I had a lovely experience last spring after a workshop with elementary teachers in a nearby school district. I thought I’d share.
It began with saying yes to a teacher’s idea, and is such a perfect example of how saying yes can plant a seed that grows into a problem people can’t let go.
Early in the workshop I shared some unit chats, which remain my favorite opener. (You can see my writeup on unit chats here, along with images). One of the unit chats I shared was from Lee Dawson; it’s an arrangement of 21st Century Pattern Blocks (from Christopher Danielson) that gives a lot of things to see and count. In unit chats, students say how many they see, which requires that they identify a unit as well. How many triangles/rhombuses/holes/blocks/etc. are there? Depending on your choice of unit, there are lots of choices of what to count.
One teacher, perhaps jokingly, said he saw “1 square” in the picture. The whole thing is a square!
It’s easy to wave away these kinds of comments. After all, I wanted people to warm up by counting something that was tricky to count. If there’s just 1 square, there’s not much counting to do.
But we should beware saying no and shutting down the contribution. Students often test teachers to see how they will or won’t accept ideas. There’s an enormous strength in taking even silly contributions seriously, because it sets the tone for what you expect, how you intend to react, and how students are expected to react. So I said: “One square… very interesting! Except now I’m wondering… is that really a square? How do you know?”
And of course, there followed a spirited debate, since determining whether this shape is actually a square is beyond the purview of elementary mathematics, meaning people had hunches but didn’t have immediate access to the kinds of tools that would convincingly answer the question. After a few minutes I cut off the conversation to focus us in another direction, but I left the question open. (See: Don’t be the answer key.) And because it was unresolved, there were a few people who couldn’t let it go.
The fundamental issue is whether the height of three rhombuses (each made from two equilateral triangles) is the same as the width of five rhombuses. A table of teachers recreated a version of the image from pattern blocks later on.
And just to be clear, we’re done with the session. All the other teachers have gone home. But there’s a group that can’t let it go. They’re pulling out mathematical tools that they haven’t used in a long time, like algebra and the Pythagorean theorem, and they’re attacking this problem with everything they’ve got. They want to know if this thing is a square or not. And they’re not leaving until they figure it out.
With their permission, I taped them as they worked through the problem. Here they are translating the problem into algebraic equations.
And here they are deciding that, as long as they’re choosing a number to set the scale in their drawings, 2 might be a better choice than 10.
I’m happy to say that they solved the problem to their satisfaction in the end. I’ll leave it open in case you’d like to try too.
The deeper takeaway for me here is that I like it when people decide that some problem is theirs. I like when it becomes personal. And look at how focused they are, and how they’re valuing and testing each others’ ideas as they try to work it out. Look how they’re having fun even as they’re struggling!
Part of doing this is to realize that this level of “stay-in-from-recess”/”don’t go home at the end of the sessions”-type focus isn’t something you can guarantee. You have to find the opportunities when they arise, and nurture those seeds as they get planted. I’ve realized that I’m always trying to notice those seeds when they get scattered in my classes, and trying to help them find soil to take root. Because when you can’t let it go, that means it belongs to you.