1. 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
Overview of the Module:
In this module, teachers will consider how Accountable Talk discussions are a means of developing students’ understanding of the CCSS for Mathematical Content and the CCSS for Mathematical Practice. Teachers will learn about the power of engaging students in Accountable Talk discussions in which evidence of accountability to the learning community, to knowledge and to rigorous thinking exists. Accountability in all three of these areas is the means by which students make sense of mathematical ideas while teachers assess student understanding of mathematical content and practice standards. Talk, (specifically Accountable Talk) is the means by which teachers can support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000).
Prior Knowledge Needed to get the Most of This Module:
This module is most effective when teachers:
know about the cognitive demands of high-level tasks. (See The Cognitive Demand of a Mathematics Task Matters.)
understand ways teachers can maintain the thinking and reasoning in lessons, specifically factors such as time, prior knowledge for solving the task, wait time, teachers’ consistent press for reasoning, etc., that make it possible for students to engage in high-level thinking and reasoning; and understand those factors that prevent students from engaging in thinking and reasoning.
know the characteristics of, and benefits of, asking students assessing and advancing questions during the lesson, most often during the Explore Phase of the lesson. (See Illuminations of Student Thinking: Supporting Student Thinking and Learning through Questioning.)
Materials:
Facilitator’s Overview of Module
Participants Handouts
The CCSS
Slides for Presentation with Notes
DVD of the Make Ten Lesson with Jennifer DiBrienza

2. 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.
Directions:
Give participants a minute to read the rationale slide. or
Paraphrase the rationale, if desired.
(SAY) Engaging students in rigorous academic tasks accompanied by Accountable Talk moves will promote effective teaching.
By seeing how students develop an understanding of mathematical ideas and strategies, teachers can prepare ways to advance student learning of the mathematical ideas.
Accountable Talk® is a registered trademark of the University of Pittsburgh

3. 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.
Directions:
Read the activities

4. 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.
Directions:
Give participants a minute to read the session activities.

5. The Structure and Routines of a Lesson
Set Up of the Task
MONITOR: Teacher selects
examples for the Share, Discuss,
and Analyze Phase based on:
Different solution paths to the
same task
Different representations
Errors
Misconceptions
Set Up the Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/
Small Group Problem Solving
Generate and Compare Solutions
Assess and advance Student Learning
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.
Share Discuss and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
Focus the Discussion on Key
Mathematical Ideas
4. Engage in a Quick Write
Directions: (Optional)
(SAY) This graphic shows the phases that a high-level task goes through as it is enacted in a classroom. It assumes we have already selected a high-level task and are aware of how the task will help students work towards or use mathematical practice standards as a means of understanding mathematical content.
This lesson structure ensures that students have individual problem solving time, small group problem solving time, and whole group discussion time. All of these opportunities are times when the teacher can assess student thinking, but they are also times when students can work out answers with each other and practice methods before they are shared with a larger group.
Optional
The Share, Discuss, and Analyze Phase of the lesson emphasizes the importance of the comparison of ideas. It also notes the importance of focusing on the mathematics. We are focusing on this phase of the lesson today.

6. Accountable Talk Features and Indicators
Directions:
(SAY) Before we watch a video lesson in which a teacher is attempting to engage students in an Accountable Talk discussion, let’s look at some features of Accountable Talk discussions.

7. 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
Directions:
(SAY) Turn and Talk with your partner about what you would expect teachers and students to say during an Accountable Talk discussion for each of the features.
Probing Facilitator Questions and Possible Responses: (5 min.)
What would an Accountable Talk discussion sound like?
Students listen to each other.
Students add on to each other.
They ask questions.
All students speak during the lesson and their talk builds on one another’s contributions.
Teacher develops appropriate questions that prompts students to interact through questions and discussions with each other.

