Discourse in the Math Classroom

Discourse in the Math Classroom
June 2017 Facilitation Notes: BREAK TIMES MAY SHIFT DEPENDING ON PARTICIPANT NEEDS.

DISCOURSE IN THE MATH CLASSROOM Introduction: Who I Am
Name 1 Name 2 Insert photo Insert photo 2 minutes *Facilitators edit slide and notes for this slide. Speaker’s Notes: I am ______ from ______. Include an interesting personal story. My experience has been… Before Common Core, I was… I was skeptical about Common Core until ______ happened. Think of an English learner you’ve known. What is her/his name? Let’s remember to keep our students in mind as we engage in this work today.

Norms That Support Our Learning
DISCOURSE IN THE MATH CLASSROOM Norms That Support Our Learning Take responsibility for yourself as a learner Honor timeframes (start, end, activity) Be an active and hands-on learner Use technology to enhance learning Strive for equity of voice Contribute to a learning environment in which it is “safe to not know” 1 minutes Speaker's Notes: Review norms.

DISCOURSE IN THE MATH CLASSROOM Today’s Agenda
Welcome! Objectives Establishing Appropriate Goals Identifying and Choosing High Cognitive-Demand Tasks Classroom Discourse: Anticipating, Monitoring & Selecting, Sequencing & Connecting Applications to Your Practice 1 minute

DISCOURSE IN THE MATH CLASSROOM I. Welcome!
Norms we just discussed: Keep these in mind during the learning today! Notetaking guide: Use this to track your learning today! Locations of Restrooms and Water 1 minute Facilitation Notes: Explain how to call the group back together after small-group conversations or think-pair-share times. [Use silent hand signal to indicate the message, “Please finish your sentence, but do not start a new paragraph,” to allow participants to complete their thoughts before turning attention back to the large group.]

DISCOURSE IN THE MATH CLASSROOM II. Objectives
Participants will… identify and create rigorous and appropriate lesson goals. identify and/or generate high cognitive-demand tasks. plan for instruction using five practices to build mathematical discussions and discourse. 1 min Speaker's Notes: Review objectives for the day.

DISCOURSE IN THE MATH CLASSROOM A comment about today’s work:
We are ramping into our conversation about classroom discourse! 1 minute Help participants understand that the early parts of our work together will focus on foundational pieces that will lead to rich classroom discourse.

DISCOURSE IN THE MATH CLASSROOM Conceptual Understanding...
What does that really mean? 5 min (including the following slides, up to the agenda) Discuss the need to have a shared understanding of what this means for our work today. We’ll begin by taking a look at a quick activity together. First, I’m going to ask you to memorize nine facts. [Show each fact for about 3-5 seconds.]

DISCOURSE IN THE MATH CLASSROOM Fact #1
5 seconds

B 5 seconds

5 seconds

G 5 seconds

5 seconds

DISCOURSE IN THE MATH CLASSROOM What Is This?
15 seconds Got it? Now how many of us can “translate” this?

DISCOURSE IN THE MATH CLASSROOM What Is This?
15 seconds Be honest, how many of us got this “right”? What made that challenging? Elicit answers: many facts, not a lot of time, more or less “random.” F A C E

DISCOURSE IN THE MATH CLASSROOM Connection: Conceptual Understanding?
B C D E F 30 seconds Now, let’s try another experiment. Instead of those nine facts, spend some time studying this diagram. G H I

DISCOURSE IN THE MATH CLASSROOM Try It Again!
1 minute What does it spell?

