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 1
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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
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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.
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No Place Like Home
Two sisters, Janet and Sandy, each represented their
travels from home by sketching their path on the graph
shown below. The x-axis represents the time of their journey
in minutes and the y-axis represents the distance from home
in miles.
Sandy
miles
Janet
minutes
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No Place Like Home
1. Decide whether you agree or disagree with each of the
following statements. Support your answer
mathematically, using specific points or time intervals
where appropriate.
a. Janet traveled mostly uphill while Sandy traveled
mostly downhill.
b. Sandy traveled at a faster rate than Janet.
c. Sandy and Janet were at the same place at the
same time once during their journeys.
d. Each girl always traveled at a constant rate.
e. Both girls were at home at some point during their
journeys.
f. Sandy stopped walking at 14 minutes.
g. Each girl’s journey represents a function.
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No Place Like Home cont.
2. Predict each of the girls’ location after 22 minutes. Justify your
solution mathematically.
3. Write a story for each girl’s journey. You may want to use the
map of their neighborhood shown below.
Park
Mall
School
store
Home
Pet Store
Extension: Assume that at 14 minutes, both girls remembered that
cookies had to be taken out of the oven so they rushed home at a
much faster rate than they were originally traveling. Sketch this
situation on the graph and explain your reasoning.
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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 – 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.★
Reasoning with Equations and Inequalities
(A-REI)
Represent and solve equations and inequalities graphically
A.-REI.D.11
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.★
★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, 201010
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
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Five Representations of Mathematical Ideas
Pictures
Manipulative
Written
Models
Symbols
Real-world
Situations
Oral & Written
Language
Adapted from
Van De Walle, 2004, p. 30
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Five Different Representations of a Function
Language
Context
Table
Graph
Equation
Van De Walle, 2004, p. 440
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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.
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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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Selecting and Sequencing Student
Work
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Monitoring Sheet
Strategy
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Who and What
Order
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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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Underlying Mathematical Ideas Related to
the Lesson (Essential Understandings)
• When all real numbers in an interval make sense for a
contextual situation, the domain will be defined on that
interval. These are functions of a continuous variable.
• The language of change and rate of change
(increasing, decreasing, constant, relative maximum or
minimum) can be used to describe how two quantities
vary together over a range of possible values.
• A rate of change describes how one variable quantity
changes with respect to another – in other words, a
rate of change describes the covariation between two
variables (NCTM, EU 2b)
• A function’s rate of change is one of the main
characteristics that determine what kinds of real-world
phenomena the function can model (NCTM, EU 2c).
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Selecting and Sequencing Student
Work continued
(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
• When all real numbers in an interval make sense for a
contextual situation, the domain will be defined on that
interval. These are functions of a continuous variable.
• The language of change and rate of change
(increasing, decreasing, constant, relative maximum or
minimum) can be used to describe how two quantities
vary together over a range of possible values
• A rate of change describes how one variable quantity
changes with respect to another – in other words, a
rate of change describes the covariation between two
variables (NCTM, EU 2b)
• A function’s rate of change is one of the main
characteristics that determine what kinds of real-world
phenomena the function can model (NCTM, EU 2c)
© 2013 UNIVERSITY OF PITTSBURGH
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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, p. 6-8, NGA Center/CCSSO
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“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: A function’s rate of
change is one of the
main characteristics
that determine what
kinds of real-world
phenomena the
function can model
(NCTM, EU 2c)
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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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