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## Finding the Area of a Rectangle [Desmathmistakes Activity]

Here’s the work of a 4th Grader named Jaden. He has a lot of interesting ideas for finding the area of a rectangle. What do you notice in his work? What do you wonder?

When I asked teachers this question as part of a Desmathmistakes activity, there were a lot of interesting responses. While all sorts of observations about student work are valuable, it can be especially valuable to transform our observations about student thinking into some next step. (Researchers look at work as an end in itself. When teachers look at student work it’s almost always to evaluate it or to figure out what to do next in class. We’re doing the latter here.)

Here were three of my favorite responses to the activity, with thanks to (in order) Mary, K, and Cindy.

In case you’re curious, here is everybody’s rectangles:

Finally, on twitter Kristin Gray is thinking in a different direction:

Kristin’s idea is for a string of area calculation problems that all total to to the same area, but are partitioned in different ways:

Some meta-questions: What were people thinking about during this activity? What were they doing? Were they learning something? Could they be learning something?

Jump into the comments if you have some thoughts about Desmathmistakes Experiment #2.

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## Centimeters to Meters

What’s an example of some feedback that you think a teacher might consider giving, but is not the ideal response?

What feedback would you give this student on the page?

If you had five minutes to work with this kid one-on-one, what would you talk about?

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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.

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## The average is whichever thing has the most

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.)