Matt Enlow shared this problem on Twitter this morning:
I had a ton of fun playing with it! It’s one of those problems that takes ideas that I think I understand — in this case, properties of equations and exponentiation — and turns them on their head, forcing me to think in new ways and helping me to better understand math I learned a long time ago. Play with it! There are more solutions than I thought at first. If you’d like a hint, check out the replies on Twitter.
My first instinct when I see something like this is to ask, “How can I engage my students with this problem?” I love math, and I love problems, and I want my students to experience the joy of solving problems. For a long time I would seek out problems like this one, problems I loved, to share with students. But many of those experiences were counterproductive, and I’d like to try to explain why. First, here’s another problem that I recently saw on Twitter and enjoyed playing with:
Give it a shot!
Interlude: Complicated vs Complex
Atul Gawande writes in The Checklist Manifesto about the difference between complicated and complex. Sending a rocket to the moon is complicated. There are lots of little things that have to be figured out and designed and built and work right and lots of people who have to collaborate to put the pieces together. But once we get one rocket to the moon successfully, we can pretty well follow those steps and get another to the moon, and another.
On the other hand, raising a child is complex. There are lots of moving pieces, and lots of nuance and judgment, and raising one child does not mean that raising the next suddenly becomes a task of copying what was done before.
Working with something complicated involves coordinating lots of little things that have to be done right and add up to one big thing. Working with complexity involves much more judgment, subtlety, and responsiveness.
Back to Problems
One reason to give students problems is to teach content. That’s important! But it’s not what I’m interested in here. The problems I give students also send messages about what it means to do mathematics. I worry that the first problem, with the factoring and exponentiation and all of the subtleties embedded in it, sends a message that practicing mathematics is complicated. It sends a message that math involves learning lots of little things and then piecing them together in unusual and contrived ways to figure out new things, but to be successful you have to remember all those little pieces and put them together in just the right way. I think problems like these play out in inequitable ways; students who already have strong skills and a disposition toward making sense of and persevering on a math problem are likely to get some positive reinforcement, and students already disaffected feel confused and left out of the conversation.
I think the dragon problem sends a different message. It invites experimentation and sense-making, and it can be represented lots of different ways, all from a very simple prompt. I think it sends a message that practicing mathematics is complex. Math isn’t easy; it takes originality, depth of thought, and a willingness to try new ideas and take risks. And it has value precisely because it’s not easy, and working through something hard can feel gratifying and fun. But that’s a very different message about the nature of mathematics, and why someone might want to pursue it in the future.
I love both of these problems, and the first problem was still fun for me. I still find it elegant and thought-provoking. I want to design some sequences of problems that get at similar ideas, where students can engage with the idea of exponentiation and the properties of equations. Those might serve a really useful purpose in helping to illuminate deep mathematical concepts that I often hurry past in the high school curriculum. But I only have so much time to engage students with problem solving for the sake of problem solving. For the purpose of helping students see themselves as potential mathematicians and illuminating the depth of what it means to practice mathematics, I think complex, inviting problems are where I want to focus my effort.