1. TEAM-Math and AMSTI Professional Mathematics Learning Communities Building Classroom Discourse

2. Goals for Today’s Session To better understand: How to analyze and organize student thinking in order to promote discourse. Teacher and student actions that support a discourse-rich classroom environment.

3. The Typical Math Classroom Discussion in groups: What is the daily routine for the typical mathematics classroom? What is the teacher’s role? What is the students’ role? What are the limitations of this organization? What might be an alternative approach?

4. Discourse What do we mean by “discourse”? From Principles to Actions: Mathematical discourse includes the purposeful exchange of ideas through classroom discussion, as well as through other forms of verbal, visual, and written communication.

5. Advantages What are the advantages to a classroom that focuses on building student discourse? From Principles to Actions: The discourse in the mathematics classroom gives students opportunities to: – share ideas and clarify understandings, – construct convincing arguments regarding why and how things work, – develop a language for expressing mathematical ideas, and – learn to see things from other perspectives.

6. Phases of a Lesson Launch – full group introduction to a worthwhile task Explore – generally in small groups Share and Summarize – full group Apply/extend – small groups or individually

7. Five Practices for effectively using student responses in classroom discourse Anticipating student responses prior to the lesson Monitoring students’ work on and engagement with the tasks Selecting particular students to present their mathematical work Sequencing students’ responses in a particular order for discussion Connecting different students’ responses and connecting the responses to key mathematical ideas

8. The Candy Jar Solve the problem in as many ways as you can. Analyze the sample student solutions Discuss the sequence in which you would have students present their solutions

9. What Happened… Mr. Donnelly monitors his students as they work in small groups on the Candy Jar task, providing support as needed and taking note of their strategies. He decides to have the groups who created solutions B, A, and D present their work (in this order), since these groups used the strategies that he is targeting (i.e., scaling up, scale factor, and unit rate). This sequencing reflects the sophistication and frequency of strategies (i.e., most groups used a version of the scaling up strategy, and only one group used the unit rate strategy).

10. What Happened… During the discussion, Mr. Donnelly asks the presenters to explain what they did and why, and he invites other students to consider whether the approach makes sense and to ask questions. He makes a point of labeling each of the three strategies, asking students which one is most efficient in solving this particular task, and he poses questions that help students make connections among the strategies and with the key ideas that he is targeting. Specifically, he wants students to see that the scale factor is the same as the number of entries in the table used for scaling up. In other words, it would take 20 candy jars with the same number of Jolly Ranchers and jawbreakers as the original jar to make the new candy jar. Mr. Donnelly then will have his students compare this result with the unit rate, which is the factor that relates the number of Jolly Ranchers and the number of jawbreakers in each column of the table in solution 1 (e.g., 5 × 2.6 = 13, just as 55 × 2.6 = 143, just as 100 × 2.6 = 260).

11. What Happened… Toward the end of the lesson, Mr. Donnelly places solution C on the document camera in the classroom and asks students to decide whether or not this is a viable approach to solving the task and to justify their answers. Mr. Donnelly gives the students five minutes to write a response, and he collects their responses as they leave the room to go to the next class. He expects their responses to give him some insight into whether they are coming to understand that for ratios to remain constant, their numerators and denominators must grow at a rate that is multiplicative, not additive.

12. Teacher and Student Actions Look at the chart: – How are the teacher and student actions connected? – Which of these do you experience on a consistent basis? – Which of these are more challenging to do? What connections to the Standards for Mathematical Practice do the Student Actions suggest?