Empowering Learners through the Standards for Mathematical Practice of the Common Core

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Empowering Learners through the Standards for
Mathematical Practice of the Common Core

• Juli K. Dixon, Ph.D.
• University of Central Florida
• juli.dixon_at_ucf.edu

2
Solve this
3 1/7
3
Perspective
When asked to justify the solution to 3 1/7
A student said this
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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.
5
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?
6
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.
7
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 the CCSSM?
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Background of the CCSSM

• Published by the National Governors 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

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Background of the CCSSM
45 States DC have adopted the Common Core State
Standards
Minnesota adopted the CCSS in ELA/literacy only
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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).

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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).
• Weve already met this challenge in Florida. How
  can we use our momentum to take us further and
  deeper?

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NGSSS Content Standards Wordle
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CCSSM Content Standards Wordle
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Content Standards

• Standards define what students should know
  and be able to do
• Clusters group related standards
• Domains group related clusters
• Critical Areas much like our big ideas

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Content Standards

• Measurement and Data K.MD
• Describe and compare measurable attributes.
• Describe measurable attributes of objects, such
  as length or weight. Describe several measurable
  attributes of a single object.
• Directly compare two objects with a measurable
  attribute in common, to see which object has
  more of/less of the attribute, and describe
  the difference. For example, directly compare the
  heights of two children and describe one child as
  taller/shorter.
• Classify objects and count the number of objects
  in each category.
• Classify objects into given categories count the
  numbers of objects in each category and sort the
  categories by count.

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Content Standards

• Measurement and Data K.MD
• Describe and compare measurable attributes.
• Describe measurable attributes of objects, such
  as length or weight. Describe several measurable
  attributes of a single object.
• Directly compare two objects with a measurable
  attribute in common, to see which object has
  more of/less of the attribute, and describe
  the difference. For example, directly compare the
  heights of two children and describe one child as
  taller/shorter.
• Classify objects and count the number of objects
  in each category.
• Classify objects into given categories count the
  numbers of objects in each category and sort the
  categories by count.

Domain
Cluster
Standard
Standard
Cluster
Standard
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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).
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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 Councils (NRC) Adding It
  Up (NRC, 2001).

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Making Sense of the Mathematical Practices
NCTM Process Standards

• Problem Solving
• Reasoning and Proof
• Communication
• Representation
• Connections

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Making Sense of the Mathematical Practices
NRC Strands of Mathematical Proficiency

• Adaptive Reasoning
• Strategic Competence
• Conceptual Understanding
• Procedural Fluency
• Productive Disposition

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Making Sense of the Mathematical Practices
NRC Strands of Mathematical Proficiency

• Adaptive Reasoning
• Strategic Competence
• Conceptual Understanding
• Procedural Fluency
• Productive Disposition

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Standards for Mathematical Practice Wordle
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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.
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Perspective

• Lead with Mathematical Practices
• Implement CCSS beginning with mathematical
  practices,
• Revise current materials and assessments to
  connect to practices, and
• Develop an observational scheme for principals
  that supports developing mathematical practices.
• (CSMC, 2010)

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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

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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.

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Impact on Depth (CCSS)
Number Operations in Base Ten NBT Use place
value understanding and properties of operations
to perform multi-digit arithmetic 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.
Domain
Cluster
Standard
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Solve this
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Solve this
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What did you do?
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Perspective
What do you think fourth grade students would
do? How might they solve 4 x 7 x 25?
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(No Transcript)
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Perspective
Are you observing this sort of mathematics talk
in classrooms? Is this sort of math talk
important?
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Perspective
What does this have to do with the Common Core
State Standards for Mathematics (CCSSM)?
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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

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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

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Impact on Depth
What does it mean to use strategies to
multiply? When do students begin to develop
these strategies?
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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.

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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...
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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...
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What does it mean to use strategies to multiply?

• Consider 6 x 7

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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?

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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?

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(No Transcript)
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Making Sense of Multiplication

• Consider 6 x 7
• How about 4 x 27?

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(No Transcript)
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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

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Reason abstractly and quantitatively
2

• 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.
• This is especially useful when the contexts are
  meaningful to the students.

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Reason abstractly and quantitatively
2

• Consider the following problems
• Jessica has 8 key chains. Calvin has 9 key
  chains. How many key chains do they have all
  together?
• Jessica has 8 key chains. Alex has 15 key chains.
  How many more key chains does Alex have than
  Jessica?

