Supporting Rigorous Mathematics Teaching and Learning

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Using Academically Productive Talk Moves: Orchestrating a Focused Discussion

Tennessee Department of EducationMiddle School MathematicsGrade 6

Rationale

Mathematics reform calls for teachers to engage students in discussing, explaining, and justifying their ideas. Although teachers are asked to use students’ ideas as the basis for instruction, they must also keep in mind the mathematics that the class is expected to explore (Sherin, 2000, p. 125).

By engaging in a high-level task and reflecting on ways in which the facilitator structured and supported the discussion of mathematical ideas, teachers will learn that they are responsible for orchestrating discussions in ways that make it possible for students to own their learning, as well as for the teacher to assess and advance student understanding of knowledge and mathematical reasoning. 2


Session Goals

Participants will:

• learn about Accountable Talk features and indicators and consider the benefit of all being present in a lesson;

• learn that there are specific moves related to each of the talk features that help to develop a discourse culture; and

• consider the importance of the four key moves of ensuring productive discussion (marking, recapping, challenging, and revoicing).

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Overview of Activities

Participants will:

• review the Accountable Talk features and indicators;

• identify and discuss Accountable Talk moves in a video; and

• align CCSS and essential understandings (EUs) to a task and zoom in for a more specific look at key moves for engaging in productive talk (marking, recapping, challenging, and revoicing).

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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

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

Linking to Research/Literature: The QUASAR Project

Setting Goals Selecting TasksAnticipating Student Responses

Orchestrating Productive Discussion• Monitoring students as they work• Asking assessing and advancing questions• Selecting solution paths• Sequencing student responses• Connecting student responses via Accountable

Talk® discussionsAccountable Talk® is a registered trademark of the University of Pittsburgh 6


Accountable Talk Features and Indicators

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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 so that the discussion is accountable to:

− the learning community;− accurate, relevant knowledge; and− standards of rigorous thinking.

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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.

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Solving and Discussing the Cognitive Demand of the

Light Bulb Task

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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 PATHCOMPARE: 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

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Engaging in a Lesson: The Light Bulb Task

• Solve the task.

• Discuss your solutions with your peers.

• Attempt to engage in an Accountable Talk discussion when discussing the solutions. Assign one person in the group to be the observer. This person will be responsible for reporting some of the ways in which the group is accountable to:

− the learning community;− accurate, relevant knowledge; and − standards of rigorous thinking.

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Engaging in a Lesson: The Light Bulb Task

Alazar Electric Company sells light bulbs to big box stores – the big chain stores that frequently buy large numbers of bulbs in one sale. They sample their bulbs for defects routinely. A sample of 96 light bulbs consisted of 4 defective ones. Assume that today’s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today. The big businesses they sell to accept no larger than a 4% rate of defective bulbs. Does today’s batch meet that expectation? Explain how you made your decision.

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Reflecting on Our Engagement in the Lesson

The observer should share some observations about the group’s engagement in an Accountable Talk discussion.

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Reflecting on Our Engagement in the Lesson

• In what ways did small groups engage in an Accountable Talk discussion?

• In what ways did we engage in an Accountable Talk discussion during the group discussion of the solutions?

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Aligning the CCSS to the Light Bulb Task

• Study the Grade 6 CCSS for Mathematical Content within the Ratio and Proportion domain.

Which standards are students expected to demonstrate when solving the task?

• Identify the CCSS for Mathematical Practice required by the written task.

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The CCSS for Mathematical Content: Grade 6

Common Core State Standards, 2010, p. 42, NGA Center/CCSSO

Ratios and Proportional Relationships 6.RPUnderstand ratio concepts and use ratio reasoning to solve problems.

6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly 3 votes.”

6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

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The CCSS for Mathematical Content: Grade 6

Common Core State Standards, 2010, p. 42, NGA Center/CCSSO

Ratios and Proportional Relationships 6.RPUnderstand ratio concepts and use ratio reasoning to solve problems.

6.RP.A.3a Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.RP.A.3b Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

6.RP.A.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

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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/CCSSO19


Determining the Cognitive Demand of the Task:The Light Bulb Task

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Determining the Cognitive Demandof the Task

Refer to 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.

How would you characterize the Light Bulb Task in terms of its cognitive demand? (Refer to the indicators on the Task Analysis Guide.)

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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. 22


The Light Bulb Task:A Doing Mathematics Task

• Requires 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).

