This is the third and final installment of my notes from the Bay Area fall meeting of Escape from the Textbook!
(Part 1) (Part 2)
We ended the meeting with a segment led by Avery Pickford. (See his notes about the meeting.)
He presented this problem:
(He didn't present it exactly like this -- this is how the problem appears in Algebra: Themes, Tools, Concepts. The whole book is available for free here.)
Since I had offered this problem many times to students and teachers, I chose to work with Bree Murray on a generalization. David Louis suggested we go 3-D, which was a great idea. How many unit cubes does a line go through as it connects (0,0,0) to (p,q,r)? (The latter is a lattice point.)
Working on this turned out to be very fun and satisfying, though in retrospect I think we didn't fully solve the problem. Still, we have a partial solution, and we had to bring to bear many habits of mind (as outlined by Avery -- habits of mind was the theme of this segment). Using translucent interlocking cubes helped us get a handle on the problem, but the main issues turned out to be about numbers. I won't say more here.
And speaking of habits of mind: we met in David Louis's classroom at the SF Friends' School. In addition to student work on various problems, the walls feature posters on "thinking like a mathematician", "how to be successful in this classroom", "Solving a problem", "Listen, Understand, Deepen". That is one way to make habits of mind explicit to students. Something I intend to suggest to my colleagues at the Urban School, even though we don't have "our own" classrooms. We might have students create such posters after a teacher-led discussion. Then the posters will be visible to other math classes who meet in the same room.
In Avery's words, habits of mind are part of "math as a verb", but our job also involves teaching "math as a noun", the various skills and understandings of mathematical subject matter that society expects. David's closing comment was about the need to develop "math as a verb" activities that directly link with "math as a noun". Such activities can serve as anchors to our units. Amen! This is precisely what much of my work as a curriculum developer has centered on.