Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Using Academically Productive Talk Moves:
Orchestrating a Focused Discussion
Tennessee Department of Education
Elementary School Mathematics
Grade 4
© 2013 UNIVERSITY OF PITTSBURGH
Rationale
Mathematics reform calls for teachers to engage students in
discussing, explaining, and justifying their ideas. Although
teachers are asked to use students’ ideas as the basis for
instruction, they must also keep in mind the mathematics
that the class is expected to explore (Sherin, 2000, p. 125).
By engaging in a high-level task and reflecting on ways in
which the facilitator structured and supported the discussion
of mathematical ideas, teachers will learn that they are
responsible for orchestrating discussions in ways that make
it possible for students to own their learning, as well as for
the teacher to assess and advance student understanding
of knowledge and mathematical reasoning.
2
Session Goals
Participants will:
• learn about Accountable Talk® features and indicators
and consider the benefit of all being present in a
lesson;
• learn that there are specific moves related to each of
the talk features that help to develop a discourse
culture; and
• consider the importance of the four key moves of
ensuring productive discussion (marking, recapping,
challenging, and revoicing).
Accountable Talk® is a registered trademark of the University of Pittsburgh.
© 2013 UNIVERSITY OF PITTSBURGH
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Overview of Activities
Participants will:
•
review the Accountable Talk features and
indicators;
•
identify and discuss Accountable Talk moves in a
video; and
•
align CCSS and essential understandings (EUs) to
a fraction task and zoom in for a more specific look
at key moves for engaging in productive talk
(marking, recapping, challenging, and revoicing).
© 2013 UNIVERSITY OF PITTSBURGH
4
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
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
Setting Goals
Selecting Tasks
Anticipating Student Responses
Accountable Talk® is a registered trademark of the
University of Pittsburgh.
Orchestrating Productive Discussion
• Monitoring students as they work
• Asking assessing and advancing questions
• Selecting solution paths
• Sequencing student responses
• Connecting student responses via Accountable
6
Talk® discussions
Accountable Talk
Features and Indicators
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Accountable Talk Discussion
• Study the Accountable Talk features and indicators.
• Turn and Talk with your partner about what you
would expect teachers and students to be saying
during an Accountable Talk discussion so that the
discussion is accountable to:
− the learning community;
− accurate, relevant knowledge; and
− standards of rigorous thinking.
© 2013 UNIVERSITY OF PITTSBURGH
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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.
© 2013 UNIVERSITY OF PITTSBURGH
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Solving and Discussing the
Cognitive Demand of the
Brownie 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
© 2013 UNIVERSITY OF PITTSBURGH
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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Engaging in a Lesson:
The Brownie Task
• Solve the task.
• Discuss your solutions with your peers.
• Attempt to engage in an Accountable Talk discussion
when discussing the solutions. Assign one person in
the group to be the observer. This person will be
responsible for reporting some of the ways in which
the group is accountable to:
− the learning community;
− accurate, relevant knowledge; and
− standards of rigorous thinking.
© 2013 UNIVERSITY OF PITTSBURGH
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The Brownie Task
There are 7 brownies. 4 people are sharing the
brownies. How much of the brownies will each person
get? What do you call that amount?
Investigations, 1998
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Reflecting on Our Engagement in the
Lesson
The observer should share some observations about
the group’s engagement in an Accountable Talk
discussion.
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Reflecting on Our Engagement in the
Lesson
• In what ways did small groups engage in an
Accountable Talk discussion?
• In what ways did we engage in an Accountable Talk
discussion during the group discussion of the
solutions?
© 2013 UNIVERSITY OF PITTSBURGH
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Aligning the CCSS to the Brownie Task
• Study the Grade 4 CCSS for Mathematical Content
within the Number and Operations – Fractions
domain.
Which standards are students expected to
demonstrate when solving the fraction task?
• Identify the CCSS for Mathematical Practice required
by the written task.
© 2013 UNIVERSITY OF PITTSBURGH
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The CCSS for Mathematical Content: Grade 4
Number and Operations – Fractions
4.NF
Extend understanding of fraction equivalence and ordering.
4.NF.A.1
Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using
visual fraction models, with attention to how the number and size of the parts
differ even though the two fractions themselves are the same size. Use this
principle to recognize and generate equivalent fractions.
4.NF.A.2
Compare two fractions with different numerators and different denominators,
e.g., by creating common denominators or numerators, or by comparing to a
benchmark fraction such as 1/2. Recognize that comparisons are valid only
when the two fractions refer to the same whole. Record the results of
comparisons with symbols >, =, or <, and justify the conclusions, e.g., by
using a visual fraction model.
Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers.
4.NF.B.3
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
4.NF.B.3a
Understand addition and subtraction of fractions as joining and separating
parts referring to the same whole.