DISCOURSE IN THE MATH CLASSROOM Memorization or Understanding?
1 minute Now, how many of us got this “right”? What made this easier to accomplish? Elicit answers: understanding the “big idea” or structure, not “just memorizing.” It is this level understanding that we hope all of our students achieve. Today we will see how carefully chosen tasks and productive discourse can increase conceptual understanding for students. D E C A D E

DISCOURSE IN THE MATH CLASSROOM Let’s watch a teacher in action.
This is Ms. Latimer with her Kindergarten students. 12 minutes Facilitation notes: Let’s watch Ms. Latimer with her students, and notice the tensions between conceptual and procedural understandings. What catches your attention? 6 minutes to watch the video, and then 6 minutes to re-watch certain sections and to discuss. NOTE the moment where the child says he knows the answer is going to be 8 (at 2:16). What are your thoughts about how she handled this? NOTE that she neutrally accepting of all answers.

DISCOURSE IN THE MATH CLASSROOM Standards for Mathematical Practices
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. 3 minutes Facilitation notes: Let’s reflect on the eight Standards for Mathematical Practices. What are the verbs and verb phrases here?

DISCOURSE IN THE MATH CLASSROOM III. Lesson Goals
Goal A: Students will learn the Pythagorean Theorem (c2 = a2 + b2 ). State the desired goal. State the desired objective. 2 minutes Facilitation notes: Now let’s turn our attention to lesson goals other teachers have written. First, what’s our verb? Second, with which of the Standards for Mathematical Practices does this connect?

DISCOURSE IN THE MATH CLASSROOM Lesson Goals
State the desired goal. State the desired objective. Goal A: Students will learn the Pythagorean Theorem (c2 = a2 + b2 ) Goal B: Students will be able to use the Pythagorean Theorem ( c2 = a2 + b2 ) to solve a series of missing value problems. 2 minutes Facilitation notes: Again, what are our verbs? Second, with which of the Standards for Mathematical Practices does this connect?

DISCOURSE IN THE MATH CLASSROOM Lesson Goals
Goal A: Students will learn the Pythagorean Theorem (c2 = a2 + b2 ) Goal B: Students will be able to use the Pythagorean Theorem ( c2 = a2 + b2 ) to solve a series of missing value problems. Goal C: Students will recognize that the areas of the square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on the legs and will conjecture that c2 = a2 + b2. State the desired goal. State the desired objective. 2 minutes Facilitation notes: Finally, what are our verbs? With which of the Standards for Mathematical Practices does this connect?

DISCOURSE IN THE MATH CLASSROOM Reflecting on These Lesson Goals
Will this goal assist the teacher in selecting a task for the lesson? Is it clear what counts as evidence of student learning? What key ideas and relationships will be highlighted during a possible discussion? What is the focus of the lesson: application, procedural skill and fluency, or conceptual understanding? 10 minutes MODEL: Discuss with an elbow partner the differences by answering the following questions: Will this goal assist the teacher in selecting a task for the lesson? Is it clear what is evidence of student learning? What key ideas and relationships will be highlighted during a possible discussion? From the goal, what is the focus of the lesson; application, procedural skill and fluency or conceptual understanding?

DISCOURSE IN THE MATH CLASSROOM Lesson Goals: Let’s Practice
Elementary: Students will understand equivalent fractions. Middle: Students will solve ratio problems. High: Students will prove theorems about triangles. Rewrite these goals to be more explicit. Choose one at your grade level. 5 minutes Facilitation notes: In pairs, identify the verbs in these goals, and revise to make them more explicit. 2-3 minutes to work and then 2-3 minutes to share thinking with the group. Having strong lesson goals will be critical to our work with the “five practices.”

DISCOURSE IN THE MATH CLASSROOM Lesson Goals: Let’s Practice
Now turn your attention to the lesson you brought with you today. How can you improve the lesson goal(s)? 15 minutes Give participants time to revise, then share, their learning goals.

Let’s pause for a break! 10 minutes

DISCOURSE IN THE MATH CLASSROOM IV. Choosing High-Quality Tasks
Cognitive Demand Guide Take a look at the tasks on your handout. Describe the characteristics of each. Order from least demanding to most demanding. 5 minutes In addition to writing appropriate lesson goals, selecting the right task is important as well. Take a look at the four tasks on the handout (A, B, C, D), and describe the characteristics of each. With your partner/ group, order them from least demanding to most demanding.

