Supporting Rigorous Mathematics Teaching and Learning
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Transcript of Supporting Rigorous Mathematics Teaching and Learning
© 2013 UNIVERSITY OF PITTSBURGH
Supporting Rigorous Mathematics Teaching and Learning
Selecting and Sequencing Based on Essential Understandings
Tennessee Department of EducationMiddle School MathematicsGrade 6
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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.
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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.
3Accountable 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
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“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
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TASKS
as they appear in curricular/ instructional materials
TASKS
as set up by the teachers
TASKS
as implemented by students
Student Learning
The Mathematical Tasks Framework
Stein, Smith, Henningsen, & Silver, 2000
Linking to Research/Literature: The QUASAR Project
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TASKS
as they appear in curricular/ instructional materials
TASKS
as set up by the teachers
TASKS
as implemented by students
Student Learning
The Mathematical Tasks Framework
Stein, Smith, Henningsen, & Silver, 2000
Linking to Research/Literature: The QUASAR Project
Setting GoalsSelecting TasksAnticipating 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 7
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Identify Goals for Instructionand Select an Appropriate Task
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The Structure and Routines of a Lesson
The Explore Phase/Private Work TimeGenerate Solutions
The Explore Phase/Small Group Problem Solving
1. Generate and Compare Solutions2. Assess and Advance Student Learning
Share, Discuss, and Analyze Phase of the Lesson1. Share and Model2. Compare Solutions3. 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 PATHCOMPARE: Students discuss similarities and difference between solution paths.FOCUS: Discuss the meaning of mathematical ideas in each representationREFLECT: Engage students in a Quick Write or a discussion of the process.
Set Up the TaskSet Up of the Task
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Contextualizing Our Work TogetherImagine 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 represent rational numbers on a number line. (6.NS.C.6, C.6a, C.6c)
• 20% of the students need additional work understanding when values in context should be represented with negative numbers (6.NS.C.5). These students also need opportunities to struggle with and make sense of the problem. (MP1)
• 5% of the students are consistently able to represent positive and negative rational numbers as points on the number line and are working on understanding absolute value as distance from 0. (6.NS.C.7c)
• 5% of the students struggle to pay attention and their understanding of numbers and operations is two grade levels below sixth grade.
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The CCSS for Mathematics: Grade 6The Number System 6.NS
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.
Common Core State Standards, 2010, p. 43, NGA Center/CCSSO 11
The CCSS for Mathematics: Grade 6The Number System 6.NS
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.
Common Core State Standards, 2010, p. 43, NGA Center/CCSSO 12
Standards for Mathematical Practice Related to the Task
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 13
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Identify Goals: Solving the Task(Small Group Discussion)
Solve the task.
Discuss the possible solution paths to the task.
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Hiking Task
Dia’Monique and Yanely picnicked together. Then Dia’Monique hiked 17 miles. Yanely hiked 14 miles in the opposite direction. What is their distance from each other? Draw a picture to show their distance from each other.
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Identify Goals Related to the Task(Whole Group Discussion)
Does the task provide opportunities for students to access the Standards for Mathematical Content and Standards for Mathematical Practice that we have identified for student learning?
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Identify Goals: Essential Understandings (Whole Group Discussion)
Study the essential understandings associated with the Number System Common Core Standards.
Which of the essential understandings are the goals of the Hiking Task?
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Essential UnderstandingPositive and Negative Numbers Can be Used to Represent Real-World QuantitiesPositive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). Rational Numbers Can be Located on a Number LineAny rational number can be modeled using a point on the number line because the real number line extends infinitely in the positive and negative directions. The sign and the magnitude of the number determine the location of the point.Absolute Value is a Measure of a Number’s Distance From 0The absolute value of a number is the number’s magnitude or distance from 0. If two rational numbers differ only by their signs, they have the same absolute value because they are the same distance from zero. Distance Between Two Values on a Number Line Can be Determined Using ArithmeticThe distance between a positive and negative value on a number line is equal to the sum of their absolute values because they are located on opposite sides of zero.
