Supporting Rigorous Mathematics
Teaching and Learning
Academically Productive Talk in Mathematics: A
Means of Making Sense of Mathematical Ideas
Tennessee Department of Education
Elementary School Mathematics
Grade 1
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Rationale
Effective teaching requires being able to support students as they
work on challenging tasks without taking over the process of thinking
for them (NCTM, 2000).
Building a practice of engaging students in academically rigorous
tasks supported by Accountable Talk® discourse facilitates effective
teaching. Students develop an understanding of mathematical ideas,
strategies, and representations and teachers gain insights into what
students know and what they can do. These insights prepare
teachers to consider ways that advance student understanding of
mathematical ideas, strategies, or connections to representations.
Today, by analyzing math classroom discussions, teachers will study
how Accountable Talk discussion supports student learning and
helps teachers to maintain the cognitive demand of the task.
Accountable Talk® is a registered trademark of the University of Pittsburgh
2
Session Goals
Participants will:
• learn a set of Accountable Talk features and indicators;
and
• recognize Accountable Talk stems for each of the
features of talk, and consider the potential benefit of
posting and practicing talk stems with students.
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Overview of Activities
Participants will:
• discuss Accountable Talk features and indicators;
• discuss students’ solution paths for a task;
• analyze and identify Accountable Talk features and
indicators in a lesson; and
• plan for an Accountable Talk discussion.
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The Structure and Routines of a Lesson
Set
Task
SetUp
Upthe
of the
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
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
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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LEARNING RESEARCH AND DEVELOPMENT CENTER
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.
5
Accountable Talk
Features and Indicators
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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 for each of the features.
− accountability to the learning community
− accountability to accurate, relevant knowledge
− accountability to discipline-specific standards
of rigorous thinking
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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Accountable Talk Discussion
Indicators for all three features must be present in order
for the discussion to be an “Accountable Talk
Discussion.”
• accountability to the learning community
• accountability to accurate, relevant knowledge
• accountability to discipline-specific standards
of rigorous thinking
Why might this be important?
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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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
LEARNING RESEARCH AND DEVELOPMENT CENTER
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Accountable Talk Starters
• Work in triads.
• On your chart paper, write talk starters for the
Accountable Talk indicators.
A talk starter is the start of a sentence that you might
hear from students if they are holding themselves
accountable for using Accountable Talk moves.
e.g., I want to add on to ______ (Community move).
The denominator of a fraction tells us _____
(Knowledge move).
The two equations are equivalent because ____
(Rigor move).
(Work for 5 minutes.)
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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Accountable Talk Starters
• What do you notice about the talk starts for the:
 accountability to the learning community
 accountability to accurate, relevant knowledge
 accountability to discipline-specific standards
of rigorous thinking
• What is the distinction between the stems for knowledge
and those for rigorous thinking?
• Why should we pay attention to this?
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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Preparing to Analyze Accountable Talk
Features and Indicators in Classroom
Practice
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LEARNING RESEARCH AND DEVELOPMENT CENTER
The Make a Ten Task
Use the interlinking cubes to make a structure of ten.
Write a number sentence to describe your structure of
ten. Draw a picture for your structure of ten.
Make a second structure of ten that looks different from
your first structure of ten. Write a different number
sentence for this structure of ten.
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Solve the Task
(Private Think Time and Small Group Time)
• Work privately. Analyze the student work for The Make a
Ten Task in your participant handout.
• Work with others at your table. Hold yourselves
accountable for engaging in an Accountable Talk
discussion when you discuss the student work.
 What do the students understand? How do you know?
 How does one solution path differ from the other?
 What might be used in the student work to prompt
students to think about and discuss the ideas
articulated in the standards?
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The Make a Ten Task: Devon’s Work
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The Make a Ten Task: David’s Work
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The Make a Ten Task: Tinesha’s Work
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Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Represent and solve problems involving addition and subtraction.
