Transcript Mathematical Tasks: The Study of Equivalence November 18
Supporting Rigorous Mathematics Teaching and Learning
Using Academically Productive Talk Moves: Orchestrating a Focused Discussion
Tennessee Department of Education Middle School Mathematics Grade 8 © 2013 UNIVERSITY OF PITTSBURGH
Rationale
Mathematics reform calls for teachers to engage students in discussing, explaining, and justifying their ideas. Although teachers are asked to use students’ ideas as the basis for instruction, they must also keep in mind the mathematics that the class is expected to explore (Sherin, 2000, p. 125). By engaging in a high-level task and reflecting on ways in which the facilitator structured and supported the discussion of mathematical ideas, teachers will learn that they are responsible for orchestrating discussions in ways that make it possible for students to own their learning, as well as for the teacher to assess and advance student understanding of knowledge and mathematical reasoning. 2
Session Goals
• • • Participants will: learn about Accountable Talk ® features and indicators and consider the benefit of all being present in a lesson; learn that there are specific moves related to each of the talk features that help to develop a discourse culture; and consider the importance of the four key moves of ensuring productive discussion (marking, recapping, challenging, and revoicing).
Accountable Talk ® is a registered trademark of the University of Pittsburgh © 2013 UNIVERSITY OF PITTSBURGH 3
Overview of Activities
• • • Participants will: review the
Accountable Talk
features and indicators; identify and discuss
Accountable Talk
moves in a video; and align CCSS and essential understandings (EUs) to a task and zoom in for a more specific look at key moves for engaging in productive talk (marking, recapping, challenging, and revoicing).
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Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework
TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning
Stein, Smith, Henningsen, & Silver, 2000 5
Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework
TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning
Stein, Smith, Henningsen, & Silver, 2000
Setting Goals Selecting Tasks Anticipating Student Responses
Accountable Talk ® is a registered trademark of the University of Pittsburgh • • • • •
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
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Accountable Talk
Features and Indicators
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Accountable Talk Discussion
• • Study the
Accountable Talk
features and indicators. Turn and talk with your partner about what you would expect teachers and students to be saying during an
Accountable Talk
discussion so that the discussion is accountable to: − the learning community ; − accurate, relevant knowledge ; and − standards of rigorous thinking .
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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.
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Solving and Discussing the Cognitive Demand of the Calling Plans 2 Task
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The Structure and Routines of a Lesson
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/ Small Group Problem Solving
1. Generate and Compare Solutions 2. Assess and Advance Student Learning
Share, Discuss, and Analyze Phase of the Lesson
1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write © 2013 UNIVERSITY OF PITTSBURGH
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.
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Engaging in a Lesson: The Calling Plans 2 Task
• Solve the task. • Discuss your solutions with your peers. • Attempt to engage in an
Accountable Talk
discussion when discussing the solutions. Assign one person in the group to be the observer. This person will be responsible for reporting some of the ways in which the group is accountable to: − the learning community ; − accurate, relevant knowledge ; and − standards of rigorous thinking .
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Engaging in a Lesson: The Calling Plans 2 Task
Long-distance Company A charges a base rate of $5 per month plus 4 cents per minute that you are on the phone. Long-distance Company B charges a base rate of only $2 per month but they charge you 10 cents per minute used.
Create a phone plan, Company C, that costs the same as Companies A and B at 50 minutes, but has a lower monthly fee than either of the plans.
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Reflecting on Our Engagement in the Lesson
The observer should share some observations about the group’s engagement in an
Accountable Talk
discussion.
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Reflecting on Our Engagement in the Lesson
• In what ways did small groups engage in an
Accountable Talk
discussion? • In what ways did we engage in an
Accountable Talk
discussion during the group discussion of the solutions?
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Aligning the CCSS to the Calling Plans 2 Task
• Study the Grade 8 CCSS for Mathematical Content within the Expressions and Equations domain. Which standards are students expected to demonstrate when solving the task?
• Identify the CCSS for Mathematical Practice required by the written task. © 2013 UNIVERSITY OF PITTSBURGH 16
The CCSS for Mathematical Content − Grade 8
Expressions and Equations 8.EE
Analyze and solve linear equations and pairs of simultaneous linear equations.
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.
17 Common Core State Standards, 2010, p. 55, NGA Center/CCSSO
The CCSS for Mathematical Content − Grade 8
Functions 8.F
Define, evaluate, and compare functions.
8.F.A.3
Interpret the equation
y = mx + b
as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4), and (3,9), which are not on a straight line.