Essential Understandings (Small Group Discussion)
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Selecting and Sequencing Student Work for the
Share, Discuss, and Analyze Phase of the Lesson
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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.
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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.
1. Which pieces of student work will allow you to address the essential understanding?
2. How will you sequence the student’s work that you have selected? Be prepared to share your rationale.
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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 RationalePositive and Negative Numbers Can be Used to Represent Real-World QuantitiesRational Numbers Can be Located on a Number LineAbsolute Value is a Measure of a Number’s Distance From 0Distance Between Two Values on a Number Line Can be Determined Using Arithmetic
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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.
Positive and Negative Numbers Can be Used to Represent Real-World QuantitiesPositive numbers represent…Rational Numbers Can be Located on a Number LineAny rational number…Absolute Value is a Measure of a Number’s Distance From 0The absolute value…Distance Between Two Values on a Number Line Can be Determined Using ArithmeticThe distance between…
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Group A
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Group B
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Group C
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Group D
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Group E
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Group F
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Group G
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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.
Positive and Negative Numbers Can be Used to Represent Real-World QuantitiesPositive numbers represent…Rational Numbers Can be Located on a Number LineAny rational number…Absolute Value is a Measure of a Number’s Distance From 0The absolute value…Distance Between Two Values on a Number Line Can be Determined Using ArithmeticThe distance between…
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Academic Rigor in a Thinking Curriculum
The Share, Discuss, and Analyze Phase of the Lesson
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Academic Rigor In a Thinking Curriculum
A teacher must always be assessing and advancing student learning.
A lesson is academically rigorous if student learning related to the essential understanding is advanced in the lesson.
Accountable Talk discussion is the means by which teachers can find out what students know or do not know and advance them to the goals of the lesson.
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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.
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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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Accountable Talk MovesTalk Move 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: 4 + 4 + 4.
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 CommunityKeeping 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? 36
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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 toRigorous 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?
Accountable Talk Moves (continued)
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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.
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An Example: Accountable Talk Discussion The Focus Essential Understanding Positive and Negative Numbers Can be Used to Represent Real-World QuantitiesPositive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). Group B Group G
• Group B, how did your group determine where to place the point for each girl?• Who understood what she said about the opposite directions? (Community)• Can you say back what he said about the sign of the numbers? (Community)• Signed numbers can be used to represent positions in opposite directions from
a set point. (Revoicing)• Group G didn’t use negative numbers. How does your model represent the
problem situation? (Knowledge)• What do the 14 and 17 in Group G’s model represent? Is it possible to walk a
negative distance? What does the -14 represent? (Rigor)39
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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 HookCompare 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-ExampleIf 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).
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An Example: Accountable Talk Discussion The Focus Essential Understanding Positive and Negative Numbers Can be Used to Represent Real-World QuantitiesPositive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). Group B Group G
• You can’t walk negative 14 miles, can you? Is it okay to use a negative number to represent Yanely? Do you have to use negative numbers? (Hook)
• Group B, how did your group determine where to place the point for each girl?• Who understood what she said about the opposite directions? (Community)• Can you say back what he said about the sign of the numbers? (Community)• Signed numbers can be used to represent positions in opposite directions from a set
point. (Revoicing) • Group G didn’t use negative numbers. How does your model represent the problem
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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.
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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.
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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.”
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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?
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Zooming In on Problematizing(Whole Group Discussion)
Do a Gallery Walk. Read each others’ problematizing “hook.”
What do you notice about the use of hooks? What role do “hooks” play in the lesson?
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Step Back and Application to Our Work
What have you learned today that you will apply when planning or teaching in your classroom?
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Summary of Our Planning ProcessParticipants:
• identify goals for instruction;– Align Standards for Mathematical Content and
Standards for Mathematical Practice with a task.– Select essential understandings that relate to the
Standards for Mathematical Content and Standards for Mathematical Practice.
• prepare for the Share, Discuss, and Analyze Phase of the lesson.– Analyze and select student work that can be used to
discuss essential understandings of mathematics. – Learn methods of problematizing the mathematics
in the lesson.
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