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Supporting Rigorous Mathematics Teaching and Learning Academically Productive Talk in Mathematics: A Means of Making Sense of Mathematical Ideas Tennessee Department of Education Elementary School Mathematics Grade 1 © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). Building a practice of engaging students in academically rigorous tasks supported by Accountable Talk® discourse facilitates effective teaching. Students develop an understanding of mathematical ideas, strategies, and representations and teachers gain insights into what students know and what they can do. These insights prepare teachers to consider ways that advance student understanding of mathematical ideas, strategies, or connections to representations. Today, by analyzing math classroom discussions, teachers will study how Accountable Talk discussion supports student learning and helps teachers to maintain the cognitive demand of the task. Accountable Talk® is a registered trademark of the University of Pittsburgh 2 Session Goals Participants will: • learn a set of Accountable Talk features and indicators; and • recognize Accountable Talk stems for each of the features of talk, and consider the potential benefit of posting and practicing talk stems with students. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 3 Overview of Activities Participants will: • discuss Accountable Talk features and indicators; • discuss students’ solution paths for a task; • analyze and identify Accountable Talk features and indicators in a lesson; and • plan for an Accountable Talk discussion. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 4 The Structure and Routines of a Lesson Set Task SetUp Upthe of the Task 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 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 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 © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 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. 5 Accountable Talk Features and Indicators © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 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 for each of the features. − accountability to the learning community − accountability to accurate, relevant knowledge − accountability to discipline-specific standards of rigorous thinking © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 7 Accountable Talk Discussion Indicators for all three features must be present in order for the discussion to be an “Accountable Talk Discussion.” • accountability to the learning community • accountability to accurate, relevant knowledge • accountability to discipline-specific standards of rigorous thinking Why might this be important? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 8 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. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 9 Accountable Talk Starters • Work in triads. • On your chart paper, write talk starters for the Accountable Talk indicators. A talk starter is the start of a sentence that you might hear from students if they are holding themselves accountable for using Accountable Talk moves. e.g., I want to add on to ______ (Community move). The denominator of a fraction tells us _____ (Knowledge move). The two equations are equivalent because ____ (Rigor move). (Work for 5 minutes.) © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 10 Accountable Talk Starters • What do you notice about the talk starts for the: accountability to the learning community accountability to accurate, relevant knowledge accountability to discipline-specific standards of rigorous thinking • What is the distinction between the stems for knowledge and those for rigorous thinking? • Why should we pay attention to this? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 11 Preparing to Analyze Accountable Talk Features and Indicators in Classroom Practice © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER The Make a Ten Task Use the interlinking cubes to make a structure of ten. Write a number sentence to describe your structure of ten. Draw a picture for your structure of ten. Make a second structure of ten that looks different from your first structure of ten. Write a different number sentence for this structure of ten. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 13 Solve the Task (Private Think Time and Small Group Time) • Work privately. Analyze the student work for The Make a Ten Task in your participant handout. • Work with others at your table. Hold yourselves accountable for engaging in an Accountable Talk discussion when you discuss the student work. What do the students understand? How do you know? How does one solution path differ from the other? What might be used in the student work to prompt students to think about and discuss the ideas articulated in the standards? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 14 The Make a Ten Task: Devon’s Work © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 15 The Make a Ten Task: David’s Work © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 16 The Make a Ten Task: Tinesha’s Work © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 17 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Represent and solve problems involving addition and subtraction. 1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 1.OA.A.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 18 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.B.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 1.OA.B.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 19 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Add and subtract within 20. 1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 20 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Work with addition and subtraction equations. 1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 21 Table 1: Common Addition and Subtraction Situations Common Core State Standards, 2010, p. 88, NGA Center/CCSSO 22 The Common Core Standards for Mathematical Practice What would have to happen in order for students to have opportunities to make use of 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 23 Accountable Talk Features and Indicators Which of the Accountable Talk Features and Indicators were illustrated in our discussion? