Supporting Rigorous Mathematics Teaching and Learning

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGHACCOUNTABLE TALK® IS A REGISTERED TRADEMARK OF THE UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Selecting and Sequencing Based on Essential Understandings

Tennessee Department of EducationElementary School MathematicsGrade 5


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Rationale

There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001).

By engaging in an analysis of a lesson-planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding.


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Session Goals

Participants will learn about:

• goal-setting and the relationship of goals to the CCSS and essential understandings;

• essential understandings as they relate to selecting

and sequencing student work;

• Accountable Talk® moves related to essential understandings; and

• prompts that problematize or “hook” students during the Share, Discuss, and Analyze phase of the lesson.

“The effectiveness of a lesson depends significantly on the care with which the lesson plan is prepared.”

Brahier, 2000

“During the planning phase, teachers make

decisions that affect instruction dramatically.

They decide what to teach, how they are going

to teach, how to organize the classroom, what

routines to use, and how to adapt instruction for

individuals.”

Fennema & Franke, 1992, p. 156

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

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 GoalsSelecting 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 discussions


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Identify Goals for Instructionand Select an Appropriate 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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Contextualizing Our Work Together

Imagine that you are working with a group of students who have the following understanding of the concepts.

• 70% of the students need to multiply fractions. (5.NF.B4 and 5.NF.B5)

• 20% of the students need additional work on fraction standards previously addressed (4.NF standards). These students also need opportunities to struggle with and make sense of the problem. (MP1)

• 5% of the students still do not recognize the importance of knowing what the “whole” is when talking about fractions. (Part of 4.NF.A2)

• 5% of the students struggle to pay attention and their understanding of mathematics is two grade levels below fifth grade.

The CCSS for Mathematics: Grade 5Number and Operations – Fractions 5.NFApply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.

5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.

5.NF.B.7c Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Common Core State Standards, 2010, p. 36 - 37, NGA Center/CCSSO

Mathematical Practice Standards Related to the Task

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


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Identify Goals: Solving the Task(Small Group Discussion)

Solve the task.

Discuss the possible solution paths to the task.


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Bobby’s Hike Task

Bobby said that he wanted to go for a four-mile hike. Bobby stops every mile for a sip of water from his water bottle. How many times does Bobby stop? Be sure to show how you found your answer with both diagrams and an explanation in words. What equations involving fractions match your diagram?

Extension: It takes Bobby hour to travel one mile. How often does Bobby stop for water? How do you know your answer is correct? Show with words, diagrams, and a fractional equation. Is there another equation that matches your diagram?


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Identify Goals Related to the Task(Whole Group Discussion)

Does the task provide opportunities for students to access the Mathematical Content Standards and Practice Standards that we have identified for student learning?


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Identify Goals: Essential Understandings (Whole Group Discussion)

Study the essential understandings associated with the Number and Operations – Fractions Common Core Standards.

Which of the essential understandings are the goals of Bobby’s Hike Task?

The CCSS for Mathematics: Grade 5Number and Operations – Fractions 5.NFApply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.

5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.

5.NF.B.7c Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Common Core State Standards, 2010, p. 36 - 37, NGA Center/CCSSO


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Essential UnderstandingEqual Size PiecesA fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.

Continuous and Discrete FiguresA fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but, regardless of the model, what remains true about all of the models is that they represent equal parts of a whole.

Meaning of the DenominatorThe larger the name of the denominator, the smaller the size of the piece.

Dividing FractionsWhen dividing a fraction by a whole number, every iteration of the unit fraction needs to be divided by the whole number.

Dividing by FractionsWhen dividing a whole number by a unit fraction, the number of times that the unit fraction fits inside the whole number is determined by the denominator.

Essential Understandings (Small Group Discussion)


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Selecting and Sequencing Student Work for the

Share, Discuss, and Analyze Phase of the Lesson


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Analyzing Student Work(Private Think Time)

• Analyze the student work. • Identify what each group knows related to the

essential understandings.

• Consider the questions that you have about each group’s work as it relates to the essential understandings.


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Prepare for the Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Small Group Discussion)

Assume that you have circulated and asked students assessing and advancing questions.Study the student work samples.

1. Which pieces of student work will allow you to address the essential understanding?

2. How will you sequence the student’s work that you have selected? Be prepared to share your rationale.


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The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work(Small Group Discussion)

In your small group, come to consensus on the work that you select, and share your rationale. Be prepared to justify your selection and sequence of student work.

Essential Understandings Group(s) Order Rationale

Equal Size Pieces

Continuous and Discrete Figures

Dividing Fractions

Dividing by Fractions


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The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work(Whole Group Discussion)

What order did you identify for the EUs and student work? What is your rationale for each selection?

Essential Understandings#1 via Gr.

#2 via Gr.

#3 via Gr.

#4 Via Gr.

Equal Size PiecesA fraction describes…

Continuous and Discrete FiguresA fraction can be continuous…

Dividing FractionsWhen dividing a fraction…

Dividing by FractionsWhen dividing a whole number…

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGHACCOUNTABLE TALK® IS A REGISTERED TRADEMARK OF THE UNIVERSITY OF PITTSBURGH 24

Group A


Group B


Group C


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Group D


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Group E


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Group F


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Group G


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The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work(Whole Group Discussion)

What order did you identify for the EUs and student work? What is your rationale for each selection?

Essential Understandings#1 via Gr.

#2 via Gr.

#3 via Gr.

#4 Via Gr.