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

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

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

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

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

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Accountable Talk Moves

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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 PATHCOMPARE: 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

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Accountable Talk Moves

Examine the ways in which the moves are grouped based on how they:

• support accountability to the learning community; • support accountability to knowledge; and • support accountability to rigorous thinking.

Consider:In what ways are the Accountable Talk categories similar? Different?

Why do you think we need a category called “To Ensure Purposeful, Coherent, and Productive Group Discussion”?

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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. 27

Accountable Talk MovesTalk Move Function Example

To Ensure Purposeful, Coherent, and Productive Group Discussion

Marking Direct attention to the value and importance of a student’s contribution.

That’s an important point.

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.

S: 4 + 4 + 4.

You said three groups of four.

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?

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? 28


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)

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Reflection Question

As you watch the short video segment, consider what students are learning and where you might focus the discussion in order to discuss mathematical ideas listed in the CCSS.

Identify:• the specific Accountable Talk moves used by the

teacher; and • the purpose that the moves served.

Mark times during the lesson when you would call the lesson academically rigorous.

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The Light Bulb Lesson Context• Visiting Teacher: Victoria Bill• Teacher: Reginald Coleman• School: Community Health Academy of the Heights

Middle School• District: New York City Schools• Principal: Ms. Vu• Grade Level: 7th Grade

The students in the video episode are in a mainstream mathematics classroom in the New York City Schools. The students are solving the Light Bulb Task. This part of the video captures the Share, Discuss, and Analyze phase of the lesson.

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Norms for Collaborative Study

The goal of all conversations about episodes of teaching (or artifacts of practice in general) is to advance our own learning, not to “fix” the practice of others.

In order to achieve this goal, the facilitator chooses a lens to frame what you look at and to what you pay attention. Use the Accountable Talk features and indicators when viewing the lesson.

During this work, we:• agree to analyze the episode or artifact from the identified

perspective;• cite specific examples during the discussion that provide

evidence of a particular claim;• listen to and build on others’ ideas; and• use language that is respectful of those in the video and in the

group.32


The Light Bulb Task

Alazar Electric Company sells light bulbs to big box stores – the big chain stores that frequently buy large numbers of bulbs in one sale. They sample their bulbs for defects routinely. A sample of 96 light bulbs consisted of 4 defective ones. Assume that today’s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today. The big businesses they sell to accept no larger than a 4% rate of defective bulbs. Does today’s batch meet that expectation? Explain how you made your decision. 33


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?

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Essential Understandings

Study the essential understandings the teacher considered in preparation for the Share, Discuss, and Analyze phase of the lesson.

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Essential UnderstandingsEssential Understanding CCSS

Comparing QuantitiesTwo quantities can be compared using addition/subtraction or multiplication/division. Forming a ratio is a way of comparing two quantities multiplicatively. Reasoning with ratios involves attending to and coordinating two quantities.

6.RP.A.1

Unit RateWhen the ratio of a/b is scaled up or down to a/b/1, a/b to 1 is referred to as a unit rate.

6.RP.A.2

Using ModelsReal-world relationships involving ratios can be modeled with a number of representations, e.g., diagram, table, graph, or ratio; however, in any such representation, both quantities in the relationship must be scaled multiplicatively. A ratio can be scaled up using multiplication because the two quantities vary in such a way that one of them is a constant multiple of the other; a ratio can be scaled down using division, since division by some number, q, is the equivalent of multiplication by the multiplicative inverse of q, 1/q.

6.RP.A.3

Solve Unit Rate ProblemsTwo unit rates are associated with a multiplicative relationship a and b: a/b to 1 and b/a to 1. Each unit rate reveals different information about real-world problems associated with the relationship.

6.RP.A.3a

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Characteristics of an Academically Rigorous Lesson

This task is a cognitively demanding task; however, it may not necessarily end up being an academically rigorous task.

What do we mean by this statement?

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Academic Rigor in a Thinking Curriculum

The principle of learning, Academic Rigor in a Thinking Curriculum, consists of three features:

• A Knowledge Core• High-Thinking Demand• Active Use of Knowledge

In order to determine if a lesson has been academically rigorous, we have to determine the degree to which student learning is advanced by the lesson.

What do we have to hear and see in order to determine if the lesson was academically rigorous?

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Essential UnderstandingsEssential Understanding CCSS

Comparing QuantitiesTwo quantities can be compared using addition/subtraction or multiplication/division. Forming a ratio is a way of comparing two quantities multiplicatively. Reasoning with ratios involves attending to and coordinating two quantities.

