common core math and technology
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Common Core Math and Technology tinyurl.com/ccmathandtech
4th Grade Teacher Apple Distinguished Educator
Google Certified Teacher @RossCoops31 rcooper@eastpennsd.org
How can I let go?
How can I let go?
Why is a change necessary? !
How can this change be made more comfortable?
!
How can technology help?
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
Pennsylvania Standards
for Mathematical Practice
Make sense of and persevere in solving complex and novel
mathematical problems
Use effective mathematical
reasoning to construct viable arguments and
critique the reasoning 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?
25 because a shepherd has the word sheepin it so you have to take away the dogs
and you just get 25. !
I dont have enough info. I cant
answer this question !
If he started out with 2 they reproduced it would take a year (about) for each to be born. And the
same with the dogs. If he started at the age of about 18
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:
Before: • 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: ModelFigure out what fraction of the flag is red
(which would be affected by shortage of red thread).
Before: ModelFigure out what fraction of the flag is red
(which would be affected by shortage of red thread). !
You will work in groups of 5-6 and will be responsible for sharing your ideas and solutions with the group.
!You may use any alternate materials to figure out the
fractional piece of red. !
First, create a list of strategies that you could use to solve the problem. Then, come to a consensus and
pick one way and get started!
Before: Discussion
How is what you experienced the same or different from your current classroom?
!
How are we making students
accountable for their own learning? !
!
How is this helping you to let go?
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will 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 students
accountable for their own learning? !
!
How is this helping you to let go?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
Questioning!
Observations do not have to be silent. Probing into student thinking through the use of questions can provide better data and more insights to inform instruction. As you circulate around the classroom to observe and evaluate students’ understanding, your use of questions is one of the most important ways to formatively assess in each lesson phase. Keep the following questions in mind (or on a clipboard, index cards, or a bookmark) 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 fraction quantities
NO
T TH
ERE
YET
ON
TA
RG
ET
AB
OVE
AN
D
BEY
ON
D
CO
MM
ENTS
FRACTIONS
MATHEMATICAL PRACTICES
Understands numerator/denominator
Area models
Set models
Use fractions in real contexts
Make sense of problems and perseveres
Models with mathematics
Uses appropriate tools
NAME: Sharon V.
Estimates fraction quantities
NO
T TH
ERE
YET
ON
TA
RG
ET
AB
OVE
AN
D
BEY
ON
D
CO
MM
ENTS
FRACTIONS
MATHEMATICAL PRACTICES
Understands numerator/denominator
Area models
Set models
Use fractions in real contexts
Make sense of problems and perseveres
Models with mathematics
Uses appropriate tools
NAME: Sharon V.
Estimates fraction quantities
NO
T TH
ERE
YET
ON
TA
RG
ET
AB
OVE
AN
D
BEY
ON
D
CO
MM
ENTS
FRACTIONS
MATHEMATICAL PRACTICES
Understands numerator/denominator
Area models
Set models
Use fractions in real contexts
Make sense of problems and perseveres
Models with mathematics
Uses appropriate tools
Used pattern blocks to show
2/3 and 3/6
Showing greater reasonableness
Stated problem in own words
Reluctant to use abstract models
Names
Lalie
Pete
Sid
Lakeshia
George
Pam
Maria
Topic: !Mental Computation Adding 2-digit numbers
Not There Yet CommentsOn Target Above and Beyond
Names
Lalie
Pete
Sid
Lakeshia
George
Pam
Maria
Topic: !Mental Computation Adding 2-digit numbers
Not There Yet CommentsOn Target Above and Beyond
Can’t do mentally
Has at least one strategy
Uses different methods with
different numbers
Names
Lalie
Pete
Sid
Lakeshia
George
Pam
Maria
Topic: !Mental Computation Adding 2-digit numbers
Not There Yet CommentsOn Target Above and Beyond
3-20
3-18
3-24
3-24
3-20
Can’t do mentally
Has at least one strategy
Uses different methods with
different numbers
3-21
Names
Lalie
Pete
Sid
Lakeshia
George
Pam
Maria
Topic: !Mental Computation Adding 2-digit numbers
Not There Yet CommentsOn Target Above and Beyond
Difficulty with regrouping
Flexible approaches used
Counts by tens, then adds ones
Beginning to add the group of tens first
Using a posted hundreds chart
3-20
3-18
3-24
3-24
3-20
Can’t do mentally
Has at least one strategy
Uses different methods with
different numbers
3-21
Observation Rubric Making Whole Given Fraction Part
Above and Beyond Clear understanding. Communicates concept in multiple representations. Shows evidence of using idea without prompting.
On Target Understands or is developing well. Uses designated models.
Not There Yet Some confusion or misunderstanding. Only models idea with help.
Observation Rubric Making Whole Given Fraction Part
Above and Beyond Clear understanding. Communicates concept in multiple representations. Shows evidence of using idea without prompting.
On Target Understands or is developing well. Uses designated models.
Not There Yet Some confusion or misunderstanding. Only models idea with help.
Fraction whole made from parts in rods and in sets. Explains easily.
Can make whole in either rod or set format (note). Hesitant. Needs prompt to identify unit fraction.
Needs help to do activity. No confidence.
Observation Rubric Making Whole Given Fraction Part
Above and Beyond Clear understanding. Communicates concept in multiple representations. Shows evidence of using idea without prompting.
On Target Understands or is developing well. Uses designated models.
Not There Yet Some confusion or misunderstanding. Only models idea with help.
Fraction whole made from parts in rods and in sets. Explains easily.
Can make whole in either rod or set format (note). Hesitant. Needs prompt to identify unit fraction.
Needs help to do activity. 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 students
accountable 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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 prior knowledge.
Be sure the problem is understood.
Establish clear expectations.
Begin with a simple version of the task; connect to students’ experiences; brainstorm approaches or solution strategies; estimate or predict whether tasks involve a single computation or are aimed at the development of a computational procedure.
Have students explain to you what the problem is asking. Go over vocabulary that may 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 they will have a choice. Tell them how they will share their solutions and reasoning.
During Let go!
Notice students’ mathematical thinking.
Provide appropriate support.
Provide worthwhile extensions.
Although it is tempting to want to step in and “help,” hold back and enjoy observing and 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. Can you 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 the problem. Ensure that students understand the problem (What do you know about the problem?); ask the student what he or she has already tried (also, Where did you get stuck?); 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 have solved?); 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 good questions for extending thinking are, What if...? or Would that same idea work for...?
AfterPromote a community of learners.
Listen actively without evaluation.
Summarize main ideas and identify future problems.
You must teach your students about your expectations for this part of the lesson and how to interact respectfully with their peers. Role play appropriate (and inappropriate) ways of responding to each other. The “Orchestrating Classroom Discourse” section provides strategies and recommendations for how to facilitate discussions.
The goal here is to notice students’ mathematical thinking and making that thinking visible to other students. Avoid judging the correctness of an answer so students are more willing to share their ideas. Support students’ thinking without evaluation by simply asking what others think about a students’ response.
Formalize the main ideas of the lesson, helping to highlight connections between strategies 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-Student Dialogue
Examples of teacher prompts for supporting classroom discussions.
How can I let go?
Technology Integration
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Forms
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4th Grade Teacher Apple Distinguished Educator
Google Certified Teacher @RossCoops31 rcooper@eastpennsd.org
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