Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Strategies for Scaffolding Student
Understanding: Academically Productive Talk
and the Use of Representations
Tennessee Department of Education
High School Mathematics
Algebra 1
© 2013 UNIVERSITY OF PITTSBURGH
Rationale
Teachers provoke students’ reasoning about
mathematics through the tasks they provide and the
questions they ask. (NCTM, 1991) Asking questions
that reveal students’ knowledge about mathematics
allows teachers to design instruction that responds to
and builds on this knowledge. (NCTM, 2000) Questions
are one of the only tools teachers have for finding out
what students are thinking. (Michaels, 2005)
Today, by analyzing a classroom discussion, teachers
will study and reflect on ways in which Accountable
Talk® (AT) moves and the use of representations
support student learning and help teachers to maintain
the cognitive demand of a task.
Accountable talk® is a registered trademark of the University of Pittsburgh.
Session Goals
Participants will learn about:
• Accountable Talk moves to support the
development of community, knowledge, and
rigorous thinking;
• Accountable Talk moves that ensure a productive
and coherent discussion and consider why moves
in this category are critical; and
• representations as a means of scaffolding student
learning.
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Overview of Activities
Participants will:
• analyze and discuss Accountable Talk moves;
• engage in and reflect on a lesson in relationship to
the CCSS;
• analyze classroom discourse to determine the
Accountable Talk moves used by the teacher and
the benefit to student learning;
• design and enact a lesson, making use of the
Accountable Talk moves; and
• learn and apply a set of scaffolding strategies that
make use of the representations.
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Review the
Accountable Talk Features
and Indicators
Learn Moves Associated With
the Accountable Talk Features
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Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
The Structure and Routines of a Lesson
Set Up
Up the
of the
Task
Set
Task
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
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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.
Accountable Talk Discussion
• Review the Accountable Talk features and indicators.
• Turn and Talk with your partner about what you recall
about each of the Accountable Talk features.
- Accountability to the learning community
- Accountability to accurate, relevant knowledge
- Accountability to discipline-specific 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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Accountable Talk Moves
Consider:
• In what ways are the Accountable Talk moves
different in each of the categories?
− Support Accountability to Community
− Support Accountability to Knowledge
− Support Accountability to Rigorous Thinking
• There is a fourth category called, “To Ensure
Purposeful, Coherent, and Productive Group
Discussion.” Why do you think we need the set of
moves in this category?
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Moves
Talk Move
Function
Example
To Ensure Purposeful, Coherent,
and Productive Group Discussion
Marking
Direct attention to the value and
importance of a student’s
contribution.
It is important to say describe to
compare the size of the pieces and
then to look at how many pieces of
that size.
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.
You said 3; yes, there are three
columns and each column is of the
whole.
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?
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Moves (continued)
Talk Move
Function
Example
To Support Accountability to
Community
Keeping the
Channels Open
Ensure that students can hear
each other, and remind them that
they must hear what others have
said.
Say that again and louder.
Can someone repeat what was just
said?
Keeping
Everyone
Together
Ensure that everyone not only
heard, but also understood, what
a speaker said.
Can someone add on to what was
said?
Did everyone hear that?
Linking
Contributions
Make explicit the relationship
between a new contribution and
what has gone before.
Does anyone have a similar idea?
Do you agree or disagree with
what was said?
Your idea sounds similar to his
idea.
Verifying and
Clarifying
Revoice a student’s contribution,
thereby helping both speakers
and listeners to engage more
profitably in the conversation.
So are you saying...?
Can you say more?
Who understood what was said?
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Moves (continued)
Talk Move
Function
Example
To Support Accountability to
Knowledge
Pressing for
Accuracy
Hold students accountable for the
accuracy, credibility, and clarity of
their contributions.
Why does that happen?
Someone give me the term for
that.
Building on
Prior
Knowledge
Tie a current contribution back to
knowledge accumulated by the
class at a previous time.
What have we learned in the
past that links with this?
To Support Accountability to
Rigorous Thinking
Pressing for
Reasoning
Elicit evidence to establish what
contribution a student’s utterance
is intended to make within the
group’s larger enterprise.
