# Supporting Rigorous Mathematics Teaching and Learning

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Supporting Rigorous Mathematics Teaching and Learning
Selecting and Sequencing Based on Essential Understandings Tennessee Department of Education Elementary School Mathematics Grade 4 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

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. Directions: (SAY) Take a minute and read the rationale for the lesson. As you can see from the rationale, “understand” is really important. The Common Core State Standards include standards that focus on understanding of mathematical concepts and the development of skills. We will engage in the lesson with the goal of deepening our understanding of concepts related to the task.

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, analyze phase of the lesson that motivate students to think productively about the mathematics. Facilitator: 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 be truly proficient with the Standards for Mathematical 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.

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

“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 of the things outside of our control. This quote reminds us that there are many decisions to be made everyday 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.

Linking to Research/Literature: The QUASAR Project

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 We know that the task matters a lot. And that, in fact, if the task chosen 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. Stein, Smith, Henningsen, & Silver, 2000 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

Identify Goals for Instruction and Select an Appropriate Task

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

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 have a fraction 𝑎 𝑏 as a sum of a parts of 1 𝑏 . (4.NF.B3) At this stage in student learning, the teacher is not as concerned that students are precise. S/he will take note of who is and who is not. The teacher, however, will be precise with his/her revoicing of student contributions. (MP6) All of the students can benefit from modeling with fractions. (MP4) Students will discuss the structure of mathematics related to the meaning of the numerator. (MP7) 20% of the students need additional work on fraction standards previously addressed (3.NF standards). These students also need opportunities to struggle with and make sense of the problem. (MP1) 5% of the students still do not recognize the importance of knowing what the “whole” is when talking about fractions. (Part of 4.NF.A2) The teacher will emphasize repeated reasoning when students find common denominators for each fraction. (MP8) Directions: (Say) This data is in your participant handout. Take a moment to read the context for the lesson we are about to consider. The language of the standards appears on the next pages of our participant materials.

The CCSS for Mathematics: Grade 4
Number and Operations – Fractions NF Extend understanding of fraction equivalence and ordering. 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Directions: (Say) These are the standards that were identified by the teacher as the goals of the lesson. The fractions standards are familiar to us from our work together on previous modules. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

The CCSS for Mathematics: Grade 4
Number and Operations – Fractions NF Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Directions: (Say) These standards are from the fractions domain. Read the standards. Probing Facilitator Questions and Possible Participant Responses (Italics): What do you notice? These standards link fractions to decimals but in a very narrow way. So there is deep understanding of the relationship. What are the implications for instruction? There is the ability to use manipulatives and demonstrate with the numbers involved. There is a lot of prior knowledge necessary with comparisons. Is every part of each of these standards addressed by the task? No. The students would need more numbers and more experiences decomposing fractional place values. Common Core State Standards, 2010, p. 31, NGA Center/CCSSO

Mathematical Practice Standards 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. Directions: (Say) The bolded standards are the Standards for Mathematical Practice identified by the teacher as goals of the lesson. Probing Facilitator Questions and Possible Participant Responses (Italics): 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. Give an example of what it means for a student to look for and make use of structure. If a student recognizes a need for and writes an equivalent fraction/decimal to solve a problem, (s)he is looking for and making use of the structure of place value, as well as the meaning of the numerator vs. the denominator. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

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.

The Pizza Task Jolla has 1 4 of a pizza. Sarah has 30 100 of a pizza.
Maria has of a pizza. Tim’s pizza is shaded on the pizza. How much pizza is Tim’s share? Jake has of a pizza. Juan has of a pizza. Show each of the student’s amount of pizza. Compare the students’ amounts of pizza. Explain with words and use the >, <, or = symbols to show who has the most pizza. Explain with words and use the >, <, or = symbols to show who has the least amount of pizza.

Jolla’s Pizza Tim’s Pizza Juan’s Pizza Sarah’s Pizza Maria’s Pizza Jake’s Pizza

Identify Goals Related to the Task (Whole Group Discussion)
Does the task provide opportunities for students to access the Mathematical Content Standards and Practice Standards that we have identified for student learning? Probing Facilitator Questions and Possible Participant Responses (Italics): How does this task address 4.NF.A.2? Students are tasked with comparing numbers less than one; because the numbers are given in fractional form. They are comparing fractions; they will have the opportunity to compare them as decimals when they use the 10x10 grid to show tenths or hundredths. How does this task address 4.NF.C.5? It is likely that many of the students will immediately write the fractional amounts as decimals. If they do not, however, the teacher can ask students to use both forms. What structures are students looking for and making use of in this task? Students will make use of the structure of fractions (meaning of the numerator and of the denominator) as well as the structure of place value with decimals.

