1. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH Review the Accountable Talk Features and Indicators Learn Moves Associated With the Accountable Talk Features

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

7. © 2013 UNIVERSITY OF PITTSBURGH The Structure and Routines of a Lesson The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving 1.Generate and Compare Solutions 2.Assess and Advance Student Learning Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3.Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: Different solution paths to the same task Different representations Errors Misconceptions 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. Set Up the Task Set Up of the Task

8. © 2013 UNIVERSITY OF PITTSBURGH 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 learning community - Accountability to accurate, relevant knowledge - Accountability to discipline-specific standards of rigorous thinking

9. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH Accountable Talk Moves Talk MoveFunctionExample To Ensure Purposeful, Coherent, and Productive Group Discussion MarkingDirect attention to the value and importance of a student’s contribution. It is important to say describe to compare the size of the pieces and then to look at how many pieces of that size. ChallengingRedirect a question back to the students or use students’ contributions as a source for further challenge or query. Let me challenge you: Is that always true? RevoicingAlign a student’s explanation with content or connect two or more contributions with the goal of advancing the discussion of the content. You said 3; yes, there are three columns and each column is of the whole. RecappingMake public in a concise, coherent form the group’s achievement at creating a shared understanding of the phenomenon under discussion. Let me put these ideas all together. What have we discovered?

12. © 2013 UNIVERSITY OF PITTSBURGH Talk Move FunctionExample To Support Accountability to Community Keeping the Channels Open Ensure that students can hear each other, and remind them that they must hear what others have said. Say that again and louder. Can someone repeat what was just said? Keeping Everyone Together Ensure that everyone not only heard, but also understood, what a speaker said. Can someone add on to what was said? Did everyone hear that? Linking Contributions Make explicit the relationship between a new contribution and what has gone before. Does anyone have a similar idea? Do you agree or disagree with what was said? Your idea sounds similar to his idea. Verifying and Clarifying Revoice a student’s contribution, thereby helping both speakers and listeners to engage more profitably in the conversation. So are you saying...? Can you say more? Who understood what was said? Accountable Talk Moves (continued)

13. © 2013 UNIVERSITY OF PITTSBURGH Talk Move FunctionExample To Support Accountability to Knowledge Pressing for Accuracy Hold students accountable for the accuracy, credibility, and clarity of their contributions. Why does that happen? Someone give me the term for that. Building on Prior Knowledge Tie a current contribution back to knowledge accumulated by the class at a previous time. What have we learned in the past that links with this? To Support Accountability to Rigorous Thinking Pressing for Reasoning Elicit evidence to establish what contribution a student’s utterance is intended to make within the group’s larger enterprise. Say why this works. What does this mean? Who can make a claim and then tell us what their claim means? Expanding Reasoning Open up extra time and space in the conversation for student reasoning. Does the idea work if I change the context? Use bigger numbers? Accountable Talk Moves (continued)

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

15. Five Different Representations of a Function What role do the representations play in a discussion? Language TableContext GraphEquation Van De Walle, 2004, p. 440

16. © 2013 UNIVERSITY OF PITTSBURGH Engage In and Reflect On a Lesson Missing Function Task

17. © 2013 UNIVERSITY OF PITTSBURGH 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. xf(x) -20 1 02 13 24

18. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH 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 Content CCSS Conceptual Category – Number and Quantity The Real Number System N-RN Extend the properties of exponents to rational exponents. N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

22. The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Seeing Structure in Expressions (A–SSE) Write expressions in equivalent forms to solve problems. A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★ ). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

23. The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Arithmetic with Polynomials and Rational Expressions (A–APR) Understand the relationship between zeros and factors of polynomials. A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

24. The CCSS for Mathematical Content CCSS Conceptual Category – Functions Building Functions (F–BF) Build a function that models a relationship between two quantities. F-BF.A.1 Write a function that describes a relationship between two quantities. ★ F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★ ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★ ). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010, p. 70, NGA Center/CCSSO

25. The CCSS for Mathematical Practice 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

26. © 2013 UNIVERSITY OF PITTSBURGH 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. Teacher:Jamie Bassham Grade Level:Algebra 2 School:Tyner Academy School District:Hamilton County School District

27. © 2013 UNIVERSITY OF PITTSBURGH 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. xf(x) -20 1 02 13 24

28. © 2013 UNIVERSITY OF PITTSBURGH 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, x 1, the point (x 1, f(x 1 ) ∙ g(x 1 )) 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. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH 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 Written Symbols Manipulative Models Real-world Situations Oral & Written Language Modified from Van De Walle, 2004, p. 30 Five Representations of Mathematical Ideas What role did tools or representations play in scaffolding student learning?

32. Five Different Representations of a Function What role did tools or representations play in scaffolding student learning? Language TableContext GraphEquation Van De Walle, 2004, p. 440

33. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH 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. xf(x) -20 1 02 13 24

35. © 2013 UNIVERSITY OF PITTSBURGH 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. © 2013 UNIVERSITY OF PITTSBURGH Reflection: The Use of Accountable Talk Moves and Tools to Scaffold Student Learning What have you learned?