1. Supporting Rigorous Mathematics Teaching and Learning
Selecting and Sequencing Based on Essential Understandings
Overview of the Module:
There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001). Equally important to tending to students’ mathematical thinking is understanding the instructional goals of the lesson and aligning these goals with student thinking. By engaging in setting goals, anticipating student responses, and considering the selection and sequencing with the intent of advancing student understanding of the goals, teachers will learn about the relationship between these parts of lesson planning.
No Prior Knowledge Necessary.
Materials:
Slides with note pages
Mathematics Common Core State Standards (CSSS) (the Standards for Mathematical Practice and the grade-level Standards for Mathematical Content)
Participant handouts
Chart paper and markers
Tennessee Department of Education
Middle School Mathematics
Grade 8

2. Rationale
There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001).
By engaging in an analysis of a lesson-planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding.
(SAY) Take a minute and read the rationale for the lesson.
It can be difficult for us to determine the role of the teacher in a classroom that is student-centered. How do we use student thinking and student work as the foundation for the discussions in the classroom while ensuring that meaningful mathematics is being learned? This will be the question that guides our work today.

3. Session Goals Participants will learn about:
goal-setting and the relationship of goals to the CCSS and essential understandings;
essential understandings as they relate to selecting and sequencing student work;
Accountable Talk® moves related to essential understandings; and
prompts that problematize or “hook” students during the Share, Discuss, and Analyze phase of the lesson.
Directions:
Read the goals on the slide.
To personalize the goals a little consider these points.
We all know goals for the lesson are critical. How often do you CLEARLY know exactly what you are targeting in the lesson? To help us with this, we will consider essential understandings or what the mathematics is in the lesson. We will also think about ways to pose questions during the Share, Discuss, and Analyze Phase of the lesson that motivate students to think productively about the mathematics.
Facilitator Note:
By the end of the session, participants should be making the following statements:
Essential understandings are the mathematical underpinnings of the Standards for Mathematical Content. They are the math that teachers and students need to get at in order to truly be proficient with the content and often the Standards for Mathematical Practice. The EUs are a finer grain size than the standards.
When selecting and sequencing student work for the SDA, a teacher considers how understanding is constructed and orders student solution paths so that students are moving closer to the mathematical target.
Accountable talk moves are used to structure discussions so that student work is used as the basis for the discussion while still moving towards important mathematical ideas.
Problematizing or hooking the students helps ensure that the discussion will be mathematically productive and not devolve into a show and tell.
Accountable Talk® is a registered trademark of the University of Pittsburgh.

4. “The effectiveness of a lesson depends significantly on the care with which the lesson plan is prepared.”
Brahier, 2000
Directions:
Read the quote.

5. “During the planning phase, teachers make decisions that affect instruction dramatically. They decide what to teach, how they are going to teach, how to organize the classroom, what routines to use, and how to adapt instruction for individuals.”
Fennema & Franke, 1992, p. 156
Directions:
Read the quote.
(SAY) As educators, it is easy to focus on all the things outside our control. This quote reminds us that there are many decisions to be made every day that are our responsibility. Focusing on these elements of lesson planning and instruction that are within our control will have a tremendous impact on student learning.

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
(SAY) This framework was developed by the QUASAR project, a large-scale study of many middle school classrooms. 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.  
The first rectangle represents the task as it appears on the paper.
The second rectangle represents how the teacher sets up the task.
The third rectangle represents how the students engage with the task. The teacher still plays an essential role in this part of the enactment of the task.
The culmination is the learning that occurs.
Stein, Smith, Henningsen, & Silver, 2000

7. Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework
TASKS
as set up by the teachers
TASKS
as implemented by students
TASKS
as they appear in curricular/ instructional materials
Student Learning
Stein, Smith, Henningsen, & Silver, 2000
(SAY) We know that the task matters a lot. In fact, if the chosen task does not have a high-level of cognitive demand and does not align to the goals set, it is difficult, if not impossible, to orchestrate an Accountable Talk discussion around the task.
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

8. Identify Goals for Instruction and Select an Appropriate Task

9. 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) This diagram represents the structures and routines of a lesson. In this session we will be focusing on the Share, Discuss, and Analyze Phase of the lesson (point to SDA box), specifically selecting and sequencing student work and orchestrating a discussion around that work.

