I attended the California Math Council meeting last Saturday. This post is a report on one talk I attended. It was given by Scott Farrand, a prof at Cal State University Sacramento. (I also reported on one of his talks last year.)
This year's talk was called "Think First", which can be interpreted a few ways, all of them relevant.
His idea is that warm-ups should not be review. Instead, they should be challenging (or at least interesting) problems, tightly focused on the day's lesson. The intention is to reveal a key idea, to generate curiosity, or in some cases to dispose of a necessary digression up front. Students are not being assessed, and they can discuss the problems with their neighbors.
Here is an example he gave:
Students are stunned to learn that everyone in the class gets the same slope. This sets the stage for proving that the slope between any two points on a given line is always the same, no matter what points you pick. This is a an example of using a warm-up to generate curiosity, and it is sure to make for a vastly superior level of engagement in the day's lesson.
He has found several benefits to this kind of warm-ups:
- It has students thinking right at the start of class
- It is a great lesson-planning tool, as it helps you think of what the key ideas are
- Sharing the warm-ups with colleagues who teach other sections of the same course is "subversive", as it helps move the culture in their classrooms with little effort on their part
In fact, this is a great professional development tool, as it puts change within reach: just coming up with five minutes' worth of exploration will get you started on the way to a more student-centered classroom. That is a lot more likely to happen than somehow making huge changes all at once.
Someone was worried that if the problems are too hard, students will not do anything. He said "Be bold -- don't worry about too hard." If it's too hard, you can provide hints, or offer an easier version of the problem, either to individuals or small groups as you circulate, or to the whole class. On the other hand, if it's too easy, you've wasted precious time and gained nothing.
One practical point he made was that if he writes the problem on the board, some students will spend the whole time just copying it down. Better hand it out on paper, and circulate to make sure everyone gets started right away. Another preactical point is that the impact is much reduced if you don't do them at the start of class.
In the Algebra: Themes, Tools, Concepts textbook I co-authored, many lessons start with an "exploration". The intent was similar to Farrand's version of the warm-up, but unfortunately, the explorations in ATTC often turned out to be too big and time consuming, and to require too much teacher leadership. Part of the beauty of Scott's concept is that while the warm-up problem is often challenging, it does not take a long time.
PS: I also attended two excellent math talks, both of them using GeoGebra:
- Agnes Tuska guided us to discover, step by step, the quadrature of an arbitrary polygon. (In other words, a strategy to construct a square that has the same area as any given polygon.)
- Ted Courant presented many results about various curves as envelopes of families of lines. This involved many beautiful images, and made me want to go back to the "spirograph" GeoGebra figures I created last summer, and see about enhancing them to explore the ideas he presented.