Transcript The National Council of Supervisors of Mathematics

The National Council of
Supervisors of Mathematics
The Common Core State Standards
Illustrating the Standards for
Mathematical Practice:
Look for and express regularity in
repeated reasoning
Construct viable arguments
www.mathedleadership.org
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Module Evaluation
Facilitator: At the end of this
Powerpoint, you will find a
link to an anonymous brief esurvey that will help us
understand how the module
is being used and how well it
worked in your setting.
We hope you will help us
grow and improve our NCSM
resources!
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
2
Common Core State Standards
Mathematics
• Standards for
Content
• Standards for
Practice
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Today’s Goals
• To explore the mathematical standards for
Content and Practice
• To consider how the Common Core State
Standards (CCSS) are likely to impact your
mathematics program and plan next steps
In particular, participants will
• Examine opportunities to help students
express regularity and repeated reasoning
and construct viable arguments (MP3, MP8)
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Standards for Mathematical Practice
“The Standards for
Mathematical Practice
describe varieties of
expertise that
mathematics educators at
all levels should seek to
develop in their students.
These practices rest on
important ‘processes and
proficiencies’ with
longstanding importance
in mathematics
education.” (CCSS, 2010)
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Content Goals
These standards of practice will be
examined in the context of the
following content standard:
Understand properties of the
operations.
Grade 1: p. 14 Grade 2: p. 18
Grade 3: p. 22 Grade 4: p. 28
Grade 5: p. 33 (paragraph #2)
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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What generalization is suggested by
these problems?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Video 1 – Will it always work?
Ms. Kaye’s
3rd Grade
CTB/McGraw-Hill; Mathematics Assessment Resource Services, 2003
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Video 2 – Will it always work?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Video 3 – Will it always work?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Adding 1 to a Factor
Writing prompt:
In a multiplication problem, if you add 1 to a
factor, I think this will happen to the product…
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Students’ Articulation of the Claim
• “The number that is not increased is the
number that the answer goes up by.”
• “The number that is staying and not
going up, increases by however many it
is.”
• “I think that the factor you increase, it
goes up by the other factor.”
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Choose which of the original equations you want
to work with. Then do one of these…
• Draw a picture for the original equation; then
change it just enough to match the new equations.
• Make an array for the original equation; then
change it just enough to match the new equations.
• Write a story for the original equation; then change
it just enough to match the new equations.
Example: Original equation 7 x 5 = 35
New equations 7 x 6 = 42
8 x 5 = 40
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Frannie’s Story Context
There are 7 jewelry boxes and each box
has 5 pieces of jewelry. There are 35
pieces of jewelry altogether.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Jewelry Boxes
7x5
Seven boxes with
five pieces of jewelry
in each box
35 pieces of jewelry
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Jewelry Boxes
7x5  8x5
Eight
Seven boxes with five
pieces of jewelry in
each box
35 pieces of jewelry
+ 5 pieces of jewelry
40 pieces of jewelry
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Jewelry Boxes
7x5
Seven boxes with
five pieces of jewelry
in each box
35 pieces of jewelry
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Jewelry Boxes
7x57x6
six
Seven boxes with five
pieces of jewelry in
each box
35 pieces of jewelry
+ 7 pieces of jewelry
42 pieces of jewelry
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Other Stories
•
•
•
•
Baskets of bouncy balls
Tanks with salmon eggs
Baskets of mozzarella sticks
Rows of chairs
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Making Sense of Multiplication
Explain how the array changes from 7 x 5 to 8 x 5
and from 7 x 5 to 7 x 6.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Video 4
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Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Video 5
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Importance of Contrasting
Operations
• We were talking about the addends
changing by 1 and what happens to
the sum.
• Now we’re talking about the factors
changing by 1 and what happens to
the product.
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Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Pause and Reflect
•
How did the mathematical practices play
out in these sessions?
•
What properties of addition and
multiplication were under consideration?
•
How did the practices of noticing
regularities and constructing arguments
support learning about these properties
and about the meaning of the
operations?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Associative Property of
Addition
a + (b + c) = (a + b) + c
a + (b + 1) = (a + b) + 1
5 + (7 + 1) = (5 + 7) + 1
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Distributive Property of
Multiplication over Addition
a (b + c) = ab + ac
a (b + 1) = ab + a
5 (7 + 1) = 35 + 5
(a + b) c = ac + bc
(a + 1) c = ac + c
(5 + 1) 7 = 35 + 7
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Importance of Contrasting
Operations
• Commutative property of addition:
a+b=b+a
• What about subtraction?
Is a – b equal to b – a?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Video 6
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Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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2nd grade
• How were the students engaged in
the 8th practice: noticing and
expressing regularity?
• How were they engaged in the 3rd
practice, creating viable arguments?
• How did the practices of noticing
regularities and constructing
arguments support learning about
properties and about the meaning of
the operations?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Today’s Goals
• To explore the mathematical standards for
Content and Practice
• To consider how the Common Core State
Standards (CCSS) are likely to impact your
mathematics program and plan next steps
In particular, participants will
• Examine opportunities to help students
express regularity and repeated reasoning
and construct viable arguments (MP3, MP8)
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
31
Reflection on Standards for
Mathematical Practice
1.
Individually review the
Standards for Mathematical
Practice.
2. Choose a partner at your
table and discuss a new
insight you had into the
Standards for Mathematical
Practice.
3. Then discuss the following
question.
What implications might the Standards for
Mathematical Practice have on your classroom?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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End of Day Reflections
1.
Are there any aspects of your
own thinking and/or practice
that our work today has
caused you to consider or
reconsider? Explain.
2. Are there any aspects of your
students’ mathematical
learning that our work today
has caused you to consider or
reconsider? Explain.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Acknowledgements
• This module was developed by Deborah Schifter
and Susan Jo Russell, based on work from the
projects Teaching to the Big Ideas, Foundations of
Algebra in the Elementary and Middle Grades, and
Using Routines as an Instructional Tool for
Developing Students’ Conceptions of Proof.
Video clips are used with permission from these
projects.
• This work was funded in part by the
National Science Foundation through
grants ESI-0242609 to EDC and ESI0550176 and DRL-1019482 to TERC. Any
opinions, findings, conclusions, or
recommendations expressed here are
those of the authors and do not necessarily
reflect the views of the National Science
Foundation.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Join us in thanking the
Noyce Foundation
for their generous grant to NCSM that
made this series possible!
http://www.noycefdn.org/
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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NCSM Series Contributors
• Geraldine Devine, Oakland Schools, Waterford, MI
• Aimee L. Evans, Arch Ford ESC, Plumerville, AR
• David Foster, Silicon Valley Mathematics Initiative, San José
State University, San José, California
• Dana L. Gosen, Ph.D., Oakland Schools, Waterford, MI
• Linda K. Griffith, Ph.D., University of Central Arkansas
• Cynthia A. Miller, Ph.D., Arkansas State University
• Valerie L. Mills, Oakland Schools, Waterford, MI
• Susan Jo Russell, Ed.D., TERC, Cambridge, MA
• Deborah Schifter, Ph.D., Education Development Center,
Waltham, MA
• Nanette Seago, WestEd, San Francisco, California
• Hope Bjerke, Editing Consultant, Redding, CA
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
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Help Us Grow!
The link below will
connect you to a
anonymous brief esurvey that will help us
understand how the
module is being used
and how well it worked in
your setting.
Please help us improve the module by
completing a short ten question survey at:
http://tinyurl.com/samplesurvey1
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice: Reasoning and Explaining
37