10. Contextualizing Our Work Together
Imagine that you are working with a group of students who have the following understanding of the concepts.
70% of the students need to make sense of what it means to solve a system of equations. (8.EE.C.8, C.8a, C.8b)
20% of the students need additional work on using linear equations to model a problem situation. (8.F.B.4)
Five percent of the students still struggle with solving linear equations in one variable. (8.EE.C.7)
Five percent of the students struggle to pay attention and their understanding of expressions and equations is two grade levels below eighth grade.
(SAY) This data is in your participant packet. Take a moment to read the context for the lesson we are about to consider. The language of the standards for mathematical content appears on the next pages of our participant materials.

11. The CCSS for Mathematics: Grade 8
Expressions and Equations EE
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.7
Solve linear equations in one variable.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8a
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.C.8b
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
8.EE.C.8c
Solve real-world and mathematical problems leading to two linear
equations in two variables. For example, given coordinates for two
pairs of points, determine whether the line through the first pair of
points intersects the line through the second pair.
(SAY) These are the standards that were identified by the teacher as the goals of the lesson. The Expressions and Equations standards are familiar to us from our work together on previous modules.
Common Core State Standards, 2010, p , NGA Center/CCSSO

12. The CCSS for Mathematics: Grade 8
Functions F
Use functions to model relationships between quantities.
8.F.B.4
Construct a function to model a linear relationship between two
quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
(SAY) This standard for mathematical content is from the Functions domain. Read the standard.
Probing Facilitator Questions and Possible Participant Responses:
What do you notice?
This standard is big. It seems to encompass writing functions, identifying rate of change from different representations, and interpreting rate of change and initial value of a linear function from different representations.
What are the implications for instruction when a standard is this dense?
It will take several tasks to unpack everything that is in this standard because there are several essential understandings that can be associated with this standard.
Which part of this standard did the teacher identify as one of the goals of the lesson?
Writing or constructing linear equations to model a situation
Common Core State Standards, 2010, p. 55, NGA Center/CCSSO

13. Standards for Mathematical Practice Related to the Task
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.
(SAY) The bold standards are the standards for Mathematical Practice identified by the teacher as goals of the lesson.
Probing Facilitator Questions and Possible Participant Responses:
What does it mean to reason abstractly and quantitatively?
Abstract values from the problem, perform calculations with them, and interpret the results in the context of the problem.
What does it mean to attend to precision?
Attending to precision means using precise mathematical notation and language, explaining the meaning of symbols, and clearly articulating the steps in a mathematical process, as well as not making errors and rounding values appropriately.
What does it mean for a student to look for and make use of structure? Give an example
If a student recognizes a need for and writes an equivalent expression to solve a problem, s/he is looking for and making use of structure. For example, in the equation 6x + 12 = 30, the expressions on both sides of the equal sign are multiples of 6. If the student recognizes this and rewrites the equation to read 6(x + 2) = 6(5) so that it is clear that x + 2 = 5 and therefore x = 3, they have looked for and made use of structure.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

14. Identify Goals: Solving the Task (Small Group Discussion)
Solve the task.
Discuss the possible solution paths to the task.
Directions:
Give participants time to solve the task and discuss solution paths. The purpose at this time is to give participants an opportunity to think about ways that students might enter into and respond to the task. Do NOT facilitate a true Share, Discuss and Analyze Phase by selecting and sequencing participant responses. Don’t work too hard to make connections between solution paths. This will come later.

