Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Selecting and Sequencing Based on Essential
Understandings
Tennessee Department of Education
Elementary School Mathematics
Grade 5
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Rationale
There is wide agreement regarding the value of
teachers attending to and basing their instructional
decisions on the mathematical thinking of their students
(Warfield, 2001).
By engaging in an analysis of a lesson-planning
process, teachers will have the opportunity to consider
the ways in which the process can be used to help them
plan and reflect, both individually and collectively, on
instructional activities that are based on student thinking
and understanding.
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Session Goals
Participants will learn about:
• goal-setting and the relationship of goals to the
CCSS and essential understandings;
• essential understandings as they relate to selecting
and sequencing student work;
• Accountable Talk® moves related to essential
understandings; and
• prompts that problematize or “hook” students during
the Share, Discuss, and Analyze phase of the
lesson.
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“The effectiveness of a lesson depends
significantly on the care with which the
lesson plan is prepared.”
Brahier, 2000
4
“During the planning phase, teachers make
decisions that affect instruction dramatically.
They decide what to teach, how they are going
to teach, how to organize the classroom, what
routines to use, and how to adapt instruction for
individuals.”
Fennema & Franke, 1992, p. 156
5
Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
6
Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
Setting Goals
Selecting Tasks
Anticipating Student Responses
Orchestrating Productive Discussion
• Monitoring students as they work
• Asking assessing and advancing questions
• Selecting solution paths
• Sequencing student responses
• Connecting student responses via Accountable
7
Talk discussions
Identify Goals for Instruction
and Select an Appropriate Task
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The Structure and Routines of a Lesson
Set Up
Up the
of the
Task
Set
Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/
Small Group Problem Solving
1. Generate and Compare Solutions
2. Assess and Advance Student Learning
Share, Discuss, and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
3. Focus the Discussion on Key
Mathematical Ideas
4. Engage in a Quick Write
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MONITOR: Teacher selects
examples for the Share, Discuss,
and Analyze Phase based on:
• Different solution paths to the
same task
• Different representations
• Errors
• Misconceptions
SHARE: Students explain their
methods, repeat others’ ideas,
put ideas into their own words,
add on to ideas and ask
for clarification.
REPEAT THE CYCLE FOR EACH
SOLUTION PATH
COMPARE: Students discuss
similarities and difference
between solution paths.
FOCUS: Discuss the meaning
of mathematical ideas in each
representation
REFLECT: Engage students
in a Quick Write or a discussion
of the process.
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Contextualizing Our Work Together
Imagine that you are working with a group of students who have the
following understanding of the concepts.
• 70% of the students need to multiply fractions. (5.NF.B4 and
5.NF.B5)
• 20% of the students need additional work on fraction
standards previously addressed (4.NF standards). These
students also need opportunities to struggle with and make
sense of the problem. (MP1)
• 5% of the students still do not recognize the importance of
knowing what the “whole” is when talking about fractions.
(Part of 4.NF.A2)
• 5% of the students struggle to pay attention and their
understanding of mathematics is two grade levels below fifth
grade.
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The CCSS for Mathematics: Grade 5
Number and Operations – Fractions
5.NF
Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions
by whole numbers and whole numbers by unit fractions.
5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute
such quotients. For example, create a story context for (1/3) ÷ 4, and use a
visual fraction model to show the quotient. Use the relationship between
multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4
= 1/3.
5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such
5.NF.B.7c
quotients. For example, create a story context for 4 ÷ (1/5), and use a visual
fraction model to show the quotient. Use the relationship between
multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) =
4.
Solve real-world problems involving division of unit fractions by non-zero
whole numbers and division of whole numbers by unit fractions, e.g., by
using visual fraction models and equations to represent the problem. For
example, how much chocolate will each person get if 3 people share 1/2 lb of
chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Common Core State Standards, 2010, p. 36 - 37, NGA Center/CCSSO
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Mathematical Practice Standards Related to
the Task
1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
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Identify Goals: Solving the Task
(Small Group Discussion)
Solve the task.
Discuss the possible solution paths to the task.
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Bobby’s Hike Task
Bobby said that he wanted to go for a four-mile hike.
1
Bobby stops every mile for a sip of water from his
3
water bottle. How many times does Bobby stop? Be
sure to show how you found your answer with both
diagrams and an explanation in words. What equations
involving fractions match your diagram?
