Embracing “Hairy” Math Problems

Recently in math class I described the lesson of the day as a “hairy” math problem. A few of the kids recognized the colloquialism, drumming up images of tangled and knotted hair after a surfing session, or those tumbleweeds found under the couch during spring cleaning. 

It’s always a joy when classwork is worthy of the “hairy math problem” moniker. Oftentimes, day-to-day math lessons are so focused or simplified that the exercises become mundane. They don’t bring a lot of pop and the energy in the room can lag. 

This isn’t the case if we allow the scope to widen a bit. As Dan Meyers framed it in his August 30 math blog, it adds life to a math concept if we replace this:

With this video of a few pallets of bricks:

To launch our year of pre-algebra, I’ve given the students a healthy dose of work that’s more dynamic perhaps than students expected. Students are responding with interest, enjoying the sense of creativity and freedom found in this kind of work. 

How are they showing up?

With so much talking, especially after they get their bearings. Initially, the conversations surround making sense of the problems. They take their first stabs at a solution. Soon, moans of exhaustion can be heard as they realize they’ve approached it all wrong. I love it because these complaints come with a smile, ones that say, Thanks for challenging us in this good way!

The video shows various ways students document their work. Hairy math problems provide students the freedom to work from different angles and use different tools. The diverse approaches deepens everyone’s understanding, as people comment Cool, I didn’t think of it like that! As professor Jo Boalar says, “My research on math learners suggests that when students think they’re in class to learn—to explore ideas and think freely—they understand more and achieve at higher levels than when they think the point is to get questions right.” (link to full article, link to more about mathematical freedom)

I also love the irreverence for tidiness that comes as kids sink deeper into the problem. Check out the whiteboard that Ginger and Ozzie filled up with gobs of calculations. 

Because of the nature of the problem, they needed a nudge to head towards a more accurate solution. The blue table on top was a scaffold for them to incorporate the step they left out. I left them to it and was pleased to see they had finished the first and then scrawled another table for the next part. They were bright eyed with the satisfaction that comes from understanding something complex.

(side note: at the end of a recent 3 day conference that the middle school math department attended, I had a similar feeling when I stepped back from my seat and took in the butcher paper that had been covering my table. Everything within arm’s reach was a mess of markings and calculations. It was a physical product of the fun I had been having with math. I could see Ozzie and Ginger felt something similar).

Lastly, I love these problems because they encourage kids to invest in mathematics. Other disciplines come with inspiration baked in: compelling narratives, fist-raising injustices, or bubbles in beakers. Inspiration in mathematics is found from within as the mind looks to explain an alternative reality, one with agreed upon rules and curious situations that puzzle us into caring. 

As evidence of this caring, Cal asked during academic advisory after class, “Hey Michael, can I study Ozzie and Ginger’s board? I want to understand what they did.” He paced in front of their work with various I’m thinking hard postures while I conferred with other students. In a bit I saw him at a table, having moved onto something else. From the bright, satisfied look he carried, it was clear that he, too, had come to a resolution with the hairy math problem of the day.