Empowering Learners through the Common Core State Standards in Grades 3-5

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Empowering Learners through the Common Core State
Standardsin Grades 3-5

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

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Solve this
3 1/7
3
Solve this
3 1/7
Tell someone near you how you solved it.
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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.
6
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?
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.
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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).
• So what do these standards look like anyway?

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CCSSM Content Standards Wordle for Grades 3-5
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Content Standards

• Define expectations for students at each grade
  level
• Use concepts from earlier grades
• Emphasize need to justify mathematical moves
• Indicate understanding and skill are equally
  important
• Include expectations that students demonstrate
  understanding of procedures

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Content Standards

• Critical Areas major areas of focus for grade
• Domains group related clusters
• Clusters group related standards
• Standards define what students should know
  and be able to do

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Content Standards
Number Operations in Base Ten 4.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.
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Content Standards
Number Operations in Base Ten 4.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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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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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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Perspective
According to a recommendation from the Center for
the Study of Mathematics Curriculum (CSMC, 2010),
we should lead with the Mathematical Practices.
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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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SMARTER Balanced Assessment Consortium
Draft Assessment Claims for Mathematics

• Concepts and Procedures
• Problem Solving
• Communicating Reasoning
• Data Analysis and Modeling
• See Draft Item Spec released January 26, 2012

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Content Standards
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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Perspective
In an effort to simplify students learning
pathways and minimize barriers (stigler, et. al.,
1999), teachers often provide students with
efficient procedures too early. When we do this
we minimize students opportunities to engage in
these practices.
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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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Content Standards
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
• What strategies can we use?

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What does it mean to use strategies to multiply?

• Consider 6 x 7
• What strategies can we use?
• How can using strategies to multiply these
  factors help students look for and make use of
  structure? (SMP7)

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(No Transcript)
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The Standards for Mathematical Practice help us
to focus on processes, not just products.
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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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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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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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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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The exploration of fractions provide excellent
opportunities for student engagement with the
Standards for Mathematical Practice.
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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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We must anticipate student misconceptions and use
them as spring boards to learning.
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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 practice
were the students engaged?
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With which practice were the fifth 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 practice were the fifth 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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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?

• There are at least three ways to think about
  this
• Where do we start as teachers and administrators?
  
• Where do we start as users of mathematics?
  Thinking mathematically.
• Where do we start with respect to grade level?

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Describing the Standards
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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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.

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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 Common Core State
Standardsin Grades 3-5

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