1
Extension: It takes Bobby hour to travel one mile. How
2
often does Bobby stop for water? How do you know
your answer is correct? Show with words, diagrams,
and a fractional equation. Is there another equation that
matches your diagram?
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Identify Goals Related to the Task
(Whole Group Discussion)
Does the task provide opportunities for students to
access the Mathematical Content Standards and
Practice Standards that we have identified for student
learning?
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Identify Goals: Essential Understandings
(Whole Group Discussion)
Study the essential understandings associated with the
Number and Operations – Fractions Common Core
Standards.
Which of the essential understandings are the goals of
Bobby’s Hike Task?
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The CCSS for Mathematics: Grade 5
Number and Operations – Fractions
5.NF
Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions
by whole numbers and whole numbers by unit fractions.
5.NF.B.7a
Interpret division of a unit fraction by a non-zero whole number, and
compute such quotients. For example, create a story context for (1/3) ÷ 4,
and use a visual fraction model to show the quotient. Use the relationship
between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because
(1/12) x 4 = 1/3.
5.NF.B.7b
Interpret division of a whole number by a unit fraction, and compute such
quotients. For example, create a story context for 4 ÷ (1/5), and use a visual
fraction model to show the quotient. Use the relationship between
multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) =
4.
Solve real-world problems involving division of unit fractions by non-zero
whole numbers and division of whole numbers by unit fractions, e.g., by
using visual fraction models and equations to represent the problem. For
example, how much chocolate will each person get if 3 people share 1/2 lb
of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
5.NF.B.7c
Common Core State Standards, 2010, p. 36 - 37, NGA Center/CCSSO
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Essential Understandings
(Small Group Discussion)
Essential Understanding
Equal Size Pieces
A fraction describes the division of a whole or unit (region, set, segment) into equal
parts. A fraction is relative to the size of the whole or unit.
Continuous and Discrete Figures
A fraction can be continuous (linear model), or a measurable quantity (area model),
or a group of discrete/countable things (set model) but, regardless of the model,
what remains true about all of the models is that they represent equal parts of a
whole.
Meaning of the Denominator
The larger the name of the denominator, the smaller the size of the piece.
Dividing Fractions
When dividing a fraction by a whole number, every iteration of the unit fraction
needs to be divided by the whole number.
Dividing by Fractions
When dividing a whole number by a unit fraction, the number of times that the unit
fraction fits inside the whole number is determined by the denominator.
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Selecting and Sequencing
Student Work for the
Share, Discuss, and Analyze
Phase of the Lesson
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Analyzing Student Work
(Private Think Time)
• Analyze the student work.
• Identify what each group knows related to the
essential understandings.
• Consider the questions that you have about each
group’s work as it relates to the essential
understandings.
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Prepare for the Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Small Group Discussion)
Assume that you have circulated and asked students
assessing and advancing questions.
Study the student work samples.
1. Which pieces of student work will allow you to
address the essential understanding?
2. How will you sequence the student’s work that you
have selected? Be prepared to share your rationale.
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The Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Small Group Discussion)
In your small group, come to consensus on the work
that you select, and share your rationale. Be prepared
to justify your selection and sequence of student work.
Essential Understandings
Group(s)
Order Rationale
Equal Size Pieces
Continuous and Discrete Figures
Dividing Fractions
Dividing by Fractions
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The Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Whole Group Discussion)
What order did you identify for the EUs and student
work?
What is your rationale for each selection?
Essential Understandings
#1 via
Gr.
#2 via
Gr.
#3 via
Gr.
#4 Via
Gr.
Equal Size Pieces
A fraction describes…
Continuous and Discrete Figures
A fraction can be continuous…
Dividing Fractions
When dividing a fraction…
Dividing by Fractions
When dividing a whole number…
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Group A
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Group B
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Group C
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Group D
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Group E
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Group F
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Group G
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The Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Whole Group Discussion)
What order did you identify for the EUs and student
work?
What is your rationale for each selection?
Essential Understandings
#1 via
Gr.
#2 via
Gr.
#3 via
Gr.
#4 Via
Gr.