15. Scuba Math Task
Serena and Trevon are taking a scuba diving course while on vacation in Hawaii. Serena begins swimming toward the surface as Trevon begins his dive. The tables below represent their depth in feet with respect to time in seconds.
At what time will Trevon and Serena be at the same depth? Show your work and explain your reasoning.
Time (seconds)
Serena’s Depth (feet)
-90 10
-85 20
-80 30
-75 40
-70
Time (seconds)
Trevon’s Depth (feet) 20
-25 40
-50 60
-75 80
-100

16. Identify Goals Related to the Task (Whole Group Discussion)
Does the task provide opportunities for students to access the Standards for Mathematical Content and the Standards for Mathematical Practice that we have identified for student learning?
Probing Facilitator Questions and Possible Participant Responses:
How does this task address 8.EE.8a?
This task will address 8.EE.8a if both the graphical and algebraic solutions are explored in the Share, Discuss and Analyze Phase. If students graph the data in the tables, they will see that the lines intersect at only one point. This point is the only point that the lines share. The coordinates of the point are a numeric solution to both equations.
What else will students learn about solving and analyzing systems of equations?
Students will learn that since a system of linear equations can be modeled by two lines, then if the lines intersect the system has a solution. If the lines are parallel, the system does not have a solution.
How does this task address 8-F.B.4?
It is likely that at least one group will write linear equations to represent the divers’ depths. Students will also determine and interpret the rate of change as the rate of the divers’ ascent or descent and the initial value as the depth at time 0.
What structures are students looking for and making use of in this task?
Students will make use of the structure of linear equations by applying the transitive property. They will realize that it makes sense to set the expressions for the divers’ depths equal to one another in order to find the x value for which the expressions are both equal to the same y value.

17. Identify Goals: Essential Understandings (Whole Group Discussion)
Study the essential understandings associated with the Expressions and Equations Common Core Standards.
Which of the essential understandings are the goals of the Scuba Math Task?
(SAY)
The last time we were together, we spent time writing essential understandings. What were some of your takeaways about the relationship between the standards and the U’s and the characteristics of EUs?
Possible Responses:
EUs break down complicated standards into smaller components.
EUs address conceptual understanding, not procedural skill.
EUs answer the question, “What is it I really want my students to understand even if they have forgotten some of the facts and information?”
(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 to drive instruction. Which of the essential understandings can be used to drive the implementation of the Scuba Math Task?

18. Essential Understandings (Small Group Discussion)
Solutions Make the Equations True
The solution(s) of a system of two linear equations is the ordered pair or pairs (x, y) that satisfy both equations.
Systems of Equations Can be Solved Graphically
The line representing a linear equation consists of all of the ordered pairs (x, y) that satisfy the equation. So, the solution of a system of linear equations is represented graphically by the intersection of the lines representing the equations because the point(s) at the intersection satisfy both equations.
Systems of Equations Can be Solved Algebraically Using Substitution
Given a true equation in two variables, the equation formed by isolating either of the variables is also true. Therefore, so is the equation you get by substituting an expression equal to one of the variables into a second true equation.
A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions
Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution. Parallel lines have no points in common. Therefore, a system of two linear equations representing distinct parallel lines has no solutions. Linear equations representing the same line have infinitely many points in common. Therefore, a system of two linear equations representing the same line has infinitely many solutions.
Probing Facilitator Questions and Possible Participant Responses:
Does the Scuba Math Task have the potential to address all of these EUs? Why or why not?
Since this system has a solution, the students will probably not independently consider scenarios in which there are zero or infinitely many solutions. The teacher can, however, drive the discussion so that students explore the idea that they can tell from the rate of change whether or not the lines will intersect.
What would students say to indicate that they have the understanding communicated by the third EU?
They would say that if we know that an expression is equal to y, then we should be able to substitute it in for y in another equation and still maintain the equality.