DISCOURSE IN THE MATH CLASSROOM Why?
Beginning with “high-level, cognitively challenging complex tasks” helps students learn to reason abstractly and problem-solve. This also leads to increased engagement. Low-level tasks rarely result in high-level engagement. 3 minutes Facilitation notes:

Discussion question--What’s not a “high level task?”
DISCOURSE IN THE MATH CLASSROOM In General: What’s a “High-Level Task?” Allows students to become engaged in the learning Uses prior knowledge Focuses on sense-making, problem solving, and skill building Makes connections Focuses on representing, explaining, and justifying ideas Not a single, set way to solve Multiple entry points/access for all Discussion question--What’s not a “high level task?” 8 minutes For the discussion question: Take a look at the “Task Analysis Guide” in the handout. This page summarizes research by Stein & Smith (1998) intended to help teachers create and select the most What do you notice as similar and different between the categories of tasks?

DISCOURSE IN THE MATH CLASSROOM Remember This?
1 minute Facilitation notes: Remember this? Now that you have this new taxonomy for thinking about tasks, how would you categorize it?

DISCOURSE IN THE MATH CLASSROOM Now Let’s Return to the Four Examples from Earlier (A, B, C and D)
2 minutes 1. Does a particular feature (e.g., writing an explanation as part of your answer, drawing a picture to explain what you did, using manipulatives to solve the task) indicate that the task has a certain level of cognitive demand? 2. Is there a difference between "level of cognitive demand" and "difficulty"? 3. What effect does context (e.g., setting in which the task is used, students' prior experience, grade level) have on the level of cognitive demand required by a task?

Procedures without Connections (A) Doing the Math (B) Memorization (C)
DISCOURSE IN THE MATH CLASSROOM Now Let’s Return to the Four Examples from Earlier (A, B, C and D) Procedures without Connections (A) Doing the Math (B) Memorization (C) Procedures with Connections (D) 2 minutes 1. Does a particular feature (e.g., writing an explanation as part of your answer, drawing a picture to explain what you did, using manipulatives to solve the task) indicate that the task has a certain level of cognitive demand? 2. Is there a difference between "level of cognitive demand" and "difficulty"? 3. What effect does context (e.g., setting in which the task is used, students' prior experience, grade level) have on the level of cognitive demand required by a task?

DISCOURSE IN THE MATH CLASSROOM Turn to Your Materials
Choose a task you have brought. Identify the level of cognitive demand in this task. How do you know? Does the task match the learning goal you revised earlier? How can you revise your task to make it more cognitively demanding? 15 minutes: NOTE: This slide / activity may be CUT if limited time available. Facilitation notes: Select a task from the lesson you brought. Then evaluate it against the cognitive demand criteria and be ready to justify your choices. After 10 minutes of work time, select 2 or 3 participants to share their task and thinking with the whole group.

Welcome Back!

Cognitive Demand Implementation Design Set-Up Student Learning
DISCOURSE IN THE MATH CLASSROOM Mathematics Task Framework Cognitive Demand Stein et al, 1998 Design Set-Up Implementation Student Learning 1 minute As we move into our focus on discourse, let’s keep in mind that central to all this work is solid student learning.

Routinizing problematic aspects of the task
DISCOURSE IN THE MATH CLASSROOM Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off- task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes. 5 minutes Facilitation notes: This is the same info as on the previous 2 slides, but in side-by- side format. Discussion questions: What does scaffolding mean? What does it look like? Decline happens when teachers scaffold the thinking out of the task for the students. When there are problematic aspects of the task, what do you do? How does what you do affect the maintenance or decline of the task. Source: Stein, Grover & Henningsen, 1996 41

