**I. **

There I was, helping Samantha with some subtraction, when I hear another kid nearby — Lena — cracking up, really losing it. Lena was laughing, and though I try to ignore her, she’s laughing persistently. Lena turns and looks at me with a huge, ridiculous smile across her tiny little third grader face.

“It’s just zero!” she says.

“Yep,” I say. I force a smile.

“It’s just zero!” she says it again. I try to grin convincingly back, as my mind races. What’s so funny?

“Haha, that’s right,” I replied, hoping that I sounded sort of like a human does when they get a joke.

For context, here is what Lena was working on: a big-fat subtraction worksheet. Here is a sampling of some of the hilarious problems I’d included on the page:

120 – 30

Also:

21 – 2

Don’t forget:

110 – 60

And this classic:

8 – 3

You may also notice that this list of uproarious* *problems seems a bit on the easier side for third graders. For Lena (and Samantha) it was not. Subtraction has been coming exceedingly slowly for these kids — much slower than their multiplication, actually. It’s February, so we’re not anywhere near the finish line. Even so, I’m beginning to start to anticipate to realize that my time with my students is, ever so slowly, slipping away. I want these kids to have a good year next year in math, to be happy about school. I don’t want this to gnaw at me over the summer.

Anyway, Lena is cracking herself up so I have to go over and see what she’s up to. I look at her page. Suddenly, I’m in on the joke.

You see Lena subtracts digit by digit, because someone taught her to do that. I don’t know exactly what to say — it’s not wrong, and she is so shaky with so much subtraction. It gets her in trouble with problems like 17 – 8, because she brings the 1 down unnecessarily. Still, it’s something to work with.

But the thing is that she really needs to focus on each digit with all her attention. She can’t yet take that step back to see the problem as a whole. So there she is, with 251 – 251. Carefully, slowly, she considers each digit:

*2 minus 2 is…0.*

*5 minus 5…0 as well.*

*1 minus 1…wait a second…*

And there you go, there’s the joke, it’s just zero.

**II.**

Ooh, by the way, Samantha is pretty interesting too.

Samantha also does that column-by-column thing, and it serves her well until she gets to problems like 125 – 50, since you can’t take away 5 from 2.

She started the year trying to borrow in these situations, but she really lost all sense of gravity as soon as she got permission to mess with the numbers. She’d do some of the weirdest things I’d ever seen with subtraction — I can’t remember them, they’re so weird. All I remember is that a bunch of times she would proudly shove a piece of paper in front of me and with, like, innocent puppy eyes, ask, “Is this right?”

And 100% of the time the paper would look like this:

125 – 50 = 972

Seriously! It was all over the place.

My take is that Samantha’s brain is just overloaded when she tries to keep track of all the parts of these problems. Every stage of it requires understanding and attention. She uses a strategy to compute 12 – 5, to take away 1, to realize that this leaves 0, to turn the 2 into a 12, to realize that this is, you know, *subtraction** *so it should make things smaller, etc., etc.

I don’t think she should be going all-in on borrowing yet, not until she has a bit more knowledge to rest on.

But what do we do for her? Samantha asks for lots of help, and until recently I’ve been a bit stumped about how to help her.

I think I might have figured it out, though. The other day Samantha comes over to me, once again stumped on a problem. Her paper looks like this:

I have a false start, going into some totally different strategy for subtracting. Whatever, she gets that far-off stare, she can’t deal with all of it. It’s another way of thinking — it’s not her way of thinking which — for better or for worse — is column-by-column subtraction.

I think, and then I have an idea. She can, I know, subtract two-digit numbers — it’s laborious, but she can do it. So I write an example next to the problem on her page. How about this, I say?

OK, this actually makes sense to her! She uses it to work on the original problem. I offer to give her some more questions to practice — she completes each, surprised that she’s handling the problems correctly.

Is there more to notice here? Sure there is. She should know that the “32” in “324” means 320, she should know how to handle 320 – 150 without drawing little lines, and down the line I sure hope that 32 – 15 doesn’t take quite so much out of her.

But has she learned something? By any fair reckoning, of course she has.

**III.**

Math class should be joyous, they say, full of laughter and insight. I agree! But it seems that a lot of people in education go further, as they’re eager to point you to the source of classroom joy. See this? It’s a picture of kids smiling while studying math. Want it? You’ve got to try instructional practice X, Y and Z.

I promise, you, though, that kids and learning are weirder than that. You’ll plan for fun, and they’ll hate it. The next day you’ll run out of fresh ideas, open a new browser window, type in www dot kuta software dot com slash free dot html, print out worksheets with answer keys, sort of just *push** them *over the desks until each kid has a sheet nearby, then mumble incoherently for a couple of minutes when all you’d really like to say is “here is this, I’m sorry, please do it” and you’ll brace for the worst…

…and that will be the day when everyone is having a blast with math, even Tobias, which is surprising because Tobias has just been sitting there quietly since October when he broke up with Julia, and like you told his mother it’s been very tricky to get him to open up, but there he is chatting about exponent rules with Harry, and he seems alive and (to be honest) happy in a way that you haven’t seen him in a long time.