Equal Size Pieces
A fraction describes…
Continuous and Discrete Figures
A fraction can be continuous…
Dividing Fractions
When dividing a fraction…
Dividing by Fractions
When dividing a whole number…
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Academic Rigor in a Thinking
Curriculum
The Share, Discuss, and Analyze
Phase of the Lesson
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Academic Rigor In a Thinking
Curriculum
A teacher must always be assessing and advancing
student learning.
A lesson is academically rigorous if student learning
related to the essential understanding is advanced in
the lesson.
Accountable Talk discussion is the means by which
teachers can find out what students know or do not
know and advance them to the goals of the lesson.
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Accountable Talk Discussions
Recall what you know about the Accountable Talk
features and indicators. In order to recall what you
know:
• Study the chart with the Accountable Talk moves.
You are already familiar with the Accountable Talk
moves that can be used to Ensure Purposeful,
Coherent, and Productive Group Discussion.
• Study the Accountable Talk moves associated
with creating accountability to:
 the learning community;
 knowledge; and
 rigorous thinking.
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Accountable Talk Features and Indicators
Accountability to the Learning Community
• Active participation in classroom talk.
• Listen attentively.
• Elaborate and build on each others’ ideas.
• Work to clarify or expand a proposition.
Accountability to Knowledge
• Specific and accurate knowledge.
• Appropriate evidence for claims and arguments.
• Commitment to getting it right.
Accountability to Rigorous Thinking
• Synthesize several sources of information.
• Construct explanations and test understanding of concepts.
• Formulate conjectures and hypotheses.
• Employ generally accepted standards of reasoning.
• Challenge the quality of evidence and reasoning.
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Accountable Talk Moves
Talk Move
Function
Example
To Ensure Purposeful, Coherent, and
Productive Group Discussion
Marking
Direct attention to the value and importance of a
student’s contribution.
That’s an important point. One factor tells
use the number of groups and the other
factor tells us how many items in the group.
Challenging
Redirect a question back to the students or use
students’ contributions as a source for further
challenge or query.
Let me challenge you: Is that always true?
Revoicing
Align a student’s explanation with content or connect
two or more contributions with the goal of advancing
the discussion of the content.
S: 4 + 4 + 4.
Make public in a concise, coherent form, the group’s
achievement at creating a shared understanding of the
phenomenon under discussion.
Let me put these ideas all together.
What have we discovered?
Recapping
You said three groups of four.
To Support Accountability to Community
Keeping the
Channels
Open
Ensure that students can hear each other, and
remind them that they must hear what others
have said.
Say that again and louder.
Can someone repeat what was just said?
Keeping
Everyone
Together
Ensure that everyone not only heard, but also
understood, what a speaker said.
Can someone add on to what was said?
Did everyone hear that?
Linking
Contributions
Make explicit the relationship between a new
contribution and what has gone before.
Does anyone have a similar idea?
Do you agree or disagree with what was
said?
Your idea sounds similar to his idea.
Verifying and
Clarifying
Revoice a student’s contribution, thereby helping
both speakers and listeners to engage more
profitably in the conversation.
So are you saying..?
Can you say more?
Who understood what was said?
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Accountable Talk Moves (continued)
To Support Accountability to Knowledge
Pressing for
Accuracy
Hold students accountable for the accuracy,
credibility, and clarity of their contributions.
Why does that happen?
Someone give me the term for that.
Building on
Prior
Knowledge
Tie a current contribution back to knowledge
accumulated by the class at a previous time.
What have we learned in the past that links
with this?
To Support Accountability to
Rigorous Thinking
Pressing for
Reasoning
Elicit evidence to establish what contribution a
student’s utterance is intended to make within
the group’s larger enterprise.
Say why this works.
What does this mean?
Who can make a claim and then tell us
what their claim means?
Expanding
Reasoning
Open up extra time and space in the
conversation for student reasoning.
Does the idea work if I change the
context? Use bigger numbers?
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The Share, Discuss, and Analyze Phase of
the Lesson: Planning a Discussion
(Small Group Discussion)
• From the list of potential EUs and its related student
work, each group will select an essential
understanding to focus their discussion.
• Identify a teacher in the group who will be in charge
of leading a discussion with the group after the
Accountable Talk moves related to the EU have been
written.
Write a set of Accountable Talk moves on chart paper
so it is public to your group for the next stage in the
process.
