1. Supporting Rigorous Mathematics Teaching and Learning
Using Academically Productive Talk Moves: Orchestrating a Focused Discussion
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 standards for mathematical content and practices. 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, and
the ways in which a task declines during instruction and ways of maintaining the cognitive demand of a high-level task during instruction. They learn about 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. Participants also learn about those factors that prevent students from engaging in thinking and reasoning. This is one of the reasons we are learning about Accountable Talk features and Accountable Talk moves, because this can be a means of maintaining the demands of a high-level task.
Materials:
Facilitator’s overview of module
Slides with note pages
DVD of The Light Bulb Lesson
Participants Handout
The CCSS
Tennessee Department of Education
Middle School Mathematics
Grade 6

2. 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.
Directions:
Give participants a minute to read the rationale slide. or
Paraphrase the rationale, if desired.

3. 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).
Directions:
Paraphrase the session goals.

4. 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 task and zoom in for a more specific look at key moves for engaging in productive talk (marking, recapping, challenging, and revoicing).
Directions:
Paraphrase the session activities.

5. Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework
TASKS
as set up by the teachers
TASKS
as they appear in curricular/ instructional materials
TASKS
as implemented by students
Student Learning
(SAY) The Framework was developed by the QUASAR project. The study recognized that math tasks pass through phases during lessons. The most important phase is the first, the selection of a high-level task. Without a high-level task, it is not possible to engage students in thinking and reasoning. In addition to the selection of high-level tasks, the QUASAR project learned that it was also important for teachers to think about how a task plays out as a teacher sets it up in the classroom and as students explore and discuss the task. 67% of high-level tasks are NOT carried out the way they are intended to be carried out. Therefore, it is important that teachers have opportunities to consider ways of maintaining the cognitive demand of tasks during implementation.  
As you recall, everything starts with the task. It is important to pick a high-level task. In fact it will be very difficult to engage students in an Accountable Talk discussion of a low-level task in the lesson.
Stein, Smith, Henningsen, & Silver, 2000

6. Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework
TASKS
as set up by the teachers
TASKS
as they appear in curricular/ instructional materials
TASKS
as implemented by students
Student Learning
Stein, Smith, Henningsen, & Silver, 2000
(SAY) Below the third rectangle, we see that the students’ engagement in the task is supported by the teacher’s facilitation. Today we will examine specific Accountable Talk moves that a teacher makes during the Share, Discuss, and Analyze phase of the lesson to ensure that the discussion is purposeful, coherent, and productive for all students in the classroom.
Setting Goals
Selecting Tasks
Anticipating Student Responses
Orchestrating Productive Discussion
Monitoring students as they work
Asking assessing and advancing questions
Selecting solution paths
Sequencing student responses
Connecting student responses via Accountable Talk® discussions
Accountable Talk® is a registered trademark of the University of Pittsburgh

7. Accountable Talk Features and Indicators
(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.

8. 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.
(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:
What would an Accountable Talk discussion sound like?
Students listen to each other.
Students add on to each other’s contributions.
Students ask questions.
All students speak during the lesson and their talk builds on one another’s contributions.
What about Accountability to Rigorous Thinking?
Students compare solutions.
Students make generalizations.
Students talk about how one solution path differs from another solution path.
What does Accountability to Knowledge sound like?
Students use math terms.
Students talk about accurate solutions.
Students challenge incorrect solution paths.

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.
Directions:
Direct participants’ attention to the Accountable Talk features and indicators in their participant handout. Give them an opportunity to independently read the features and indicators.
Challenge the participants to engaged in an Accountable Talk discussion when solving and discussing the solution paths to the task. Ask them to talk in small groups about what this might sound like. After a few minutes, use some of the probing questions below to elicit what the talk would sound like in an Accountable Talk discussion.
Possible Probing Questions and Possible Responses:
These are indicators of each of the three features of an Accountable Talk discussion. Remember that indicators of all three features must be present. Why is it important that discussions be accountable to the learning community, to knowledge, and to rigorous thinking?
If the discussion is not accountable to the community, then not all students are learning. Students may not be understanding the concepts if they are not asked to elaborate and build on each others’ ideas.
If the accountability to accurate and relevant knowledge is not present, the students may be thinking deeply and reasoning, but their calculations may be inaccurate.
If the accountability to accurate and relevant knowledge is not present, the students may not be referring to an established body of knowledge, tools, and resources as a foundation for constructing new learning.
If the discussion is not accountable to rigorous thinking, the students may be just memorizing and regurgitating the ideas of the teacher, the text, or their peers.