6.RP.A.1

Unit RateWhen the ratio of a/b is scaled up or down to a/b/1, a/b to 1 is referred to as a unit rate. 6.RP.A.2Using ModelsReal-world relationships involving ratios can be modeled with a number of representations, e.g., diagram, table, graph, or ratio; however, in any such representation, both quantities in the relationship must be scaled multiplicatively. A ratio can be scaled up using multiplication because the two quantities vary in such a way that one of them is a constant multiple of the other; a ratio can be scaled down using division, since division by some number, q, is the equivalent of multiplication by the multiplicative inverse of q, 1/q.

6.RP.A.3

Solve Unit Rate ProblemsTwo unit rates are associated with a multiplicative relationship a and b: a/b to 1 and b/a to 1. Each unit rate reveals different information about real-world problems associated with the relationship.

6.RP.A.3a

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Five Different Representations of a Function

Van De Walle, 2004, p. 440

Language

TableContext

Graph Equation

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Focusing on Key Accountable Talk Moves

The Light Bulb Task

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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.

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Accountable Talk MovesTalk Move Function Example

To Ensure Purposeful, Coherent, and Productive Group Discussion

Marking Direct attention to the value and importance of a student’s contribution.

That’s an important point.

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.

S: 4 + 4 + 4.

You said three groups of four.

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?

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?

43


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)

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Focusing on Accountable Talk Moves

Read the description of each move and study the example that has been provided for each move.

What is distinct about each of the moves?• Revoice student contributions; • mark significant contributions; • challenge with a counter-example; or • recap the components of the lesson.

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Revoicing

• Extend a student’s contribution.• Connect a student’s contribution to the text or to

other students’ contributions. Align content with an explanation. Add clarity to a contribution. Link student contributions to accurate

mathematical vocabulary. Connect two or more contributions to advance

the lesson.

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An Example of Revoicing

S: —and it gives you 12 or you multiply 4 times 3 because there’s 3 boxes.

T: All right, he said you could do 4 times 3 and then—how would you get from here to here, though? (Points to other side of table, the total number of bulbs.)

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Marking

Explicitly talk about an idea.• Highlight features that are unique to a situation.• Draw attention to an idea or to alternative ideas.

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An Example of Marking

S: It looks like they wrote fractions. Like, broken bulbs over total bulbs. 4 over 96 is equal to 250 over 6,000.

T: Hmm—did everyone hear what Selena just said? She noticed that when we write the fraction of defective bulbs out of total bulbs, they are both equivalent.

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Recapping

Summarize or retell.• Make explicit the large idea.• Provide students with a holistic view of the

concept.

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Challenging

Redirect a question back to the students, or use students’ contributions as a source for further challenge or query.

• Share a counter-example and ask students to compare problems.

• Question the meaning of the math concept.

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An Example of Recapping and Challenge

S: Multiply by – by 4 – 62 times 5 – no, 62.5 times 4. T: 62 and times 4, 4 defective? S: Uhm, I get 250. T: 250! Clap your hands if everyone got that.

[Applause] So, why did you have to do – why did you multiply 96 by 62 and and also 4 by 62 and ?

S: What? T: Why did you have to do it on both sides? When we

worked up here, we did 92 x 2 and on the other side we did 4 x 2. When we multiplied by 3 on this side we multiplied by 3 on the other side. Why do we have to do this in order for it to be proportional?

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Appropriation

The process of appropriation is reciprocal andsequential.

• If appropriation takes place, the child transforms the new knowledge or skill into an action in a new and gradually understood activity.

• What would this mean with respect to classroom discourse? What should we expect to happen in the classroom?

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Orchestrating Discussions

Read the segments of transcript from the lesson.Decide if examples 1 – 3 illustrate marking, recapping, challenging, or revoicing.

Be prepared to share your rationale for identifying a particular discussion move.

Write the next discussion move for examples 4 and 5 and be prepared to share your move and your rationale for writing the move.

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Reflecting on Talk Moves

What have you learned about:• marking;• recapping;• challenging; and• revoicing?

Why are these moves important in lessons?

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Application to Practice

• What will you keep in mind when attempting to use Accountable Talk moves during a lesson? What role does talk play?

• What does it take to maintain the demands of a cognitively demanding task during the lesson so that you have a rigorous mathematics lesson?

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Supporting Rigorous Mathematics Teaching and Learning

Documents

Transcript of Supporting Rigorous Mathematics Teaching and Learning