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. Common Core State Standards, 2010, p. 55, NGA Center/CCSSO 18
The CCSS for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 19
Determining the Cognitive Demand of the Task: The Calling Plans 2 Task
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Determining the Cognitive Demand of the Task
Refer to the Mathematical Task Analysis Guide.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A., 2000. Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.
How would you characterize the Calling Plans 2 Task in terms of its cognitive demand?
(Refer to the indicators on the Task Analysis Guide.)
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The Mathematical Task Analysis Guide
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press .
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The Calling Plans 2 Task: A Doing Mathematics Task
• Requires complex and non-algorithmic thinking (i.e., there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked-out example).
• Requires students to explore and to understand the nature of mathematical concepts, processes, or relationships.
• Demands self-monitoring or self regulation of one’s own cognitive processes.
• Requires students to access relevant knowledge and experiences and make appropriate use of them in working through the task.
• Requires students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
• Requires considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.
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Accountable Talk Moves
© 2013 UNIVERSITY OF PITTSBURGH
The Structure and Routines of a Lesson
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/ Small-Group Problem Solving
1. Generate and Compare Solutions 2. Assess and Advance Student Learning
Share, Discuss, and Analyze Phase of the Lesson
1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write © 2013 UNIVERSITY OF PITTSBURGH
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.
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Accountable Talk Moves
Examine the ways in which the moves are grouped based on how they: • • • support accountability to the learning community; support accountability to knowledge; and support accountability to rigorous thinking. Consider: In what ways are the
Accountable Talk
categories similar? Different?
Why do you think we need a category called “To Ensure Purposeful, Coherent, and Productive Group Discussion”?
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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.
© 2013 UNIVERSITY OF PITTSBURGH 27
Accountable Talk Moves
Talk Move Function To Ensure Purposeful, Coherent, and Productive Group Discussion
Marking Challenging Revoicing Direct
attention
to the value and importance of a student’s contribution.
Redirect a question back to the students or use students’ contributions as a source for further challenge or query.
Align a student’s explanation with content or connect two or more contributions with the goal of advancing the discussion of the content. Recapping Keeping the Channels Open Keeping Everyone Together Linking Contributions Make public in a concise, coherent form, the group’s achievement at creating a shared understanding of the phenomenon under discussion.
To Support Accountability to Community
Ensure that students can hear each other, and remind them that they must hear what others have said.
Ensure that everyone not only heard, but also understood, what a speaker said.
Make explicit the relationship between a new contribution and what has gone before.
Verifying and Clarifying Revoice a student’s contribution, thereby helping both speakers and listeners to engage more profitably in the conversation.
Example
That’s an important point.
Let me challenge you: Is that always true?
S: 4 + 4 + 4.
You said three groups of four. Let me put these ideas all together.
What have we discovered?
Say that again and louder.
Can someone repeat what was just said?
Can someone add on to what was said?
Did everyone hear that?
Does anyone have a similar idea?
Do you agree or disagree with what was said?
Your idea sounds similar to his idea. So are you saying..?
Can you say more? Who understood what was said?
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Accountable Talk Moves
(continued)
Pressing for Accuracy Building on Prior Knowledge
To Support Accountability to Knowledge
Hold students accountable for the accuracy, credibility, and clarity of their contributions.
Tie a current contribution back to knowledge accumulated by the class at a previous time.
Pressing for Reasoning
To Support Accountability to Rigorous Thinking
Elicit evidence to establish what contribution a student’s utterance is intended to make within the group’s larger enterprise.
Expanding Reasoning Open up extra time and space in the conversation for student reasoning.
Why does that happen?
Someone give me the term for that.
What have we learned in the past that links with this?
Say why this works.
What does this mean?
Who can make a claim and then tell us what their claim means?
Does the idea work if I change the context? Use bigger numbers?
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Reflection Question
As you watch the short video segment, consider what students are learning and where you might focus the discussion in order to discuss mathematical ideas listed in the CCSS.
Identify: • the specific
Accountable Talk
moves used by the teacher; and • the purpose that the moves served.
Mark times during the lesson when you would call the lesson academically rigorous.
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The Calling Plans 2 Lesson Context
• Teacher: Elizabeth Brovey • School: Pittsburgh Classical Academy • District: Pittsburgh Public Schools • Principal: Ms. Merlo • Grade Level: 7 th Grade The students in the video episode are in a mainstream mathematics classroom in the Pittsburgh Public Schools. The students have just discussed one solution to the Calling Plans 2 Task,
y
= 1 + .12
x.
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Norms for Collaborative Study
The goal of all conversations about episodes of teaching (or artifacts of practice in general) is to advance our own learning, not to “fix” the practice of others.
