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 2
© 2013 UNIVERSITY OF PITTSBURGH
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 can do. These insights prepare teachers to
consider ways to advance student learning.
Today, by analyzing math classroom discussions, teachers will study
how Accountable Talk (AT) discussions support student learning and
help teachers maintain the cognitive demand of the task.
Accountable Talk® is a registered trademark of the University of Pittsburgh
Session Goals
Participants will:
• learn a set of Accountable Talk features and indicators;
and
• recognize Accountable Talk stems for each of the
features 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 Structures and Routines of a Lesson
Set
Task
Set Up
Up the
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
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 by engaging students in a
quick write or a discussion of the
process.
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 for each of the features.
− Accountability to the learning community
− Accountability to accurate, relevant knowledge
− Accountability to discipline-specific standards
of rigorous thinking
© 2013 UNIVERSITY OF PITTSBURGH
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?
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Features and Indicators
Accountability to the Learning Community
• Active participation in classroom talk.
• Listen attentively.
• Elaborate and build on each other’s 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 Starters
• Work in triads.
• On 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.)
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Talk Starters
What do you notice about the talk starters 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?
© 2013 UNIVERSITY OF PITTSBURGH
Using the Accountable Talk
Features and Indicators to Analyze
Classroom Practice
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Strings Task
Solve the set of addition expressions. Each time you solve a
problem, try to use the previous equation to solve the problem.
7 + 3 = ___
17 + 3 = ___
27 + 3 = ___
37 + 3 = ___
47 + 3 = ___
Solve each problem two different ways. Make a drawing or show
your work on a number line.
What pattern do you notice? If the pattern continues, what would
the next three equations be?
© 2013 UNIVERSITY OF PITTSBURGH
13
Lesson Context
Teacher: Jennifer DiBrienzo
Grade: 2
School: School #41
School District: NYC, District 2
Jenniefer DiBrienza is engaging students in solving and
discussing the Strings Task. She will engage the class as a
whole in discussing the Strings Task and then they will do
several problems independently.
Jennifer is also showing students how to use a new tool,
the open number line.
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Reflecting on the Lesson
Watch the video.
What are students learning in the Strings Task?
Which Accountable Talk features and indicators were
illustrated in the lesson?
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Accountable Talk Features and Indicators
Which of the Accountable Talk features and indicators
were illustrated in the classroom video?
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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
Thinking Through a Lesson: The Strings
Task
(Private Think Time and Small Group Time)
Work with others at your table. Hold yourselves accountable
for engaging in an Accountable Talk discussion when you
think through the lesson.
• What do students need to understand?
• Which solution paths might students use when solving
the task? How does one solution path differ from the
other?
• What questions will you have to ask to address the
ideas in the Standards for Mathematical Content?
© 2013 UNIVERSITY OF PITTSBURGH
The CCSS for Mathematical Content: Grade 2
Operations and Algebraic Thinking
2.OA
Represent and solve problems involving addition and subtraction.
2.OA.A.1
Use addition and subtraction within 100 to solve one- and
two-step word problems involving situations of adding to,
taking from, putting together, taking apart, and comparing,
with unknowns in all positions, e.g., by using drawings and
equations with a symbol for the unknown number to
represent the problem.
Add and subtract within 20.
2.OA.B.2
Fluently add and subtract within 20 using mental strategies.
By end of Grade 2, know from memory all sums of two onedigit numbers.
Common Core State Standards, 2010
The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten
2.NBT
Understand place value.
2.NBT.A.1 Understand that the three digits of a three-digit number represent
amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds,
0 tens, and 6 ones. Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens—called a
“hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900
refer to one, two, three, four, five, six, seven, eight, or nine
hundreds (and 0 tens and 0 ones).
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten
2.NBT
Understand place value.
2.NBT.A.2 Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.A.3 Read and write numbers to 1000 using base-ten numerals, number
names, and expanded form.
2.NBT.A.4 Compare two three-digit numbers based on meanings of the
hundreds, tens, and ones digits, using >, =, and < symbols to record
the results of comparisons.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten
2.NBT
Use place value understanding and properties of operations to add and
subtract.
2.NBT.B.5 Fluently add and subtract within 100 using strategies based on
place value, properties of operations, and/or the relationship
between addition and subtraction.
2.NBT.B.6 Add up to four two-digit numbers using strategies based on place
value and properties of operations.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten
2.NBT
Use place value understanding and properties of operations to add and
subtract.
2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings
and strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction; relate the
strategy to a written method. Understand that in adding or
subtracting three-digit numbers, one adds or subtracts hundreds
and hundreds, tens and tens, ones and ones; and sometimes it is
necessary to compose or decompose tens or hundreds.
2.NBT.B.8 Mentally add 10 or 100 to a given number 100–900, and mentally
subtract 10 or 100 from a given number 100–900.
2.NBT.B.9 Explain why addition and subtraction strategies work, using place
value and the properties of operations.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
Common Core State 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
24
Essential Understandings
Review the essential understandings for the lesson. Which essential
understandings will students be left with after the lesson?
• Counting strategies are based on order and hierarchical inclusion of
numbers. (NCTM)
• Counting includes one-to-one correspondence, regardless of the kind of
objects in the set and the order in which they are counted. (NCTM)
• Counting tells how many items there are altogether. When counting, the
last number tells the total number of items. (NCSM Journal/Van de Walle
spring-summer 2012)
• Sets of ten (and tens of tens) can be perceived as single entities. These
sets can then be counted and used as a means of describing quantities.
• The positions of digits in numbers determine what they represent–which
size group they count. This is the major principle of place value
numeration.
• There are patterns to the way that numbers are formed. For example, each
decade has a symbolic pattern reflective of the 1-9 sequence.
• The groupings of ones, tens, and hundreds can be taken apart in different
ways. For example, 256 can be 1 hundred, 14 tens, and 16 ones but also
250 and 6. Taking numbers apart and recombining them in flexible ways is
a significant skill for computation.
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Strings Task
Solve the set of addition expressions. Each time you solve a
problem, try to use the previous equation to solve the problem.
7 + 3 = ___
17 + 3 = ___
27 + 3 = ___
37 + 3 = ___
47 + 3 = ___
Solve each problem two different ways. Make a drawing or show
your work on a number line.
What pattern do you notice? If the pattern continues, what would
the next three equations be?
© 2013 UNIVERSITY OF PITTSBURGH
26
Giving It a Go
In the video, students use several strategies when solving
the task. Students count on or add the ones place and then
the tens place.
You will plan the lesson for the following strings of numbers:
147 + 3 = ___
157 + 3 = ___
167 + 3 = ___
Use the open number line in the lesson.
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Reflection on the Lesson
Common Core State Standards (CCSS)
Examine the second grade CCSS for Mathematics.
• Which CCSS for Mathematical Content did we
discuss?
• Which CCSS for Mathematical Practice did we use
when solving and discussing the task?
© 2013 UNIVERSITY OF PITTSBURGH
Five Representations of Mathematical Ideas
Pictures
Manipulative
Written
Models
Symbols
Real-world
Situations
Oral & Written
Language
Modified from Van De Walle, 2004, p. 30
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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Reflection
• 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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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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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 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.
© 2013 UNIVERSITY OF PITTSBURGH