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An Example: Accountable Talk Discussion
The Focus Essential Understanding
Continuous and Discrete Figures
A fraction can be continuous (linear model), or a measureable quantity (area
model), or a group of discrete/countable things (set model) but, regardless of the
model, what remains true about all of the models is that they represent equal parts
of a whole.
Group F
Group G
•
•
•
•
•
•
Explain how your model shows the problem.
Who understood what he said about the number line? (Community)
Can you say back what he said how the model shows the hike? (Community)
Who can add on and talk about the section of the number line? (Community)
The denominator tells the number of equal parts in the whole. (Marking)
1
Do we see 3 in both models? (Rigor)
•
Tell us how you found 3 in your picture (Group G). (Rigor)
1
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Problematize the Accountable Talk Discussion
(Whole Group Discussion)
Using the list of essential understandings identified earlier, write
Accountable Talk discussion questions to elicit from students a
discussion of the mathematics.
Begin the discussion with a “hook” to get student attention focused on
an aspect of the mathematics.
Type of Hook
Example of a Hook
Compare and
Contrast
Compare the half that has two equal pieces with the
figure that has three pieces.
Insert a Claim and
Ask if it is True
Three equal pieces of the six that are on one side of
the figure show half of the figure. If I move the three
pieces to different places in the whole, is half of the
figure still shaded?
Challenge
You said two pieces are needed to create halves. How
can this be half; it has three pieces?
A Counter-Example
If this figure shows halves (a figure showing three
sixths), tell me about this figure (a figure showing three
sixths but the sixths are not equal pieces).
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An Example: Accountable Talk Discussion
The Focus Essential Understanding
Continuous and Discrete Figures
A fraction can be continuous (linear model), or a measurable quantity (area
model), or a group of discrete/countable things (set model) but, regardless of the
model, what remains true about all of the models is that they represent equal
parts of a whole.
Group F
Group G
•
•
•
•
•
One group used a number line and one group used an area model. How
can this be? Can they both model the problem? (Hook)
Can Group F explain where the whole and where the stops are?
Who understood what they said about the divisions of the line? (Community)
Can you say back what they said about the meaning of the numerator and
denominator for Bobby’s hike? (Community)
Each group made statements about the model being accurate to the context.
Where do we see division in each of the models? (Rigor)
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Revisiting Your Accountable Talk Prompts
with an Eye Toward Problematizing
Revisit your Accountable Talk prompts.
Have you problematized the mathematics so as to draw
students’ attention to the mathematical goal of the
lesson?
• If you have already problematized the work, then
underline the prompt in red.
• If you have not problematized the lesson, do so
now. Write your problematizing prompt in red at
the bottom and indicate where you would insert it
in the set of prompts.
We will be doing a Gallery Walk after we role play.
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Role Play Our Accountable Talk Discussion
• You will have 15 minutes to role play the discussion of
one essential understanding.
• Identify one observer in the group. The observer will
keep track of the discussion moves used in the lesson.
• The teacher will engage you in a discussion. (Note:
You are well-behaved students.)
The goals for the lesson are:
 to engage all students in the group in developing
an understanding of the EU; and
 to gather evidence of student understanding
based on what the student shares during the
discussion.
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Reflecting on the Role-Play: The
Accountable Talk Discussion
• The observer has 2 minutes to share observations
related to the lessons. The observations should be
shared as “noticings.”
• Others in the group have 1 minute to share their
“noticings.”
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Reflecting on the Role Play: The
Accountable Talk Discussion
(Whole Group Discussion)
Now that you have engaged in role playing, what are
you now thinking about regarding Accountable Talk
discussions?
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Zooming In on Problematizing
(Whole Group Discussion)
Do a Gallery Walk. Read each others’ problematizing
“hook.”
What do you notice about the use of hooks? What role
do “hooks” play in the lesson?
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Step Back and Application to Our
Work
What have you learned today that you will apply when
planning or teaching in your classroom?
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Summary of Our Planning Process
Participants:
• identify goals for instruction;
– Align Content Standards and Mathematical Practice
Standards with a task.
– Select essential understandings that relate to the
Content Standards and Mathematical Practice
Standards.
• prepare for the Share, Discuss, and Analyze phase of
the lesson.
– Analyze and select student work that can be used to
discuss essential understandings of mathematics.
– Learn methods of problematizing the mathematics in
the lesson.
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