1.OA.A.1
Use addition and subtraction within 20 to solve word problems
involving situations of adding to, taking from, putting together,
taking apart, and comparing, with unknowns in all positions,
e.g., by using objects, drawings, and equations with a symbol
for the unknown number to represent the problem.
1.OA.A.2
Solve word problems that call for addition of three whole
numbers whose sum is less than or equal to 20, e.g., by using
objects, drawings, and equations with a symbol for the unknown
number to represent the problem.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
18
Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Understand and apply properties of operations and the relationship
between addition and subtraction.
1.OA.B.3 Apply properties of operations as strategies to add and subtract.
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known.
(Commutative property of addition.) To add 2 + 6 + 4, the second
two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10
= 12. (Associative property of addition.)
1.OA.B.4 Understand subtraction as an unknown-addend problem. For
example, subtract 10 – 8 by finding the number that makes 10
when added to 8.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
19
Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Add and subtract within 20.
1.OA.C.5
Relate counting to addition and subtraction (e.g., by counting on
2 to add 2).
1.OA.C.6
Add and subtract within 20, demonstrating fluency for addition
and subtraction within 10. Use strategies such as counting on;
making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing
a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 =
9); using the relationship between addition and subtraction
(e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and
creating equivalent but easier or known sums (e.g., adding 6 +
7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
20
Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Work with addition and subtraction equations.
1.OA.D.7
Understand the meaning of the equal sign, and determine if
equations involving addition and subtraction are true or false.
For example, which of the following equations are true and
which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
1.OA.D.8
Determine the unknown whole number in an addition or
subtraction equation relating three whole numbers. For
example, determine the unknown number that makes the
equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 +
6 = ?.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
21
Table 1: Common Addition and
Subtraction Situations
Common Core State Standards, 2010, p. 88, NGA Center/CCSSO
22
The Common Core Standards for
Mathematical Practice
What would have to happen in order for students to have
opportunities to make use of 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
23
Accountable Talk Features and
Indicators
Which of the Accountable Talk Features and Indicators
were illustrated in our discussion?
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Using the Accountable Talk
Features and Indicators to Analyze
Classroom Practice
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LEARNING RESEARCH AND DEVELOPMENT CENTER
Reflecting on the Lesson
Watch the video.
What are students learning in the Make a Ten Lesson?
Which Accountable Talk features and indicators were
illustrated in the video of the Make a Ten lesson?
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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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
LEARNING RESEARCH AND DEVELOPMENT CENTER
27
Context for the Make a Ten Lesson
Teacher: Jennifer DiBrienza
School: PS 116
District: New York City, District 2
Grade Level: First Grade
The lesson was conducted in a first grade classroom. The
students have worked with interlocking cubes many times.
They have explored ways in which they can use the blocks to
make different shapes. They have described the figures that
they can make with the interlocking cubes.
In this lesson, we observe the Share, Discuss, and Analyze
phase of the lesson. The students have worked independently
to create two different structures and to write number
sentences to describe their structures. Now the students come
together to share and discuss each other’s structures.
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The Structures and Routines of a Lesson
Set
Task
SetUp
Upthe
of the
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
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
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
LEARNING RESEARCH AND DEVELOPMENT CENTER
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.
29
The Make a Ten Lesson
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Common Core State Standards (CCSS)
Examine the first grade CCSS for Mathematics in your
participant handout.
– Which CCSS for Mathematical Content did we
discuss?
– Which CCSS for Mathematical Practice did we
use when solving and discussing the task?
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LEARNING RESEARCH AND DEVELOPMENT CENTER
31
Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Represent and solve problems involving addition and subtraction.
1.OA.A.1 Use addition and subtraction within 20 to solve word problems
involving situations of adding to, taking from, putting together,
taking apart, and comparing, with unknowns in all positions, e.g.,
by using objects, drawings, and equations with a symbol for the
unknown number to represent the problem.
