I was hanging out with some 3rd Graders today. Their task was to find the area and perimeter of some shapes. This girl was working with a shape like this one, except not quite this one:

In particular, the shape she was working with had a perimeter of 22, and an area of 21. She, however, had counted an area of 21 for both. She had already called me over once to ask whether you count each of the sides of a corner square in the perimeter. My Spider Sense went flaring, but I wasn’t sure how to help, so I told her that each side did count for the perimeter. She seemed suspicious, but went with it. Then she called me over again.

Girl: Wait…how is it possible for the perimeter and the area to be the same?

Me: That’s a really interesting question. I’m curious: what’s perimeter, anyway?

Girl: Well… Perimeter…It’s a thing, but it’s outside of something.

Me: Hehe. That’s not a bad start…

Girl: OK, so perimeter is the space outside of a shape. Area is the space inside a shape.

Me: Cool, that’s very interesting!

At this point I’ve sort of constructed a theory about her initial question. Maybe she’s thinking of perimeter as 3D space, instead of lined space. That could explain her confusion about the space around the shape being equal to the space in the shape. Maybe that’s also the source of her doubt about double counting the sides of a square in the perimeter. (Though that doesn’t fit in super-well.)

I decided to push on her definition.

Me: So, perimeter is the space outside of a shape. So is this all the perimeter? [I drew a shape and shaded in the area around it.]

Girl: No! No, it couldn’t be, because then that would go on for all of this space. It’s more like this:

Me: Interesting! I have a question about your picture. Is it important that the lines stick out of the shape? Could you have drawn it where the lines don’t extend out of it?

Girl: No, it’s important that they stick out.

At this point I didn’t know exactly what to do, so I just tried to explain that perimeter and area measure different things. I gave concrete examples of perimeter (“It’s like a fence”), but I didn’t really feel like the explanations stuck with her.

I’m curious to hear all of your thoughts on this interaction. My takeaway is a curricular one. Area and perimeter are different concepts, and they don’t necessarily benefit from being presented together and in contrast with each other. Maybe it’s better to introduce each individually, and only play them off each other after students have a solid notion of each concept’s meaning.

Open thread. Talk about what you find interesting here, or anything. I’ll kick things off in the comments.

I think that this might be my favorite conversation since I started hanging out (as a math assistant) with 3rd Graders. File this one under “place value.”

The question that this kid was grappling with was to write “8,000 + 500 + 20 + 6″ in standard form. I’ve forgotten most of the details of the conversation, but all of the interesting stuff is right there on the page.

First, note how she first writes 25,03 to that first question, at the top of the image. And though it’s harder to see, she’s done that for 60,47 as well. When I came over to her, I thought that she was just confused about the convention, and I offered a correction.

Then she started working on the “8,000 + 500 + 20 + 6″ problem. First she wrote 26, and then she tried to figure out how much more it was. She ended up with what you see in the image, at which point I realized that there was something about the way she was thinking that I hadn’t anticipated. She was chunking the number into 26 and everything else.

She really struggled to figure out what to do with 8,000 and 500. The boxes around the numbers are part of my attempt to push her to tell me what each numeral represented (“5 whats? Shouldn’t this just be 8 plus 5 plus 2 plus 6, so 21?”).

But there was no way that was going to work. The fact that she was chunking “26″ together means that she doesn’t really get place value for 10s either. Those of you with more experience will hopefully help me out in the comments, but I’d imagine she sees 26 as a single number, not as composed of any parts.

This was confirmed when I asked her what the 2 in the 26 meant. She thought I was nuts. She said that 26 is the number right after 25. I repeated the question, and she thought it was ridiculous.

To me, this really speaks to the value of activities that defamiliarize place value for students. See Anna’s Ba-na-na or Christopher’s Orpda for activities that do this.

Summer’s over. Let’s get back to work here.

A sprinkling of thoughts:

• Nicora, on a  recent post, came out against using “x out of y” to introduce fractions. She writes: ”One of the most common ways to introduce fractions to young students is to talk about ¾ as “3 out of 4. And it’s part of the reason why students have so much trouble with fractions later on.” I wonder whether this is the sort of mistake that you’d see more of if you use this sort of language in class.
• I bet that this student would have trouble if asked to place these fractions on the number line.
• I don’t really get the task. Are these supposed to add up to 1? What’s the unit here?
• I don’t think that the student believes that 3/5 is more than 2/3. I’d bet that this student just isn’t paying care to the relative sizes of boxes. This aspect of the representation just doesn’t strike her as significant.
• I also notice, though, that within each activity there’s an attempt to make the boxes the same size.

I think that this last observation might be a way into a line of questioning that could help. I’d point to a shaded in box (maybe the kickball one) and ask, “What does this mean?” And then I’d point to another box and ask the same question. This would force us to bring out the unit, and the comparisons between the shaded boxes would force us to have a conversation about the relative amount of time spent at each activity. This would naturally bring us into ranking, which I think would be a good follow-up activity.

There are a bunch of interesting things here — please comment on them — but one moral I’ll take out of this is that learning math often involves becoming sensitive to nuances that would otherwise seem irrelevant.