This yielded a few really interesting mistakes, all pointing me to the same conclusion: my 3rd Graders need to learn how large numbers work. Though, as always, the student thinking itself was interesting.

Sorry that this kid used the “highlighter.” But they got 6000 for those 30 fifty dollar bills.

Their count: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000. Thousands are stupid, we should just eliminate them entirely and count “11 hundred, 12 hundred, etc.” It would solve a lot of problems! We’ll have to wait until I’m King of Math to do that one.

This same student made another mistake while working on this problem. They asked me if 1000 was made of three 50s. I asked, “wait what?” They clarified that since two 50s make 100, maybe three 50s make 1000?

Students don’t know how to read equations, and when they see two numbers they habitually add them together. Any blank is there as the result of an operation. The equals sign just means “make sure that you do this operation.”

How I Addressed It

My goal is to help students connect equations to the notion of equivalence — something that students in my experience already come into my classes with a decent understanding of, whether from experience or school.

I show this image, and talk about how we know that the pairs of apple buckets have the same number of apples (you can move one apple from one bucket to the other).

Then, we move to puzzles. I tried to leave different buckets “missing,” because I know that these are really four different types of problems. In particular, students are the most confused when the third bucket is missing (since they just tend to sum the first two numbers and put that in the third bucket).

Then, I want to nudge students towards connecting the arrow symbol to the equals sign and buckets to boxes of missing numbers:

This might also be a good time for this activity:

For an extra challenge, I ask students to only use the digits 0-9 each once.

Commentary

Every year I see this mistake in 3rd Grade. I’ve tangled with it over and over again, and I’ve also tangled with the research on the equals sign:

Ultimately, I don’t think the problem is in interpreting the equals sign exactly. It’s more about reading an equation, which is hard.

That said, I don’t want to ram directly into their rigid understanding of equals signs and equations, so I use arrows and buckets to help describe equivalence. Then, I just casually slide into using the equals signs and abstract equations in a similar way. That’s my current approach to making a change.

I used to have a big conversation in class about what the equals sign meant, but ultimately I became dissatisfied with that and moved to this approach.

I wasn’t able to turn all of the ideas into activities, but here are the follow-up activities I came up with. If I were addressing this error in class I think this could be a progression of activities that help address the thinking in this mistake.

This mistake brings up the concept of teaching with keywords to me. I asked the student to tell me how he got his answer; he pointed to the word “total” and said that he needed to add. I’d like to know what ideas educators have to intervene when a student is already clearly looking for keywords and is not making sense of the problem.

What do we mean by “make sense of a problem”?

Are we imagining an all-math skill, tools that can be used to make sense of any math problem no matter the topic or age of a student? Something like “read the problem carefully!” or “draw a picture!”?

Or are we imagining a local skill, some way to make sense of this problem and problems like it? Something like…well, I’m not exactly sure what would help someone make sense of this problem. Maybe, “if you see names, you might be comparing!” or “if you see a lot of numbers in a problem, rewrite them in a list so you can focus on what the problem is asking”?

I suppose that I’m inclined to think of keyword mistakes as a lot like applying the distributive property where it doesn’t belong or other “over-extending” mistakes. Generalizations are smart things to do, and a keyword generalization is a smart thing to think also, and it’s usually correct. Knowing that this mistake exists, I might create a set of problems all that contain the word “total” with some being summing up problems and others being compare problems and so on. My reasoning is sort of simple: this kid thinks that “total” means add ’em all up, so let’s provide him with counterexamples and then urge him to make a new generalization.

In this case, the mistake (or whatever we call it) isn’t about what the student wrote, but what he said.

At the end of class, I asked my 3rd Graders to write a story problem for 13 x 2 and hand it in. As he was leaving, a boy handed me this slip and apologized for it.

“Why are you apologizing?”

“Because my story is for 2 x 13, not for 13 x 2.”

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Commentary:

The big lesson here is that the order matters in multiplication, as it does with addition (for most young kids 9+2 is much easier than 2+9) and as it does for algebra (4 + 2x = 10 is not the same as 10 = 2x + 4). Each of these problems has a different flavor for people who are beginning to get comfortable with these types of problems. Saying that two problems are “the same” is a substantive mathematical claim, and it needs to be taken with the seriousness that all mathematical claims require.

When I was a kid, a friend asked what my dad does for a living. “He’s a dank,” 18-year old Michael said. What I meant to say was that my dad worked at a bank, but I was distracted or tired and I mixed up the two words.

I thought about this while looking through a 3rd grader’s addition work. “He’s a dank” seems a lot like saying “46+30+2=58” to me.