19. Selecting and Sequencing Student Work for the Share, Discuss, and Analyze Phase of the Lesson

20. Analyzing Student Work (Private Think Time)
Analyze the student work on pages of the your handout.
Identify what each group knows related to the essential understandings.
Consider the questions that you have about each group’s work as it relates to the essential understandings.
(SAY) Imagine that these are students in your class. You have identified the EUs we discussed as your learning goals. As you look at student work, think about what these students know with respect to the EUs and what is not clear based on the work on their papers.
Facilitator Note:
Remind participants that this analysis should be completed individually for now and that they will have opportunity shortly to discuss their thinking.

21. Study the student work samples.
Prepare for the Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Small Group Discussion)
Assume that you have circulated and asked students assessing and advancing questions.
Study the student work samples.
Which pieces of student work will allow you to address the essential understanding?
How will you sequence the student’s work that you have selected? Be prepared to share your rationale.
(SAY) At your tables identify which EUs you will use to focus your whole class discussion and which pieces of student work you will use to focus the discussion on the essential understandings. Use the table on the next slide/page to record your group decisions.

22. The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Small Group Discussion)
In your small group, come to a consensus on the work that you select, and share your rationale. Be prepared to justify your selection and sequence of student work.
Essential Understandings
Group(s)
Order
Rationale
Solutions Make the Equations True
Systems of Equations Can be Solved Graphically
Systems of Equations Can be Solved Algebraically Using Substitution
A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions
(SAY) There are many rules of thumb for selecting and sequencing student work. Today we are going to select and sequence with the intent of moving the whole class on a trajectory toward our learning goals/essential understandings. Decide which student work you will use to focus the discussion on which EU and in what order. Be prepared to share your rationale.

23. The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Whole Group Discussion)
What order did you identify for the EUs and student work? What is your rationale for each selection?
Essential Understandings
#1 via Gr.
#2 via Gr.
#3 via Gr.
#4 Via Gr.
Solutions Make the Equations True
The solution(s)…
Systems of Equations Can be Solved Graphically
The line representing…
Systems of Equations Can be Solved Algebraically Using Substitution
Given two true equations…
A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions
Two distinct lines…
Directions:
Create a chart of this table to display at the front of the room.
(SAY) In order to display and compare the decisions made by each group, we will all record our selection and sequencing of student work on this chart. Each group has a different color marker. Using your group’s color, place the letter corresponding to the student work you are using in the appropriate box.
For example, if I want to begin my whole class discussion by exploring the second EU with Group C’s work, I will put a C (for Group C) in the box corresponding to the second EU in the first column.
Facilitator Note:
Have at least two participants repeat these directions to make sure that everyone understands the process.

24. Group A Facilitator Information: Noticings and Wonderings:
These students have indicated the time at which each diver will be at a depth of -75 feet. The students have indicated that this is the “same time,” perhaps because they are both represented in the fourth row in their respective tables.
The students have calculated the ratio of time:depth for each data set in each table. Trevon’s depth is proportional to the time while Serena’s depth is not.
Evidence of Essential Understandings:
The students looked for values in common to both divers, so they may understand that the ordered pair (x, y) that satisfies both equations is the solution.

25. Group B Facilitator Information: Noticings and Wonderings:
These students have graphed the data from the tables and have estimated a point of intersection from the graphs.
Do they know that their solution is approximate? Do they know what the point of intersection represents in the problem context?
Evidence of Essential Understandings:
The students understand that systems of equations can be solved graphically and that the point of intersection is the solution of the system. They probably understand that the solution is the ordered pair (x, y) that satisfies both equations. This work could be used to develop that understanding for the class.
This work can be used to develop the understanding that a system of linear equations may have zero, one, or infinitely many solutions.

26. Group C Facilitator Information: Noticings and Wonderings:
These students make several observations of the data in the table including calculating the rate of change of each set of data and writing equations to represent the relationship between time and depth of each diver. The students incorrectly conclude “they wouldn’t meet because they are at different paces.”
At the bottom of the page, the students substitute (0, -90) into the equation representing Serena’s depth and simplify the equation to -90 = Why did they substitute these values in and what does it mean that the equation simplified to this true numeric statement?
Evidence of Essential Understandings:
This student work could be used to develop the understanding that a system of linear equations may have one, zero, or infinitely many solutions. The students understand that the rate of the change of the functions has some bearing on whether or not the functions will have points in common.