Identify Goals: Essential Understandings (Whole Group Discussion)
Study the essential understandings associated with the Number and Operations – Fractions Common Core Standards. Which of the essential understandings are the goals of The Pizza Task? Directions: (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 EUs and the characteristics of EUs? Possible Responses: EUs break down complicated standards into smaller components. EUs address conceptual understanding, not procedural skill. They 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 Pizza Task?

The CCSS for Mathematics: Grade 4
Number and Operations – Fractions NF Extend understanding of fraction equivalence and ordering. 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

The CCSS for Mathematics: Grade 4
Number and Operations – Fractions NF Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Common Core State Standards, 2010, p. 31, NGA Center/CCSSO

Essential Understandings (Small Group Discussion)
Equal Size Pieces A fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit. Meaning of the Denominator The larger the name of the denominator, the smaller the size of the piece. Use of Benchmarks Comparison to known benchmark quantities can help students determine the relative size of a fractional piece because the benchmark quantity can clearly be seen as smaller or larger than the piece. One significant benchmark quantity is one-half. Equivalency A fraction can be named in more than one way and the fractions will be equivalent as long as the same portion of the set or area of the figure is represented. Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator. Probing Facilitator Questions and Possible Participant Responses (Italics): Does The Pizza Task have the potential to address all of these EUs? Why or why not? The base ten denominator, which defines the size of the pieces in the fraction, relates to the place value of the decimal equivalents. What would students say to indicate that they have the understanding communicated by the 3rd EU? They could say that if we know that half is also 0.5 and 0.05, everything can be compared to that benchmark for starters.

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

Analyzing Student Work (Private Think Time)
Analyze the student work. 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. Directions: (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. Remind participants that this analysis should be completed individually for now and that they will have the opportunity shortly to discuss their thinking.

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. Directions: (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.

The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Small Group Discussion) In your small group, come to 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 Meaning of the Denominator Use of Benchmarks Equivalency Creating Equivalent Fractions (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.

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. Meaning of the Denominator The larger the name of… Use of Benchmarks Comparison to known benchmark… Equivalency A fraction can be named in more… Creating Equivalent Fractions When the denominator is multiplied… 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. Have at least two participants repeat these directions to make sure that everyone understands the process.

Group A Noticings and Wonderings:
These students have indicated each of the fractions with base ten denominators and stated the relationship between the denominators 10 and 100 (there are 10 hundredths in one tenth). The 0.02 is incorrect but a wondering maybe if the students understand the place value system in numbers less than one. Evidence of Essential Understandings:- The students looked for equivalencies with base ten denominators.

Group B Noticings and Wonderings:
These students worked to state equivalencies for base ten denominator fractions. Do they see where 3/10 cannot equal 3/100? Do they see the numerators as separate from the entire value of the fraction (since they wrote 3 < 20)? Evidence of Essential Understandings: The students understand that fractions have a multitude of equivalent forms and that some consistent operation needs to be applied to the numerator and denominator.

Group C Noticings and Wonderings:
These students make all of the fractions into equivalent forms that have the same denominator, perhaps to make comparisons more convenient. Do these students see the decimal form equivalencies? Evidence of Essential Understandings: This student work has evidence (symbolically) of the meaning of the denominator. It remains to be seen if that understanding translates to the pictorial representation.

Group D Noticings and Wonderings:
The students use known benchmark equivalencies (1/4 = 25/100). How did they know that if they found a common denominator they only have to compare the numerators? Evidence of Essential Understandings: This work has evidence of understanding of equivalencies as well as of comparisons (to benchmarks and to each other when there is a common denominator).

Group E Noticings and Wonderings:
This group finds an accurate equivalency as well as an inaccurate one. Can the students identify that the denominator determines the size of the piece/portion, and therefore two fractions with common numerators but uncommon denominators cannot be equal? Evidence of Essential Understandings: The student work can be used to develop the understanding of equivalency and of the meaning of the denominator, perhaps by having the students represent their work on the 10x10 grids.

Group F Noticings and Wonderings:
These students have determined that the 10x10 grid needs to be broken into equal pieces in order to shade a fraction that does not have a denominator of 100. Do these students see that where you shade a portion, as long as it is the same size, does not change the equivalence? (The 25/100 is now represented/shaded 3 different ways – are they all equivalent?) Evidence of Essential Understandings: The student work can be used to discuss the fact that an area that is shaded is equivalent if it takes up the same amount of space.