1.OA.A.2 Solve word problems that call for addition of three whole
numbers whose sum is less than or equal to 20, e.g., by using
objects, drawings, and equations with a symbol for the unknown
number to represent the problem.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
32
Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Understand and apply properties of operations and the relationship
between addition and subtraction.
1.OA.B.3 Apply properties of operations as strategies to add and subtract.
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known.
(Commutative property of addition.) To add 2 + 6 + 4, the second
two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10
= 12. (Associative property of addition.)
1.OA.B.4 Understand subtraction as an unknown-addend problem. For
example, subtract 10 – 8 by finding the number that makes 10
when added to 8.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
33
Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Add and subtract within 20.
1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on
2 to add 2).
1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition
and subtraction within 10. Use strategies such as counting on;
making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing
a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 =
9); using the relationship between addition and subtraction (e.g.,
knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating
equivalent but easier or known sums (e.g., adding 6 + 7 by
creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
34
Common Core State Standards for
Mathematics: Grade 1
Operations and Algebraic Thinking
1.OA
Work with addition and subtraction equations.
1.OA.D.7
Understand the meaning of the equal sign, and determine if
equations involving addition and subtraction are true or false.
For example, which of the following equations are true and
which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
1.OA.D.8
Determine the unknown whole number in an addition or
subtraction equation relating three whole numbers. For
example, determine the unknown number that makes the
equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 +
6 = ?.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
35
Table 1: Common Addition and Subtraction
Situations
Common Core State Standards, 2010, p. 88, NGA Center/CCSSO
36
Common Core Standards for
Mathematical Practice
What would have to happen in order for students to have
opportunities to make use of 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
37
Linking to Research/Literature
Connections Between Representations
Pictures
Manipulative
Models
Written
Symbols
Real-world
Situations
Oral
Language
Adapted from Lesh, Post, & Behr, 1987
38
Accountable Talk Discussion
Successful teachers are skillful in building shared
contexts of the mind (not merely assuming them) and
assuring that there is equity and access to these
experiences. Talk about these experiences for all
members of the classroom are a necessary part of the
experience. Over time, these contexts of the mind and
collective experiences with talk lead to the development
of a "discourse community"—with shared
understandings, ways of speaking, and new discursive
tools with which to explore and generate knowledge. In
this way, an intellectual "commonwealth" can be built
on a base of tremendous sociocultural diversity.
Accountable Talk℠ Sourcebook: For Classroom Conversation that Works (IFL, 2010)
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39
Giving it a Go: Planning for an
Accountable Talk Discussion
• Identify a person who will be the teacher of the lesson.
• Others in the group will engage in the lesson once the
lesson has been planned.
• Plan the lesson together. Actually write questions that the
teacher will ask and anticipate participant responses.
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40
Focus of Lesson
Students share two structures. The teacher’s goal for the
lesson is to help students understand the connections
between the two structures. (Associative Property of
Addition)
(2 + 5) + 3 and 2 + (5 + 3).
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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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42
A Wondering…
• What will you keep in mind when attempting to
engage students in Accountable Talk discussions?
• What does it take to maintain the demands of a
cognitively demanding task during the lesson so
that you have a rigorous mathematics lesson?
• What role does talk play?
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43
Bridge to Practice
• Plan a lesson with colleagues. Select a high-level task.
• Anticipate student responses. Discuss ways in which you will engage
students in talk that is accountable to the learning community, to
knowledge, and to standards of rigorous thinking. Specifically, list the
moves and the questions that you will ask during the lesson.
• Engage students in an Accountable Talk discussion. Ask a colleague
to scribe a segment of your lesson, or audio or video tape your own
lesson and transcribe it later.
• Analyze the Accountable Talk discussion in the transcribed segment
of the talk. Identify talk moves and the purpose that the moves served
in the lesson. Have a segment of the transcript so you can identify
specific moves.
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44