Say why this works.
What does this mean?
Who can make a claim and then
tell us what their claim means?
Expanding
Reasoning
Open up extra time and space in
the conversation for student
reasoning.
Does the idea work if I change
the context? Use bigger
numbers?
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Five Representations of Mathematical Ideas
What role do the representations play in a discussion?
Pictures
Written
Manipulative
Models
Symbols
Real-world
Situations
Oral & Written
Language
Modified from Van De Walle, 2004, p. 30
Five Different Representations of a Function
What role do the representations play in a discussion?
Language
Context
Table
Graph
Van De Walle, 2004, p. 440
Equation
Engage In and Reflect On a Lesson
Bike and Truck Task
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Bike and Truck Task
Distance from start of road (in feet)
A bicycle traveling at a steady rate and a truck are
moving along a road in the same direction. The graph
below shows their positions as a function of time. Let
B(t) represent the bicycle’s distance and K(t) represent
the truck’s distance.
Time (in seconds)
© 2013 UNIVERSITY OF PITTSBURGH
Bike and Truck Task
1. Label the graphs appropriately with B(t) and K(t).
Explain how you made your decision.
2. Describe the movement of the truck. Explain how
you used the values of B(t) and K(t) to make
decisions about your description.
3. Which vehicle was first to reach 300 feet from the
start of the road? How can you use the domain
and/or range to determine which vehicle was the first
to reach 300 feet? Explain your reasoning in words.
4. Jack claims that the average rate of change for both
the bicycle and the truck was the same in the first 17
seconds of travel. Explain why you agree or disagree
with Jack and why.
© 2013 UNIVERSITY OF PITTSBURGH
The Cognitive Demand of the Task
Why is this considered to be a cognitively demanding
task?
© 2013 UNIVERSITY OF PITTSBURGH
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.
The Common Core State Standards
(CCSS)
Solve the task.
Examine the CCSS for Mathematics.
• Which CCSS for Mathematical Content will
students discuss when solving the task?
• Which CCSS for Mathematical Practice will
students use when solving and discussing the
task?
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The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Creating Equations★
(A–CED)
Create equations that describe numbers or relationships.
A-CED.A.1 Create equations and inequalities in one variable and use them to
solve problems. Include equations arising from linear and quadratic
functions, and simple rational and exponential functions.
A-CED.A.2 Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels
and scales.
A-CED.A.3 Represent constraints by equations or inequalities, and by systems
of equations and/or inequalities, and interpret solutions as viable or
nonviable options in a modeling context. For example, represent
inequalities describing nutritional and cost constraints on
combinations of different foods.
A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. For example, rearrange
Ohm’s law V = IR to highlight resistance R.
★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling
standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a
star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Reasoning with Equations and Inequalities
(A–REI)
Solve equations and inequalities in one variable.
A-REI.B.3
Solve linear equations and inequalities in one variable,
including equations with coefficients represented by letters.
A-REI.B.4
Solve quadratic equations in one variable.
A-REI.B.4a Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q
that has the same solutions. Derive the quadratic formula from
this form.
A-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49),
taking square roots, completing the square, the quadratic
formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a
and b.
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Reasoning with Equations and Inequalities
(A–REI)
Represent and solve equations and inequalities graphically.
A-REI.D.10
A-REI.D.11
A-REI.D.12
Understand that the graph of an equation in two variables is the set of
all its solutions plotted in the coordinate plane, often forming a curve
(which could be a line).
Explain why the x-coordinates of the points where the graphs of the
equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.★
Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and
graph the solution set to a system of linear inequalities in two
variables as the intersection of the corresponding half-planes.
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling
standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star,
each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Functions
Interpreting Functions
(F–IF)
Interpret functions that arise in applications in terms of the context.
F-IF.B.4
For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.★
F-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the
positive integers would be an appropriate domain for the function.★
F-IF.B.6
Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is
marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 69, NGA Center/CCSSO
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
Analyzing a Lesson: Lesson Context
Teacher:
Grade Level:
Shalunda Shackelford
Algebra 1
School:
School District:
Tyner Academy
Hamilton County School District
The students and the teacher in this school have been
working to make sense of the Common Core State
Standards for the past two years.