10. Solving and Discussing the Cognitive Demand of the Light Bulb Task
(SAY) Before we get to the specifics of the talk, like any good lesson planning, it is important that we solve and discuss the solution paths to the task.

11. The Structure and Routines of a Lesson
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, 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
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.
Set Up the Task
Set Up of the Task
(SAY) You will have time to work privately, then time to work in small groups. One person at each table will serve as an observer. The observer will engage in the task and will also take notes on the ways in which your small group engages in an Accountable Talk discussion. After you engage in a small group discussion, we will engage in a whole group discussion. At this time our observers will keep track of some of the Accountable Talk moves used by the “teacher.”

12. Engaging in a Lesson: The Light Bulb 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.
Directions:
Read the directions and the tasks to participants and give them private time to solve the task. Suggest that the observer actually record some of the dialogue as the group is discussing the solution to the task.

13. Engaging in a Lesson: The Light Bulb Task
Alazar Electric Company sells light bulbs to big box stores – the big chain stores that frequently buy large numbers of bulbs in one sale. They sample their bulbs for defects routinely. A sample of 96 light bulbs consisted of 4 defective ones. Assume that today’s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today. The big businesses they sell to accept no larger than a 4% rate of defective bulbs. Does today’s batch meet that expectation? Explain how you made your decision.
Directions:
Circulate while participants are working in small groups and ask assessing and advancing questions based on where the participants are in the problem-solving process.
Possible Solution Paths
Assessing Questions
Advancing Questions
A participant can’t get started.
Tell me about the light bulbs. What are you trying to figure out?
Would it help to make a diagram of the situation or to organize our data in some way?
Using a table—table does not show exact solution.
What patterns do you see in the table?
How can you use this table to determine the number of defective light bulbs in a batch of 6000? How can you use this table to determine the percent of light bulbs that are defective?
Setting up and solving a proportion using cross multiplication = x/6000
Tell me about your work.
How does multiplying 4 by 6000 relate to the problem situation?
Estimating – 4 out of 96 is close to 4 out of There are 60 groups of 100 in 6000, so there are about 240 (4*60) defective bulbs in a batch of 6000.
Why did you round up to 100?
Do we have to use “nice” numbers to reason in this way?
Divides 6000 by 96 and multiplies the quotient by 4.
Why did you divide 6000 by 96? What does that number tell you?
Could you use this method with any size batch?

14. Reflecting on Our Engagement in the Lesson
The observer should share some observations about the group’s engagement in an Accountable Talk discussion.
(SAY) The observer should share some observations about the groups’ engagement in an Accountable Talk discussion.
Possible Observations:
Small Group:
Our small group talk was accountable to the community. People listened attentively to one another and elaborated and built on each others’ ideas.
Our small group talk was accountable to knowledge. People provided evidence for their claims and arguments and worked to construct clear and correct reasoning.
Our small group talk was accountable to rigorous thinking. People in the group formulated hypotheses and others challenged their hypotheses or pushed for reasoning.
Whole Group:
The facilitator used community moves. S/he asked participants to say more and to add on to each others’ ideas. S/he asked if others agree or disagree with an idea under discussion.
The facilitator used knowledge moves. S/he encouraged use of appropriate vocabulary.
The facilitator used moves to support accountability to rigorous thinking. S/he asked participants to explain why the pattern changed the way it does and what that means in the context of the problem.
The facilitator marked contributions made by participants and recapped the process by which we came to consensus on the number of defective bulbs.

