A lovely third grader pointed to her page.
—
Her: Is this how you write a quarter?
Me: How many quarters are there in a whole?
Her: Four. How do you write a quarter?
—
Isn’t that interesting? What do you think is going on here?
Category: Elementary School
I’m stumped. Ideas?
(Thanks to Avery for the submission!)
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)
Tina reflects:
“Kids seem to forget their skills when the level is increased. They revert back to intuition like “subtract numerators, subtract denominators” when faced with trig functions, but as soon as I ask “how do you subtract fractions?” they immediately recall “common denominators.” How do we get them to stop and think when the cognitive load is higher?”
That’s a great question.
Eric writes in with this really cool mistake.
Do you see the problem?
Of course you don’t, because you don’t know the original question.
Eric writes, “The problem was originally 12 2/3 + 8.”
Explain? EXPLAIN?
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.
One of the joys of teaching fractions for the first time is seeing mistakes from my students that, prior to now, I’ve only read about.
(Does that make me a dork?)
(Yes, Michael, that makes you a dork.)
Anyway, the first mistake is an absolute classic. With numbers that show any sort of structural complexity, students regularly treat the components individually. See, decimals, complex numbers, polynomials, and other things I’m sure. (If you have links or suggestions of other sorts of common mistakes that fit this paradigm, drop a comment in the comment hole.)
The second mistake pushes on their part-whole understanding in a direct way. I’m not sure how this mistake plays itself out beyond the geometric context. Thoughts, people?
The third mistake is an interesting one, and one that I suspect is about being used to seeing number lines only of unit length before. Other than that, I don’t have much interesting to say about that. But maybe you do?
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.
You can’t say that the kid is incapable of understanding what the box means here. Still, in the space of one line, it slipped through her fingers.
Is this connected to the way kids inconsistently treat exponents? I’m struggling to articulate a general principle, but it goes something like “Operations defined in terms of others are strongly associated with their parent operation, to the point that students often perform the parent in place of the derivative operation. As a result, students should always be introduced to a new operation in its own context, not in terms of other operations, whenever possible.”
Thoughts?
Open thread. Talk about what you find interesting here, or anything. I’ll kick things off in the comments.