In order to achieve this goal, the facilitator chooses a lens to frame what you look at and to what you pay attention. Use the
Accountable Talk
features and indicators when viewing the lesson. During this work, we: • agree to analyze the episode or artifact from the identified perspective; • cite specific examples during the discussion that provide evidence of a particular claim; • • listen to and build on others’ ideas; and use language that is respectful of those in the video and in the group.
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The Calling Plans 2 Task
Long-distance Company A charges a base rate of $5 per month plus 4 cents per minute that you are on the phone. Long-distance Company B charges a base rate of only $2 per month but they charge you 10 cents per minute used.
Create a phone plan, Company C, that costs the same as Companies A and B at 50 minutes, but has a lower monthly fee than either of the plans.
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Reflecting on the Accountable Talk Discussion
Step back from the discussion. What are some patterns that you notice?
What mathematical ideas does the teacher want students to discover and discuss? © 2013 UNIVERSITY OF PITTSBURGH 34
Essential Understandings
Study the essential understandings the teacher considered in preparation for the Share, Discuss, and Analyze phase of the lesson.
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Essential Understandings
Essential Understanding Solutions Make Equations True
The solution of a system of linear equations in two variables is the ordered pair of pairs (
x, y
) that satisfies all of the equations in the system. The solution of a system of two or more linear equations can be represented algebraically, graphically, in a table, and in a context.
CCSS
8.EE.C.8
Solving Systems Graphically
The solution of a system of two or more linear equations is represented graphically by the intersection of the lines representing the solution sets of each of the inequalities in the system because the point(s) at that intersection satisfy all of the equations in the system. 8.EE.C.8a
Two distinct lines will intersect if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes will have one solution.
Linear Models
A linear relationship can be represented by an equation of the form
y = mx + b
where
m
and
b
have a regular and predictable meaning in the context, table, equation, and graph. 8.F.A.3
Connecting Representations
The graph of a linear relationship is a line whose coordinates form the solution set to the associated linear equation and establishes the relationship between the variables in the context.
© 2013 UNIVERSITY OF PITTSBURGH 8.F.B.4
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Characteristics of an Academically Rigorous Lesson
This task is a cognitively demanding task; however, it may not necessarily end up being an academically rigorous task. What do we mean by this statement? © 2013 UNIVERSITY OF PITTSBURGH 37
Academic Rigor in a Thinking Curriculum
The principle of learning,
Academic Rigor in a Thinking Curriculum,
consists of three features: • • •
A Knowledge Core High-Thinking Demand Active Use of Knowledge
In order to determine if a lesson has been academically rigorous, we have to determine the degree to which student learning is advanced by the lesson What do we have to hear and see in order to determine if the lesson was academically rigorous? © 2013 UNIVERSITY OF PITTSBURGH 38
Essential Understandings
Essential Understanding Solutions Make Equations True
The solution of a system of linear equations in two variables is the ordered pair of pairs (
x, y
) that satisfies all of the equations in the system.
CCSS
8.EE.C.8
The solution of a system of two or more linear equations can be represented algebraically, graphically, in a table, and in a context.
Solving Systems Graphically
The solution of a system of two or more linear equations is represented graphically by the intersection of the lines representing the solution sets of each of the inequalities in the system because the point(s) at that intersection satisfy all of the equations in the system. 8.EE.C.8a
Two distinct lines will intersect if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes will have one solution.
Linear Models
A linear relationship can be represented by an equation of the form
y = mx + b
where
m
and
b
have a regular and predictable meaning in the context, table, equation, and graph. 8.F.A.3
Connecting Representations
The graph of a linear relationship is a line whose coordinates form the solution set to the associated linear equation and establishes the relationship between the variables in the context.
© 2013 UNIVERSITY OF PITTSBURGH 8.F.B.4
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Five Different Representations of a Function
Language Context Table Graph Equation Van De Walle, 2004, p. 440 40
Focusing on Key Accountable Talk Moves The Calling Plans 2 Task
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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.
© 2013 UNIVERSITY OF PITTSBURGH 42
Accountable Talk Moves
Talk Move
Marking Challenging Revoicing Recapping Keeping the Channels Open Keeping Everyone Together Linking Contributions
Function To Ensure Purposeful, Coherent, and Productive Group Discussion
Direct
attention
to the value and importance of a student’s contribution.
Redirect a question back to the students or use students’ contributions as a source for further challenge or query.
Align a student’s explanation with content or connect two or more contributions with the goal of advancing the discussion of the content. Make public in a concise, coherent form, the group’s achievement at creating a shared understanding of the phenomenon under discussion.
To Support Accountability to Community
Ensure that students can hear each other, and remind them that they must hear what others have said.
Ensure that everyone not only heard, but also understood, what a speaker said.