15. 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?
Possible Probing Question and Possible Responses:
What did you observe about the talk in small groups?
What did it sound like when a group was accountable to the community?
People adding on to others’ ideas and asking each other questions instead of the facilitator.
Did the group do any rigorous thinking when talking about the task? What did this sound like?
Connecting the various visual and numeric models required us to engage in productive struggle and more than one of us had an “aha” moment as we made sense of another group’s work.
Tell us about the talk in the whole group. What did observers note?
They noted that the facilitator shaped and directed the discussion.
Was there accountability to the community, to knowledge, and to rigorous thinking?
See previous slide.

16. Aligning the CCSS to the Light Bulb Task
Study the Grade 6 CCSS for Mathematical Content within the Ratio and Proportion domain.
Which standards are students expected to demonstrate when solving the task?
Identify the CCSS for Mathematical Practice required by the written task.
Directions:
Read the directions on the slide.

17. The CCSS for Mathematical Content: Grade 6
Ratios and Proportional Relationships RP
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly 3 votes.”
6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Possible Probing Questions and Possible Responses:
The task does not specify that students have to use ratios to represent the relationship. So is 6.RP.A.1 addressed by this task?
Even if students never use the word ratio, they will inevitably use ratio language and will use the multiplicative relationship between the two quantities to scale up and down. It is up to the teacher to make sure that the vocabulary is introduced if it is not already familiar to the students.
Do you have to use a unit rate to solve this problem? If not, is 6.RP.A.2 addressed by this task?
Some groups used the unit rate and others did not, but we all had to identify it and make sense of it during the whole class discussion. So, this standard might not be addressed if there is no discussion.
Common Core State Standards, 2010, p. 42, NGA Center/CCSSO

18. The CCSS for Mathematical Content: Grade 6
Ratios and Proportional Relationships RP
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.3a
Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.A.3b
Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
6.RP.A.3c
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
6.RP.A.3d
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Possible Probing Questions and Possible Responses:
Does The Light Bulb Task address all parts of all of these standards and if not, what are the implications for the teacher?
If part or all of a standard is not addressed by this task, then the teacher will have to use another task to teach and/or formatively assess student understanding of these concepts.
Common Core State Standards, 2010, p. 42, NGA Center/CCSSO

19. 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.
Directions:
Show the list of standards for mathematical practice. Ask participants which of these standards for mathematical practice students will have an opportunity to use when solving the task.
Possible Probing Questions and Possible Responses:
What does it mean to look for and make use of structure with respect to this problem?
The structure that we are looking for and making use of is that of the ratio relationship. Recognizing that the relationship between the number of bulbs and the number of defective bulbs is a multiplicative relationship and that the ratio used to represent that relationship can be scaled up and down.
What does it mean to reason abstractly and quantitatively?
Abstract numeric values from the problem situation, operate on them quantitatively using arithmetic or algebra, and then interpret the results in the context of the problem.
Attend to precision means more than just getting the correct answer. What would it mean in this problem?
In this problem precision applies to clearly identifying the quantities in the ratio, using the words ratio and fraction appropriately, labeling columns on a table or axes on a graph correctly, and performing calculations correctly.
When might students use repeated reasoning when doing this problem?
If students create a table of values to represent and organize various quantities of defective bulbs and total bulbs, they may generalize a relationship that can be used to determine the number of defective bulbs in the batch of 6,000 bulbs.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 19

20. Determining the Cognitive Demand of the Task: The Light Bulb Task

21. 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., Implementing standards-based mathematics instruction: A casebook for professional development, p New York: Teachers College Press.
How would you characterize the Light Bulb Task in terms of its cognitive demand? (Refer to the indicators on the Task Analysis Guide.)
(SAY) Is The Light Bulb Task a cognitively demanding task, according to Stein, Smith, Henningsen and Silver, 2000? Refer to the TAG when analyzing the demand of the task.
Possible Probing Questions and Possible Responses:
The task is a “Doing Mathematics” task provided that students have not learned and practiced a procedure for doing the task. What makes this task a “Doing Mathematics” task”?
The task does not suggest a pathway or give students a worked-out example. There are multiple pathways that students can use to figure out how many defective bulbs there will be in a batch of 6000 bulbs.
Do any parts of the task require students to demonstrate their understanding of mathematical ideas? What is the underlying mathematical idea in the task?
Students have to use ratio reasoning to solve this task. They have to use the fact that 4 out of every 96 bulbs (or 1 out of 24) is defective and use this ratio relationship to answer the questions. (See additional responses on the next slide.)

