EPSD

How can I let go?

Solve problems

Manage Oneself

Adapt to Change

Analyze/Conceptualize

Reflect on/Improve Performance

Communicate

Work in Teams

Create/Innovate/Critique

Engage in Lifelong Learning

Purpose for Shifts in Mathematical Practice

4 sC

C

C

CC

ollaboration

ritical Thinking

reative Thinking

ommunication

PennsylvaniaStandards

for MathematicalPractice

Make sense of and persevere in solving complex and novel

mathematicalproblems

Use effective mathematical

reasoning to construct viable arguments and

critique thereasoning of others

Apply mathematical knowledge to analyze and model situations/relationships using

multiple representations and appropriate tools in

order to make decisions, solve

problems, and draw conclusions

Communicate precisely when making

mathematical statements and

express answers with a degree of precision

appropriate for the context of the

problem/situation

Make use of structure and repeated

reasoning to gain a mathematical

perspective and formulate generalized

problem solving strategies

1 2

43 5

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There are 25 sheep and 5 dogs in a flock. How old is

the Shepherd?

There are 25 sheep and 5 dogs in a flock. How old is

the Shepherd?

Three out of four students will give a numerical answer

to this problem.

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There are 25 sheep and 5 dogs in a flock. How old is

the Shepherd?

How can I let go?

How can I let go?

How can I let go?

Before

How can I let go?

Before

During

How can I let go?

Before

During

After

Before

During

After

Getting Ready

* Get students mentally ready to work on task.* Be sure all expectations for products are clear.

Student Work

* Let go!* Listen carefully.* Provide hints.* Observe and assess.

Class Discourse

* Accept student solutions without evaluation.* Conduct discussion as students justify and evaluate results and methods.

Before

During

After

Getting Ready

* Get students mentally ready to work on task.* Be sure all expectations for products are clear.

Student Work

* Let go!* Listen carefully.* Provide hints.* Observe and assess.

Class Discourse

* Accept student solutions without evaluation.* Conduct discussion as students justify and evaluate results and methods.

Before

During

After

Before

During

After

Before

During

After

Model

Discussion

Research

Before

During

After

Model

Discussion

Research

Before

During

After

Model

Discussion

Research

Before

During

After

Before

Before: Model

Before: Model

Before: Model• Connect to student

experiences

• Students explain what the question is asking

• Let go!

• Avoid telling them how to solve the problem

• Role play appropriate answers

• Lay the groundwork for future activities

Before: Model

Before: Model

Due to the government shutdown, factories have been told that there is a

shortage of red thread in the United States. What kinds of products will this affect?

Before: Discussion

How is what you experienced the same or different from your current classroom?

How are we making studentsaccountable for their own learning?

How is this helping you to let go?

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

Before

During

After

During

During: Model

During: Model

Create a list of strategies that you could use to solve this problem.

Come to a consensus and pick one strategy - get started!

During: Discussion

How is what you experienced the same or different from your current classroom?

How are we making studentsaccountable for their own learning?

How is this helping you to let go?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

QuestioningObservations do not have to be silent. Probing into student thinking through the use ofquestions can provide better data and more insights to inform instruction. As youcirculate around the classroom to observe and evaluate students’ understanding, youruse of questions is one of the most important ways to formatively assess in each lessonphase. Keep the following questions in mind (or on a clipboard, index cards, or abookmark) as you move about the classroom to prompt and probe students’ thinking:

• What can you tell me about [today’s topic]?• How can you put the problem in your own words?• What did you do that helped you understand the problem?• Was there something in the problem that reminded you of another problem we’ve

done?• Did you find any numbers or information you didn’t need? How did you know that

the information was not important?• How did you decide what to do?• How did you decide whether your answer was right?• Did you try something that didn’t work? How did you figure out it was not going to

work?• Can something you did in this problem help you solve other problems?

NAME: Sharon V.

