Transcript DixonFCTM2011Handout

Empowering Young
Learners through the
Standards for
Mathematical Practice
Juli K. Dixon, Ph.D.
University of Central Florida
[email protected]
Solve this…
Perspective…
What do you think fourth grade
students would do?
How might they solve 4 x 7 x 25?
Perspective…
Are you observing this sort of
mathematics talk in classrooms?
Is this sort of math talk important?
Perspective…
What does this have to do with the
Common Core State Standards for
Mathematics (CCSSM)?
Background of the CCSSM
• Published by the National Governor’s
Association and the Council of Chief State
School Officers in June 2010
• Result of collaboration from 48 states
• Provides a focused curriculum with an
emphasis on teaching for depth
Background of the CCSSM
“… standards must address the problem of a
curriculum that is ‘a mile wide and an inch
deep.’ These Standards are a substantial
answer to that challenge” (CCSS, 2010, p. 3).
We’ve already met this challenge in Florida.
How can we use our momentum to take us
further and deeper?
NGSSS Content Standards
Wordle
CCSSM Content Standards
Wordle
Background of the CCSSM
The CCSSM consist of Content Standards and
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” (CCSS),
2010, p. 6).
Making Sense of the
Mathematical Practices
The Standards for Mathematical Practice
are based on:
• The National Council of Teachers of
Mathematics’ (NCTM) Principles and
Standards for School Mathematics
(NCTM, 2000), and
• The National Research Council’s (NRC)
Adding It Up (NRC, 2001).
Making Sense of the
Mathematical Practices
NCTM Process Standards:
• Problem Solving
• Reasoning and Proof
• Communication
• Representation
• Connections
Making Sense of the
Mathematical Practices
NRC Strands of Mathematical Proficiency:
• Adaptive Reasoning
• Strategic Competence
• Conceptual Understanding
• Procedural Fluency
• Productive Disposition
Standards of Mathematical
Practice Wordle
Perspective…
According to a recommendation from
the Center for the Study of
Mathematics Curriculum (CSMC,
2010), we should lead with the
Mathematical Practices. Florida is
positioned well to do this.
Making Sense of the
Mathematical Practices
The 8 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
We will only address 4 today
The 8 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
Impact on Depth… (NGSSS)
Grade 4 Big Idea 1: Develop quick recall of
multiplication facts and related division facts
and fluency with whole number multiplication.
MA.4.A.1.2: Multiply multi-digit whole numbers
through four digits fluently, demonstrating
understanding of the standard algorithm, and
checking for reasonableness of results,
including solving real-world problems.
Impact on Depth… (CCSS)
Grade 4 Cluster: Use place value understanding
and properties of operations to perform multidigit arithmetic.
4.NBT.5: Multiply multi-digit numbers using
strategies based on place value and the
properties of operations. Illustrate and explain
the calculations by using equations, rectangular
arrays, and/or area models.
Impact on Depth…
What does it mean to use
strategies to multiply?
When do students begin to
develop these strategies?
Impact on Depth… (NGSSS)
Grade 3 Big Idea 1: Develop understanding of
multiplication and division and strategies for
basic multiplication facts and related division
facts.
MA.3.A.1.2: Solve multiplication and division fact
problems by using strategies that result form
applying number properties.
Impact on Depth… (CCSS)
Grade 3 Cluster: Understand properties of
multiplication…
3.OA.5: Apply properties of operations as
strategies to multiply and divide.
Grade 3 Cluster: Multiply and divide within
100
3.OA.7: Fluently multiply within 100, using
strategies such as the relationship between
multiplication and division or properties of
operations.
What does it mean to use
strategies to multiply?
Consider 6 x 7
How can using strategies to multiply these
factors help students look for and make use of
structure? (SMP7)
What strategies can we use?
How might this sort of thinking influence the
order in which facts are introduced in grade 3?
Now solve 4 x 7 x 25…
The Standards for Mathematical
Practice help us to focus on
processes, not just products.
Empowering Young
Learners
Reasoning abstractly and quantitatively
often involves making sense of
mathematics in real-world contexts.
Word problems can provide examples of
mathematics in real-world contexts.
We need to help students make sense of
them. Not just solve them.
Empowering Young
Learners
Consider the following problems:
Jessica has 7 key chains. Calvin has 8 key
chains. How many key chains do they have all
together?
Jessica has 7 key chains. Alex has 15 key
chains. How many more key chains does Alex
have than Jessica?
Key words seem helpful, or are they….
Empowering Young
Learners
Now consider this problem:
Jessica has 7 key chains. How many more key
chains does she need to have 15 key chains
all together?
How would a child who has been conditioned
to use key words solve it?
Empowering Young
Learners
We need students to make sense of problem
situations as well as each other’s thinking.
Consider these students as they reason
about division.
Empowering Young
Learners
We need students to make sense of problem
situations as well as each other’s thinking.
Consider these students as they reason
about division.
Notice how the teacher’s questions focus on
making sense of the problem.
Empowering Young
Learners
We need students to make sense of problem
situations as well as each other’s thinking.
Consider these students as they reason
about remainders.
Notice how they need support to construct
viable arguments and critique the reasoning
of others.
How might you change your
practice to address these now?
The 8 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
How do we support this
empowerment?
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Teachers need content knowledge for
teaching mathematics to know the tasks to
provide, the questions to ask, and how to
assess for understanding.
Math Talk needs to be supported in the
classroom.
Social norms need to be established in
classroom and professional development
settings to address misconceptions in
respectful ways.