Equal Size PiecesA fraction describes…

Continuous and Discrete FiguresA fraction can be continuous…

Dividing FractionsWhen dividing a fraction…

Dividing by FractionsWhen dividing a whole number…


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Academic Rigor in a Thinking Curriculum

The Share, Discuss, and Analyze Phase of the Lesson


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Academic Rigor In a Thinking Curriculum

A teacher must always be assessing and advancing student learning.

A lesson is academically rigorous if student learning related to the essential understanding is advanced in the lesson.

Accountable Talk discussion is the means by which teachers can find out what students know or do not know and advance them to the goals of the lesson.


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Accountable Talk Discussions

Recall what you know about the Accountable Talk features and indicators. In order to recall what you know:

• Study the chart with the Accountable Talk moves. You are already familiar with the Accountable Talk moves that can be used to Ensure Purposeful, Coherent, and Productive Group Discussion.

• Study the Accountable Talk moves associated with creating accountability to:

the learning community; knowledge; and rigorous thinking.


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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 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. One factor tells use the number of groups and the other factor tells us how many items in the group.

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?


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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The Share, Discuss, and Analyze Phase of the Lesson: Planning a Discussion (Small Group Discussion)

• From the list of potential EUs and its related student work, each group will select an essential understanding to focus their discussion.

• Identify a teacher in the group who will be in charge of leading a discussion with the group after the Accountable Talk moves related to the EU have been written.

Write a set of Accountable Talk moves on chart paper so it is public to your group for the next stage in the process.


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An Example: Accountable Talk Discussion The Focus Essential Understanding Continuous and Discrete FiguresA fraction can be continuous (linear model), or a measureable quantity (area model), or a group of discrete/countable things (set model) but, regardless of the model, what remains true about all of the models is that they represent equal parts of a whole. Group F Group G

• Explain how your model shows the problem. • Who understood what he said about the number line? (Community)• Can you say back what he said how the model shows the hike? (Community)• Who can add on and talk about the section of the number line? (Community)• The denominator tells the number of equal parts in the whole. (Marking)• Do we see in both models? (Rigor)• Tell us how you found in your picture (Group G). (Rigor)


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Problematize the Accountable Talk Discussion(Whole Group Discussion)

Using the list of essential understandings identified earlier, write Accountable Talk discussion questions to elicit from students a discussion of the mathematics.Begin the discussion with a “hook” to get student attention focused on an aspect of the mathematics. Type of Hook Example of a HookCompare and Contrast

Compare the half that has two equal pieces with the figure that has three pieces.

Insert a Claim and Ask if it is True

Three equal pieces of the six that are on one side of the figure show half of the figure. If I move the three pieces to different places in the whole, is half of the figure still shaded?

Challenge You said two pieces are needed to create halves. How can this be half; it has three pieces?

A Counter-ExampleIf this figure shows halves (a figure showing three sixths), tell me about this figure (a figure showing three sixths but the sixths are not equal pieces).


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An Example: Accountable Talk Discussion The Focus Essential Understanding Continuous and Discrete FiguresA fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but, regardless of the model, what remains true about all of the models is that they represent equal parts of a whole. Group F Group G

• One group used a number line and one group used an area model. How can this be? Can they both model the problem? (Hook)

• Can Group F explain where the whole and where the stops are?• Who understood what they said about the divisions of the line? (Community)• Can you say back what they said about the meaning of the numerator and

denominator for Bobby’s hike? (Community)• Each group made statements about the model being accurate to the context.

Where do we see division in each of the models? (Rigor)


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Revisiting Your Accountable Talk Prompts with an Eye Toward Problematizing

Revisit your Accountable Talk prompts.Have you problematized the mathematics so as to draw students’ attention to the mathematical goal of the lesson?

• If you have already problematized the work, then underline the prompt in red.

• If you have not problematized the lesson, do so now. Write your problematizing prompt in red at the bottom and indicate where you would insert it in the set of prompts.

We will be doing a Gallery Walk after we role play.


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Role Play Our Accountable Talk Discussion

• You will have 15 minutes to role play the discussion of one essential understanding.

• Identify one observer in the group. The observer will keep track of the discussion moves used in the lesson.

• The teacher will engage you in a discussion. (Note: You are well-behaved students.)

The goals for the lesson are: to engage all students in the group in developing

an understanding of the EU; and to gather evidence of student understanding

based on what the student shares during the discussion.


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Reflecting on the Role-Play: The Accountable Talk Discussion

• The observer has 2 minutes to share observations related to the lessons. The observations should be shared as “noticings.”

• Others in the group have 1 minute to share their “noticings.”


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Reflecting on the Role Play: The Accountable Talk Discussion(Whole Group Discussion)

Now that you have engaged in role playing, what are you now thinking about regarding Accountable Talk discussions?


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Zooming In on Problematizing(Whole Group Discussion)

Do a Gallery Walk. Read each others’ problematizing “hook.”

What do you notice about the use of hooks? What role do “hooks” play in the lesson?


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Step Back and Application to Our Work

What have you learned today that you will apply when planning or teaching in your classroom?


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Summary of Our Planning ProcessParticipants:

• identify goals for instruction;– Align Content Standards and Mathematical Practice

Standards with a task.– Select essential understandings that relate to the

Content Standards and Mathematical Practice Standards.

• prepare for the Share, Discuss, and Analyze phase of the lesson.– Analyze and select student work that can be used to

discuss essential understandings of mathematics. – Learn methods of problematizing the mathematics

in the lesson.

Documents

Supporting Rigorous Mathematics Teaching and Learning