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.

–Henri

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Henri,How did you work on the 3-d solution? I mean did you have any physical models you were working with? I am just curious.-RC

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David has translucent interlocking centimeter cubes in his classroom, so yes, we made models and discussed them. In 2D, it is helpful to have a method for counting how many square boundary lines you cross on your way to (p,q). The 3D equivalent is to sort out how many cube faces you cross on your way to (p,q,r). In 2D, you have to pay attention to the situation where p and q have a common factor. For example, on your way to (10,6), you will pass through (5,3). In 3D the plot thickens considerably, because all three may have a common factor, but even if they don't, any pair of them may have a common factor. Not only that, but you can have a situation where p and q have a common factor, and q and r have a different common factor. Each of those situations will reflect a geometric situation involving the crossing of faces. For example, in the case where all three have a common factor, your diagonal passes through a cube vertex, and crosses three boundary planes at once.–Henri

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