Supporting Rigorous Mathematics Teaching and Learning

Supporting Rigorous Mathematics Teaching and Learning • Strategies for Scaffolding Student Understanding: Academically Productive Talk and the Use of Representations Tennessee Department of Education Elementary School Mathematics Grade 1

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

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 • the use of representations to scaffold talk and, ultimately, learning.

Overview of Activities Participants will: • analyze and discuss Accountable Talk moves; • engage in and reflect on our engagement in 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.

Review the Accountable Talk Features and Indicators Learn Moves Associated with the Accountable Talk Features

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

The Structure and Routines of a Lesson Set Up of the Task • 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 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

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

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

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?

Accountable Talk Moves

Accountable Talk Moves (continued) 12

Accountable Talk Moves (continued)

Pictures Manipulative Models Written Symbols Real-world Situations Oral Language Five Representations of Mathematical IdeasWhat role do the representations play in a discussion? Adapted from Lesh, Post, & Behr, 1987

Engage and Reflect on a LessonBags of Candy Task

Bags of Candy Task Tyler has 9 candies in his bag. He puts some more candies in his bag. Now there are 16 candies in his bag. How many more candies did Tyler put in his bag? Draw a picture and write an equation that shows Tyler’s candy. Mary has some candies in a bag. She puts 8 more candies in the bag. Now she has 16 candies in her bag. How many candies did she have in her bag? Draw a picture and write an equation that shows Mary’s candy. Explain how both students can have 16 candies if they added different amounts.

Analyzing the Demands of the Tasks Why is the task considered a high-level task?

The Mathematical Task Analysis Guide Stein and Smith, 1998; Stein, Smith, Henningsen, & Silver, 2000 and 2008.

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?

Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

Table 1: Common Addition and Subtraction Situations Common Core State Standards, 2010, p. 88, NGA Center/CCSSO

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.

Analyzing a Lesson: Lesson Context Teacher: Erica Wilkins Grade: 1 School: Sam Houston Elementary School District: Lebanon School District 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 Mathematical Content Standards in very deliberate ways during the lesson.

Instructional Goals Erica’s instructional goals for the lesson are: • students will make sense of “Adding To” situations with the start unknown and the change unknown; • students will understand the relationship between subtraction and missing addend problems; and • students will understand that doubles can be used to solve other problems or amounts in either of the addends can moved, but the sum will remain the same.

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.

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?

Characteristics of an Academically Rigorous Lesson(Whole Group Discussion) In what ways was the lesson academically rigorous? What does it mean for a lesson to be academically rigorous?

Academic Rigor in a Thinking Curriculum Academic Rigor in a Thinking Curriculum consists of indicators that students are accountable to: • A Knowledge Core • High-Thinking Demand • Active Use of Knowledge Most importantly, some indication that student learning/understanding is advancing from its current state needs to be seen. Did we see evidence of rigor via the Accountable Talk discussion?

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

Giving it a Go: Planning for An Accountable Talk Discussion of a Mathematical Idea Identify a person who will teach 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.

Focus of the Discussion Suppose John has 9 + __ = 18. How many candies does John add? Goals: • The relationship between subtraction and missing addend tasks • Counting on, use of known facts, or compensation can be used to solve a problem Students think about this in a variety of ways: • Some students use addition and counting on. • Some students just know that 9 + 9 is 18. • Some students think about 9 as 10 and add 8 but then subtract 1 because 10 is one more than 9. • One student uses subtraction to determine the missing addend. You want some students in the class to understand how counting on relates to the known fact of 9 + 9 = 18, how compensation can be used to solve the problem, and the relationship between subtraction and missing addends.

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

Bridge to Practice • Plan a lesson with colleagues. Create a high-level task that we didn’t use in this session. • Anticipate student responses prior to the lesson. Discuss ways in which you will engage students in talk that is accountable to community, to knowledge, and to standards of rigorous thinking. Specifically, list questions that you will ask during the lesson. Check that you have thought about all of the moves. • Engage students in an Accountable Talk discussion. Ask a colleague to scribe a segment of your lesson, or audio or videotape your own lesson and transcribe it later. • Analyze the Accountable Talk discussion in the transcribed segment of the talk. Identify questions and anticipated student responses. Bring a segment of the transcript so you can share specific moves. BRING to the next session: • A high-level task, your script, and your written reflection about the way the classroom discussion was accountable to the community, to knowledge, and to rigorous thinking. Bring a segment of the transcribed lesson so you can talk about specific moves that you made in the lesson and how students benefited from the moves.

Supporting Rigorous Mathematics Teaching and Learning