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Reason abstractly and quantitatively
2

• Consider the following problems
• Jessica has 8 key chains. Calvin has 9 key
  chains. How many key chains do they have all
  together?
• Jessica has 8 key chains. Alex has 15 key chains.
  How many more key chains does Alex have than
  Jessica?
• Key words seem helpful

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Reason abstractly and quantitatively
2

• Consider the following problems
• Jessica has 8 key chains. Calvin has 9 key
  chains. How many key chains do they have all
  together?
• Jessica has 8 key chains. Alex has 15 key chains.
  How many more key chains does Alex have than
  Jessica?
• Key words seem helpful, or are they.

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Reason abstractly and quantitatively
2

• Now consider this problem
• Jessica has 8 key chains. How many more key
  chains does she need to have 13 key chains all
  together?

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Reason abstractly and quantitatively
2

• Now consider this problem
• Jessica has 8 key chains. How many more key
  chains does she need to have 13 key chains all
  together?
• How would a child who has been conditioned to use
  key words solve it?

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Reason abstractly and quantitatively
2

• Now consider this problem
• Jessica has 8 key chains. How many more key
  chains does she need to have 13 key chains all
  together?
• How would a child who has been conditioned to use
  key words solve it?
• How might a child reason abstractly and
  quantitatively to solve these problems?

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Reason abstractly and quantitatively
2

• Consider this problem
• Jessica has 8 key chains. Calvin has 9 key
  chains. How many key chains do they have all
  together?
• I know that 8 8 16, so

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Reason abstractly and quantitatively
2

• Consider this problem
• Jessica has 8 key chains. Alex has 15 key chains.
  How many more key chains does Alex have than
  Jessica?
• I know that 8 8 16, so

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Reason abstractly and quantitatively
2

• Now consider this problem
• Jessica has 8 key chains. How many more key
  chains does she need to have 13 key chains all
  together?
• 8 __ 13
• (How might making a ten help?)

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Which Practices Have We Addressed?
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

59
Which Practices Have We Addressed?
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

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Use appropriate tools strategically

• This practice will be very difficult to capture
  in textbook-driven instruction.

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Use appropriate tools strategically
5

• This practice supports hands-on learning
• Tools must include technology
• Tools manipulatives, number lines, and paper and
  pencil
• Mathematically proficient students know which
  tool to use for a given task.

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Use appropriate tools strategically
5

• Consider this Kindergarten class.

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Use appropriate tools strategically
5

• Consider this Kindergarten class.
• What did you notice?

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The exploration of fractions provide excellent
opportunities for student engagement with the
Standards for Mathematical Practice.
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Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get?
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Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get?
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Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get? Solving this wouldnt require much
perseverance but what if we said
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Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get? Give each person the biggest
unbroken piece of cookie possible to start.
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Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get? Give each person the biggest
unbroken piece of cookie possible to start.
71
Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get? Give each person the biggest
unbroken piece of cookie possible to start.
72
Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get? Give each person the biggest
unbroken piece of cookie possible to start.
73
Engaging Students in Reasoning and Sense Making

• Consider this

A student is asked to share 4 cookies equally
among 5 friends. How much of a cookie should each
friend get? Give each person the biggest
unbroken piece of cookie possible to start.
74
Engaging Students in Reasoning and Sense Making

• Consider this

So how much of a cookie would person A get?
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Engaging Students in Reasoning and Sense Making

• Consider this

So how much of a cookie would person A get?
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Engaging Students in Reasoning and Sense Making

• Consider this

So how much of a cookie would person A get?
77
Engaging Students in Reasoning and Sense Making

• Consider this

So how much of a cookie would person A get?
78
Engaging Students in Reasoning and Sense Making

• Consider this

So how much of a cookie would person A get?
79
Engaging Students in Reasoning and Sense Making

• Consider this

So how much of a cookie would person A get?
80
Engaging Students in Reasoning and Sense Making

• Consider this

So how much of a cookie would person A get? -
How much is this all together?
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Engaging Students in Reasoning and Sense Making

• Consider this

What is important here is that the problem
requires diligence to solve and yet with
perseverance the solution is within reach.
Students are reasoning
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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).

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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?

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Engaging Students in Reasoning and Sense Making

• 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.

85
Consider this 5th grade class.
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(No Transcript)
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What was the misconception?
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What was the misconception?With which practices
were the students engaged?
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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

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Where do we start?
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How do we support this empowerment?

• What needs to occur at the administrative level?
• What needs to occur to support teachers?
• What needs to occur to support students?

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Advice to help parents support their children

• Teach procedures only after they are introduced
  in school. Ask your child to explain his or her
  thinking to you. Discuss this with your teacher.
• Drill addition/multiplication facts only after
  your child explores strategies.
• Help your child become more proficient in using
  mathematics at home.

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How do we support this empowerment?

• What we know best might be the most difficult to
  change.

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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.

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Empowering Learners through the Standards for
Mathematical Practice of the Common Core

• Juli K. Dixon, Ph.D.
• University of Central Florida
• juli.dixon_at_ucf.edu