Math Class Discussion
Posted: February 8th, 2012 | Author: Michael Goldstein | | 4 Comments »
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.
Of course I would respond to this.
I totally agree with Will; this is good stuff.
However, I think you cut off the most important point. Discussion can only happen when there is something meaningful to discuss.
In the humanities, this can be a debatable question, like “Should the US have used the atomic bomb to end WWII?” It’s hard to have a debate on a closed question like “How did WWII end in the Pacific?”
In math, as the authors so kindly note, the best way to have a discussion is to present a challenging problem that students have to grapple with independently first. (It can also have a debatable question, where appropriate.)
Of course, writing these kinds of questions is hard. The ideal question has multiple entry points and is challenging enough to keep students working, but not too hard that students give up quickly. If the question is perfect, then the discussion needs to be managed well. But since most of the time, there is some flaw to the author’s grand scheme, the teacher needs to be able think on the fly. For example, what do you do if every kid is on a wild goose chase? Or, what happens if they all get the right answer?
Another, unstated, issue is time. Discussions take time and are less linear than traditional lessons, so more time is spent getting to the same point. And then there still needs to be practice, summarization / codification, etc. That’s pretty tough to do in a 42 minute period, if not impossible.
I aim to start every class with a discussion; this method doesn’t always fit with the point of a lesson, but I personally think it should be the default for math teachers.
Looking forward to trying it out with my classes.
@mathteacher
I’m not sure a provocative question is necessary. Present multiple ways of solving an otherwise mundane problem and debate + connections + discussion about multiple representations will occur naturally.
Take something like 5(x+3) =15. On on side of the page, you solve it in the traditional way: distribute, subtract, divide. On the other side of the page, you divide both sides by 5 first, leaving you with x + 3 = 5. Students debate the merits of both ways- which is more efficient, which will always work smoothly, etc. Here, you’re changing the theme of the class from “here’s how you solve this” to “what’s the best way to approach this?”
This type of comparison is typically done with, say, systems of equations. Students will compare substitution and elimination. This is not enough, though, and there’s good evidence that it can be done with every skill within a content domain. What’s interesting about this strategy is that it requires very little adjustment on the teacher’s side. You’re still teaching the same concepts, only *constantly* coercing students to consider multiple methods. The questions that Mike posted above arise naturally. Students become more procedurally (an in turn, conceptually) flexible.
Bethany Rittle-Johnson (Vanderbilt) and Jon Star (Harvard) have done phenomenal work on this:
http://gseacademic.harvard.edu/contrastingcases/index.html
@Sean
Thanks for sharing. I don’t disagree with you.
“What’s the best way to solve this problem?” can be provocative, meaningful question, because just like a problem with multiple entry points it forces students to evaluate and synthesize.
What I was trying to make clear is that you can’t just take a traditional “I do, we do, you do” format and think that you can stop someplace in the middle and have a discussion. There has to be a point at which there is something meaningful to discuss. Otherwise the discussion will fall flat and feel useless in the flow of the lesson, and the teacher will reject using the method in the future.
Which method is better is a perfectly valid question to discuss…