I will often bring these rich tasks
to our class for students and I have found that students not only enjoy them, but that these tasks have also been great for developing and reinforcing the "habits of a mathematician."
I usually title these tasks "POWs" (for Problem of the Week). Yesterday, as part of the progression in figuring out the area of the "infinite star" (aka Koch Snowflake) I gave students this problem hoping that we could emphasize the importance of 'taking things apart and putting them back together':
I was pretty surprised by the difficulties students had with this problem. In fact, I had even included a more difficult one in anticipating students blowing right through this one. About 50% of the class was completely stumped by the original problem (which I appropriately named the 'Crooked House'…in case you haven't noticed, I appreciate a cheesy title). I asked them at the end of class to reflect on and discuss strategies that were helpful and even asked them why this problem was so difficult. They said things you might expect:
"we didn't have a formula"
"I needed the distance formula"
"I didn't know where to start"
A little piece of my soul died. We have spent all year de-emphasizing formulas and talking about habits that give you a place to start. Then, this happened:
student: "It was like a POW, but not."
me: "What do you mean?"
student: "Well, it seemed like a POW-type problem but it wasn't one."
me: "So, just because it didn't say 'POW' at the top of the page you didn't approach it in the same way?"
It was clear that students see math compartmentalized into separate worlds. I haven't helped them see that these habits ARE mathematics. They aren't just useful for puzzle-like problems, these habits are at the foundation of the creation and authoring of ALL mathematics. With my last two units I have started to include 1 or 2 of our habits as the central focus. Today's prompt looked like this:
It was a great reminder of the importance of redefining mathematics for students, helping them see connections, and making sure the message we send is clear and consistent in ALL that we do.