When kids are learning to give fractions meaning, I think they often struggle to figure out how the numerator and denominator are coordinated. Here we see a middle step in understanding, maybe: it’s not that the numerator and denominator are totally disconnected. They’re just coordinated in a way that doesn’t really correspond to how they actually work together (i.e. denominator tells you the “unit” and the numerator tells you the “quantity.”)

Maybe the progression of learning looks like this:

• 2/3 means “2 and 3,” nothing to do with each other. Totally baffling notation.
• 2/3 means “2 by 3” or “2 times 3,” some more familiar situation where two numbers can be coordinated in a relation.
• 2/3 means “2 thirds,” which is a productive way to coordinate the numerator and denominator.

Thoughts? Am I overinterpreting this as a middle step in a progression, when it’s actually just a totally uncoordinated interpretation of the fraction?

Thanks for this, Graham!

What’s interesting about this to me is the mental connection between division and subtraction. I doubt that this kid has anything like an explicit model of division that involves “taking away,” but it makes sense to me that the ideas of subtraction/division would be associated much in the way that addition/multiplication are.

All the more reason to make sure that there’s a robust understanding of multiplication that goes beyond “repeated addition,” no?

Nathan sends along a really lovely fraction mistake.

If I’ve got this right, the kid added 3 and 7 to get the numerator, and added 1 and 2 to get the numerator? This is a way of thinking about fractions that’s new to me. Can anyone offer a better theory or some helpful context for this kids’ thinking?

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.

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?

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?

I’ve been reading some constructivist stuff lately, so…

What resources does this student have for refining their understanding of fractions?

(Thanks for the submission, Dionn!)

See the title of the post.