Scaffolding of student thinking and reasoning
DISCOURSE IN THE MATH CLASSROOM Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore 5 minutes Facilitation notes: This is the same info as on the previous 2 slides, but in side-by- side format. Discussion questions: What does scaffolding mean? What does it look like? Decline happens when teachers scaffold the thinking out of the task for the students. When there are problematic aspects of the task, what do you do? How does what you do affect the maintenance or decline of the task. Source: Stein, Grover & Henningsen, 1996 42

DISCOURSE IN THE MATH CLASSROOM Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Decline Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes Maintenance Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore 5 minutes Facilitation notes: This is the same info as on the previous 2 slides, but in side-by- side format. Discussion questions: What does scaffolding mean? What does it look like? Decline happens when teachers scaffold the thinking out of the task for the students. When there are problematic aspects of the task, what do you do? How does what you do affect the maintenance or decline of the task. Source: Stein, Grover & Henningsen, 1996 43

Make sense of problems and persevere in solving them.
DISCOURSE IN THE MATH CLASSROOM Revisiting the SMPs 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. 3 minutes Facilitation notes: Now that we have talked about setting goals and selecting rich tasks, let us turn our attention back to the Standards for Mathematical Practices. What is the connection to discourse in these standards? Turn and talk and then share.

What happens when students share their work with the class?
DISCOURSE IN THE MATH CLASSROOM V. Classroom Discourse In your own classroom, what’s the state of discourse? What’s going well? What could be better? What happens when students share their work with the class? 5 minutes Facilitation notes: Before we dive into the “five practices,” let’s think about our own students, who are the reason we’re actually here. What is the state of discourse currently in our classrooms? Highlight any responses to 1. In response to question 2, highlight responses that indicate that currently, sharing student work is not a powerful source of student dialogue. Today, we will see how student work can improve the quality of discussions in class.

DISCOURSE IN THE MATH CLASSROOM Why is discussion important?
Talk with your table groups. What comes to mind? 5 minutes Facilitation notes: 3 minutes to talk, 2 minutes to share out key ideas

DISCOURSE IN THE MATH CLASSROOM The Importance of Discussion
Mathematical discussions are a key part of current visions of effective mathematics teaching: To encourage student construction of mathematical ideas To make student’s thinking public so it can be guided in mathematically sound directions To learn mathematical discourse practices 3 minutes Facilitation notes: ”Here are some other ideas I heard you discuss.”

DISCOURSE IN THE MATH CLASSROOM The Leaves and Caterpillar Problem
A fourth-grade class needs five leaves each day to feed its 2 caterpillars. How many leaves would they need each day for 12 caterpillars? Use drawings, words, or numbers to show how you got your answer. Solve this problem in as many ways as you can, both correct and incorrect. Consider what a 4th grader might do. 8 minutes Facilitation notes: Using your handout, let’s brainstorm and generate some different ways students might respond to this problem. Compare and share with your neighbors to help generate a larger range of possible approaches. Use the “Workspace” section of your handout. ALSO: What are your thoughts on the level of rigor in this problem? Does this count as “doing mathematics” based on what we recently discussed? What evidence do you have for this?

DISCOURSE IN THE MATH CLASSROOM A Dramatic Reading
As you read and listen, consider: What is promising about Mr. Crane’s instruction? In what ways might Mr. Crane improve? 10 minutes Facilitation notes: Now let’s read about a teacher who used this task with his learners. We need a few dramatic readers! Read aloud, a few lines or paragraph at a time. Please read with dramatic enthusiasm! As you listen and consider the questions, keep in mind the Factors Associated with the Maintenance and Decline of High-Level Cognitive Tasks. Discuss responses to the prompts.

DISCOURSE IN THE MATH CLASSROOM The Leaves and Caterpillar Vignette
What is promising? What teachers commonly mention: Students are working on a mathematical task that appears to be both appropriate and worthwhile Students are encouraged to provide explanations and use strategies that make sense to them Students are working with partners and publicly sharing their solutions and strategies with peers Students’ ideas appear to be respected 10 minutes Facilitation notes: Address points that participants in the room have not previously noted. Ball has argued that the field needs to take responsibility for helping teachers learn how to continually “size up” whether important mathematical ideas are being developed and when and how to step in and redirect the conversation if needed.

