Will Austin is C.O.O. of Uncommon Schools Boston. He moonlights teaching a math methods course for our teacher residency. Our trainees heart him. However, they haven't seen him since his new, more grown-up haircut. I don't know if this will slice into his popularity. It may just be they heart him because he's unbelievably good at teaching novice math teachers.
Will sent along this article -- summarized by Kim Marshall of The Marshall Memo -- which written by a team from University of Michigan. I've blogged before about UM; they're cooking up some good stuff there.
From Kim's summary:
Leading Effective Math Discussions with Students
“Discussions are a central component of mathematics instruction,” say University of Michigan researchers Timothy Boerst, Laurie Sleep, Deborah Ball, and Hyman Bass in this Teachers College Record article. “Successful discussions require substantial teaching skill. This is because students must be helped to engage in complex mathematical practices such as giving explanations, making connections, and using representations, and, at the same time, teachers’ moves must be contingent on what students say and do. Furthermore, leading a discussion requires mathematical knowledge for teaching, given that teachers need to size up mathematical ideas flexibly, frame strategic questions, and keep an eye on core mathematical points.”
"Discussions" in math class are not easy to come by. A lot of traditional math classes:
1. Teacher explains procedure (like how to find area of a circle)
2. Ask kids many questions with short answers ("What is the radius? Joe? What happens when we square it? Keisha?")
3. Grind out some problems.
The UM team suggests questions which generate discussion:
• Launch and orchestrate the discussion: initial eliciting of students’ thinking: - Does anyone have a solution they would like to share? - How did you begin working on this problem? - Does someone have a different idea? - What have you found so far? - Did anyone approach the problem in a different way? • Prove students’ answers, try to figure out what a student means or is thinking, check whether right answers are supported by correct understanding, probe wrong answers to understand student thinking: - How do you know? - So what you’re saying is ____ - When you say ____, do you mean ____? - Could you explain a little more about what you are thinking? - Why did you ____? - How did you get ____? - Could you use some concrete materials to show us how that works? • Focus students to listen and respond to others’ ideas: - What do other people think? - How does what ____ said go along with what you were thinking? - Who can explain this using ____’s idea? - Would someone be willing to add on to what ____ said? • Support students to make connections, for example, between a model and a mathematical idea or a specific notation: - How is ____’s method similar to (or different from) ____’s? - How does one representation correspond to another representation? - Can you think of another problem that is similar to this one? - How does that match what you wrote on the board? • Guide students to reason mathematically – making conjectures, stating definitions, generalizing, proving: - Can you explain why this is true? - Does this method always work? - What do these solutions have in common? - Have we found all the possible answers? - How do you know it works in all cases? • Extend students’ current thinking and assess how far it can be stretched: - Can you think of another way to solve this problem? - What would happen if the numbers were changed to ____? - Can you use this same method to solve ____?
As Will says, this is a good cheat sheet. File it away.