That’s weird, right?

They clearly get the visual model. Now, granted, it’s hard to apply this visual model when multiplying by “one and a half.” Still, there’s a clear attempt to work it out with pies, and then they wrote four. I mean, what’s going on?

Maybe the kid was just adding instead of multiplying. After all, 2 and a half and 1 and a half makes 4. Maybe he forgot what operation he was working on. He was confused.

But then you work your way through the stack of papers, and you see this mistake coming up a bunch.

Why do kids that clearly get that we’re multiplying end up adding?

You might say, hey, these were just guesses from students who were unable to grapple with a difficult problem. They just wrote anything down. You’re going to have to trust me, because I was there, that this wasn’t the case. These were kids who were, like, I’ve got this, what else you got?

If you’ve followed my work for the last year or so, you know that I’m really into exponent mistakes. I’m inclined to connect this multiplying fractions error with some of the things that I’ve shown you all about exponents. This seems, to me, to be another situation where kids default to a computationally easier operation when faced with applying an operation in difficult context. Sometimes that’s exponentiation defaulting to multiplication, but here it’s multiplication defaulting to addition.

This student — let’s call her Alice — is in 4th Grade. She did some work with fractions in 3rd Grade, but clearly isn’t comfortable with them.

I went over to Alice and noticed that she wrote “0.5” for point A. I asked her to read that number, and she said “a half.” Then I drew a half-filled circle and I asked Alice to tell me what fraction of the circle was filled in. She said “a half.”

Me: Can you write “a half” as a fraction?

Alice: Why do you have to? This way is so much easier.

[I show her how I write a half.]

Alice: Oh, a one and a two.

[I draw two more circles, one with a quarter filled in, the other with three quarters filled in.]

Me: What part of the circle is filled in in these two circles?

Alice: A quarter. Three quarters.

Me: How would you write those numbers down.

Alice: Umm…so this would be one-four?

Me: Yes, though I’d read this as one-fourth.

Alice: And this would be one-three.

This is interesting in all sorts of ways. First, because you can really see in Alice’s work the difference between written and spoken language. Alice can tell you what a half is. She can even tell you how much is shaded in on the other circles, but she can’t write it. Attention needs to be given to both verbal and written language, and we teachers tend to focus on our students written work.

Also, “one-four” and “one-three”? That’s so interesting. Alice sees “three” as the most important part of “three quarters,” and tentatively thinks that fractions are just always “one-something.” That’s a pretty strong tell.

The other remarkable thing is how strongly Alice prefers decimal representations to fractions. Alice showed this preference consistently in her problem solving.

The kindly Professor Danielson argues that, in a curriculum, fractions ought to precede decimals. But it’s also true that decimals are addictive. In my high school classes, kids use their calculators to transform fractions to decimals as a defensive measure. You know the easiest way to help (most) kids solve equations with fractions? Point out that they can convert those fractions to decimals.

Decimals are absolutely enticing to people, even to this kid who is just getting started in this whole mess.

Based on the first of these, I’d think that the student was mistakenly adding instead of subtracting. But how could that also explain the second mistake?

On the other hand, it’s hard to imagine that the student is subtracting in the second case, since they end up with a number that’s larger than what they started with.

Ideas?

A lovely third grader pointed to her page.

Her: Is this how you write a quarter?

Me: How many quarters are there in a whole?

Her: Four. How do you write a quarter?

Isn’t that interesting? What do you think is going on here?

Tina reflects:

“Kids seem to forget their skills when the level is increased. They revert back to intuition like “subtract numerators, subtract denominators” when faced with trig functions, but as soon as I ask “how do you subtract fractions?” they immediately recall “common denominators.” How do we get them to stop and think when the cognitive load is higher?”

That’s a great question.

Eric writes in with this really cool mistake.

Do you see the problem?

Of course you don’t, because you don’t know the original question.

Eric writes, “The problem was originally 12 2/3 + 8.”

Explain? EXPLAIN?

One of the joys of teaching fractions for the first time is seeing mistakes from my students that, prior to now, I’ve only read about.

(Does that make me a dork?)

(Yes, Michael, that makes you a dork.)

Anyway, the first mistake is an absolute classic. With numbers that show any sort of structural complexity, students regularly treat the components individually. See, decimals, complex numbers, polynomials, and other things I’m sure. (If you have links or suggestions of other sorts of common mistakes that fit this paradigm, drop a comment in the comment hole.)

The second mistake pushes on their part-whole understanding in a direct way. I’m not sure how this mistake plays itself out beyond the geometric context. Thoughts, people?

The third mistake is an interesting one, and one that I suspect is about being used to seeing number lines only of unit length before. Other than that, I don’t have much interesting to say about that. But maybe you do?

Lots of good stuff going on here. But I don’t think I entirely understand where 1/8 came from, though I get how that gets turned into 5.8.

[I never know whether to include all the mistakes from a class set or just a few. I feel as if it’s helpful to include more mistakes, but sometimes overwhelming. My solution today is to post one especially cool mistake largely, and the others smallerly. Let me know whether that works.]

I’m not sure that I agree with the teacher’s diagnosis that this sort of mistake is procedural. Presumably, she means that the mistake was the result of remembering some sort of algorithm, a wrong algorithm. But what sort of division algorithm would lead you to multiply 2 and 3? in this situation?

On twitter, someone suggested (was it you?) that the issue here was that 1 divided by 1/3 is 3. And then twice that would be 6.  This would then be an instance of a more common pattern of error, the “it’s always linear” error. (Was it Dave who said this? Maybe this was Dave.)

This actually fits pretty well with the student’s explanation. It’s not a bad take.

Do you agree? The first three times that I wrote this post I said something like “The ‘linearity’ hypothesis is a pretty good one, but it doesn’t quite fit with what the student said. I’d suggest that the student had an association between the numbers 1, 2, 3 and 6, and it’s that instant judgement that her explanation is aiming to justify.” But now I’m not sure if I can really find any fault with the linearity hypothesis.

People who have more experience with fractions than I have: is the linearity explanation one that resonates with your experience of kids learning to divide fractions?

(P.S. I’ve got 4th and a 5th grade assignments, along with my high school classes this year. I’m excited to bulk up my understanding of little person stuff. Any of your submissions would help me test the theory that analyzing student work is a solid way to help build pedagogical content knowledge. SUBMIT!)
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