I wonder if the student would’ve made the same mistake with “scaling by 2.” In any event, this isn’t necessarily a nasty misconception to fix, but I’m always interested by the circumstances when addition and multiplication get tangled up.
What is this student doing? Where is their misunderstanding? @mpershan#mathmustakespic.twitter.com/7YQNGnopwi
— Lisa Bejarano (@lisabej_manitou) January 20, 2014
Let’s help Lisa out in the comments, mmk?
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.”)
You guys actually about this girl and her trouble with perimeter already.
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?
What is this student thinking? @mpershan #mathmistakes pic.twitter.com/ytTFS4sDV6
— Lisa Bejarano (@lisa_bej) January 12, 2014
What interesting mistakes! Let’s make everything that’s puzzling about these explicit.
“6, 8 are equal, but 10 isn’t equal.”
- Does this mean that 6 and 8 are equal to each other? Or that 6 is equal, 8 is equal, but 10 is not equal. (To what???)
- What on Earth does it mean that 10 isn’t equal!
- What exactly does this student think “right triangle” means? Does he think it means that all the sides are equal?
“Yes, because all the sides are equal.”
- How? You drew a picture showing that the triangle has sides of length 6, 8 and 10!
- So a right triangle needs to have all equal sides?
This is mysterious to me, but what’s important is to not dismiss these students as hopelessly confused. Take the second mistake. What we’ve discovered is that you can know a lot and still think that a 6, 8, 10 triangle has all equal sides. That’s really cool!
As far as shedding light on these mistakes, I’m really having trouble coming up with anything that makes sense. I’d say that the top student is not saying that 6 and 8 are equal to each other, but then what is that student saying?
Tina asks:
Why do kids have such a hard time distinguishing between sides and angles? They are so different in my mind, I don’t even know how to explain the difference.
Thoughts?
The submitter asks: What would you write as feedback here? What would actually help this kid?
I ask: What’s the role of written feedback, more generally?
(Thanks again, Tina!)