© 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Selecting and Sequencing Based on Essential Understandings Tennessee

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Selecting and Sequencing Based on Essential Understandings

Tennessee Department of Education

Elementary School Mathematics

Grade 4

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.

3© 2013 UNIVERSITY OF PITTSBURGH

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.

Accountable Talk is a registered trademark of the University of Pittsburgh.

“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


Identify Goals for Instructionand Select an Appropriate Task


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 Lesson

1. Share and Model

2. Compare Solutions

3. 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


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 make sense of what it means to have a fraction as a sum of a parts of . (4.NF.B3)

At this stage in student learning, the teacher is not as concerned that students are precise. S/he will take note of who is and who is not. The teacher, however, will be precise with his/her revoicing of student contributions. (MP6)

All of the students can benefit from modeling with fractions. (MP4) Students will discuss the structure of mathematics related to the meaning

of the numerator. (MP7) 20% of the students need additional work on fraction standards

previously addressed (3.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)

• The teacher will emphasize repeated reasoning when students find common denominators for each fraction. (MP8)

The CCSS for Mathematics: Grade 4

Number and Operations – Fractions 4.NF

Extend understanding of fraction equivalence and ordering.

4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Common Core State Standards, 2010, p. 30, NGA Center/CCSSO



Understand decimal notation for fractions, and compare decimal fractions.

4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction withdenominator 100, and use this technique to add two fractions withrespective denominators 10 and 100. For example, express 3/10 as30/100, and add 3/10 + 4/100 = 34/100.

4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. Forexample, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size.Recognize that comparisons are valid only when the two decimalsrefer to the same whole. Record the results of comparisons with thesymbols >, =, or <, and justify the conclusions, e.g., by using a visualmodel.


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


Identify Goals: Solving the Task(Small Group Discussion)

Solve the task.

Discuss the possible solution paths to the task.


The Pizza TaskJolla has of a pizza.

Sarah has of a pizza.

Maria has of a pizza.

Tim’s pizza is shaded on the pizza. How much pizza is Tim’s share?

Jake has of a pizza.

Juan has of a pizza.

1. Show each of the student’s amount of pizza.

2. Compare the students’ amounts of pizza. Explain with words and use the >, <, or = symbols to show who has the most pizza.

3. Explain with words and use the >, <, or = symbols to show who has the least amount of pizza.


The Pizza Task (continued)

Jolla’s Pizza Tim’s Pizza Juan’s Pizza

Sarah’s Pizza Maria’s Pizza Jake’s Pizza


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?


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 The Pizza Task?



Extend understanding of fraction equivalence and ordering.

4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.




Understand decimal notation for fractions, and compare decimal fractions.

4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction withdenominator 100, and use this technique to add two fractions withrespective denominators 10 and 100. For example, express 3/10 as30/100, and add 3/10 + 4/100 = 34/100.

4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. Forexample, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size.Recognize that comparisons are valid only when the two decimalsrefer to the same whole. Record the results of comparisons with thesymbols >, =, or <, and justify the conclusions, e.g., by using a visualmodel.



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.

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

Use of BenchmarksComparison to known benchmark quantities can help students determine the relative size of a fractional piece because the benchmark quantity can clearly be seen as smaller or larger than the piece. One significant benchmark quantity is one-half.EquivalencyA fraction can be named in more than one way and the fractions will be equivalent as long as the same portion of the set or area of the figure is represented.Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator.

Essential Understandings (Small Group Discussion)


Selecting and Sequencing Student Work for the

Share, Discuss, and Analyze Phase of the Lesson


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.


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.


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

Meaning of the Denominator

Use of Benchmarks

Equivalency

Creating Equivalent Fractions


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.

Meaning of the DenominatorThe larger the name of…

Use of BenchmarksComparison to known benchmark…

EquivalencyA fraction can be named in more…

Creating Equivalent Fractions When the denominator is multiplied…

© 2013 UNIVERSITY OF PITTSBURGH 27

Group A


Group B


Group C


Group D


Group E


Group F


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.

Meaning of the DenominatorThe larger the name of…

Use of BenchmarksComparison to known benchmark…

EquivalencyA fraction can be named in more…

Creating Equivalent Fractions When the denominator is multiplied…


Academic Rigor in a Thinking Curriculum

The Share, Discuss, and Analyze Phase of the Lesson


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.


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.


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

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 Community

Keeping 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)


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.


An Example: Accountable Talk Discussion The Focus Essential Understanding

Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator.Group A Group B

• Explain your set of equivalencies.• Who understood what he said about the 100 and the 10? (Community)• Can you say back what he said how the model shows ? (Community)• Who can add on and talk about the 3 and the 30? (Community)• The denominator tells the number of equal parts in the whole. (Marking) • Do we see in both pieces of work? (Rigor) • Tell us how you found in your picture (Group A). (Rigor)


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 Hook

Compare 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?

ChallengeYou 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).


An Example: Accountable Talk Discussion The Focus Essential Understanding Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator.Group A Group B

• Both groups say that is equal to . How can this be when the fractions use different numbers? (Hook)

• Can Group B explain why = = ?• Who understood what they said about the denominators? (Community)• Can you say back what they said about the numerator changing?

(Community)• Each group made statements about equivalency. How does the visual

model differ from/support the symbolic model? (Rigor)


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.


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.


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.”


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?


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?


Step Back and Application to Our Work

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


Summary of Our Planning Process

Participants:

• 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

© 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Selecting and Sequencing Based on Essential Understandings Tennessee