DISCOURSE IN THE MATH CLASSROOM The Leaves and Caterpillar Vignette
What can be improved? What teachers commonly mention: Beyond having students use different strategies, Mr. Crane’s goal for the lesson is not clear for students as they work. There is a “show and tell” feel to the presentations not clear what each strategy adds to the discussion different strategies are not related key mathematical ideas are not discussed no evaluation of strategies for accuracy, efficiency, etc. 10 minutes Facilitation notes: Address points that participants in the room have not previously noted. Many aspects of the instruction in Mr. Crane’s class that need work…. One aspect of his instruction that we might want to help him think about is the whole class discussion. The sharing out of solutions has the feel of “show and tell” where there is little filtering on the part of the teacher regarding what was selected for presentation; no assistance is provided with respect to drawing connections among methods or tying them to widely shared disciplinary methods or concepts. Ball has argued that the field needs to take responsibility for helping teachers learn how to continually “size up” whether important mathematical ideas are being developed and when and how to step in and redirect the conversation if needed.

DISCOURSE IN THE MATH CLASSROOM What now?
How can we improve our skills with orchestrating productive mathematics discussion? 10 minutes Facilitation notes: Address points that participants in the room have not previously noted. Many aspects of the instruction in Mr. Crane’s class that need work…. One aspect of his instruction that we might want to help him think about is the whole class discussion. The sharing out of solutions has the feel of “show and tell” where there is little filtering on the part of the teacher regarding what was selected for presentation; no assistance is provided with respect to drawing connections among methods or tying them to widely shared disciplinary methods or concepts. Ball has argued that the field needs to take responsibility for helping teachers learn how to continually “size up” whether important mathematical ideas are being developed and when and how to step in and redirect the conversation if needed.

DISCOURSE IN THE MATH CLASSROOM The Role of the Teacher
Worthwhile tasks alone are not sufficient for effective teaching. Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge. NCTM, 2000 2 minutes: Facilitation notes: This quote comes from the NCTM Principles & Standards for School Mathematics emphasizes the role of teacher in orchestrating discussion. NOTE that this is essential to our work of supporting independent thinkers!

DISCOURSE IN THE MATH CLASSROOM The Role of Communication
Communication is an essential part of mathematics and mathematics education. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public. NCTM, 2000 2 minutes Facilitation notes: How do you see this quote (from the year 2000) as connected to the Standards for Mathematical Practice? This is a strong rationale for a share-and-discuss phase of the lesson that is more than just a show-and-tell of answers. Not only are students sharing the mathematics and making connections, but they are also creating permanence for the mathematical ideas.

DISCOURSE IN THE MATH CLASSROOM What is “expert facilitation”?
Skillful improvisation... Interpret and synthesize students’ thinking on the fly Fashion responses that guide students to evaluate each other’s thinking, and promote building of mathematical content over time ...requires deep knowledge of: Relevant mathematical content Student thinking about content and how to frame it Subtle pedagogical moves How to rapidly apply all of this in specific circumstances 3 minutes Facilitation notes: Review these points. Now we’ll do “fist to five.” Show a fist if this set of ideas is going to be extremely challenging for you to implement. Show five fingers if this is something you already do regularly with great skill, ease, and dexterity. Show 1, 2, 3 or 4 fingers if you’re somewhere in between. Take 90 seconds to chat with someone with the same number a you.

DISCOURSE IN THE MATH CLASSROOM Some Challenges
Lack of familiarity Reduces teachers’ perceived level of control Requires complex, split-second decisions Requires flexible, deep, and interconnected knowledge of content, pedagogy, and students 2 minutes Facilitation notes: NOTE that these responsibilities are tough! We are tasked with complex, challenging work that demands the best from us!

