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 2
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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
• Actively participate 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?
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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?
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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?
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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
Missing Function Task
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Missing Function Task
If h(x) = f(x) · g(x), what can you determine about g(x)
from the given table and graph? Explain your reasoning.
x
-2
-1
0
1
2
f(x)
0
1
2
3
4
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The Cognitive Demand of the Task
Why is this considered to be a cognitively demanding
task?
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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
.
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 – Number and Quantity
The Real Number System
N-RN
Extend the properties of exponents to rational exponents.
N-RN.A.1 Explain how the definition of the meaning of rational
exponents follows from extending the properties of
integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For
example, we define 51/3 to be the cube root of 5
because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must
equal 5.
N-RN.A.2 Rewrite expressions involving radicals and rational
exponents using the properties of exponents.
Common Core State Standards, 2010, p. 60, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Seeing Structure in Expressions
(A–SSE)
Write expressions in equivalent forms to solve problems.
A-SSE.B.3
Choose and produce an equivalent form of an expression to
reveal and explain properties of the quantity represented by
the expression.★
A-SSE.B.3c Use the properties of exponents to transform expressions for
exponential functions. For example the expression 1.15t can
be rewritten as (1.151/12)12t ͌ 1.01212t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.
A-SSE.B.4
Derive the formula for the sum of a finite geometric series
(when the common ratio is not 1), and use the formula to solve
problems. For example, calculate mortgage payments.★
★ 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. 64, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Arithmetic with Polynomials and Rational Expressions (A–APR)
Understand the relationship between zeros and factors of
polynomials.
A-APR.B.2
Know and apply the Remainder Theorem: For a polynomial
p(x) and a number a, the remainder on division by x – a is
p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A-APR.B.3
Identify zeros of polynomials when suitable factorizations are
available, and use the zeros to construct a rough graph of the
function defined by the polynomial.
Common Core State Standards, 2010, p. 64, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Functions
Building Functions
(F–BF)
Build a function that models a relationship between two quantities.
F-BF.A.1
F-BF.A.1a
Write a function that describes a relationship between two
quantities.★
Determine an explicit expression, a recursive process, or steps
for calculation from a context.
F-BF.A.1b
Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model.
F-BF.A.2
Write arithmetic and geometric sequences both recursively
and with an explicit formula, use them to model situations, and
translate between the two forms.★
★ 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. 70, 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:
Jamie Bassham
Algebra 2
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
Missing Function Task
If h(x) = f(x) · g(x), what can you determine about g(x)
from the given table and graph? Explain your reasoning.
x
-2
-1
0
1
2
f(x)
0
1
2
3
4
© 2013 UNIVERSITY OF PITTSBURGH
Instructional Goals
Jamie’s instructional goals for the lesson are:
• students will multiply two linear functions using their
graphs or tables of values and recognize that, given
two functions f(x) and g(x) and a specific x-value,
x1, the point (x1, f(x1) ∙ g(x1)) will be on the graph of
the product f(x) ∙ g(x); and
• students will recognize that the product of two or
more linear functions is a polynomial function
having the same x-intercepts as the original
functions because the original functions are factors
of the polynomial.
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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?
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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
Missing Function Task
If h(x) = f(x) · g(x), what can you determine about g(x)
from the given table and graph? Explain your reasoning.
x
-2
-1
0
1
2
f(x)
0
1
2
3
4
© 2013 UNIVERSITY OF PITTSBURGH
Focus of the Discussion
Plan to engage students in
a discussion focused on
recognizing that the product
of any two linear functions,
not just functions that
produce parallel lines, must
be a parabola and a
polynomial of degree 2 by
using graphical and
algebraic representations.
You may choose to use the
student work to the right 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?
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