27. Group D Facilitator Information: Noticings and Wonderings:
The students use these tables to interpolate the data to determine when the divers are at the same depth.
Both tables look at the times when the divers are between 59 and 75 feet deep. The exact solution is not visible on this table. Why did the students choose this interval for both tables?
What is the next step? Will the students graph this data? Will they determine linear equations to represent each set of data?
Evidence of Essential Understandings:
This work can be used to develop the understanding that the solution to a system is the ordered pair that makes both equations true. This work could also be used to explore the idea that solutions can be found graphically either by plotting the points in these tables or confirming that they match the graphs created by Group B and discussing what the solution will look like on the table and graph.

28. Group E Facilitator Information: Noticings and Wonderings:
This group finds an approximate solution using a table. Do they know their solution is imprecise? They also find an exact time algebraically. It is unclear why they divided -90 by It is also unclear whether or not they noticed and how they will reconcile the different values they determined for the time at which the divers are at the same depth. Do they understand what this x-value represents?
Evidence of Essential Understandings:
The students in this group understand that the point of intersection on the graph represents the ordered pair (x, y) that satisfies both solutions. They also understand that a system of linear equations can be solved algebraically. (They have not shown all of their steps, but appear to have used substitution. We need more evidence of student thinking.)

29. Group F Facilitator Information: Noticings and Wonderings:
These students have determined that one (or both?) of the data sets does not represent a proportional relationship. They have written equations for both divers. The students have not found a solution, but have indicated that they are thinking about graphing to see if they get the “same line.”
Evidence of Essential Understandings:
The students understand that each scenario can be represented by an equation and that the values in the tables are solutions to the equations. They understand as well that the equations can be modeled using lines and (probably) that the points on the line are solutions to the equations. This evidence can be used to build the understanding that the system of equations can be solved graphically and that a system of linear equations may have zero, one, or infinitely many solutions.

30. The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Whole Group Discussion)
What order did you identify for the EUs and student work? What is your rationale for each selection?
Essential Understandings
#1 via Gr.
#2 via Gr.
#3 via Gr.
#4 Via Gr.
Solutions Make the Equations True
The solution(s)…
Systems of Equations Can be Solved Graphically
The line representing…
Systems of Equations Can be Solved Algebraically Using Substitution
Given two true equations…
A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions
Two distinct lines…
Directions:
Once all groups have recorded the pieces of student work, they will use to focus the discussion and the order in which they will sequence them, facilitate a whole group discussion of similarities, differences, and big ideas and insights gained from engaging in the process. If most of the groups chose the same EUs or student work, start the discussion by asking for the rationale from groups that differed from the rest.
Probing Facilitator Questions and Possible Responses:
Why did you choose to sequence the EUs in the order that you did?
Our group thought that the first EU was the most important one, so we selected and sequenced work so that the conversation will culminate with discussion of this EU. (Some groups may choose to start the conversation with their “big” EU)
Did your group feel that one of these EUs was more important than the others? If so, which one and what pieces of student work did you use to address that EU?
We thought that the last EU is very important, because if students have this understanding, they will be able to determine if the system has a solution before they do a lot of work to find that solution. We thought Group C was a good group to start with for this EU because they notice that the divers are moving at different paces, but they draw an incorrect conclusion from this. So that misconception can be clarified by looking at Group B’s work so we can see how the different paces are visible in the graph and then how that actually means that they will intersect.
How did the context of the lesson (slide 10) factor into your decision making for selecting and sequencing student work?
Since 70% of the class needs to make sense of what it means to solve a system of equations, we decided that the first EU is the most important one because that is the EU that makes sense of what the solution means in terms of the functions that describe the divers’ depths.

