This child made it clear that

• She knew that an array was a rectangle
• That this was technically a rectangle
• These super-long folks were not arrays, or at least she didn’t think they were, because they didn’t look like a rectangle
• The 2 x 17 was an array

To what do you attribute this perception? (You can check your answers in the back of the book.)

Two 4th Grade girls were playing “Guess My Rule.” All the green shapes fit Molly’s rule, all the red ones were excluded by her rule. Christi was trying to figure out the rule.

Christi: Oh, I’ve got it! They’re all triangles.

Molly: Nope!

Christi:  But, look, it’s true! They’ve all got three sides.

Molly:  They all have three sides, but they’re not all triangles.

Christi:  WTF?

Molly:  Yeah, L has three sides, but it’s not a triangle.

Why not? What makes this seem untriangley to Molly?

(Relevant: Young Children’s Ideas About Shapes)

I gave the 4th Graders meter sticks today, and (of course) they did all sorts of weirdo things with them. Drumming, whacking things, marching while cradling the meter stick like a rifle.

They were supposed to be measuring the perimeter of the classroom.

One kiddo seemed to be trying to poke the ceiling, but he seemed to be doing it with enough care that I thought he might be measuring something.

Me: Wha?

Him: I’m measuring that jut in the ceiling.

Me: Why?

Him: Because it’s in the way, you’d have to follow it if you were walking on the ceiling.

(Note: I’d previously described perimeter in terms of path. The perimeter is the path you take around some region.)

Me: But you wouldn’t have to go around that thing if you were walking around the room this way, while standing on the floor.

Him: But you would if you were on the ceiling.

Me: But we’re not measuring the path on the ceiling, we’re going on the floor.

Him: Oh, I thought that we were measuring the perimeter of the whole room.

Now, maybe he was just being a punk because he wanted an excuse to poke the ceiling with a long stick. Maybe, though, he had a really interesting interpretation of perimeter, as all the paths that you take around a room. After all, there’s some ambiguity in the way I talked about the perimeter of the room, since the room is a 3D object, and perimeter is usually applied to objects in the plane.

That ambiguity, though, is a feature, not a flaw of the task assigned. Too many perimeter problems that I see young kids do only take place around rectangles or other polygonal shapes. This conversation with the kid was a really interesting one because it pushed on the messy process of finding 2D ways of seeing our 3D world.

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.