I’m not sure what to call this sort of mistake. I’m tempted to call it a memory overload error, but I have no idea if that’s (a) psychologically apt or (b) meaningful to other people.

The crucial thing, though, is not to simply disregard these sorts of mistakes as silly errors, or as a sign that the student is lacking some general cognitive skill like “attention to detail” or “being careful with their work.” That would be a bad misdiagnosis.

To start building the case for why, pay attention to the “stupid” arithmetic mistakes that adults (and teachers and mathematicians) make while they’re working on a problem. Here’s one I made last summer while trying a matrix multiplication, when I did 1*2+2*3 and ended up with 10.

Do I suffer from a general sloppiness in my work? A lack of attention to detail? Nah, I was just distracted by making sure that I kept track of a bunch of others things that weren’t automatic for me. My attention was elsewhere.

What causes these sorts of errors? Any sort of distraction, but it’s important not to trivialize distraction. Distraction can come from any number of places.

Distraction can come from various non-mathematical things, like friends, chatting, not caring about the problem, etc.

Distraction can also come from mathematical factors. If I were better at the matrix multiplication part of matrix multiplication, I would be less likely to mess up some quick arithmetic that I’d otherwise get right.

What about my 3rd grader? There are two possibilities, and both are worth considering:

The kid might have been distracted by whatever non-mathematical thing happened to be drawing her attention away at the moment.

She might have found keeping track of the tens and ones difficult, and paying attention to the decomposition used up the mental resources that were needed to keep track of everything. She ends up adding 2 and 3 for the tens digit, 6 and 2 for the ones digit.

One of the themes of this blog has been a desire to dig deeper than “stupid mistake.” This is one sort of error that teachers often identify as a “silly” mistake, but labeling it as “silly” probably misses out on some truth about a kid’s mathematical thinking.

Questions:

What do memory overload mistakes look like in geometry? In non-computational contexts?

What other categories of “silly” errors are there? (I’d toss “mathematical habits” into the mix. Or maybe we should call that “fluency with a falsity”?)

What sort of feedback would you give my 3rd grader?

Part of what makes learning fractions tricky is that there at least three unnatural things to learn:

The written language of fractions

The spoken language of fractions

The math of fractions

I work in a third grade classroom right now and I’ve heard a bunch of kids say something like the following:

“This is a third.”

Why? There’s an enormous mushing that goes around with “fourth” and “four,” with “third” and “three.”

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Related(?) mistake: 4/6 is equivalent to 1/3

Maybe that isn’t related, but I heard it out of a kid who thought a third was 3/4, so it’s probably connected somehow. Maybe you guys can figure out how.

Yes, yes, kids multiply the base and the power. Here’s what’s remarkable about this:

They do know the definition of exponents. It’s written a line above. They did it a line above.

They’re doing this with confidence. There aren’t erased numbers. This isn’t slow thinking. This is just what kids think seven squared ought to be.

By defining exponents in terms of multiplication while offering no other images or models for what exponentiation does, we create a default model for exponents that sticks with people forever. When mentally taxed — either with a tough multiplication, or with an unusual power — kids revert back to this default model. They’ll do this especially in high school, and they’ll get questions wrong on tests and all sorts of other things not because they’re being sloppy, but because this default model is constantly lurking in their minds.

Incidentally, I asked these kids why they think about multiplication when they see powers, and this is what they said:

I’ve written about a lot of this stuff before. See here, especially, where I shared the high school versions of this mistake.

Now that all of this has been established, the next step needs to be finding a curricular approach that doesn’t rely as heavily on the “repeated multiplication” model for exponents. We need to build a distinctive set of images and intuitions that are native to exponents so that our kids aren’t always defaulting into multiplication when they have to think hard about math.

This is work that I’ve started, in a post titled “Exponents Without Repeated Multiplication.” I’ll send you there for the details, but I stake out two major claims about exponents education:

Much in the way that arrays support early multiplication work, geometric notions of area and volume can serve as the bedrock of an exponents education

We tend to think of four, not five, major operations of arithmetic, but we need to start thinking about exponents as on par with all the others and taking care to build them thoughtfully throughout the entire curriculum.

Beyond all of this, these exponents mistakes serve as a big reminder about the nature of learning, teaching and knowledge. The big, big lesson of all of this is that knowing/not-knowing is not clean and it’s not binary. There are degrees of knowing something. Would you say that these students don’t yet understand what exponents mean? What does that even mean, given the contradictory evidence we have in front of us.

But, then, what does it mean to understand something at all?