31. Academic Rigor in a Thinking Curriculum The Share, Discuss, and Analyze Phase of the Lesson

32. Academic Rigor In a Thinking Curriculum
A teacher must always be assessing and advancing
student learning.
A lesson is academically rigorous if student learning related to the essential understanding is advanced in the lesson.
Accountable Talk discussion is the means by which teachers can find out what students know or do not know and advance them to the goals of the lesson.
(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 if students figure something out?
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?
Students struggling to make sense of ideas.
Students having “aha” moments and sharing their reasoning related to the mathematical idea, relationship, or process.
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.

33. Accountable Talk Discussions
Recall what you know about the Accountable Talk features and indicators. In order to recall what you know:
Study the chart with the Accountable Talk moves. You are already familiar with the Accountable Talk moves that can be used to Ensure Purposeful, Coherent, and Productive Group Discussion.
Study the Accountable Talk moves associated with creating accountability to:
the learning community;
knowledge; and
rigorous thinking.
Directions:
Direct participants to read the Accountable Talk Moves in their participant packets.
Probing Facilitator Questions and Possible Participant Responses:
How are the Accountable Talk Moves associated with accountability to community, knowledge, and rigorous thinking similar? How are they different? Do you disagree with how any of the moves are classified? If so, why?
The community moves are all made to make sure that everyone is involved in the discussion and understands the ideas being discussed.
The knowledge moves are used to make sure that we are discussing mathematics that is accurate and that we are attending to precision of language.
The rigorous thinking moves are made to press for reasoning and meaning.
Let’s think more about Academic Rigor… How do the Accountable Talk Moves to support accountability to rigorous thinking relate to the characteristics of academically rigorous lessons we identified?
If students are not prompted to articulate and expand their reasoning, there are no opportunities for “aha” moments.
Pressing students for reasoning ensures that they are carrying the cognitive load instead of doing the thinking for them. That is where the rigor lies in the lesson.
Expanding reasoning moves prompt students to make connections. “Aha” moments often occur when we see how ideas are related to one another.

34. 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 engage 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 the use some of the probing questions below to elicit what the talk would sound like in an Accountable Talk discussion.
Probing Facilitator Questions and Possible Responses:
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?
The commitment to getting it right will be mentioned because participants will talk about the teacher PRESS for accuracy or detail. Why might this be important to do? 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 Accountable Talk features 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?
Ask them to make a table, so they see a repeating pattern. Link to the context, 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.

35. 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. One factor tells use the number of groups and the other factor tells us how many items in the group.
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:
These Accountable Talk moves slides are to be used as a reference for the discussions that will occur around the questions on the previous slides.

36. 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?
Directions:
These Accountable Talk moves slides are to be used as a reference for the discussions that will occur around the questions on the previous slides.

37. The Share, Discuss, and Analyze Phase of the Lesson: Planning a Discussion (Small Group Discussion)
From the list of potential EUs and its related student work, each group will select an essential understanding to focus their discussion.
Identify a teacher in the group who will be in charge of leading a discussion with the group after the Accountable Talk moves related to the EU have been written.
Write a set of Accountable Talk moves on chart paper so it is public to your group for the next stage in the process.
Directions:
Explain to participants that their job is to plan a discussion focusing on one or more pieces of student work that targets an essential understanding.
(SAY) On a piece of chart paper, indicate the essential understanding and the piece or pieces of student work you will be discussing. Scribe several Accountable Talk moves or questions you will use to facilitate the discussion. Classify the moves according to the Accountable Talk feature they support. An example is provided on the next slide.