DISCOURSE IN THE MATH CLASSROOM The Five Practices
Anticipating (e.g., Fernandez & Yoshida, 2004; Schoenfeld, 1998) Monitoring (e.g., Hodge & Cobb, 2003; Nelson, 2001; Shifter, 2001) Selecting (Lampert, 2001; Stigler & Hiebert, 1999) Sequencing (Schoenfeld, 1998) Connecting (e.g., Ball, 2001; Brendehur & Frykholm, 2000) 2 minutes Facilitation notes: Luckily, we can draw from the “5 practices” to support our work! Each of these concepts is drawn from a deep and robust research base.

DISCOURSE IN THE MATH CLASSROOM The Five+ Practices
01. Establishing the Learning Goal 02. Selecting the Task Anticipating (e.g., Fernandez & Yoshida, 2004; Schoenfeld, 1998) Monitoring (e.g., Hodge & Cobb, 2003; Nelson, 2001; Shifter, 2001) Selecting (Lampert, 2001; Stigler & Hiebert, 1999) Sequencing (Schoenfeld, 1998) Connecting (e.g., Ball, 2001; Brendehur & Frykholm, 2000) 1 minute Facilitation notes: However, note that it does not just begin with anticipating student responses. The first part of our day together was invested in the first two points—0 sub 1 and 0 sub 2.

DISCOURSE IN THE MATH CLASSROOM Purpose of the Five Practices
To make student-centered instruction more manageable by moderating the degree of improvisation required by the teachers and during a discussion. 1 minute Facilitation notes: The practices we have identified are meant to make student-centered instruction more manageable by moderating the degree of improvisation required by the teacher during a discussion. Rather than focusing on in-the-moment responses to students contributions, the practices instead emphasize the importance of planning. Through planning, teachers can anticipate likely student contributions, prepare responses they might make to them, and make decisions about how to structure students’ presentations to further their mathematical agenda for the lesson.

DISCOURSE IN THE MATH CLASSROOM Preparing to Teach
3 minutes: Facilitation notes: Doing the task in advance allows the teacher to fill out the strategy column on the Tool for Monitoring. This work helps the teacher quickly identify the student work that demonstrates different strategies for solving the problem. It also raises the awareness of strategies that students are not using to solve. (Participants did this when they solved the problem earlier.) Note that we might not always have the “label” for each method at the ready (like unit rate, scale factor, etc.) and this is fine.

Listen, observe, identify key strategies. Keep track of approaches.
DISCOURSE IN THE MATH CLASSROOM VII. Monitoring Listen, observe, identify key strategies. Keep track of approaches. Ask questions of students to get them back on track or to think more deeply. 5 minutes Facilitation notes: What does this LOOK LIKE in a classroom? What strategies could YOU use to “monitor” the work of your students?

DISCOURSE IN THE MATH CLASSROOM Preparing to Teach
Column 3: Teachers Sequence the order in which selected student work will be shared. Column 1: Teachers Anticipate the strategies and and approaches students may take. Column 2: Teachers Monitor the work of specific students. (during and/or after the lesson) Column 4: Teachers consider how to Connect the thinking of multiple students to draw out the big math ideas. 3 minutes Facilitation notes: While students are working the task, the teacher can MONITOR student responses and record student names and work in the Work of Specific Students column in the appropriate strategy row. Based on what teachers see in student work, then they can SELECT specific student solutions the teacher would like to have shared. Lastly, the teacher will need to choose the sequence for students to share their work and the connections between the types of student solutions. The connections give the teacher ideas for the types of questions that he/she would like to ask of the students. The idea is to move the students toward the big ideas of mathematics. Column 2: Teachers Select which student work to highlight in a class discussion.