22. The Mathematical Task Analysis Guide
(SAY) We used this earlier to analyze the cognitive demand of tasks. Recall that it is used to classify tasks as they are written as high-level or low-level and then more specifically within those two categories. Remember also that we want to classify in only one category. That is, the task is either a “Procedures with Connections to Meaning” task or it is a “Doing Mathematics” task.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction:
A casebook for professional development, p New York: Teachers College Press.

23. The Light Bulb 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.
Directions:
Show this slide after participants have identified the task as a high-level task. This slide is optional. It is an enlargement of the “Doing Mathematics” task category so that you can be sure to point to specific bullets after participants talk about the cognitive demand of the task but do NOT make explicit connections to the bullets on the cognitive demand tool. This slide is here for this purpose.
Possible Questions and Possible Responses:
What part(s) of the task will be challenging or require some level of anxiety for students and why?
The solution to this task cannot be found by iterating by multiples of one box. Since the batch (6000) is a non-integer multiple of the starting value (96) students are unlikely to stumble upon the correct answer, and guess and check is not likely to result in a correct solution.
What does the tool mean when it read “explore and to understand the nature of mathematical concepts, processes, or relationships?” Did we deal with relationship within the concept or in the process?
We explored the concept of equivalent ratios via the process of scaling up and down using different representations. If students leave the room with only the “answer,” they have missed the opportunity to learn about why and how we use equivalent ratios.
What you said is interesting. Which of these characteristics does this refer to on the tool?

24. Accountable Talk Moves
(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 the Accountable Talk moves.

25. The Structure and Routines of a Lesson
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, 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
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.
Set Up the Task
Set Up of the Task
(SAY) The teacher that you will be observing engaged students in the Explore Phase prior to the start of the video. We will be analyzing the Accountable Talk moves used in the Share, Discuss, and Analyze Phase of the lesson.

26. 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”?
Directions:
Give individuals 5 minutes to study the Accountable Talk moves. Then engage the group in a discussion of the questions on the slide.
Possible Probing Questions and Possible Responses:
In what ways are the Accountable Talk categories similar? Different?
All of the moves are talk moves.
The community moves prompt students to listen to each other.
The category related to encouraging productive talk seems different from the other three categories.
Do you agree or disagree with the way the moves are grouped?
It is really helpful to have all of the talk moves related to a feature together.
It is easier to compare the community moves because they are grouped.
We can see that there are two reasoning prompts. One is a press for reasoning and one expands the reasoning. One precedes the other.
Why do you think we need a category called “To Ensure Purposeful, Coherent, and Productive Group Discussion?”
Challenging is a move that a teacher or student will make later in the lesson when the knowledge is sound enough that students can agree or disagree with the challenge.
Marking is a critical move because it is the teacher’s method of making sure students know the knowledge or reasoning that is correct or accurate.
Recapping, when done by the teacher or the student, is a means of summarizing or bringing ideas together in a more concise way.

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.
(SAY) The moves are organized under each of the Accountable Talk features. There is one EXTRA category on the chart. It is located at the top of the chart. Read the name given to this category and study the moves under the category. Consider why these moves were grouped and given their own name.