Estimates fractionquantities

NO

T TH

ERE

YET

ON

TA

RG

ET

AB

OVE

AN

DB

EYO

ND

CO

MM

ENTS

FRACTIONS

MATHEMATICAL PRACTICES

Understands numerator/denominator

Area models

Set models

Use fractions inreal contexts

Make sense of problems and perseveres

Models withmathematics

Uses appropriatetools

NAME: Sharon V.

Estimates fractionquantities

NO

T TH

ERE

YET

ON

TA

RG

ET

AB

OVE

AN

DB

EYO

ND

CO

MM

ENTS

FRACTIONS

MATHEMATICAL PRACTICES

Understands numerator/denominator

Area models

Set models

Use fractions inreal contexts

Make sense of problems and perseveres

Models withmathematics

Uses appropriatetools

NAME: Sharon V.

Estimates fractionquantities

NO

T TH

ERE

YET

ON

TA

RG

ET

AB

OVE

AN

DB

EYO

ND

CO

MM

ENTS

FRACTIONS

MATHEMATICAL PRACTICES

Understands numerator/denominator

Area models

Set models

Use fractions inreal contexts

Make sense of problems and perseveres

Models withmathematics

Uses appropriatetools

Used patternblocks to show

2/3 and 3/6

Showing greaterreasonableness

Stated problemin own words

Reluctant to useabstract models

Names

Lalie

Pete

Sid

Lakeshia

George

Pam

Maria

Topic:

Mental ComputationAdding 2-digit numbers

Not There Yet CommentsOn Target Above and Beyond

Names

Lalie

Pete

Sid

Lakeshia

George

Pam

Maria

Topic:

Mental ComputationAdding 2-digit numbers

Not There Yet CommentsOn Target Above and Beyond

Can’t domentally

Has at leastone strategy

Uses differentmethods with

different numbers

Names

Lalie

Pete

Sid

Lakeshia

George

Pam

Maria

Topic:

Mental ComputationAdding 2-digit numbers

Not There Yet CommentsOn Target Above and Beyond

3-20

3-18

3-24

3-24

3-20

Can’t domentally

Has at leastone strategy

Uses differentmethods with

different numbers

3-21

Names

Lalie

Pete

Sid

Lakeshia

George

Pam

Maria

Topic:

Mental ComputationAdding 2-digit numbers

Not There Yet CommentsOn Target Above and Beyond

Difficulty withregrouping

Flexible approachesused

Counts by tens, then adds ones

Beginning to add thegroup of tens first

Using a postedhundreds chart

3-20

3-18

3-24

3-24

3-20

Can’t domentally

Has at leastone strategy

Uses differentmethods with

different numbers

3-21

Observation RubricMaking Whole Given Fraction Part

Above and BeyondClear understanding. Communicates concept in multiplerepresentations. Shows evidenceof using idea without prompting.

On TargetUnderstands or is developingwell. Uses designated models.

Not There YetSome confusion ormisunderstanding. Only modelsidea with help.

Observation RubricMaking Whole Given Fraction Part

Above and BeyondClear understanding. Communicates concept in multiplerepresentations. Shows evidenceof using idea without prompting.

On TargetUnderstands or is developingwell. Uses designated models.

Not There YetSome confusion ormisunderstanding. Only modelsidea with help.

Fraction whole made fromparts in rods and in sets.Explains easily.

Can make whole in eitherrod or set format (note).Hesitant. Needs prompt toidentify unit fraction.

Needs help to doactivity. No confidence.

Observation RubricMaking Whole Given Fraction Part

Above and BeyondClear understanding. Communicates concept in multiplerepresentations. Shows evidenceof using idea without prompting.

On TargetUnderstands or is developingwell. Uses designated models.

Not There YetSome confusion ormisunderstanding. Only modelsidea with help.

Fraction whole made fromparts in rods and in sets.Explains easily.

Can make whole in eitherrod or set format (note).Hesitant. Needs prompt toidentify unit fraction.

Needs help to doactivity. No confidence.

