On January 11, Internet celebrity and Stanford grad student Dan Meyer worked with the Math Dept at the Urban School, plus a handful of members of Escape from the Textbook! He presented his thoughts about how to design and present modeling problems, but in fact his ideas have broader applicability.
I had already seen Dan's TED talk, and heard him give a keynote presentation at an NCTM institute. (See some notes I took about the latter here.) So I more or less knew what to expect. Still, it was a very worthwhile afternoon, as I collected some important and powerful ideas, which will help me as I teach and design curriculum.
The biggest idea for me was that when we map out problems, and more generally lessons, we should give them a three-act dramatic structure.
Act 1 is largely about generating perplexity. It is often visual -- the words and numbers come later. A successful Act 1 yields many student questions. (Hopefully, the main questions are the ones you intend to tackle.) The example he used with us was a photo of a museum exhibit where the walls were entirely covered with dollar bills. This triggered many great questions, and obviously the main one was how much money was there. If students are genuinely curious, they will be much more engaged, and more likely to learn something from the lesson. (To quote myself: "you cannot answer questions they don't have.")
Act 2 involves building a strategy to solve the problem: figuring out what information will be needed, and using it on the way to a solution.
Act 3 is an opportunity to debrief, figure out what was learned, and perhaps to pursue some extensions.
This is to some extent the structure of many of the lessons I have created over the years. For example, in Algebra: Themes, Tools, Concepts, many of the lessons are launched with an Exploration, which is intended to be worked on with the book closed. The rest of the lesson typically offered some scaffolding to help students head to a solution, after they have grappled with the problem on their own, in groups, or in a whole-class discussion. In that book, my Act 3's were strong on extensions, but weak on making the mathematical structures and techniques explicit.
Later, in Geometry Labs, I came up with a strong structure for Act 3 (the Discussion Questions), but many of the labs were weak on Act 1, and directly plunged into strongly guided activities. Guiding too much and too early ("by the nose") is the Achilles' heel of guided discovery, the curriculum style I usually favor.
More generally, organizing lessons from whatever source into the 3-act structure is likely make them more effective. Learn more about this: Dan Meyer's workshop is archived here.
Break a leg!
PS: One other thing to think about, also raised by Dan in his workshop, is
whether paper is the best medium for all math lessons. He points out
that for some activities, there are benefits to putting the computer at
the center. Doing so allows the use of video (and of course, math
software), and moreover it gives us the possibility of making more data
available off the Web and other sources.