Transcript Supporting Rigorous Instruction in Mathematics

Supporting Rigorous Mathematics
Teaching and Learning
Selecting and Sequencing Students’ Solution
Paths to Maximize Student Learning
Tennessee Department of Education
High School Mathematics
Algebra 2
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Session Goals
Participants will:
• learn what to monitor in student work when circulating
during the Explore Phase of the lesson;
• learn about guidelines or “rules of thumb” for selecting
and sequencing student work that target essential
understandings of the lesson; and
• learn about focus questions that target the essential
understandings.
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Overview of Activities
Participants will:
• discuss the content standards and identify the related
essential understandings of a lesson;
• analyze samples of student work;
• select and sequence student work for the Share,
Discuss, and Analyze Phase of the lesson;
• identify “rules of thumb” for selecting and sequencing
student work; and
• write focus questions that target essential
understandings.
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Rationale
Orchestrating discussions that build on students’
thinking places significant pedagogical demands on
teachers and requires an extensive and interwoven
network of knowledge. Teachers often feel that they
should avoid telling students anything, but are not
sure what they can do to encourage rigorous
mathematical thinking and reasoning.
(Stein, M.K., Engle, R., Smith, M., Hughes, E. 2008. Orchestrating Productive Mathematical Discussions: Five
Practices for Helping Teachers Move Beyond Show and Tell)
In this session, we will focus on monitoring, selecting,
and sequencing student work so you can assess and
advance student learning during the Share, Discuss,
and Analyze Phase of the lesson.
Triple Trouble
Consider the two functions graphed below. Let ℎ 𝑥 = 𝑓(𝑥) ∙ 𝑔(𝑥)
1. David, Theresa, Manuel, and Joy are working in a
group together to determine the key
characteristics of h(x). They each make a
prediction. Decide whether you agree or disagree
with each student’s prediction. Use mathematics
to justify your position.
• David: h(x) will be a parabola.
• Theresa: h(x) will have a y-intercept at (0, 12).
• Manuel: h(x) will have negative y-values over
the interval −2 ≤ 𝑥 ≤ −1.
• Joy: h(x) will have three x-intercepts.
2. Sketch a graph of h(x) on the coordinate plane.
Then identify key characteristics of the graph
(zeros, y-intercept, max/min values, end behavior)
and explain how each key characteristic results
from key characteristics of f(x) and g(x).
3. Determine the equation of h(x). Justify your
answer in terms of f(x) and g(x).
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The Task: Discussing Solution Paths
• Solve the task in as many ways as you can.
• Discuss the solution paths with colleagues at your
table.
• If only one solution path has been used, work together
to create others.
• Consider possible misconceptions or errors that we
might see from students.
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Linking the Standards to Student
Solution Paths
The task has been selected with specific Standards
for Mathematical Content and Practice in mind.
Where do you see the potential to work on these
standards in the written task or the solution paths?
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The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra 2
Building Functions
(F-BF)
Build a function that models a relationship between two quantities
F-BF.A.1
Write a function that describes a relationship between two
quantities.★
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.
Arithmetic with Polynomials and Rational Expressions
(A-APR)
Understand the relationship between zeros and factors of polynomials
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.
★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
Common Core Standards for Mathematical
Practice
What must happen in order for students to have opportunities to make
use of the Standards 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
Five Representations of Mathematical Ideas
Pictures
Manipulative
Written
Models
Symbols
Real-world
Situations
Adapted from
Van De Walle, 2004, p. 30
Oral & Written
Language
Five Different Representations of a Function
Language
Context
Table
Graph
Equation
Van De Walle, 2004, p. 440
The Structure and Routines of a Lesson
Set Up of the 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 by engaging students
in a quick write or a discussion
of the process.
Analyzing Student Work
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Analyzing Student Work
Use the student work to further your understanding of
the task.
Consider:
• What do the students know?
• How did the students solve the task?
• How do their solution paths differ from each other?
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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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19
Selecting and Sequencing Student
Work
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Monitoring Sheet
Strategy
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Who and What
Order
Selecting and Sequencing Student Work
(Small Group Discussion)
• Examine the students’ solution paths.
• Determine which solution paths you want to share
during the class discussion; keep track of your
rationale for selecting the pieces of student work.
• Determine the order in which work will be shared;
keep track of your rationale for choosing a particular
order for the sharing the work.
Record the group’s decision on the chart in
your participant handouts.
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Standards and Essential Understandings
• The product of two or more linear functions is a
polynomial function. The resulting function will have the
same x-intercepts as the original functions because the
original functions are factors of the polynomial.
• Two or more polynomial functions can be multiplied using
the algebraic representations by applying the distributive
property and combining like terms.
• Two or more polynomial functions can be multiplied using
their graphs or tables of values because 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).
• Making meaningful use of algebraic symbols means one
can choose variables and construct expressions and
equations from a context, table, or graph.
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Selecting and Sequencing Student
Work
(Small Group Discussion)
• Each team should record their group’s sequence of
solution paths on the chart.
• Identify the student’s solution path that would be shared
and discussed first, second, third, and so on.
• Be prepared to justify your response.
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Selecting and Sequencing Student Work
(Group Discussion)
• Listen to each group’s rationale for selecting and
sequencing student work.
• As you listen to the rationale, come up with a general
“rule of thumb” that can be used to guide you when
selecting and sequencing work for the Share,
Discuss, and Analyze Phase of the lesson.
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Reflecting On Essential
Understandings
Which of the sequences of student work were driven by the
standards and essential understandings?
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Reflecting on the Standards and the
Essential Understandings
• The product of two or more linear functions is a
polynomial function. The resulting function will have the
same x-intercepts as the original functions because the
original functions are factors of the polynomial.
• Two or more polynomial functions can be multiplied using
the algebraic representations by applying the distributive
property and combining like terms.
• Two or more polynomial functions can be multiplied using
their graphs or tables of values because 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).
• Making meaningful use of algebraic symbols means one
can choose variables and construct expressions and
equations from a context, table, or graph.
© 2013 UNIVERSITY OF PITTSBURGH
Common Core Standards 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
“Rules of Thumb” for Selecting and
Sequencing Student Work
What are the benefits of using the “rules of thumb” as
a guide when selecting and sequencing student work
for the Share, Discuss, and Analyze Phase of the
lesson?
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Pressing for Mathematical
Understanding
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Pressing for Mathematical
Understanding
Let’s focus on one piece of student work for the
Share, Discuss, and Analyze Phase of the lesson.
Assume that a student has explained the work and
others in the class have repeated the ideas and
asked questions. Now it is time to “FOCUS” the
discussion on an important mathematical idea.
What questions might you ask the class as a whole
to focus the discussion? Write your questions on
chart paper to be posted for a gallery walk.
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Pressing for Mathematical
Understanding
 EU: Two or more
polynomial functions
can be multiplied
using their graphs or
tables of values
because 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).
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Pressing for Mathematical
Understanding
Do a gallery walk. Review other groups’ questions.
• What are some similarities among the questions?
• What are some differences between the questions?
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Reflecting on Our Learning
What have you learned today that you will think about
and make use of next school year? Take a few minutes
and jot your thoughts down.
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