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Common Core State Standards, 2010, p. 30, NGA Center/CCSSO
The CCSS for Mathematical Content: Grade 4
Number and Operations – Fractions
4.NF
Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers.
4.NF.B.4
Apply and extend previous understandings of multiplication to multiply a
fraction by a whole number.
4.NF.B.4a
Understand a fraction a/b as a multiple of 1/b. For example, use a visual
fraction model to represent 5/4 as the product 5 × (1/4), recording the
conclusion by the equation 5/4 = 5 × (1/4).
4.NF.B.4b
Understand a multiple of a/b as a multiple of 1/b, and use this
understanding to multiply a fraction by a whole number. For example,
use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing
this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
4.NF.B.4c
Solve word problems involving multiplication of a fraction by a whole
number, e.g., by using visual fraction models and equations to represent
the problem. For example, if each person at a party will eat 3/8 of a
pound of roast beef, and there will be 5 people at the party, how many
pounds of roast beef will be needed? Between what two whole numbers
does your answer lie?
Common Core State Standards, 2010, p. 30, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 5
Number and Operations—Fractions
5.NF
Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.A.1
Add and subtract fractions with unlike denominators (including
mixed numbers) by replacing given fractions with equivalent
fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators. For example, 2/3 +
5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.A.2
Solve word problems involving addition and subtraction of
fractions referring to the same whole, including cases of unlike
denominators, e.g., by using visual fraction models or equations
to represent the problem. Use benchmark fractions and number
sense of fractions to estimate mentally and assess the
reasonableness of answers. For example, recognize an incorrect
result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Common Core State Standards, 2010, p. 36, NGA Center/CCSSO
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The CCSS for Mathematical Content: 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.3
Interpret a fraction as division of the numerator by the denominator (a/b
= a ÷ b). Solve word problems involving division of whole numbers
leading to answers in the form of fractions or mixed numbers, e.g., by
using visual fraction models or equations to represent the problem. For
example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4
multiplied by 4 equals 3, and that when 3 wholes are shared equally
among 4 people each person has a share of size 3/4. If 9 people want to
share a 50-pound sack of rice equally by weight, how many pounds of
rice should each person get? Between what two whole numbers does
your answer lie?
Common Core State Standards, 2010, p. 36, NGA Center/CCSSO
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The CCSS for Mathematical Content: 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.4
Apply and extend previous understandings of multiplication to multiply a
fraction or whole number by a fraction.
5.NF.B.4a
Interpret the product (a/b) × q as a parts of a partition of q into b equal
parts; equivalently, as the result of a sequence of operations a × q ÷ b.
For example, use a visual fraction model to show (2/3) x 4 = 8/3, and
create a story context for this equation. Do the same with (2/3) x (4/5) =
8/15. (In general, (a/b) x (c/d) = ac/bd.)
5.NF.B.4b
Find the area of a rectangle with fractional side lengths by tiling it with
unit squares of the appropriate unit fraction side lengths, and show that
the area is the same as would be found by multiplying the side lengths.
Multiply fractional side lengths to find areas of rectangles, and represent
fraction products as rectangular areas.
Common Core State Standards, 2010, p. 36, NGA Center/CCSSO
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The CCSS for Mathematical Practice
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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Determining the Cognitive
Demand of the Task:
The Brownie Task
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Determining the Cognitive Demand
of the Task
Refer to the Mathematical Task Analysis Guide.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A., 2000. Implementing standards-based mathematics
instruction: A casebook for professional development, p. 16. New York: Teachers College Press.
How would you characterize the Brownie Task in terms
of its cognitive demand? (Refer to the indicators on the
Task Analysis Guide.)
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The Mathematical Task Analysis Guide
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction:
A casebook for professional development, p. 16. New York: Teachers College Press
.
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The Brownie Task:
A Doing Mathematics Task
• Requires complex and non-algorithmic thinking (i.e., there is not a
predictable, well-rehearsed approach or pathway explicitly suggested by
the task, task instructions, or a worked-out example).
• Requires students to explore and to understand the nature of
mathematical concepts, processes, or relationships.
• Demands self-monitoring or self-regulation of one’s own cognitive
processes.
• Requires students to access relevant knowledge and experiences and
make appropriate use of them in working through the task.
• Requires students to analyze the task and actively examine task
constraints that may limit possible solution strategies and solutions.
• Requires considerable cognitive effort and may involve some level of
anxiety for the student due to the unpredictable nature of the solution
process required.
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Accountable Talk Moves
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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
© 2013 UNIVERSITY OF PITTSBURGH
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.
