Supporting Rigorous Mathematics Teaching and Learning

1. Supporting Rigorous Mathematics Teaching and Learning Using Academically Productive Talk Moves: Orchestrating a Focused Discussion Tennessee Department of Education Middle School Mathematics Grade 6

2. Rationale Mathematics reform calls for teachers to engage students in discussing, explaining, and justifying their ideas. Although teachers are asked to use students’ ideas as the basis for instruction, they must also keep in mind the mathematics that the class is expected to explore (Sherin, 2000, p. 125). By engaging in a high-level task and reflecting on ways in which the facilitator structured and supported the discussion of mathematical ideas, teachers will learn that they are responsible for orchestrating discussions in ways that make it possible for students to own their learning, as well as for the teacher to assess and advance student understanding of knowledge and mathematical reasoning.

3. Session Goals • Participants will: • learn about Accountable Talk features and indicators and consider the benefit of all being presentin a lesson; • learn that there are specific moves related to each of the talk features that help to develop a discourse culture; and • consider the importance of the four key moves of ensuring productive discussion (marking, recapping, challenging, and revoicing).

4. Overview of Activities • Participants will: • review the Accountable Talk features and indicators; • identify and discuss Accountable Talk moves in avideo; and • align CCSS and essential understandings (EUs) to a task and zoom in for a more specific look at key moves for engaging in productive talk (marking, recapping, challenging, and revoicing).

5. Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as implemented by students TASKS as they appear in curricular/ instructional materials Student Learning Stein, Smith, Henningsen, & Silver, 2000

6. Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as implemented by students TASKS as they appear in curricular/ instructional materials Student Learning Stein, Smith, Henningsen, & Silver, 2000 Setting Goals Selecting Tasks Anticipating Student Responses • Orchestrating Productive Discussion • Monitoring students as they work • Asking assessing and advancing questions • Selecting solution paths • Sequencing student responses • Connecting student responses via Accountable Talk® discussions Accountable Talk®is a registered trademark of the University of Pittsburgh

7. Accountable Talk Features and Indicators

8. Accountable Talk Discussion • Study the Accountable Talk features and indicators. • Turn and Talk with your partner about what you would expect teachers and students to be saying during an Accountable Talk discussion so that the discussion is accountable to: • the learningcommunity; • accurate, relevant knowledge; and • standards of rigorous thinking.

9. Accountable Talk Features and Indicators Accountability to the Learning Community • Active participation in classroom talk. • Listen attentively. • Elaborate and build on each others’ ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge. • Appropriate evidence for claims and arguments. • Commitment to getting it right. Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

10. Solving and Discussing the Cognitive Demand of the Light Bulb Task

11. The Structure and Routines of a Lesson • MONITOR: Teacher selects • examples for the Share, Discuss, • and Analyze Phase based on: • Different solution paths to the • same task • Different representations • Errors • Misconceptions Set Up of the Task Set Up the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: Engage students in a Quick Write or a discussion of the process. Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

12. Engaging in a Lesson: The Light Bulb Task • Solve the task. • Discuss your solutions with your peers. • Attempt to engage in an Accountable Talk discussion when discussing the solutions. Assign one person in the group to be the observer. This person will be responsible for reporting some of the ways in which the group is accountable to: • the learning community; • accurate, relevant knowledge; and • standards of rigorous thinking.

13. Engaging in a Lesson: The Light Bulb Task Alazar Electric Company sells light bulbs to big box stores – the big chain stores that frequently buy large numbers of bulbs in one sale. They sample their bulbs for defects routinely. A sample of 96 light bulbs consisted of 4 defective ones. Assume that today’s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today. The big businesses they sell to accept no larger than a 4% rate of defective bulbs. Does today’s batch meet that expectation? Explain how you made your decision.

14. Reflecting on Our Engagement in the Lesson The observer should share some observations about the group’s engagement in an Accountable Talk discussion.

15. Reflecting on Our Engagement in the Lesson • In what ways did small groups engage in an Accountable Talk discussion? • In what ways did we engage in an Accountable Talk discussion during the group discussion of the solutions?

16. Aligning the CCSS to the Light Bulb Task • Study the Grade 6 CCSS for Mathematical Content within the Ratio and Proportion domain. Which standards are students expected to demonstrate when solving the task? • Identify the CCSS for Mathematical Practice required by the written task.

17. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 42, NGA Center/CCSSO

18. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 42, NGA Center/CCSSO

19. The CCSS for Mathematical Practice Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

20. Determining the Cognitive Demand of the Task:The Light Bulb Task

21. Determining the Cognitive Demandof the Task Refer to the Mathematical Task Analysis Guide. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A., 2000. Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press. How would you characterize the Light Bulb Task in terms of its cognitive demand? (Refer to the indicators on the Task Analysis Guide.)

22. The Mathematical Task Analysis Guide Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

23. The Light Bulb Task:A Doing Mathematics Task • Requires complex and non-algorithmic thinking (i.e., there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked-out example). • Requires students to explore and to understand the nature of mathematical concepts, processes, or relationships. • Demands self-monitoring or self-regulation of one’s own cognitive processes. • Requires students to access relevant knowledge and experiences and make appropriate use of them in working through the task. • Requires students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions. • Requires considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.