28. Accountable Talk Moves
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:
You said three groups of four.
Recapping
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?
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?
Directions: It is okay if participants do not say all of these things at this time. The first four moves will be discussed in depth after the video.
Possible Probing Questions and Possible Responses:
What do you notice about that first group of moves?
Why do you think we need a category called “To Ensure Purposeful, Coherent, and Productive Group Discussion?”
Challenging is a move that a teacher or student will make later in the lesson when the knowledge is sound enough that students can agree or disagree with the challenge.
Marking is a critical move because it is the teacher’s method of making sure students know the knowledge or reasoning that is correct or accurate.
Recapping, when done by the teacher or the student, is a means of summarizing or bringing ideas together in a more concise way.
How are recapping and revoicing different?
Revoicing is done to connect a student’s contribution to another or to add some clarity and precision to a student’s contribution.
Recapping is a way of retracing the steps the class went through to figure something out.
What does challenging mean as a move?
Challenging doesn’t necessarily mean giving a student something difficult to think about.
Asking the student to try to answer their own question is an example of challenging.
Do these moves fit in any of the other categories? Why are they by themselves?
Revoicing is a lot like some of the community moves, but I think it serves a different purpose here. These moves are to highlight the important ideas that the teacher wants to surface. So, a community move might be made to make sure everyone understands a procedure that was done, but revoicing would be done to identify a key mathematical understanding, not just a procedure or fact.

29. To Support Accountability to Knowledge To Support Accountability to
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?
Possible Probing Questions and Possible Responses:
Are there some moves that are more likely to occur later in the lesson rather than earlier in the lesson?
The press for reasoning occurs after students have time to make sense of each other’s solution paths.
Are there times in the lesson when you don’t want to mark contributions? Why might it be important to ask students if they agree with ideas and to have them restate contributions before you “mark” the contributions?
If a teacher marks contributions too soon, then the teacher becomes the evaluator of responses. Students also learn that they do not have to listen because the teacher is the person in charge of validating ideas.

30. 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.
(SAY) Analyze the Share, Discuss, and Analyze Phase of the lesson for Accountable Talk moves used by the teacher. You will be working in small groups later to continue to plan for the discussion. Which moves are being used by the teacher in this part of the lesson and what purpose do they serve? Make note of the line numbers as you note teacher moves.

31. The Light Bulb Lesson Context
Visiting Teacher: Victoria Bill
Teacher: Reginald Coleman
School: Community Health Academy of the Heights Middle School
District: New York City Schools
Principal: Ms. Vu
Grade Level: 7th Grade
The students in the video episode are in a mainstream mathematics classroom in the New York City Schools. The students are solving the Light Bulb Task. This part of the video captures the Share, Discuss, and Analyze phase of the lesson.
Directions:
Read the context of the classroom lesson.

32. 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.
(SAY) This teacher is offering her practice for the purpose of others’ learning. It is easy to say what we would do or could have done better; however, this is an evaluative stance. We ask that you talk about evidence of what you see rather than what you do not see. We ask you to remember that the lesson was 1.5 hours in length and you are ONLY seeing a snapshot of the lesson. So, when you do not see something or you feel compelled to suggest a different way of working, then ask a question or wonder about what might have happened in the classroom.

33. The Light Bulb Task
Alazar Electric Company sells light bulbs to big box stores – the big chain stores that frequently buy large numbers of bulbs in one sale. They sample their bulbs for defects routinely. A sample of 96 light bulbs consisted of 4 defective ones. Assume that today’s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today. The big businesses they sell to accept no larger than a 4% rate of defective bulbs. Does today’s batch meet that expectation? Explain how you made your decision.
Directions:
The students in the video are discussing The Light Bulb Task we solved earlier.

34. 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.
Possible Probing Questions and Possible Responses:
What patterns do you notice?
The teacher used a lot of community moves.
The teacher did not press for the underlying meaning of why both quantities have to be scaled multiplicatively until the end of the lesson. 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 asked this question, participants often 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.

35. Essential Understandings
Study the essential understandings the teacher considered in preparation for the Share, Discuss, and Analyze phase of the lesson.
(SAY) We have identified the standards related to the task. These, however are not specific enough for planning questions that we might ask during the lesson. So, we have identified essential understandings related to the task. Study the essential understandings and the related standards. Did the teacher on the video press for any of these essential understandings?

