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 don’t have a picture for this, but every single one of my 4th Graders thought that “average” meant “the most common thing.”

(a) Where do they get this idea from?

(b) Is it a big deal misconception?

(c) How do you create a need for something besides “most common”?

(I think I have my own answers for (a) and (c), but I’m more curious to know what you guys all think.)