24. Accountable Talk Moves

25. The Structure and Routines of a Lesson • MONITOR: Teacher selects • examples for the Share, Discuss, • and Analyze Phase based on: • Different solution paths to the • same task • Different representations • Errors • Misconceptions Set Up of the Task Set Up the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: Engage students in a Quick Write or a discussion of the process. Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

26. Accountable Talk Moves Examine the ways in which the moves are grouped based on how they: • support accountability to the learning community; • support accountability to knowledge; and • support accountability to rigorous thinking. Consider: In what ways are the Accountable Talk categories similar? Different? Why do you think we need a category called “To Ensure Purposeful, Coherent, and Productive Group Discussion”?

27. Accountable Talk: Features and Indicators Accountability to the Learning Community • Active participation in classroom talk. • Listen attentively. • Elaborate and build on each others’ ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge. • Appropriate evidence for claims and arguments. • Commitment to getting it right. Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

28. Accountable Talk Moves

29. Accountable Talk Moves (continued)

30. Reflection Question As you watch the short video segment, consider what students are learning and where you might focus the discussion in order to discuss mathematical ideas listed in the CCSS. Identify: • the specific Accountable Talk moves used by the teacher; and • the purpose that the moves served. • Mark times during the lesson when you would call the lesson academically rigorous.

31. The Light Bulb Lesson Context Visiting Teacher: Victoria Bill Teacher: Reginald Coleman School: Community Health Academy of the Heights Middle School District: New York City Schools Principal: Ms. Vu Grade Level: 7th Grade The students in the video episode are in a mainstream mathematics classroom in the New York City Schools. The students are solving the Light Bulb Task. This part of the video captures the Share, Discuss, and Analyze phase of the lesson.

32. Norms for Collaborative Study The goal of all conversations about episodes of teaching (or artifacts of practice in general) is to advance our own learning, not to “fix” the practice of others. In order to achieve this goal, the facilitator chooses a lens to frame what you look at and to what you pay attention. Use the Accountable Talk features and indicators when viewing the lesson. During this work, we: • agree to analyze the episode or artifact from the identified perspective; • cite specific examples during the discussion that provide evidence of a particular claim; • listen to and build on others’ ideas; and • use language that is respectful of those in the video and in the group.

33. The Light Bulb Task Alazar Electric Company sells light bulbs to big box stores – the big chain stores that frequently buy large numbers of bulbs in one sale. They sample their bulbs for defects routinely. A sample of 96 light bulbs consisted of 4 defective ones. Assume that today’s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today. The big businesses they sell to accept no larger than a 4% rate of defective bulbs. Does today’s batch meet that expectation? Explain how you made your decision.

34. Reflecting on the Accountable Talk Discussion Step back from the discussion. What are some patterns that you notice? What mathematical ideas does the teacher want students to discover and discuss?

35. Essential Understandings Study the essential understandings the teacher considered in preparation for the Share, Discuss, and Analyze phase of the lesson.

36. Essential Understandings

37. Characteristics of an Academically Rigorous Lesson This task is a cognitively demanding task; however, it may not necessarily end up being an academically rigorous task. What do we mean by this statement?

38. Academic Rigor in a Thinking Curriculum The principle of learning, Academic Rigor in a Thinking Curriculum, consists of three features: • A Knowledge Core • High-Thinking Demand • Active Use of Knowledge In order to determine if a lesson has been academically rigorous, we have to determine the degree to which student learning is advanced by the lesson. What do we have to hear and see in order to determine if the lesson was academically rigorous?

39. Essential Understandings

40. Language Context Table Graph Equation Five Different Representations of a Function Van De Walle, 2004, p. 440

41. Focusing on Key Accountable Talk MovesThe Light Bulb Task

42. Accountable Talk: Features and Indicators Accountability to the Learning Community • Active participation in classroom talk. • Listen attentively. • Elaborate and build on each others’ ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge. • Appropriate evidence for claims and arguments. • Commitment to getting it right. Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

43. Accountable Talk Moves

44. Accountable Talk Moves (continued)

45. Focusing on Accountable Talk Moves Read the description of each move and study the example that has been provided for each move. What is distinct about each of the moves? • Revoice student contributions; • mark significant contributions; • challenge with a counter-example; or • recap the components of the lesson.

46. Revoicing • Extend a student’s contribution. • Connect a student’s contribution to the text or to other students’ contributions. • Align content with an explanation. • Add clarity to a contribution. • Link student contributions to accurate mathematical vocabulary. • Connect two or more contributionsto advance the lesson.

47. An Example of Revoicing S: —and it gives you 12 or you multiply 4 times 3 because there’s 3 boxes. T: All right, he said you could do 4 times 3 and then—how would you get from here to here, though? (Points to other side of table, the total number of bulbs.)

48. Marking Explicitly talk about an idea. • Highlight features that are unique to a situation. • Draw attention to an idea or to alternative ideas.

49. An Example of Marking S: It looks like they wrote fractions. Like, broken bulbs over total bulbs. 4 over 96 is equal to 250 over 6,000. T: Hmm—did everyone hear what Selena just said? She noticed that when we write the fraction of defective bulbs out of total bulbs, they are both equivalent.

50. Recapping Summarize or retell. • Make explicit the large idea. • Provide students with a holistic view of the concept.