36. Essential Understandings
CCSS
Comparing Quantities
Two quantities can be compared using addition/subtraction or multiplication/division. Forming a ratio is a way of comparing two quantities multiplicatively. Reasoning with ratios involves attending to and coordinating two quantities.
6.RP.A.1
Unit Rate
When the ratio of a/b is scaled up or down to a/b/1, a/b to 1 is referred to as a unit rate.
6.RP.A.2
Using Models
Real-world relationships involving ratios can be modeled with a number of representations, e.g., diagram, table, graph, or ratio; however, in any such representation, both quantities in the relationship must be scaled multiplicatively. A ratio can be scaled up using multiplication because the two quantities vary in such a way that one of them is a constant multiple of the other; a ratio can be scaled down using division, since division by some number, q, is the equivalent of multiplication by the multiplicative inverse of q, 1/q.
6.RP.A.3
Solve Unit Rate Problems
Two unit rates are associated with a multiplicative relationship a and b: a/b to 1 and b/a to 1. Each unit rate reveals different information about real-world problems associated with the relationship.
6.RP.A.3a
Possible Probing Questions and Possible Responses:
Did the teacher press for any of these essential understandings? If so which?
She pressed for using models EU – that the values in the table were being scaled multiplicatively.
Why is it important to know the EUs?
If you don’t have the mathematical goals in mind, students may walk out of the room thinking they learned about light bulbs instead of ratio and proportion.
Do you think having these in her mind impacted the teacher’s decisions?
Yes. It would have been easy just to have different groups show their process of solving the problem, but because she had the EUs in mind, she asked questions about the multiplicative relationship to press students to make sense of it.

37. 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?
(SAY) 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? Turn and Talk with a colleague.
Facilitator Note:
See the next slide for some possible responses.

38. 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?
(SAY) 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?
Possible Probing Questions and Possible Responses: (To be used if participants are not able to respond to the question on the slide.)
What might you hear as students work to understand something?
Students struggling to make sense of ideas.
What might you hear if students figure something out?
An “aha!” moment and then an explanation of what they now understand.
How will you know if the students understood the underlying mathematics in the task? What would you have to hear or see in the next few days of lessons?
We may not truly know if the lesson was rigorous until the next day or several days later when the student uses what he or she has learned or makes connections between past learning and new learning. If this is true, then listening and watching what students do in the next set of related lessons is critical. This is how we determine if “we have done to much of the heavy lifting” during the lesson.

39. Essential Understandings
CCSS
Comparing Quantities
Two quantities can be compared using addition/subtraction or multiplication/division. Forming a ratio is a way of comparing two quantities multiplicatively. Reasoning with ratios involves attending to and coordinating two quantities.
6.RP.A.1
Unit Rate
When the ratio of a/b is scaled up or down to a/b/1, a/b to 1 is referred to as a unit rate.
6.RP.A.2
Using Models
Real-world relationships involving ratios can be modeled with a number of representations, e.g., diagram, table, graph, or ratio; however, in any such representation, both quantities in the relationship must be scaled multiplicatively. A ratio can be scaled up using multiplication because the two quantities vary in such a way that one of them is a constant multiple of the other; a ratio can be scaled down using division, since division by some number, q, is the equivalent of multiplication by the multiplicative inverse of q, 1/q.
6.RP.A.3
Solve Unit Rate Problems
Two unit rates are associated with a multiplicative relationship a and b: a/b to 1 and b/a to 1. Each unit rate reveals different information about real-world problems associated with the relationship.
6.RP.A.3a
Possible Probing Questions and Possible Responses:
How does thinking about the EUs help ensure that the lesson will be academically rigorous? Refer back to the features of an academically rigorous lesson on the last slide.
The EUs articulate underlying mathematics the students need to learn by engaging in the lesson. By keeping these in mind, the teacher keeps the lesson accountable to the knowledge core that is one of the features of academically rigorous lessons.
Since the EUs include “because” statements, students need to not just “know” a mathematical truth, but also understand why it is true and be able to explain and justify their positions. This helps the lesson maintain a high-thinking demand.