38. An Example: Accountable Talk Discussion
The Focus Essential Understanding
A System of Equations May Have Zero, One, or Infinitely Many Solutions
Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution.
Group C Group F
How did your group determine that Trevon and Serena are moving at different paces?
Who understood what she said about Trevon’s and Serena’s paces? (Community)
Can you say back what he said about the rate of change? (Community)
Who can add on and talk about where we see the pace in the equation and graph? (Community)
The slope is visible in the table, the equation, and the graph. (Marking)
If the pace or slope is different for each person, what does that mean about whether or not Trevon and Serena are ever at the same depth? (Rigor)
What will be true about the slope and the equations if Trevon’s and Serena’s data are the same line (second piece of work)? (Rigor)
(SAY) This is an example of an Accountable Talk discussion script. Let’s read through it together.
Directions:
It may be helpful to role-play this discussion. The facilitator asks the questions on the slide and participants can predict what students might say in response to the questions. Keep this brief. It is meant to give the participants an idea of what they are expected to produce.

39. Problematize the Accountable Talk Discussion (Whole Group Discussion)
Using the list of essential understandings identified earlier, write Accountable Talk discussion questions to elicit from students a discussion of the mathematics. Begin the discussion with a “hook” to get student attention focused on an aspect of the mathematics.
Type of Hook
Example of a Hook
Compare and Contrast
Compare the half that has two equal pieces with the figure that has three pieces.
Insert a Claim and Ask if it is True
Three equal pieces of the six that are on one side of the figure show half of the figure. If I move the three pieces to different places in the whole, is half of the figure still shaded?
Challenge
You said two pieces are needed to create halves. How can this be half; it has three pieces?
A Counter-Example
If this figure shows halves (a figure showing three sixths), tell me about this figure (a figure showing three sixths but the sixths are not equal pieces).
Directions:
AFTER the groups have written their scripts, introduce the idea of a hook. The examples on this slide are not specific to our task. They are intended as general examples of the different types of hooks that can be used to problematize the discussion.
(SAY) We know that one way that the cognitive demand of a task can decline is if the Share, Discuss and Analyze Phase of the lesson is used as a show and tell instead of a true Share, Discuss, and Analyze. Often, students are willing to share their own work, but don't know how to discuss and analyze the work of others.
Probing Facilitator Questions and Possible Responses:
How might beginning the discussion with a “hook” prompt students to discuss and analyze the work being shared? What are the characteristics of a strong hook?
They focus the students’ attentions on mathematical reasoning instead of just process.
They give the students an active role to play in the discussion instead a passive audience role.
A hook poses a problem to be solved.
Facilitator Note:
It is okay if the participants are unable to articulate the function and characteristics of the hook at this time. They will have another opportunity to do this soon.

40. An Example: Accountable Talk Discussion
The Focus Essential Understanding
A System of Equations May Have Zero, One, or Infinitely Many Solutions
Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution.
Group C Group F
Both groups are talking about the pace or the slope. Are they both saying the same thing? (Hook)
Can Group C talk about how you determined Serena and Trevon are not moving at the same pace?
Who understood what she said about Trevon’s and Serena’s paces? (Community)
Can you say back what he said about the rate of change? (Community)
If the pace or slope is different for each person, what does that mean about whether or not Trevon and Serena are ever at the same depth? (Rigor)
What will be true about the slope and the equations if Trevon’s and Serena’s data are the same line (second piece of work)? (Rigor)
(SAY) Here is the same Accountable Talk discussion we looked at before, but this time with a “hook” at the beginning of the discussion.
Probing Facilitator Questions and Possible Responses:
How does adding this hook impact the discussion that will follow?
It gives the audience something specific to consider as they listen to the group share their reasoning.
The hook draws students’ attentions to the meaning of the numbers in the problem.

41. Revisiting Your Accountable Talk Prompts with an Eye Toward Problematizing
Revisit your Accountable Talk prompts.
Have you problematized the mathematics so as to draw students’ attention to the mathematical goal of the lesson?
If you have already problematized the work, then underline the prompt in red.
If you have not problematized the lesson, do so now. Write your problematizing prompt in red at the bottom and indicate where you would insert it in the set of prompts.
We will be doing a Gallery Walk after we role-play.
Directions:
Read the slide. Have one participant restate the directions and provide an opportunity for participants to ask questions. Note that the “hooks” provided on slide 39 were non- specific examples. Participants will write a specific hook that pertains to the discussion they are planning on chart paper.