DISCOURSE IN THE MATH CLASSROOM Selecting, Sequencing, and Connecting
As a table, complete the Tool for Monitoring using the student work from Leaves and Caterpillars. Monitor which students used which strategy and write names and samples from their work in column 2 Select the strategies/solution paths that you would want to have shared during a whole group discussion. Specify the sequence in which they would be shared and explain why you selected the particular responses and how you determined the ordering of the presentations. Determine the strategies you would want to specifically connect and explain the connection you would want students to see. 10 minutes Facilitation notes: Remind that we completed the anticipation when we solved the task earlier. Some things to consider when selecting and sequencing: Correct answers Different strategies Different representations Incorrect answer Correct pathway Pairs that work together (e.g., student #1 and Student #2 might be thought of as a pair for comparison of similarities and differences. Why might you do this?) Involve justification and generalization Etc.

DISCOURSE IN THE MATH CLASSROOM Thinking Through a Lesson
In what ways is this process the same or different from other lesson planning formats you have used? 10 minutes Faciitation notes: Possible points of discussion may include: *Planning in lesson study *Launch, explore, summarize questions from CMP *Research on Mathematics tasks (e.g., what it takes to keep tasks at a high level)

DISCOURSE IN THE MATH CLASSROOM VI. Anticipating
Brainstorm ways the problem might be solved. What are students likely to produce? Which problems will most likely be the most useful in addressing the mathematics? How will you elicit deeper student thinking? 5 minutes Facilitation notes: Now let’s extend this by actively anticipating what students will think, record, and do. Discussion whole group regarding what Anticipating means for the teacher. Work through the 1st and 2nd column on misconception / strategy template for Speedy Mart Problem. Then turn attention to column 3, questions and responses, for the Speedy Mart Problem. What questions could you ask to elicit student thinking and deepen understandings?

DISCOURSE IN THE MATH CLASSROOM Anticipating
S + 17 Let S = the number of gallons of juice sold by Speedy Mart Student understands how to write an expression and recognizes 17 needs to be added to Speedy Mart if the solution is from the perspective of Marathon. They also understand how to correctly define a variable. So if Speedy Mart sold 12 gallons, how many did Marathon sell? If I were to give you a value for S, such as 20, what would the solution be? What does it represent? 3 minutes Facilitation notes: So what questions did I ask myself to write the middle column? What do I know that the student understands? What am I unsure whether the student understands? What do I know the student does not understand? How did I come up with the questions in the 3rd column? What is the level of depth of this correct strategy? Can the student connect the “answer” to the problem?

DISCOURSE IN THE MATH CLASSROOM Monitoring
What do I know that the student understands? What am I unsure whether the student understands? What do I know the student does not understand? What is the level of depth for this strategy? What questions can I ask to determine where the understanding stops? 17 + 17 34 5 minutes Facilitation notes: Let’s practice with this student response. Can the student connect the “answer” to the problem?

DISCOURSE IN THE MATH CLASSROOM Anticipating & Monitoring
Strategy Work of specific students Sequence Model Expression, Variable Other Discuss the Who & What Table in the context of Speedy Mart Problem

Work out the task yourself.
DISCOURSE IN THE MATH CLASSROOM Anticipating & Monitoring: Let’s Practice Pick one lesson you brought with you and select the main task for that lesson. Work out the task yourself. Fill in as many rows of the Misconception Strategy Template as you can. Exchange with your partner, and offer other strategies. 20 minutes Facilitation notes: Depending on group size, consider mixing the group to have participants talk with someone new.

