One nice thing about NCTM was the opportunity to connect with friends and colleagues who I don’t see often enough. Another is running into fans of my work, which happened with some regularity. My fans are few, but they’re enthusiastic, and seeing them in person is a reminder that my work in curriculum development and teacher training is actually worth doing.
Here are some incomplete notes from one of the sessions I attended. Geoff Krall presented suggestions on how to improve textbook problems. (You can get the slides and the handouts on his Web site.) He started by pointing out the well-known advantages of textbooks: mostly that they are already written, and that they are organized by topics. Unfortunately, the problems they offer are often not great. Here are some of his suggestions on how to improve them. (Note: I filtered and interpreted his ideas.)
1. Reverse the problem. Instead of “graph this equation”, ask “what equation could produce this graph”? Instead of “solve this equation”, ask “what are some equations whose solution is 5?” And so on. I’ve long been an advocate of this technique, because it is easy to implement, and yields great activities. It is a guiding principle in any curriculum I design. (See, for example, Make These Designs.)
2. Turn it into a “which would you rather?” problem. (There’s a whole Web site of such problems!) As I see it, this is a subtle improvement on problems such as Comparing Cell Phone Plans, because it’s a better question than “which is cheaper?” For certain questions, it can yield more complicated answers, such as “I’d rather take a cab rather than drive and pay for parking, even though it is more expensive”. It is also more like “real life”, where there are often multiple criteria at play.
3. Given a whole page of boring drill in the book, ask students to pick five problems to do, along with an explanation of why they chose them. And five problems that they really do not want to do, along with an explanation of why. I had not come across this idea before, but I love it. It will be popular, for obvious reasons, but it will also get the students to look at the problems in a meta sort of way — a valuable skill, and possibly the germ of a great discussion.
4. “Spill coffee” on some of the givens, making them illegible. Students choose their own givens, and do their own version of the problem, possibly tailored to their level of expertise.
5. In multistep problems, remove the intermediary steps. (This, as Dan Meyer points out, makes the purpose of the problem a lot clearer.) But don’t throw them away! Use them as hint cards, or as debrief prompts.
6. Ask students to make their own version of the problem. I’ve had some luck, for example, asking students to make their own simultaneous equations word problems. You can pick some for the whole class to solve. Some of the problems will be, well, problematic, and can be the springboard for a discussion.
Nice ideas, no?
PS: I was talked into joining Twitter, where I’m @hpicciotto. Follow me!