1. Iris M. RiggsKelli Wasserman CSUSBMathematics Consultant Orchestrating Discussion: Fractions on the Number Line California Mathematics Conference Palm Springs, California November 7, 2015 Support for this work was provided by the National Science Foundation under grant Due-0962778.

2. Session Outcomes Participants will: Learn how to facilitate productive student-student conversations about math. Gain understanding of SMP 3: Construct viable arguments and critique the reasoning of others. Experience a lesson on using benchmarks to order fractions that embeds SMP 3. Consider how to incorporate talk moves into planning for discussions in mathematics.

3. Agenda Introductions Engage in lesson Debrief What are students expected to be able to do in discussions? What did the teacher do? A closer look at SMP 3

4. Focus Questions How can benchmarks help students make sense of and reason about fractions as a quantity on the number line? How can discussions engage students in the practices of SMP 3? How can productive math discussions promote math understanding? How can you utilize talk moves to facilitate language development?

5. The Lesson

6. Rights and Responsibilities Every student has the right to...Every student has the responsibility to... ●Make a contribution to an attentive audience ●Ask questions ●Be treated civilly ●Have his/her ideas discussed, not him/her ●Speak loudly enough for others to hear ●Listen for understanding ●Treat others civilly at all times ●Consider others’ ideas, then explain agreement or disagreement From Classroom Discussions, by Chapin, O’Connor and Anderson

7. Today’s Discussion Focus Understand how other people are thinking mathematically by restating and confirming your understanding of what they said.

8. Group Work Arrange yourselves in groups of 3 or 4. One of you will act as an observer, and report back to the whole group your observations. Take 2-3 minutes to look at the fractions on the next slide, and independently devise a plan for ordering them. Record your plan. No calculators! Get ready to share your plan. What will you say?

9. Ordering Fractions

10. Share your plans. Share your initial thinking from your plans. Make sure everyone participates. How will you go about placing the fractions on the number line? Consider that you will need to try to place them as close to the point that represents their value as you can.

11. Group Work—Place Your Fractions on the Number Line Distribute the post-its amongst your group members, so that every member has an opportunity to place fractions. Work with your group members to place the fractions on your number line. Be as accurate as you can! As you place your fractions, think out loud to help your group understand how you decided where to place the fractions. What strategies helped you decide? Problem adapted from Activity 13.19, page 301, Guiding Children’s Learning of Mathematics (12th Edition) by Tipps, Johnson, & Kennedy.

12. Share with another Group Once you are satisfied with your solution, collaborate with another group. Next, affix your fractions (index cards) on the string provided using the clothespins. Discussion Practice: Restate what you heard the person before you say. Ask the speaker if what you understood is correct.

13. Ordering Fractions

14. Gallery Walk Take a clean sheet of paper, and fold it into four sections. Walk around and inspect the other groups’ solutions. Use a different section for each group to record comments or questions for our discussion. Consider: What is similar to/different than your solution? Are the fractions in the same order? Are the fractions spaced similarly? Can you identify what strategies were used?

15. Our Math Discussion Focus One speaker from each group will explain how your group decided where to place the fractions. Listen to hear how others reasoned about their fractions. Be ready to restate what you understood someone else to say and then confirm if you understood correctly. Ask questions when you don’t understand someone else’s reasoning.

16. Prompts for Guiding Discussion I understood that you placed the fraction at that point because...Is that right? I don’t understand...Can you repeat your explanation to help me understand? I agree that...but I’m not agreeing with… Did you consider…?

17. Closure What strategies were used to place the fractions? What is a benchmark? What did we learn about the number line and fractions? How did the repeat move work in your discussions? How did the discussion support your understanding of the content?

18. Lesson Debrief

19. Student Discourse Includes: Ways of using language that will allow students to engage more fully in thinking and reasoning. Ways of structuring interactions to support learning goals. Classroom Discussions by Chapin, O’Connor and Anderson

20. What Happened? What kinds of communication did the “students” use? Restatements Extensions Comparisons Agreements/Disagreements Questions

22. Talk Moves Revoice Repeat Apply reasoning Extend Wait time

24. Ask students to restate someone else’s reasoning: The teacher asks a student to repeat or rephrase what another student has said, then follows up with the first student to confirm. “Can you repeat what he just said in your own words?”

28. What was done by the teacher to promote student dialogue? Stated expectations for communication? Modeled or gave examples of restating and confirming Gave verbal reminders regarding communication Responded to individual students? Questioned students about their communications? Waited before calling on students or responding?

30. How did this lesson support development of SMP 3? Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

31. Language Development Benefits English language learners acquire language in two phases: Conversational language Academic language How might the Benchmark Lesson Promote Language Development?

32. Focus Questions How can benchmarks help students make sense of and reason about fractions as a quantity on the number line? How can discussions engage students in the practices of SMP 3? How can productive math discussions promote math understanding? How can you utilize talk moves to facilitate language development?

33. Resources Fraction Comparisons on a Clothesline, TCM Blog, Shelby P. Morge, October 26, 2015: http://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Fraction-Comparisons-on-a-Clothesline/ http://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Fraction-Comparisons-on-a-Clothesline/ Chapin, O’Connor, & Anderson (2009). Classroom Discussions: Using Math Talk to Help Students Learn, Grades K-6, 2 nd Edition Lisa Ann de Garcia, How to Get Students Talking: Generating Math Talk That Supports Math Learning, http://www.mathsolutions.com/documents/How_to_Get_Students_Talking.pdf http://www.mathsolutions.com/documents/How_to_Get_Students_Talking.pdf Chapin, O’Connor, Nancy Anderson (Fall 2003) Classroom Discussions: Using Math Talk in Elementary Classrooms, From Online Newsletter Issue No. 11. http://www.mathsolutions.com/documents/0-941355-53-5_L.pdf http://www.mathsolutions.com/documents/0-941355-53-5_L.pdf

34. Thank you! Please complete your evaluation! Iris Riggs iriggs@csusb.edu Kelli Wasserman kwasserman@sbcglobal.net

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