Make explicit the relationship between a new contribution and what has gone before.
Verifying and Clarifying Revoice a student’s contribution, thereby helping both speakers and listeners to engage more profitably in the conversation.
© 2013 UNIVERSITY OF PITTSBURGH
Example
That’s an important point.
Let me challenge you: Is that always true?
S: 4 + 4 + 4.
You said three groups of four. Let me put these ideas all together.
What have we discovered?
Say that again and louder.
Can someone repeat what was just said?
Can someone add on to what was said?
Did everyone hear that?
Does anyone have a similar idea?
Do you agree or disagree with what was said?
Your idea sounds similar to his idea. So are you saying..?
Can you say more? Who understood what was said?
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Accountable Talk Moves
(continued)
Pressing for Accuracy Building on Prior Knowledge Pressing for Reasoning
To Support Accountability to Knowledge
Hold students accountable for the accuracy, credibility, and clarity of their contributions.
Tie a current contribution back to knowledge accumulated by the class at a previous time.
To Support Accountability to Rigorous Thinking
Elicit evidence to establish what contribution a student’s utterance is intended to make within the group’s larger enterprise.
Expanding Reasoning Open up extra time and space in the conversation for student reasoning.
Why does that happen?
Someone give me the term for that.
What have we learned in the past that links with this?
Say why this works.
What does this mean?
Who can make a claim and then tell us what their claim means?
Does the idea work if I change the context? Use bigger numbers?
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Focusing on Accountable Talk Moves
Read the description of each move and study the example that has been provided for each move. What is distinct about each of the moves?
• revoice student contributions; • mark significant contributions; • challenge with a counter-example; or • recap the components of the lesson.
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Revoicing
• Extend a student’s contribution.
• Connect a student’s contribution to the text or to other students’ contributions.
Align content with an explanation.
Add clarity to a contribution.
Link student contributions to accurate mathematical vocabulary. Connect two or more contributions to advance the lesson.
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An Example of Revoicing
S: Like, .11, .12, .13, .14. T: So —put that into words. What’s happening to the rate when you do that?
S: It’s going up by 1. T: It’s going up by—it’s increasing by 1 cent, right?
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Marking
Explicitly talk about an idea.
• Highlight features that are unique to a situation.
• Draw attention to an idea or to alternative ideas.
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An Example of Marking
S: Um, well, we were looking —that it had to be—the monthly rate had to be cheaper than the other two and the only other one it could be was $1. So, you had to automatically set that at $1. And, um, I — T: I want to interrupt you for a second. Does anybody have anything to say about that? Just a quick comment and then we are going to go back to her. Jake?
S: It doesn’t have to be $1. It’s part of the rule. T: You could have, like, $1.50. Keep going; how did you get to the rest of this?
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Recapping
Summarize or retell.
• Make explicit the large idea.
• Provide students with a holistic view of the concept.
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An Example of Recapping
T: So, she said she decided on a dollar, on a head start, and then she tried some numbers for the rate until — until what?
S: It had to be the same cost at 50 minutes.
T: Until she found one that had a cost of $7 at 50 minutes. And that equation was —what was it?
S:
y
= 1 + .12
x
T:
y
= 1 + .12
x
and she knew this worked, because she tested it at 50 minutes. © 2013 UNIVERSITY OF PITTSBURGH 51
Challenging
Redirect a question back to the students, or use students’ contributions as a source for further challenge or query. • Share a counter-example and ask students to compare problems.
• Question the meaning of the math concept.
© 2013 UNIVERSITY OF PITTSBURGH
An Example of Challenging
T: I want to know why it worked. Why do all of those work? Why are they all working?
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Appropriation
The process of appropriation is reciprocal and sequential. • If appropriation takes place, the child transforms the new knowledge or skill into an action in a new and gradually understood activity.
• What would this mean with respect to classroom discourse? What should we expect to happen in the classroom?
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Orchestrating Discussions
Read the segments of transcript from the lesson.
Decide if examples 1 – 3 illustrate marking, recapping, challenging, or revoicing.
Be prepared to share your rationale for identifying a particular discussion move. Write the next discussion move for examples 4 and 5 and be prepared to share your move and your rationale for writing the move. © 2013 UNIVERSITY OF PITTSBURGH 55
Reflecting on Talk Moves
What have you learned about: • marking; • recapping; • challenging; and • revoicing?
Why are these moves important in lessons? © 2013 UNIVERSITY OF PITTSBURGH 56
Application to Practice
• What will you keep in mind when attempting to use
Accountable Talk
moves during a lesson? What role does talk play? • What does it take to maintain the demands of a cognitively demanding task during the lesson so that you have a rigorous mathematics lesson?
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