’t a cure. How do we continue to grow our teaching practice so that it’s not an endeavor of fixing children’s deficits, but one of building on children’s strengths. Maybe this is a linguistic spin, but it rings true to me.

    Liked by 2 people

  4. I think this thing with the puzzle puts the misconceptions in a useful light. So much of the time it’s not so much that we have wrong ideas about a mathematical idea, it’s that we don’t have an idea about it yet, or we have a hazy or sketchy beginning idea. Sometimes it’s almost like that boundary in your field of vision between what you can see and what you can’t see; even the boundary isn’t there, perceptually. I suppose for all of us, including mathematicians, most of maths falls into this category (assuming maths to be a big territory which no-one can hope to travel through in its entirety).

    Another thing, puzzles. They should make them so that, at least for a few pieces, either an edge piece or a middle piece would have a picture that connects nicely: that might force the distinction into the field of vision.

    Liked by 1 person

  5. I am deeply curious what might be a child’s mathematical misconception–as opposed to an un-thought-out answer. child: “2 + 3 = 6” adult: “show me” What misconception would cause the child to demonstrate to himself, and the adult, that “2 + 3 = 6.” Maybe as opposed to the lack of a conception of early number concepts (i.e. no sense beyond 1, 2, many). Similar: child: “when i double the sides I double the area. Adult: “show me.” If the child persists in showing me her conclusion, I suspect it may be due to a lack of a conception of area, or maybe simply of the what I meant when I said the word “area.” I truly struggle to conceive what might be an example of a mathematical misconception.

    And maybe this is because I so strongly believe that the ways in which we know are completely viable in our experiences / experiential world. And we don’t make changes to those ways of knowing until we run into something that creates a disequilibrium. The adult’s “show me” questions may provoke such disequilibrium.

    It strikes me that many of my teacher colleagues cannot conceive the idea that children (people) do not have misconceptions.

    Liked by 1 person

  6. Super interesting stuff here. There definitely needs to be a whole academic field devoted to adults’ conceptions of children’s minds. This piece would be a foundational case example.

    I’m not super worried about whether people use “misconception”, so let’s assume for a moment that this is a word that’s useful and OK to use. I still don’t understand what misconceptions you think the boy has.

    Is it that “edge pieces don’t matter”? I don’t think he has that conception (whether mis- or not).

    Rather, I think you point to a bunch of interesting things that he—as a novice—doesn’t know about or notice yet. For me, this is an essay about differences between experts and novices.

    It seems to me that misconceptions (still operating under above assumption), come into play between novice and expert. You have to know something (have conceptions) to have misconceptions also.

    Liked by 2 people

    1. I think we’re on the same page. Here’s the part of the piece where (I think) I agree with you:

      Now, it doesn’t seem entirely accurate to me to say that Yosef has misconceptions about puzzles or shapes. It’s not like he actively thinks that edges don’t matter — he just doesn’t see the difference between edge and interior pieces yet.

      I do a bit of cheating with the title of this piece, focusing as I do on “misconceptions” even though (as you note) this doesn’t quite apply to the story I tell. The thing is that Brian and Rochelle’s objections aren’t just with seeing kids as having misconceptions — they object to seeing a child’s conceptions as lacking in any way. So whatever issues we have with misconceptions should also apply to other “deficit talk,” like saying that Yosef has a novice’s conception of shape. That’s what justified the headline, I thought, but I apologize for any confusion I’ve caused.


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