(In case you missed it, we moved from third to eighth grade with that last bit.)

All of this is to say that joy and humor in a classroom can come from where you’d least expect it — depending on what you expect.

And Samantha? Well, people will also tell you that you need to listen to the ideas of students, to truly build on their thinking, not to override their thinking but to build on it.

I agree. But what does it look like to build on how your students think? What if your student thinks about a problem in a way that isn’t just wrong, but wrong in the wrong way? It’s not just that her technique is incomplete, but it feels like a trick, like a machine that was designed to perform half the job, like a car that can only turn left?

I’m not always sure that I understand the difference between procedural and conceptual knowledge, but I think Samantha’s case is clear. She has a (half-working) procedure married with a not-quite-there-yet conceptual grounding. Is this a time to accept what she knows and to develop it? Or to dismiss her approach and bring her back to square one, conceptually speaking? Is this an exception to the rule — a time when we *shouldn’t *build on what she knows, but should instead sort of veer around her structures and start construction on a new lot?

Learning is weird — it will surprise you. Procedures can be a start. Subtraction can be hilarious. Go ahead, come up with a theory about how all of this works, but be ready to find out that something entirely different gets the same results. Share what you’ve found, and then also have the humility to know that something quite different might work as well.

I love being able to laugh about math with kids, and learning how kids think is just about my favorite part of this job. I love that so many people in education want classrooms to be joyous places where children feel understood — I want that too. But if you find yourself setting terms on how this can happen or what this looks like, please proceed with caution: it doesn’t look just one way.

I enjoyed this. I myself am an ardent “concept-before-procedure” teacher, but I find my dogma repeatedly battered and tested by the wrecking ball of classroom experience. I sometimes suspect I teach this way not because it’s the only way that works for *them*, but because it’s the only way that works for *me*.

I especially appreciated the line: “She’d do some of the weirdest things I’d ever seen with subtraction — I can’t remember them, they’re so weird.” Rings very true to me: student thinking lives in this crazy infinite-dimensional vector space, and I’m constantly projecting it down into the lower-dimensional space (3D on a good day, 1D on a bad one) where all of my theories about learning live.

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Great post! Love the analogy of a car that can only turn left. And the realization “Ha! It’s just 0!” is a terrific example of CLT. A perfect illustration of missing the forest for the trees.

I have a grand unified theory of student enjoyment of learning, where “learning” is defined as forming a connection between fact (or concept) A and fact (or concept) B. Enjoyment is proportional to:

(conceptual distance) * (conviction that the connection is real) * (personal importance) * 1/(complexity of explanation)

The conceptual distance is the surprise factor: if you can explain fact B by some seemingly-unrelated fact, A, that’s surprising and it hits you like the punch line of a joke. The further apart A and B are in my mind before I realize the connection, the more dopamine is released when the connection is made. This is literally the structure of a lot of jokes. For example: Q. Why can’t you run through a campground? A. You can only ran, because it’s past tents. It takes a half-second to make the connection between tents and tense, because those words are not typically in the same schema in our minds. If you made a joke connecting two already-related words, it wouldn’t be funny at all.

In my mind, that’s what happened when Samantha realized the answer was 0. It was quite literally a punch line for her. She was thinking about the steps in her algorithm (“tents”) and then realized the connection to what subtraction means (“tense”).

The conviction factor means that you can’t just tell students a connection exists — you have to convince them of it.

Personal importance is my nod to pseudocontext, because I think there is a small positive pseudocontext effect, when you use a context kids are actually interested in. The problem is that students don’t care about cell phone plans, etc. Personal importance also includes perceived academic importance: if students have come to believe that it’s really important for them to graph lines, then they’ll be interested in a good explanation of it, whether it’s intrinsically interesting or not.

The complexity means that if you have to explain the joke in order for someone to understand it, it’s not really funny.

——————

Another thought about your post is just that what students are thinking about is so much richer than merely processing our words and gestures. So a good teacher has to be relaxed enough to really see (and enjoy) the students’ thinking and emotions. Sometimes I get so stressed by workload that this is a challenge for me. I always dream that some day, my curriculum will be so great that I’ll just be able to be in the moment all the time at school.

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See Martha’s paper here.

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I’m curious if you’ve done any work with the open number line and subtraction? It’s a useful model that can help students who aren’t ready for the formalization of the algorithm to really think about the numbers involved. Pam Harris’s numeracy book has some great strings that develop some subtraction strings, and there are some other problems in the Young Mathematicians at Work books by Cathy Fosnot (obviously, in this case, on addition and subtraction).

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