Not really a mistake, but my kids have started doing this.

Misconceptions surrounding fractions are so well-studied that I feel a bit ridiculous sharing anything about them. Anyway…

I was chatting with this kid who was having a bunch of trouble with written fraction notation. She had been correctly solving problems that involved language such as “shade in four out of seven pieces” or “divide this shape into eighths,” but got stuck when she reached a problem that asked her to “shade in 4/6 of the shape.”

Alice: Oh, so that’s 5.

Me: Can you explain why?

Alice: Because it’s not six sixths.

Me: So, not quite.

Alice: Oh, it’s 2. Because that’s 6-4.

Me:

Alice: Or it’s 10?

Me: See…

Alice: I’m really confused here. What’s the answer?

There’s no puzzles or misunderstandings here. Alice thought that the fraction symbol was an operation between the numbers 4 and 6. And of course she did. Every other time that she’s seen two numbers and a symbol before she’s been asked to produce a third number. This is new ground for her.

I’ve been taking the advice of Brilliant Commenters Fawn, Jenny and Avery and using the language of “out of” to bridge the gap for this kid.

This mistake seems to say something significant about the way this student sees fractions, but I’m not able to piece it all together. Maybe you can?

Decoder Ring: “P” stands for “perimeter.” (“PS” stands for “Point Symmetry” and “S” stands for “lines of symmetry.”)

This is a cool mistake, and when I asked her about it she said it was because of the way the teacher was tracing the path of the perimeter on the board. He was emphasizing the points as he moved along the edge, and, well, you can see how she interpreted that.

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?

I’m pretty sure that I know what’s going on in this Geometry pre-assessment. (Thanks for the submission, RG!)

Notice that line there? The kid is counting squares. The dimensions of this rectangle are 10 by 6, and the kid is adding up 10, 10, 6 and 6 to get 32.

To me this is directly related to another perimeter/area mistake that I documented a few months ago. (Also related might be this.)

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?