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. Meaning of the Denominator The larger the name of… Use of Benchmarks Comparison to known benchmark… Equivalency A fraction can be named in more… Creating Equivalent Fractions When the denominator is multiplied… Directions: Once all groups have recorded the pieces of student work that they will use to focus the discussion, and the order in which they will sequence them, facilitate a whole group discussion of similarities, differences, big ideas, and insights gained from engaging in the process. If most of the groups chose What? Probing Facilitator Questions and Possible Participant Responses (Italics): 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. (NOTE: 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 first EU was very important, because if students do not understand portion size, they cannot create equivalencies or comparisons. 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 understand non-unit fractions, we decided that the third EU is the most important one.

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

Academic Rigor In a Thinking Curriculum

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 packet. Probing Facilitator Questions and Possible Participant Responses (Italics): How are the AT 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 designed 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 AT moves 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 us doing the thinking for them. That is where the rigor lies in the lesson. Expanding reasoning moves prompts students to make connections. “Aha” moments often occur when we see how ideas are related to one another.

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 packets. Give them an opportunity to independently read the Accountable Talk Features and Indicators. Ask them to talk in small groups about how these features and indicators support the implementation of academically rigorous lessons. Probing Facilitator Questions and Possible Participant Responses (Italics): What is the benefit of asking students to elaborate or build on each others’ 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. They are part of the struggle that students must engage in before they can make a connection triggering an “aha” moment. Why do you think the authors claim that evidence of all three features of AT must be present? Whoever talks the most learns the most. Students need to be the ones talking so that the teacher can assess what they know or don’t know. It is the students’ knowledge and reasoning that we need to hear, not just any kind of talk. What does it mean when you press students for an explanation, but they can’t explain the reasoning underlying the concept? Students often don’t understand a concept and they can’t share their mathematical reasoning because it is hard work. Students need to have a deep enough understanding of why the mathematical ideas are working the way they are working. How might you scaffold student learning in order to make it possible for students to share their mathematical reasoning? Make a table so they see a repeating pattern, link to the context, ask them if the problem reminds them of other similar problems, provide students with manipulatives, share your reasoning so they have opportunities to hear what it sounds like, invite students to “try to share their reasoning” and permit others to add on. All of these things will help students make connections between representations and to prior learning. These connections are characteristic of what it means to move thinking and to have an academically rigorous experience.

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 AT moves slides are to be used as a reference for the discussions that will occur around the questions on the previous slides.

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 AT moves slides are to be used as a reference for the discussions that will occur around the questions on the previous slides.

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 target 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. Write several AT moves or questions you will use to facilitate the discussion. Classify the moves according to the AT feature they support. An example is provided on the next slide.

An Example: Accountable Talk Discussion
The Focus Essential Understanding Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator. Group A Group B Explain your set of equivalencies. Who understood what he said about the 100 and the 10? (Community) Can you say back what he said how the model shows ? (Community) Who can add on and talk about the 3 and the 30? (Community) The denominator tells the number of equal parts in the whole. (Marking) Do we see in both pieces of work? (Rigor) Tell us how you found in your picture (Group A). (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.

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 Participant Responses (Italics): 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 of a passive audience role. A hook poses a problem to be solved. 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.

An Example: Accountable Talk Discussion
The Focus Essential Understanding Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator. Group A Group B Both groups say that 𝟑 𝟏𝟎 is equal to 𝟑𝟎 𝟏𝟎𝟎 . How can this be when the fractions use different numbers? (Hook) Can Group B explain why = = ? Who understood what they said about the denominators? (Community) Can you say back what they said about the numerator changing? (Community) Each group made statements about equivalency. How does the visual model differ from/support the symbolic model? (Rigor) (SAY) Here is the same AT discussion we looked at before, but this time with a hook at the beginning of the discussion. (Ask) How does adding this hook impact the discussion that will follow? Possible Participant Responses: It gives the students something specific to consider as they listen to the others in the group share their reasoning. The hook draws students’ attentions to a particular mathematical relationship.

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. Provide an opportunity for participants to ask questions. Note that the hooks provided on slide 41 were non- specific examples. Using chart paper, participants will write a specific hook that pertains to the discussion they are planning.

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 its discussion at its table. 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.

Reflecting on the Role-Play: The Accountable Talk Discussion
The observer has 2 minutes to share observations related to the lessons. The observations should be shared as “noticings.” Others in the group have 1 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.

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 Participant 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 AT 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 that we know where we want student thinking to move. Then we have to plan hooks and AT moves and questions to facilitate that movement. This is hard work. AT discussions will not just happen. I have to think about the moves I will make in the classroom when I am planning the lesson.

Zooming In on Problematizing (Whole Group Discussion)

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.

Summary of Our Planning Process
Participants: identify goals for instruction; Align Content Standards and Mathematical Practice Standards with a task. Select essential understandings that relate to the Content Standards and Mathematical Practice Standards. 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. Directions: (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 lessons—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.