28
Accountable Talk Moves
Examine the ways in which the moves are grouped based on
how they:
• support accountability to the learning community;
• support accountability to knowledge; and
• support accountability to rigorous thinking.
Consider:
In what ways are the Accountable Talk categories
similar? Different?
Why do you think we need a category called “To Ensure
Purposeful, Coherent, and Productive Group Discussion”?
© 2013 UNIVERSITY OF PITTSBURGH
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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.
© 2013 UNIVERSITY OF PITTSBURGH
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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.
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?
© 2013 UNIVERSITY OF PITTSBURGH
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Reflection Question
As you watch the short video segment, consider what
students are learning and where you might focus the
discussion in order to discuss mathematical ideas listed
in the CCSS.
Identify:
• the specific Accountable Talk moves used by the
teacher; and
• the purpose that the moves served.
 Mark times during the lesson when you would call
the lesson academically rigorous.
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The Brownie Task
Overview:
• A fraction lesson.
• Pre-conference (teacher and math coach plan lesson).
• Introduce the task to students.
• Students make predictions.
• Students work in pairs (a team) to solve the problem.
• Teacher and coach observe and push the students’ thinking.
• One team shares a solution with class.
7
• Group discussion about the quantity and the meaning of
4
“bottom number.”
Teacher: Katherine Casey
Coach: Lucy West
Grade 4, P.S. 116
Principal: Anna Marie Carrillo
New York City Community School
District 2
May 1996
© 2013 UNIVERSITY OF PITTSBURGH
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Norms for Collaborative Study
The goal of all conversations about episodes of teaching (or
artifacts of practice in general) is to advance our own learning, not
to “fix” the practice of others.
In order to achieve this goal, the facilitator chooses a lens to frame
what you look at and to what you pay attention. Use the
Accountable Talk features and indicators when viewing the lesson.
During this work, we:
• agree to analyze the episode or artifact from the identified
perspective;
• cite specific examples during the discussion that provide
evidence of a particular claim;
• listen to and build on others’ ideas; and
• use language that is respectful of those in the video and in the
group.
© 2013 UNIVERSITY OF PITTSBURGH
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The Brownie Task
There are 7 brownies. 4 people are sharing the
brownies. How much of the brownies will each person
get? What do you call that amount?
Investigations, 1998
36
Reflecting on the Accountable Talk
Discussion
Step back from the discussion. What are some patterns
that you notice?
What mathematical ideas does the teacher want
students to discover and discuss?
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Essential Understandings
Study the essential understandings the teacher
considered in preparation for the Share, Discuss, and
Analyze phase of the lesson.
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Essential Understandings
Essential Understanding
Iterations of Unit Fractions
When decomposing a fraction into iterations of the unit fraction, the
number of iterations is the same as the value of the numerator.
Operating with Fractions
Fractions with the same size pieces or common denominators can be
compared, combined, and the difference between them determined
because the size of the pieces are alike so only the number of pieces
(the numerator) needs to be considered.
Equivalence of Fractions
Fractions are equivalent if the same portion of the set or area of the
figure is represented or they represent the same point on the number
line.
Making Equivalent Fractions
When you multiply a fraction by a fraction equivalent to one, a/a (a >
1), the denominator is partitioned into a new designated number of
pieces that are smaller in size but larger in the number of pieces than
the original; as a result of partitioning all of the pieces, those pieces
referenced by the numerator end up being partitioned in the same
way.
© 2013
UNIVERSITY OF PITTSBURGH
CCSS
4.NF.B.4a
4.NF.B.3a
4.NF.A.1
4.NF.A.1
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Characteristics of an Academically
Rigorous Lesson
This task is a cognitively demanding task; however, it
may not necessarily end up being an academically
rigorous task.
What do we mean by this statement?
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Academic Rigor in a Thinking Curriculum
The principle of learning, Academic Rigor in a Thinking
Curriculum, consists of three features:
• A Knowledge Core
• High-Thinking Demand
• Active Use of Knowledge
In order to determine if a lesson has been academically
rigorous, we have to determine the degree to which
student learning is advanced by the lesson.
What do we have to hear and see in order to determine
if the lesson was academically rigorous?
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© 2013 UNIVERSITY OF PITTSBURGH
Essential Understandings
Essential Understanding
Iterations of Unit Fractions
When decomposing a fraction into iterations of the unit fraction, the
number of iterations is the same as the value of the numerator.
Operating with Fractions
Fractions with the same size pieces or common denominators can be
compared, combined, and the difference between them determined
because the size of the pieces are alike so only the number of pieces
(the numerator) needs to be considered.
Equivalence of Fractions
Fractions are equivalent if the same portion of the set or area of the
figure is represented or they represent the same point on the number
line.