8. 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?
Directions:
Show the slide and give participants a few minutes to read the information on the slide.
(SAY) There are three features of Accountable Talk discussions. (Show the next slide.)
(SAY) The Institute for Learning defines Accountable Talk discussions as those in which indicators from all three of these features exist. In other words, if the discussion does not involve all students in the discussion, then the teacher cannot assess student learning, but the students also do not have opportunities to make sense of ideas and share their reasoning with others; therefore, the students are not engaged in an Accountable Talk discussion.
(SAY) Each of these features--Accountability to Community, Accountability to Knowledge and Accountability to Rigorous Thinking—has a set of indicators. Your participant handout lists indicators that go with each of the features. Please read them and consider these questions.
How do the indicators align with your description of Accountable Talk discussions?
What does it sound like when students are accountable to the community?
What does it sound like when students are accountable to rigorous thinking?
What does it sound like when students are accountable to knowledge?

9. 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.
Probing Facilitator Questions and Possible Responses: (10 min.)
What is the benefit of asking students to elaborate or build on each other’s ideas?
Elaborations on ideas is a means by which the group co-constructs a solution path that they might not have been able to do independently.
Why is it important for the teacher to press students for accuracy? Why might this be important to do? (The commitment to getting it right will be mentioned because participants will talk about the teacher PRESS for accuracy or detail.)
As the teacher presses for accuracy or a degree of specificity, students hear the ideas over and over and in a variety of ways. They may not be able to make sense of ideas the first time or understand them when said one way, but they might when they hear them said another way.
Why do you think the authors claim that evidence of all three features of AT must be present?
Whoever talks the most learns the most. Students need to be the ones talking because the teacher can assess what they know or don’t know. It is the students’ knowledge and reasoning that we need to hear, not just any kind of talk.
What does it mean when you press students for an explanation, but they can’t explain the reasoning underlying the concept?
Students often don’t understand a concept and they can’t share their mathematical reasoning because it is hard work. Students need to have a deep enough understanding of why the mathematical ideas are working the way they are working.)
How might you scaffold student learning in order to make it possible for students to share their mathematical reasoning?
Make a table so they see a repeating pattern, link to the context, ask them if the problem reminds them of other similar problems, provide students with manipulatives, share your reasoning so they have opportunities to hear what it sounds like, invite students to “try to share their reasoning” and permit others to add on.

10. 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.)
Directions:
Read the directions on the slide. Assign one of the features to each group. Give the example that appears on the slide. Only give groups about five minutes to jot the talk starters for their assigned features.
Some examples of talk starters will include:

11. 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?
Directions:
Read the questions on the slide. Have participants Turn and Talk with a partner and share out ideas.
Additional Talk starters:

12. Preparing to Analyze Accountable Talk Features and Indicators in Classroom Practice
Directions:
(SAY) Before we watch a video in which a teacher is attempting to engage students in an Accountable Talk discussion, we willll take a look at the task that students are being asked to solve. We will look at student work and consider the ways that students will have an opportunity to think about math while solving this task.

13. 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.
Directions
This is the task that students in the class were solving. Let’s look at student work from the task.

14. 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?
Directions:
Read the directions.

15. The Make a Ten Task: Devon’s Work
Directions:
Possible Facilitator Questions and Possible Responses:
What does Devon understand?
He can make different structures.
He can write number sentences that describe his structures and both equal 10.
Knows that he can use more than one addend.
He MIGHT be working off his first model when creating the second model. The teacher can use this solution path to make connections between and The teacher could also discuss associativity with students.
What might the teacher do to make these connections?
The teacher could ask about the connection between and ?
What would the teacher need to ask in order to help students understand associativity?
How is = 2 + (2 + 6)? Is it equal?
How can we prove that the two expressions are the same?

16. The Make a Ten Task: David’s Work
Directions:
Possible Facilitator Questions and Possible Responses:
What does David understand?
He can make different structures.
He can write number sentences that describe his structures, and both equal 10.
He knows that he can use more than one addend.
If students haven’t worked on skip counting by 2s, this would be a good task to discuss. This is also a standard in 2nd grade so it would be a good idea to discuss it. The teacher can also make the distinction between and being addition with different addends but addition of five twos as addition of the same addend.

17. The Make a Ten Task: Tinesha’s Work
Direction:
Did students have opportunities to work on understanding these standards?
Possible Facilitator Questions and Possible Responses:
What does Tinesha understand?
She can make different structures.
She can write number sentences that describe her structures, and both equal 10.
Knows that she can use more than one addend.

18. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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.
Directions:
Which standards will students have opportunities to use when solving the Make a Ten Task?
When working on the Counting Houses Task, students will have an opportunity to work on:
1.OA.A.1: Word problem—putting together—solving situations up to 20 (This is not a situational problem.
1.OA.A.2: Word problem with three whole number addends (This problem can have three addends depending on the structure that students make.)
Which standard aligns with this task? In what way does it align?
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

19. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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 = 11 is known, then = 11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 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.
Which standards will students have opportunities to use when solving the Make a Ten Task?
1.OA.B.3: Commutative Property of Addition can be discussed but this depends on the teacher initiating the discussion.
1.OA.B.3: Associative Property of Addition can be discussed because the design of this problem does lead to the use of three addends. Students might arrange the cubes in a way to lead to three addends. However, this depends on the teacher initializing the discussion.
1.OA.B.3: The way the task is written will not require students to talk about compensation. However, it could come up if the teacher initiates the conversation.
1.OA.B.4: NOT ADDRESSED because this task does not involve subtraction.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

20. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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., = = = 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 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13).
Which standards will students have opportunities to use when solving the Make a Ten Task?
1.OA.C.5: Relating counting to addition or subtraction is not directly addressed in this task, but it could be a discussion facilitated by the teacher.
1.OA.C.6: NOT ADDRESSED because this task does not address fluency for adding, even though it does support the development of strategies used for adding.
NOTE: The CCSS talks about fluency as those tasks that students must do with speed. Although the task does not require that students do it with speed, many strategies can be discussed when solving the task. For example—counting on (starting with first length and counting on), make 10 (add the 9 and 1 to make 10 and then add 4), and decompose/compensation (break the 5 into 1 and 4, then add the 9 and 1 to make 10, then add 4)
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

21. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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, = 2 + 5, =
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, = ?.
Which standards will students have opportunities to use when solving the Make a Ten Task?
1.OA.D.7: The task asks students to create structures that equal 10, but this standard can only be addressed if the teacher sets task equal to each other (6+4 = ).
1.OA.D.8: NOT ADDRESSED because students are writing equations, not solving for unknowns in equations.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

22. Table 1: Common Addition and Subtraction Situations
(SAY)
Does this task fit anywhere on the “situations” chart?
No, this is not a situational task.
Common Core State Standards, 2010, p. 88, NGA Center/CCSSO

23. 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?
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
(SAY) Which Standards for Mathematical Practice will students have opportunities to work on and why?
Make sense of problems and persevere in solving them. Students must figure out how to construct a figure with 10 cubes. A teacher might ask students to construct two different figures that equal 10 and this would increase the demand of the task.
Reason abstractly and quantitatively. This practice does not apply to this task.
Construct viable arguments and critique the reasoning of others. Students must convince us that their structure has 10 cubes and write a number sentence that describes the cubes.
Model with mathematics. Since students have to construct a model and write an equation, they are modeling with mathematics.
Use appropriate tools strategically. Students are given the cubes; it did not appear that they had a choice of tools.
Attend to precision. The model and the equation must have a total of 10 cubes.
Look for and make use of structure. The discussion of commutativity or associativity could lead to a discussion of the structure, but as the task is written this is not likely to happen without the teacher student engagement.
Look for and express regularity in repeated reasoning. Since students only have to make one structure they are not asked to show repeated reasoning. The teacher could incorporate work on this standard by discussing the commutative property and then asking students to show how they can apply this reasoning to their figure.

24. Accountable Talk Features and Indicators
Which of the Accountable Talk Features and Indicators were illustrated in our discussion?
(SAY) In what way was your talk in your groups an Accountable Talk discussion? If the talk was not accountable to the Community, to knowledge and to rigorous thinking then what would you need to do more of?
We were accountable to each other.
We talked about where the teacher might press for connections and relationships so this is rigorous thinking.
We talked about using correct terms - sum and addends.
Probing Questions
In your discussion of the student work did you build on each other’s contributions? Is so then you were Accountable to the Community.
Did anyone work to make connections to the standards? What did that discussion sound like?

25. Using the Accountable Talk Features and Indicators to Analyze Classroom Practice
Directions:
(SAY) Now we will watch a video and analyze it through the lens of the Accountable Talk features and indicators.

26. 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?
Directions:
Read the directions to participants.

27. 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.
Directions Let’s use the features and indicator to analyze the video.

