Supporting Rigorous Mathematics Teaching and Learning

1. Supporting Rigorous Mathematics Teaching and Learning Strategies for Scaffolding Student Understanding: Academically Productive Talk and the Use of Representations Tennessee Department of Education High School Mathematics Algebra 2

2. Rationale Teachers provoke students’ reasoning about mathematics through the tasks they provide and the questions they ask. (NCTM, 1991) Asking questions that reveal students’ knowledge about mathematics allows teachers to design instruction that responds to and builds on this knowledge. (NCTM, 2000) Questions are one of the only tools teachers have for finding out what students are thinking. (Michaels, 2005) Today, by analyzing a classroom discussion, teachers will study and reflect on ways in which Accountable Talk® (AT) moves and the use of representations support student learning and help teachers to maintain the cognitive demand of a task. Accountable talk® is a registered trademark of the University of Pittsburgh.

3. Session Goals Participants will learn about: • Accountable Talk moves to support the development of community, knowledge, and rigorous thinking; • Accountable Talk moves that ensure a productive and coherent discussion and consider why moves in this category are critical; and • representations as a means of scaffolding student learning.

4. Overview of Activities Participants will: • analyze and discuss Accountable Talk moves; • engage in and reflect on a lesson in relationship to the CCSS; • analyze classroom discourse to determine the Accountable Talk moves used by the teacher and the benefit to student learning; • design and enact a lesson, making use of the Accountable Talk moves; and • learn and apply a set of scaffolding strategies that make use of the representations.

5. Review theAccountable Talk Features and IndicatorsLearn Moves Associated With the Accountable Talk Features

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

7. 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

8. Accountable Talk Discussion • Review the Accountable Talk features and indicators. • Turn and Talk with your partner about what you recall about each of the Accountable Talk features. • Accountability to the learningcommunity • Accountability to accurate, relevant knowledge • Accountability to discipline-specific standards of rigorous thinking

9. Accountable Talk Features and Indicators Accountability to the Learning Community • Actively participate 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. Accountable Talk Moves Consider: • In what ways are the Accountable Talk moves different in each of the categories? • Support Accountability to Community • Support Accountability to Knowledge • Support Accountability to Rigorous Thinking • There is a fourth category called, “To Ensure Purposeful, Coherent, and Productive Group Discussion.” Why do you think we need the set of moves in this category?

11. Accountable Talk Moves

12. Accountable Talk Moves (continued)

13. Accountable Talk Moves (continued)

14. Pictures Manipulative Models Written Symbols Real-world Situations Oral & Written Language Five Representations of Mathematical IdeasWhat role do the representations play in a discussion? Modified from Van De Walle, 2004, p. 30

15. Language Context Table Graph Equation Five Different Representations of a FunctionWhat role do the representations play in a discussion? Van De Walle, 2004, p. 440

16. Engage In and Reflect On a LessonMissing Function Task

17. Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

18. The Cognitive Demand of the Task Why is this considered to be a cognitively demanding task?

19. 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.

20. The Common Core State Standards (CCSS) Solve the task. Examine the CCSS for Mathematics. • Which CCSS for Mathematical Content will students discuss when solving the task? • Which CCSS for Mathematical Practice will students use when solving and discussing the task?

21. The CCSS for Mathematical ContentCCSS Conceptual Category – Number and Quantity Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

22. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

23. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

24. The CCSS for Mathematical ContentCCSS Conceptual Category – Functions Common Core State Standards, 2010, p. 70, NGA Center/CCSSO

25. 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.

26. Analyzing a Lesson: Lesson Context The students and the teacher in this school have been working to make sense of the Common Core State Standards for the past two years. The teacher is working on using the Accountable Talk moves and making sure she targets the mathematics standards in very deliberate ways during the lesson.

27. Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

28. Instructional Goals Jamie’s instructional goals for the lesson are: • students will multiply two linear functions using their graphs or tables of values and recognize that, given two functions f(x) and g(x) and a specific x-value, x1, the point (x1, f(x1) ∙ g(x1)) will be on the graph of the product f(x) ∙ g(x); and • students will recognize that the product of two or more linear functions is a polynomial function having the same x-intercepts as the original functions because the original functions are factors of the polynomial.

29. Reflection Question(Small Group Discussion) As you watch the video segment, consider what students are learning about mathematics. Name the moves used by the teacher and the purpose that the moves served.

30. Reflecting on the Accountable Talk Discussion(Whole Group 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? How does talk scaffold student learning?

31. Pictures Manipulative Models Written Symbols Real-world Situations Oral & Written Language Five Representations of Mathematical IdeasWhat role did tools or representations play in scaffolding student learning? Modified from Van De Walle, 2004, p. 30

32. Language Context Table Graph Equation Five Different Representations of a FunctionWhat role did tools or representations play in scaffoldingstudent learning? Van De Walle, 2004, p. 440

33. Giving it a Go: Planning for an Accountable Talk Discussion of a Mathematical Idea Identify a person who will be teaching the lesson to others in your small group. Plan the lesson together. Anticipate student responses.  Write Accountable Talk questions/moves that the teacher will ask students to advance their understanding of a mathematical idea. 

34. Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

35. Focus of the Discussion Plan to engage students in a discussion focused on recognizing that the product of any two linear functions, not just functions that produce parallel lines, must be a parabola and a polynomial of degree 2 by using graphical and algebraic representations. You may choose to use the student work to the right to begin the discussion.

36. Reflection: The Use of Accountable Talk Moves and Tools to Scaffold Student Learning What have you learned?