CCSS
4.NF.B.4a
4.NF.B.3a
4.NF.A.1
Making Equivalent Fractions
When you multiply a fraction by a fraction equivalent to one, a/a (a >
4.NF.A.1
1), the denominator is partitioned into a new designated number of
pieces that are smaller in size but larger in the number of pieces than
the original; as a result of partitioning all of the pieces, those pieces
referenced by the numerator end up being partitioned in the same way.
© 2013 UNIVERSITY OF PITTSBURGH
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Five Representations
Adapted from Lesh, Post, & Behr, 1987
43
Focusing on Key
Accountable Talk Moves
The Brownie Task
© 2013 UNIVERSITY OF PITTSBURGH
44
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.
© 2013 UNIVERSITY OF PITTSBURGH
45
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.
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?
© 2013 UNIVERSITY OF PITTSBURGH
46
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?
© 2013 UNIVERSITY OF PITTSBURGH
47
Focusing on Accountable Talk Moves
Read the description of each move and study the
example that has been provided for each move.
What is distinct about each of the moves?
• Revoice student contributions;
• mark significant contributions;
• challenge with a counter-example; or
• recap the components of the lesson.
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Revoicing
• Extend a student’s contribution.
• Connect a student’s contribution to the text or to
other students’ contributions.
 Align content with an explanation.
 Add clarity to a contribution.
 Link student contributions to accurate
mathematical vocabulary.
 Connect two or more contributions to advance
the lesson.
© 2013 UNIVERSITY OF PITTSBURGH
49
An Example of Revoicing
S6:
They said that it would be easier to split them into fourths
and all together they got one brownie and three fourths.
T:
So are you saying that you took each brownie -- like this
was one of your brownies -- and you split this into
quarters?
S5:
Yeah, and we labeled each person. Like, this one's
Person #1, so we would write under it, "This is for Person
#1, this is for Person #2, this is for Person #3, and Person
#4.”
T:
Okay, so if they did that, they split all seven brownies into
little quarters, how many of these quarters did each
person get? What do you think? Six plus seven. What is
that fraction, I wonder? Seven quarters, written out in a
line.
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Marking
Explicitly talk about an idea.
• Highlight features that are unique to a situation.
• Draw attention to an idea or to alternative ideas.
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An Example of Marking
S3: I think the piece that is extra, you're going to have
to cut it into fourths, because there are four people.
T:
Okay. So you're thinking one and a fourth: a whole
and a fourth.
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Recapping
Summarize or retell.
• Make explicit the large idea.
• Provide students with a holistic view of the
concept.
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An Example of Recapping
S5: So we split them in fourths and we decided that
the answer would be, each person would have
one brownie and three fourths.
T:
Who could repeat back what they did to go about
solving this problem?
S6: They said that it would be easier to split them into
fourths and all together they got one brownie and
three fourths.
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Challenging
Redirect a question back to the students, or use
students’ contributions as a source for further challenge
or query.
• Share a counter-example and ask students to
compare problems.
• Question the meaning of the math concept.
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An Example of Challenge
T:
But now you also have a quarter. So what would
you call all three pieces together?
S4:
I would call it a whole and a half plus a quarter.
T:
A whole and a half plus a quarter. But I'm
wondering if there's one number you could give to
all three pieces. In other words, you could say,
"All together, this is called—I would call this—”
something. Any idea?
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Appropriation
The process of appropriation is reciprocal and
sequential.
• If appropriation takes place, the child transforms
the new knowledge or skill into an action in a new
and gradually understood activity.
• What would this mean with respect to classroom
discourse? What should we expect to happen in
the classroom?
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Orchestrating Discussions
Read the segments of transcript from the lesson.
Decide if examples 1 – 4 illustrate marking, recapping,
challenging, or revoicing.
Be prepared to share your rationale for identifying a
particular discussion move.
Write the next discussion move for examples 5 and 6
and be prepared to share your move and your rationale
for writing the move.
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Reflecting on Talk Moves
What have you learned about:
• marking;
• recapping;
• challenging; and
• revoicing?
Why are these moves important in lessons?
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Application to Practice
• What will you keep in mind when attempting to use
Accountable Talk moves during a lesson? What role
does talk play?
• What does it take to maintain the demands of a
cognitively demanding task during the lesson so that
you have a rigorous mathematics lesson?
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Bridge to Practice
• Choose a high-level task.
• Identify the key mathematical ideas that are the goals of
the lesson.
• Anticipate student responses.
• Plan an Accountable Talk discussion that utilizes the
moves of marking, challenging, recapping, and
revoicing.
• Record or have a colleague scribe the Share, Discuss,
and Analyze phase of the lesson.
• Write a reflection on how planning and using these
moves impacted student learning.
• Bring back: a copy of the task, your planning notes, and
written reflection.
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