Sally

Latania

Greg

John S. Mary

Lavant (rod)

Julie (rod)

George (set)

Maria (set)

Tanisha (rod)

Lee (rod)

J.B. (set)

John H. (set)

Before

During

After

After

After: Model

After: Model

Why did your group choose this strategy to solve the problem?

Is there a different strategy you would use if you did the problem again?

After: Discussion

How is what you experienced the same or different from your current classroom?

How are we making studentsaccountable for their own learning?

How is this helping you to let go?

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

Lesson Phase Teacher Actions in a Teaching Mathematics through Problem-Solving Lesson

Three-Phase Lesson StructureTable 2.1

BeforeActivate priorknowledge.

Be sure theproblem isunderstood.

Establish clearexpectations.

Begin with a simple version of the task; connect to students’ experiences; brainstormapproaches or solution strategies; estimate or predict whether tasks involve a singlecomputation or are aimed at the development of a computational procedure.

Have students explain to you what the problem is asking. Go over vocabulary thatmay be troubling. Caution: This does not mean that you are explaining how to do a problem, just that students should understand what the problem is about.

Tell students whether they will work individually, in pairs, or small groups, or if theywill have a choice. Tell them how they will share their solutions and reasoning.

During Let go!

Notice students’mathematicalthinking.

Provideappropriatesupport.

Provideworthwhileextensions.

Although it is tempting to want to step in and “help,” hold back and enjoy observingand learning from students.

Base your questions on students’ work and their responses to you. Use prompts like:Tell me what you are doing; I see you have started to [multiply] these numbers. Canyou tell me why you are [multiplying]? [Substitute any process/strategy]; Can you tell me more about...? Why did you...? How does your work connect to the problem?

Look for ways to support students’ thinking and avoid telling them how to solve theproblem. Ensure that students understand the problem (What do you know about theproblem?); ask the student what he or she has already tried (also, Where did you getstuck?); suggest that the student use a different strategy (Can you draw a diagram?What if you used cubes to act out this problem? Is this like another problem we havesolved?); create a parallel problem with simpler values (Jacobs & Ambrose, 2008).

Challenge early finishers in some manner that is related to the problem just solved.Possible questions: I see you found one way to do this. Are there any other solutions?Are any of the solutions different or more interesting than others? Some goodquestions for extending thinking are, What if...? or Would that same idea work for...?

AfterPromote a community oflearners.

Listen activelywithoutevaluation.

Summarize mainideas and identifyfuture problems.

You must teach your students about your expectations for this part of the lesson andhow to interact respectfully with their peers. Role play appropriate (and inappropriate)ways of responding to each other. The “Orchestrating Classroom Discourse” sectionprovides strategies and recommendations for how to facilitate discussions.

The goal here is to notice students’ mathematical thinking and making that thinkingvisible to other students. Avoid judging the correctness of an answer so students aremore willing to share their ideas. Support students’ thinking without evaluation bysimply asking what others think about a students’ response.

Formalize the main ideas of the lesson, helping to highlight connections betweenstrategies or different ideas. This is the time to reinforce appropriate terminology,definitions, and symbols. Possibly lay the groundwork for future tasks and activities.

“You used the red trapezoid as your whole?”“So, first you recorded your measurements in a table?”“What parts of your drawing relate to the numbers from the story problem?”“Who can share what Ricardo just said, but using your own words?”

Clarify Students’ Ideas

“Why does it make sense to start with that particular number?”“Explain how you know that your answer is correct.”“Can you give an example?”“Do you see a connection between Julio’s idea and Rhonda’s idea?”“What if...?”“Do you agree or disagree with Johanna? Why?”

Emphasize Reasoning

“Who has a question for Vivian?”“Turn to your partner and explain why you agree or disagree with Edwin.”“Talk with Yerin about how your strategy relates to hers.”

Encourage Student-StudentDialogue

Examples of teacher prompts for supporting classroom discussions.

Persistence inProblem Solving

How can I let go?

THANK YOU