28. 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.
Facilitator Notes: Remind participants that the teacher is sharing her practice and, that as viewers, we should be respectful of her practice. We are watching just a portion of the discussion, which represents just a small portion of the whole lesson. We are watching the specifically for the facilitation of an Accountable Talk discussion.
Directions:
Ask participants to read the slide or paraphrase the content.

29. The Structures and Routines of a Lesson
Set Up of the Task
MONITOR: Teacher selects
examples for the Share, Discuss,
and Analyze Phase based on:
Different solution paths to the
same task
Different representations
Errors
Misconceptions
Set Up the Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/
Small Group Problem Solving
Generate and Compare Solutions
Assess and advance Student Learning
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.
Share Discuss and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
Focus the Discussion on Key
Mathematical Ideas
4. Engage in a Quick Write
Directions: (Optional)
(SAY) The segment that we will be watching is the Share, Discuss, and Analyze Phase of the lesson.

30. The Make a Ten Lesson Directions:
Participants will watch the video and take notes or mark the transcript as it plays. Give participants time privately to review their notes and the transcript, then have them work in pairs to discuss the video. After that, share out as a whole group. Remember to keep the discussion on the ways the teacher facilitated an Accountable Talk discussion.
Name some of the things you noticed.
The teacher is emphasizing listening to each other. She is trying to get students to listen and she does this by saying “listen, repeat because I know __ didn’t hear, Say it again so others can hear.”
The teacher invited questions.
The teacher accepted some of the student responses that might have felt like they were going to take her off track. ( and 6 + 6) validate the student responses then redirect with a question to get students back on track.
What counts as Accountability to Knowledge?
All of the facts equal 10.
What questions might you ask related to Accountability to Knowledge?
I wonder io the teacher will name the as addends?
I wonder if the teacher will call the answer the sum?)
What counts as Accountability to Rigorous Thinking?
The teacher asked if all of the facts equaled 10.
The teacher pressed students to tell her how was 10 and how was ten.
What questions might you ask related to Rigorous Thinking?
I wonder if the teacher will challenge students to apply their thinking in a new way; maybe by writing situational stories to go with their blocks, to increase the sum by 2, or to use the “making 10” as a strategy in adding.
What role did the representations play?
The teacher consistently touched the cubes.
What questions might you ask about the use of representation?
I wonder if there would have been benefits to writing all of the equations on the board?
I wonder if an elmo was used if students could have seen the model better?
I wonder if students coming up and pointing helps them follow the lesson more?

31. 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?
Directions: (10 min.)
(SAY) Which of the CCSS for Mathematical Content would students need to demonstrate when solving the task?

32. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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.
Which standards will students have opportunities to use when solving the Counting Houses Task?
When working on the Counting Houses Task, students will have an opportunity to work on:
1.OA.A.1: NOT ADDRESSED because this is not a situation word problem. (This task does not imply that an action is taking place, no portion of the path is changing in length, the lengths are just being “put together” to find the total.)
1.OA.A.2: NOT ADDRESSED because even though a model or equation showing 3 whole number addends is possible with this task as it is written, the task is not a situation word problem.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

33. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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 = 11 is known, then = 11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 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.
When working on the Counting Houses Task, students will have an opportunity to work on:
1.OA.B.3: Commutative property of addition can be discussed, but this depends on the teacher initiating the discussion.
1.OA.B.3: Associative Property of Addition can be discussed because there could be 3 addends, but this depends on the teacher initiating the discussion.
1.OA.B.3: Compensation can be discussed, but this depends on the teacher initiating the discussion.
1.OA.B.4: NOT ADDRESSED because this task does not involve subtraction.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

34. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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., = = = 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 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13).
When working on the Counting Houses Task, students will have an opportunity to work on:
1.OA.C.5: Relating counting to addition or subtraction is not directly addressed in this task, but it could be a discussion facilitated by the teacher.
1.OA.C.6: NOT ADDRESSED because this task does not work on developing fluency for adding, even though it does support the development of strategies used for adding.
NOTE: The CCSS talks about fluency as those tasks that students must do with speed. Although the task does not require that students do it with speak many strategies can be discussed when solving the task. For example—counting on (starting with first length and counting on), making 10 (add the 9 and 1 to make 10 and then add 4), and decompose/compensation (break the 5 into 1 and 4, then add the 9 and 1 to make 10, then add 4)
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