40. Five Different Representations of a Function
Language
Table
Context
Graph
Equation
Possible Probing Question and Possible Responses:
How does movement between representations of a function support academic rigor?
Movement between representations provides opportunities for students to make connections and have the “aha” moment that characterizes rigorous learning.
Use of multiple representations provides access to the problem for all students (e.g., those that are not fluid equation solvers can still generate a table of values from the context). Having to connect solution paths and make sense of how they are the same will cause some students to struggle. Rigorous thinking cannot occur without struggle.
Van De Walle, 2004, p. 440

41. Focusing on Key Accountable Talk Moves The Light Bulb Task

42. 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.
(SAY) Recall the Accountable Talk features. There are three of them. Remember all three must be present in order for the discussion to be referred to as an Accountable Talk discussion.

43. Accountable Talk Moves
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:
You said three groups of four.
Recapping
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?
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?
(SAY) Now let’s focus on ONE of the Accountable Talk categories on the chart. The first category is “To Ensure Purposeful, Coherent, and Productive Group Discussion.” Let’s look specifically at these first four moves that are intended to ensure purposeful, coherent, and productive group discussions. We are going to look at instances of each of these from the class we just observed. Most of the examples were in the portion of the video you just saw, but some are taken from the pre-edited video.

44. To Support Accountability to Knowledge To Support Accountability to
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?

45. 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.
Directions:
Read the directions and give participants private time to read the moves associated with the first category on the Accountable Talk chart. Give participants time to discuss these moves.
Possible Probing Questions and Possible Responses:
When does a teacher recap or ask students to recap during a lesson?
Recapping is done at the end of a discussion after several contributions have been shared.
Tell me about marking?
Marking is done when something significant is said.
Tell me about challenging?
Challenging can be done anytime.
Is this true? Can students be challenged at anytime in their learning of a concept?
No, some of the students should have enough of a grasp of the concept that they can argue back or disagree with the claims.
What do you know about revoicing?
Revoicing seems really important. It seems like this is the teachers means of elaborating on ideas, and adding vocabulary.
(SAY) So how do the moves differ from each other? Recapping occurs at the end of the lesson and marking occurs throughout the lesson. Challenging seem to test student’s knowledge of a concept. Revoicing inserts information into the discussion.

46. 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.
Directions:
The presenter can decide if he or she wants to walk through and read the description of each of the moves prior to letting participants read the examples of each move. Then step out and comment on the moves. Either way, there should be some exchanges about the different moves.

47. An Example of Revoicing
S: —and it gives you 12 or you multiply 4 times 3 because there’s 3 boxes. T: All right, he said you could do 4 times 3 and then—how would you get from here to here, though? (Points to other side of table, the total number of bulbs.)
Facilitator Note:
In this example the student explains how s/he determined that there are 12 defective bulbs in 3 boxes. The teacher revoices the strategy used by the student and then asks a question to prompt students to apply that strategy to the other quantity.

48. Marking Explicitly talk about an idea.
Highlight features that are unique to a situation.
Draw attention to an idea or to alternative ideas.

49. An Example of Marking
S: It looks like they wrote fractions. Like, broken bulbs over total bulbs. 4 over 96 is equal to 250 over 6,000. T: Hmm—did everyone hear what Selena just said? She noticed that when we write the fraction of defective bulbs out of total bulbs, they are both equivalent.
Facilitator Note:
In this example, the teacher marks the student’s contribution that the quantities can be represented using fraction notation and that fractions used to represent the same ratio relationship are equivalent.

50. Recapping Summarize or retell. Make explicit the large idea.
Provide students with a holistic view of the concept.

51. 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.