DISCOURSE IN THE MATH CLASSROOM Selecting
This is the process of choosing what and the who to focus on during the discussion. Why is selecting a critical step in the process? What should we think about as we are selecting student work? 5 minutes Facilitation notes: Have groups discuss and chart their responses to these questions Do a quick gallery walk so that all work is visible. Whole group – goal is to surface most important ideas What do you want to highlight? Purposefully select those that will advance mathematical ideas

DISCOURSE IN THE MATH CLASSROOM Selecting
STRATEGY WHO & WHAT ORDER Model Expression, Variable Other 3 minutes Facilitation notes: Use Speedy Mart student work to “think aloud” on how to select the strategies to be presented Make sure to identify the what and the why for each strategy Share a second selection possibility and have participants compare and contrast

DISCOURSE IN THE MATH CLASSROOM Selecting: Let’s Practice
Use the Misconception/Strategy template you completed for your lesson to select the student work you would like presented whole group, explain your thinking for each decision (mathematically), and be prepared to exchange papers with a partner or another group. 5 minutes Facilitation notes: Participants will be working individually but there will be times called out for partner feedback. Walk around during this time asking questions and giving feedback as well.

DISCOURSE IN THE MATH CLASSROOM VIII. Sequencing
In what order do you want to present the student examples? Do you want the most common / frequently occurring? Do you want to present misconceptions first? How will students share their work? Illustrate on board? Document camera? 5 minutes Facilitation notes: Discuss pros and cons for each question. Suggest not starting with “best” solutions and instead “working towards” these.

DISCOURSE IN THE MATH CLASSROOM Connecting
Often most challenging for us as teachers! Questions must make the mathematics explicit and transparent. Focus must be on mathematical meaning and relationships, tying together mathematical concepts and representations. 5 minutes

DISCOURSE IN THE MATH CLASSROOM Connecting
Generate questions to make the mathematics visible. Compare and contrast 2 or 3 students’ work. What are the mathematical relationships? What do parts of student’s work represent in the original problem? The solution? Work done in the past? 10 minutes Facilitation notes: Make sure to include: connections among correct solutions; connections between the solution methods and important grade level ideas

DISCOURSE IN THE MATH CLASSROOM VI. Applying to Your Practice
Find an appropriate task within your lesson. Determine the cognitive demand of the task and justify your reasoning. Use the Misconception/Strategy Template to develop questions and add some additional student work (correct or incorrect). Select the student work to be presented—explain your reasoning. Sequence student work—explain reasoning. Connect the responses and math ideas. Present to the group! 30 minutes Facilitation notes: Participants work in small groups. IF GROUP IS > THAN 15: Two or three individuals will present to the whole group. IF GROUP IS< THAN 15: Small groups will work together and share within their groups.

DISCOURSE IN THE MATH CLASSROOM Reflections on the Day: Choose 1
Reflect back on the day; identify something that had an impact on you or will have an impact on your practice and write about it. What was the most challenging concept for you today? Is there a topic you would have appreciated more time with? Less time? 5 Minutes Facilitation notes: Have participants record their thinking on a sticky note which they will leave at a designated location.

DISCOURSE IN THE MATH CLASSROOM Sources
Slides 1–2; 26–41 Adapted from Smith, Hughes, Engle, and Stein. “Orchestrating Discussions.” Mathematics Teaching in the Middle School 3 (May 2009): 548–56. Slides 22–24 Adapted from Smith, Margaret Schwan, and Mary Kay Stein. “Selecting and Creating Mathematical Tasks: From Research to Practice.” Mathematics Teaching in the Middle School 3 (February 1998): 344–50. Slides Adapted from Smith & Stein (2013), 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics. Slides Adapted from Smith, Stein, Arbaugh, Brown, & Mossgrove. “Characterizing the Cognitive Demands of Mathematical Tasks: A Sorting Activity.” In Professional Development Guidebook for Perspectives on the Teaching of Mathematics: Companion to the Sixty-sixth Yearbook, p. 45–72. Reston, VA: National Council of Teachers of Mathematics.

DISCOURSE IN THE MATH CLASSROOM Image References
Slide # Name and Photographer 5 “Welcome” by Janelle (Flickr) 31 “Snack-time” by little-nutbrown-hare (Flickr) 39 52 “Classroom 2” by misskprimary (Flickr) NOTE that the slide numbers need to be updated once the slides are finalized.

Discourse in the Math Classroom

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Discourse in the Math Classroom

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