Supporting Rigorous Mathematics Teaching and Learning

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER

Supporting Rigorous Mathematics Teaching and Learning

Strategies for Scaffolding Student Understanding: Academically Productive Talk and the Use of Representations Tennessee Department of EducationElementary School MathematicsGrade 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.

2Accountable 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.

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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.

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Review the Accountable Talk Features and Indicators

Learn Moves Associated with the Accountable Talk Features

Linking to Research/Literature: The QUASAR Project

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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


The Structure and Routines of a Lesson

The Explore Phase/Private Work TimeGenerate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions2. Assess and advance Student Learning

Share Discuss and Analyze Phase of the Lesson1. Share and Model2. Compare Solutions3. 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 representationREFLECT: Engage students in a Quick Write or a discussion of the process.

Set Up the TaskSet Up of the Task


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

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9© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER

Accountable Talk Features and IndicatorsAccountability 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?

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Accountable Talk Moves

Talk Move Function ExampleTo Ensure Purposeful, Coherent, and Productive

Group Discussion

Marking Direct 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.

Challenging Redirect 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?

Revoicing Align 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 1/3 of the whole.-

Recapping Make 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?


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To Support Accountability to CommunityKeeping 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)


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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 toRigorous 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)

Five Representations of Mathematical IdeasWhat role do the representations play in a discussion?

Pictures

Written Symbols

ManipulativeModels

Real-world Situations

OralLanguage

Adapted from Lesh, Post, & Behr, 1987


Engage and Reflect on a Lesson

Bags 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?

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The Mathematical Task Analysis GuideLower-Level Demands

Memorization Tasks• involve either producing previously learned

facts, rules, formulae, or definitions OR committing facts, rules, formulae, or definitions to memory.• cannot be solved using procedures because

a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure.• are not ambiguous – such tasks involve exact

reproduction of previously seen material and what is to be reproduced is clearly and directly stated.• have no connection to the concepts or

meaning that underlie the facts, rules, formulae, or definitions being learned or reproduced.

Procedures Without Connections Tasks• are algorithmic. Use of the procedure is

either specifically called for or its use is evident based on prior instruction, experience, or placement of the task.• require limited cognitive demand for

successful completion. There is little ambiguity about what needs to be done and how to do it.• have no connection to the concepts or

meaning that underlie the procedure being used.• are focused on producing correct answers

rather than developing mathematical understanding.• require no explanations, or explanations that

focus solely on describing the procedure that was used.

Higher-Level DemandsProcedures With Connections Tasks

• focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas.

• suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.

• usually are represented in multiple ways (e.g., visual diagrams, manipulatives, symbols, problem situations). Making connections among multiple representations helps to develop meaning.

• require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding.

Doing Mathematics Tasks• require 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).

• require students to explore and to understand the nature of mathematical concepts, processes, or relationships.

• demand self-monitoring or self-regulation of one’s own cognitive processes.

• require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.

• require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.

• require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.

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?

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Common Core State Standards for Mathematics: Grade 1

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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


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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.



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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).



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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 = ?.


Table 1: Common Addition and Subtraction Situations

24Common Core State Standards, 2010, p. 88, NGA Center/CCSSO

The CCSS for Mathematical Practice1. 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.

25Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO


Analyzing a Lesson: Lesson ContextTeacher: Erica WilkinsGrade: 1School: Sam Houston Elementary SchoolDistrict: 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.

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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.


Bags of Candy Task





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.

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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?

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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?

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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?

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Five Representations of Mathematical IdeasWhat role did tools or representations play in scaffolding student learning? Pictures

Written Symbols

ManipulativeModels

Real-world Situations

Oral & WrittenLanguage

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.

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Bags of Candy Task




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Focus of the DiscussionSuppose 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?

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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.

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Documents

Supporting Rigorous Mathematics Teaching and Learning