52. An Example of Recapping and Challenge
S: Multiply by – by 4 – 62 times 5 – no, 62.5 times 4. T: 62 and 5 10 times 4, 4 defective? S: Uhm, I get 250. T: 250! Clap your hands if everyone got that. [Applause] So, why did you have to do – why did you multiply 96 by 62 and 5 10 and also 4 by 62 and 5 10 ? S: What? T: Why did you have to do it on both sides? When we worked up here, we did 92 x 2 and on the other side we did 4 x 2. When we multiplied by 3 on this side we multiplied by 3 on the other side. Why do we have to do this in order for it to be proportional?
Facilitator Information:
In this example, the teacher recaps (in the form of a question) the method used to scale up the ratio. She then challenges students to reason about why it is necessary to scale both quantities in the ratio relationship multiplicatively.

53. 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?
(SAY) What would this mean with respect to classroom discourse? What should we expect to happen in the classroom?
Possible Probing Questions and Possible Responses:
With respect to the Accountable Talk moves, what should begin to happen in the classroom?
Students should begin to use the moves in the classroom.
Students will mark each other’s comments that are accurate.
Students will challenge incorrect reasoning or facts.
Students will add on or revoice their peer’s ideas.
What might be going on in the classroom if this does not happen?
There are several reasons why this does not always happen in the classroom.
Sometimes the teacher is always talking and students have no opportunity to use the talk moves.
Sometimes when the teacher is revoicing contributions, the contributions are not close enough in how they are stated for students to recognize that their statement is being revoiced.
Sometimes all of the moves are NOT distinct enough for students to recognize that different moves are being used by the teacher. Some teachers will actually say, “I want to mark that. Let me revoice what you are saying. Are you saying….”

54. Orchestrating Discussions
Read the segments of transcript from the lesson. Decide if examples 1 – 3 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 4 and 5 and be prepared to share your move and your rationale for writing the move.
Directions:
Read the directions on the slide. Give participants 5 minutes of private time to read the scripts and to write the next talk move. Only the four talk moves on the slide should be used. Tell participants to write their question and to be prepared to share their rational for using the next talk move.
Whole Group Discussion:
Example 1 is challenging. The teacher is using Juan’s contribution as a source for further challenge.
Example 2 is marking and revoicing. The teacher draws attention to a student’s contribution and uses mathematically precise language to align a student’s explanation with content.
Example 3 is challenging. The teacher challenges the students to figure out why a procedure yields the correct answer.
For examples 4 and 5:
Possible Probing Questions and Possible Responses:
Did you all use the same move for each example? Why do you think this is? Is there one “right” move to make? What factors do we consider when deciding which move to make?
Without the context of what came before it is hard to know if this is an opportunity to recap, revoice, mark or challenge. We have to consider whether this is the first time the idea has surfaced, where we are in the lesson, how much time we have left, and what other ideas students have articulated to determine what to move to make.

55. Reflecting on Talk Moves
What have you learned about:
marking;
recapping;
challenging; and
revoicing?
Why are these moves important in lessons?
Directions:
Read the question on the slide.
Possible Responses:
These moves make sure that we all understand the point of the discussion. Without them, the discussion is not focused on the essential understandings and students may leave class not knowing what they were supposed to have learned.

56. 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?
Directions:
Have participants do a Turn and Talk or a quick write and then have 3–5 volunteers share their responses to the questions on the slide.
Possible Participant Responses:
Talk is a vehicle for student learning, but in order for students to learn through talk, the teacher has to plan for the talk.
I will try to articulate the essential understandings before I plan my lesson so that I know what my math learning goals are. I will plan that will help me focus the class discussion on the EUs.
I will think about what I want to mark and will look for opportunities to revoice and challenge.
Maintaining the cognitive demand of a lesson requires preparation.
The teacher needs to find as many solution paths as possible and consider possible misconceptions and errors that students might make.
The teacher needs to plan assessing and advancing questions to ask during the explore phase.
The teacher needs to identify the math learning goals of the lesson.
The teacher needs to plan for the whole class discussion by considering what solution paths will be shared, in what order and what questions s/he will ask to make sure that the discussion uses student work to surface mathematical understandings.
The facilitator can also talk about other decisions that were made or not made, and what impact they had on the lesson.