Planning a mathematical discussion
Many teachers are now working to involve students in more meaningful mathematical discussions, where students are expected to challenge each other's comments, justify their reasoning and make mathematical connections. This work, however, can be difficult. In an article in Mathematics Teaching in the Middle School, Mary Kay Stein says that a key element of leading a good discussion is finding tasks that are likely to promote disagreement among students. She goes on to say that “Even with good tasks, however, some teachers have found that classroom discussions can fall into a rut. Teachers always ask students to ‘explain their thinking'; students always ask one another 'why?' and students know that their answers are correct when the teacher stops asking questions.”
Stein suggests that a teacher's job in generating a meaningful discussion is to find places where students are likely to disagree about the mathematics, to frame the discussion to highlight those disagreements and to help students take ownership of various solutions. This tool will help you plan a discussion by considering the appropriateness of the task, possible sources of disagreement and interventions you might make to highlight that disagreement.
Planning a mathematical discussion
The task
 Are students likely to come up with different answers to or interpretations of the problem?
 What answers or explanations are students most likely to produce?
 How could the task be modified to encourage students to produce different interpretations?

Sources of disagreement
 What are the most likely sources of disagreement about answers or method?
 Which disagreements might be most mathematically productive to pursue?
 What big ideas will you want students to take away from the discussion?

Framing the discussion
 How will you monitor student work so you know what potential solutions are available before the discussion starts?
 How will you decide which solutions to present to the class?
 What language will you use to help students take ownership of solutions?
 How will you know when the discussion is over? What will you say to wrap it up?

Some helpful language...
For setting up the discussion:
 Did everyone understand the argument? Can someone resay the argument?
 Does everyone agree with ___? If not, I should see your hand up.
 How sure are you100%, 40%? What would it take to convince you?
 Put names with various solutions so ownership is distributed across the class.
For moving it along:
 Would someone like to show a different way?
 Can both these solutions be correct?
 Who can justify one of the claims?
For wrapping up:
 How many of you agree with this claim? Can you say why?
 Why did you change your mind?
 Which argument was most convincing? Why?

Source: Stein, M.K. (2001). Mathematical argumentation: Putting umph into classroom discussions. Mathematics Teaching in the Middle School, 7(2), 110112.