35. Common Core State Standards for Mathematics: Grade 1
Operations and Algebraic Thinking 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, = 2 + 5, =
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, = ?.
When working on the Counting Houses Task, students will have an opportunity to work on:
1.OA.D.7: The task asks students to create structures that equal 10, but this standard can only be addressed if the teacher sets equations to equal to each other (6 + 4 = )
1.OA.D.8: NOT ADDRESSED because students are writing equations, not solving for unknowns in equations.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

36. Table 1: Common Addition and Subtraction Situations
(SAY)
Does this task fit anywhere on the “situations” chart?
No, this is not a situational task.
Common Core State Standards, 2010, p. 88, NGA Center/CCSSO

37. 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?
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
Directions:
Lead a group discussion related to the Standards for Mathematical Practice. If time permits, participants benefit from an opportunity to Turn and Talk prior to a group discussion.
(SAY) Consider the following questions:
Which Standards for Mathematical Practice do students demonstrate in their work?
Which Standards for Mathematical Practice do the written tasks require that they demonstrate?
Which practice standards are we assessing and advancing?
Probing Facilitator Questions and Possible Responses:
Which CCSS for Mathematical Practice did students use and how do you know? What will we see if students work abstractly and quantitatively?
The students had to figure out an approach to the problem.
What regarding the design of the task requires students to construct a viable argument?
Students have to convince us that their figure equals 10 or that =
Do any of the students recognize the structure of mathematics and why or why not?
Student who recognize that two structures can both have 10 cubes but look different and have different number sentences.

38. Linking to Research/Literature Connections Between Representations
Pictures
Written
Symbols
Manipulative
Models
Real-world
Situations
Oral
Language
Directions:
(SAY) This representation model shows the 5 areas of mathematical representations. These are the category headings for all of the different ways we represent mathematics. Research shows that students who are able to move between representations are likely to be more skilled problem solvers. Why might this be?
Probing Questions:
What role does moving between the context, the concrete models, and symbols play in supporting learning?
How can referencing a context during the discussion help students?
Is there any benefit in drawing pictures? If so, what are they?
This task requires that students explore several algorithms. What is the benefit of producing a set of equations?
Adapted from Lesh, Post, & Behr, 1987

39. 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)
Directions:
Read or paraphrase the quote, then have a brief discussion that focuses on the phrase “shared context of the mind” and how it relates to an Accountable Talk discussion.

40. 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.
Directions:
Read the slide.

41. 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).
Directions:
Read the slide.

42. 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?
Directions:
Give participants time to Turn and Talk with peers about the patterns that they notice.
Probing Facilitator Questions and Possible Responses:
What patterns do you notice?
The teacher used a lot of community moves.
What might have happened if the teacher pressed for reasoning earlier?
Only a few students might be able to engage in the conversation.
Is there anything that you wonder about?
When asking this question, often participants attempt to “fix” the teacher. This is not the goal, so if someone says, “The teacher should have… or, I would have….”, Then reword their contribution and say, “So, you are wondering if the teacher will… or if she did…?” If needed, remind teachers that we don’t want to be judgmental.

43. 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?
Directions:
Facilitate a group discussion using the questions on the slide. Below are some thoughts that should come out of the discussion.
In the planning of the lesson, some of the following should be brought out:
The facilitator considered the intended audience and their prior mathematical knowledge and adapted the task for that audience. Refer participants to the facilitation guide.
The facilitator solved the task in as many ways as possible. Refer participants to the facilitation guide.
The facilitator considered what misconceptions might occur during the lesson.
The facilitator identified key questions that would be asked during the lesson.
During the implementation of the lesson, some of the following should be brought out:
As the facilitator circulated among the groups, certain solution paths were identified and questions asked so that those solutions could be extended.
The facilitator identified the solution paths and the order in which they would be shared as participants worked.
The facilitator had certain mathematical concepts and connections between them in mind during the whole group discussion.
The facilitator can also talk about other decisions that were made or not made, and what impact they had on the lesson.

44. 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.
Directions:
Read the Bridge to Practice.