The teacher is working on using the Accountable Talk
moves and making sure she targets the mathematics
standards in very deliberate ways during the lesson.
© 2013 UNIVERSITY OF PITTSBURGH
27
Bike and Truck Task
Distance from start of road (in feet)
A bicycle traveling at a steady rate and a truck are
moving along a road in the same direction. The graph
below shows their positions as a function of time. Let
B(t) represent the bicycle’s distance and K(t) represent
the truck’s distance.
Time (in seconds)
© 2013 UNIVERSITY OF PITTSBURGH
Bike and Truck Task
1. Label the graphs appropriately with B(t) and K(t).
Explain how you made your decision.
2. Describe the movement of the truck. Explain how
you used the values of B(t) and K(t) to make
decisions about your description.
3. Which vehicle was first to reach 300 feet from the
start of the road? How can you use the domain
and/or range to determine which vehicle was the first
to reach 300 feet? Explain your reasoning in words.
4. Jack claims that the average rate of change for both
the bicycle and the truck was the same in the first 17
seconds of travel. Explain why you agree or disagree
with Jack and why.
© 2013 UNIVERSITY OF PITTSBURGH
Instructional Goals
Shalunda’s instructional goals for the lesson are:
• students will use the language of change and rate
of change (increasing, decreasing, constant,
relative maximum or minimum) to describe how two
quantities vary together over a range of possible
values; and
• students will describe how one quantity changes
with respect to another.
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Reflection Question
(Small Group Discussion)
As you watch the video segment, consider what
students are learning about mathematics.
Name the moves used by the teacher and the purpose
that the moves served.
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Reflecting on the Accountable Talk
Discussion
(Whole Group 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?
• How does talk scaffold student learning?
© 2013 UNIVERSITY OF PITTSBURGH
Five Representations of Mathematical Ideas
What role did tools or representations play in scaffolding student
learning?
Pictures
Manipulative
Written
Models
Symbols
Real-world
Situations
Oral & Written
Language
Modified from Van De Walle, 2004, p. 30
Five Different Representations of a Function
What role did tools or representations play in scaffolding
student learning?
Language
Context
Table
Graph
Van De Walle, 2004, p. 440
Equation
Giving it a Go: Planning for an
Accountable Talk Discussion of a
Mathematical Idea
• Identify a person who will be teaching the lesson to
others in your small group.
• Plan the lesson together. Anticipate student
responses.
• Write Accountable Talk questions/moves that the
teacher will ask students to advance their
understanding of a mathematical idea.
© 2013 UNIVERSITY OF PITTSBURGH
Bike and Truck Task
Distance from start of road (in feet)
A bicycle traveling at a steady rate and a truck are
moving along a road in the same direction. The graph
below shows their positions as a function of time. Let
B(t) represent the bicycle’s distance and K(t) represent
the truck’s distance.
Time (in seconds)
© 2013 UNIVERSITY OF PITTSBURGH
Bike and Truck Task
1. Label the graphs appropriately with B(t) and K(t).
Explain how you made your decision.
2. Describe the movement of the truck. Explain how
you used the values of B(t) and K(t) to make
decisions about your description.
3. Which vehicle was first to reach 300 feet from the
start of the road? How can you use the domain
and/or range to determine which vehicle was the first
to reach 300 feet? Explain your reasoning in words.
4. Jack claims that the average rate of change for both
the bicycle and the truck was the same in the first 17
seconds of travel. Explain why you agree or disagree
with Jack and why.
© 2013 UNIVERSITY OF PITTSBURGH
Focus of the Discussion
The average rate of change is the change in the
dependent variable over a specified interval in the
domain. Linear functions are the only family of functions
for which the average rate of change is the same on
every interval in the domain.
Plan to engage students in a discussion of average rate
of change. You may choose to use the student work
below to begin the discussion.
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Reflection: The Use of Accountable
Talk Moves and Tools to Scaffold
Student Learning
What have you learned?
© 2013 UNIVERSITY OF PITTSBURGH