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 24 Using the Accountable Talk Features and Indicators to Analyze Classroom Practice © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Reflecting on the Lesson Watch the video. What are students learning in the Make a Ten Lesson? Which Accountable Talk features and indicators were illustrated in the video of the Make a Ten lesson? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 26 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. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 27 Context for the Make a Ten Lesson Teacher: Jennifer DiBrienza School: PS 116 District: New York City, District 2 Grade Level: First Grade The lesson was conducted in a first grade classroom. The students have worked with interlocking cubes many times. They have explored ways in which they can use the blocks to make different shapes. They have described the figures that they can make with the interlocking cubes. In this lesson, we observe the Share, Discuss, and Analyze phase of the lesson. The students have worked independently to create two different structures and to write number sentences to describe their structures. Now the students come together to share and discuss each other’s structures. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 28 The Structures and Routines of a Lesson Set Task SetUp Upthe of the Task 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 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 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 © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 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. 29 The Make a Ten Lesson © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 30 Common Core State Standards (CCSS) Examine the first grade CCSS for Mathematics in your participant handout. – Which CCSS for Mathematical Content did we discuss? – Which CCSS for Mathematical Practice did we use when solving and discussing the task? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 31 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Represent and solve problems involving addition and subtraction. 1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 1.OA.A.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 32 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.B.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 1.OA.B.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 33 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Add and subtract within 20. 1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 34 Common Core State Standards for Mathematics: Grade 1 Operations and Algebraic Thinking 1.OA Work with addition and subtraction equations. 1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 35 Table 1: Common Addition and Subtraction Situations Common Core State Standards, 2010, p. 88, NGA Center/CCSSO 36 Common Core Standards for Mathematical Practice What would have to happen in order for students to have opportunities to make use of 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 37 Linking to Research/Literature Connections Between Representations Pictures Manipulative Models Written Symbols Real-world Situations Oral Language Adapted from Lesh, Post, & Behr, 1987 38 Accountable Talk Discussion Successful teachers are skillful in building shared contexts of the mind (not merely assuming them) and assuring that there is equity and access to these experiences. Talk about these experiences for all members of the classroom are a necessary part of the experience. Over time, these contexts of the mind and collective experiences with talk lead to the development of a "discourse community"—with shared understandings, ways of speaking, and new discursive tools with which to explore and generate knowledge. In this way, an intellectual "commonwealth" can be built on a base of tremendous sociocultural diversity. Accountable Talk℠ Sourcebook: For Classroom Conversation that Works (IFL, 2010) © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 39 Giving it a Go: Planning for an Accountable Talk Discussion • Identify a person who will be the teacher of the lesson. • Others in the group will engage in the lesson once the lesson has been planned. • Plan the lesson together. Actually write questions that the teacher will ask and anticipate participant responses. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 40 Focus of Lesson Students share two structures. The teacher’s goal for the lesson is to help students understand the connections between the two structures. (Associative Property of Addition) (2 + 5) + 3 and 2 + (5 + 3). © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 41 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? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 42 A Wondering… • What will you keep in mind when attempting to engage students in Accountable Talk discussions? • What does it take to maintain the demands of a cognitively demanding task during the lesson so that you have a rigorous mathematics lesson? • What role does talk play? © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 43 Bridge to Practice • Plan a lesson with colleagues. Select a high-level task. • Anticipate student responses. Discuss ways in which you will engage students in talk that is accountable to the learning community, to knowledge, and to standards of rigorous thinking. Specifically, list the moves and the questions that you will ask during the lesson. • Engage students in an Accountable Talk discussion. Ask a colleague to scribe a segment of your lesson, or audio or video tape your own lesson and transcribe it later. • Analyze the Accountable Talk discussion in the transcribed segment of the talk. Identify talk moves and the purpose that the moves served in the lesson. Have a segment of the transcript so you can identify specific moves. © 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER 44