42. Role-Play Our Accountable Talk Discussion
You will have 15 minutes to role-play the discussion of one essential understanding.
Identify one observer in the group. The observer will keep track of the discussion moves used in the lesson.
The teacher will engage you in a discussion. (Note: You are well-behaved students.)
The goals for the lesson are:
to engage all students in the group in developing an understanding of the EU; and
to gather evidence of student understanding based on what the student shares during the discussion.
Directions:
Each group will role-play their discussion at their tables. Explain to participants that the moves they scribed are the starting point, but that it may be necessary to insert other moves and questions in order to engage all of the “students” and to collect evidence of student understanding. The discussion should be focused on the piece(s) of student work that the group chose to target the essential understandings.

43. Reflecting on the Role-Play: The Accountable Talk Discussion
The observer has two minutes to share observations related to the lessons. The observations should be shared as “noticings.”
Others in the group have one minute to share their “noticings.”
Directions:
These observations are being shared at the individual tables so that all participants will be able to participate in the whole group discussion that follows.

44. Reflecting on the Role-Play: The Accountable Talk Discussion (Whole Group Discussion)
Now that you have engaged in role-playing, what are you now thinking about regarding Accountable Talk discussions?
Directions:
Read the question on the slide.
Possible Responses:
Asking a hook with an EU in mind helps students know what to attend to in the discussion.
Keeping the EUs in mind allows me to craft Accountable Talk moves and questions that keep the discussion productive and focused.
If we want to have academically rigorous lessons characterized by movement in student thinking, we have to know what the EUs are so we know where we want to move student thinking toward. Then we have to plan hooks and Accountable Talk moves and questions to facilitate that movement.
This is hard work. Accountable Talk discussions will not just happen. I have to think about the moves I will make in the classroom when I am planning the lesson.

45. Zooming In on Problematizing (Whole Group Discussion)
Do a Gallery Walk. Read each others’ problematizing “hook.” What do you notice about the use of hooks? What role do “hooks” play in the lesson?
Directions:
Read the questions on the slides.
Possible Responses:
The hooks used relate to the essential understanding, but don’t ask too much too soon. They give students something to think about, but not something so complicated that the students tune out.
They present a problem for the students to consider beyond the “problem” posed by the task. They take the students beyond answer getting and toward unearthing mathematical truths that have meaning beyond the context of the task.
Hooks don’t ask for an answer to the problem or for a procedural response. Students have already solved the problem. The hooks push the students’ thinking in a direction they may not have gone on their own, such as considering a counter-example or critiquing the reasoning of a claim set forth by the teacher.

46. Step Back and Application to Our Work
What have you learned today that you will apply when planning or teaching in your classroom?
Directions:
Give participants a few minutes to write down their thoughts in response to this question. Ask for volunteers to share. Do not discuss participant responses. Simply allow 3–5 people to share and thank them for sharing. The purpose is for participants to hear some of what their colleagues are taking away from this module.

47. Summary of Our Planning Process
Participants:
Identify goals for instruction;
Align Standards for Mathematical Content and Standards for Mathematical Practice with a task.
Select essential understandings that relate to the Standards for Mathematical Content and Standards for Mathematical Practice.
Prepare for the Share, Discuss, and Analyze Phase of the lesson.
Analyze and select student work that can be used to discuss essential understandings of mathematics.
Learn methods of problematizing the mathematics in the lesson.
(SAY) Recall that our rationale for today’s work indicated that we would analyze a lesson-planning process. This is a summary of the process we engaged in. Today we only planned for one phase of the lesson, the Share Discuss, and Analyze Phase. The Share, Discuss, and Analyze Phase is a difficult phase to plan for and is the phase most often overlooked in planning and implementation. We know, however, that a good deal of the learning occurs in this phase. When students have opportunities to analyze each others’ thinking and reasoning, their own learning is advanced and academically rigorous thinking occurs.