Transcript K-8 Mathematics Standards

Empowering Learners
through the Common
Core State Standards
Juli K. Dixon, Ph.D.
University of Central Florida
[email protected]
Solve this…
3 ÷ 1/7
Solve this…
3 ÷ 1/7
Tell someone near you how you solved it.
Perspective…
When asked to justify the solution to 3 ÷ 1/7
A student said this…
Perspective…
When asked to justify the solution to 3 ÷ 1/7
A student said this…
“Just change the division sign to
multiplication and flip the fraction after the
sign. 3 ÷ 1/7 becomes 3 x 7/1. So I find
3/1 x 7/1 which is 21/1 or 21.”
Perspective…
When asked to justify the solution to 3 ÷ 1/7
A student said this…
“Just change the division sign to
multiplication and flip the fraction after the
sign. 3 ÷ 1/7 becomes 3 x 7/1. So I find
3/1 x 7/1 which is 21/1 or 21.”
Is this an acceptable justification?
Perspective…
When asked to justify the solution to 3 ÷ 1/7
Another student said this…
“I know there are 7 groups of 1/7 in one
whole. Since there are three wholes, I
have 3 x 7 or 21 groups of 1/7 in 3
wholes so 3 ÷ 1/7 = 21.”
Perspective…
When asked to justify the solution to 3 ÷ 1/7
Another student said this…
“I know there are 7 groups of 1/7 in one
whole. Since there are three wholes, I
have 3 x 7 or 21 groups of 1/7 in 3
wholes so 3 ÷ 1/7 = 21.”
How is this justification different and what
does it have to do with State Standards?
Background of the CCSSM
(Common Core State Standards)
• 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
45 States + DC have adopted the Common Core State Standards
Minnesota adopted the CCSS in ELA/literacy only
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).
CCSSM Content Standards
Wordle
Content Standards
• Critical Areas – much like our big ideas
• Domains – group related clusters
• Clusters – group related standards
• Standards – define what students should know
and be able to do
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
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 for Mathematical
Practice Wordle
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
Impact on Depth… (CCSS)
Domain
Number & Operations in Base Ten
4.NBT
Cluster
Use place value understanding and properties of operations to
perform multi-digit arithmetic
Standard
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.
Solve this…
Solve this…
What did you do?
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)?
With which practices were the
fourth grade students engaged?
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
With which practices were the
fourth grade students engaged?
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…
What does it mean to use
strategies to multiply?
When do students begin to
develop these strategies?
Impact on Depth… (CCSS)
Operations & Algebraic Thinking
3.OA
Understand properties of multiplication and the relationship
between multiplication and division.
5.
Apply properties as strategies to multiply and divide…
Multiply and divide within 100.
7.
Fluently multiply within 100, using strategies such as the
relationship between multiplication and division or properties of
operations...
Impact on Depth… (CCSS)
Operations & Algebraic Thinking
3.OA
Understand properties of multiplication and the relationship
between multiplication and division.
5.
Apply properties as strategies to multiply and divide…
Multiply and divide within 100.
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
What does it mean to use
strategies to multiply?
Consider 6 x 7
What strategies can we use?
What does it mean to use
strategies to multiply?
Consider 6 x 7
What strategies can we use?
How might this sort of thinking influence the
order in which facts are introduced in grade 3?
With which practice were the
third grade students engaged?
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
With which practice were the
third grade students engaged?
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?
“… a lack of understanding [of mathematical
content] effectively prevents a student from
engaging in the mathematical practices”
(CCSS, 2010, p. 8).
When and how do we develop this
understanding?
Where do we start?
….in Kindergarten.
Making Sense of Numbers
to 20
How do we help students to
truly make sense of number
patterns?
Making Sense of Numbers
to 20
How do we help students to
truly make sense of number
patterns?
Consider number pairs for 8.
Describing the Standards
K.OA.3: Decompose numbers less than or equal
to 10 into pairs in more than one way, e.g., by
using objects or drawings, and record each
decomposition by a drawing or equation (e.g., 5
= 2 + 3 and 5 = 4 + 1).
Describing the Standards
Watch the following video of a Kindergarten
class as they explore Number Pairs for 8.
Describing the Standards
Watch the following video of a Kindergarten
class as they explore Number Pairs for 8.
Look for how patterns are addressed.
Which practice was of focus here?
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
Which practice was of focus here?
The 8 Standards for Mathematical Practice:
8 Look for and express regularity in repeated reasoning
Engaging Students in
Reasoning and Sense Making

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We need to question students when they are
wrong and when they are right.
We need to create an environment where
students are expected to share their thinking.
We need to look for opportunities for
students to reason about and make sense of
mathematics.
What is your biggest take away
regarding the practices and your role?
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?



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.
How do we support this
empowerment?
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What we know best might be the most
difficult to change.
We need to be aware of the changes so
that we can prepare ourselves to meet
the needs of our students.
Empowering Learners
through the Common
Core State Standards
Juli K. Dixon, Ph.D.
University of Central Florida
[email protected]