Empowering Learners
through the Standards
for Mathematical
Practice of the
Common Core
Juli K. Dixon, Ph.D.
University of Central Florida
juli.dixon@ucf.edu
Solve this…
3 ÷ 1/7
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 the 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
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).
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
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
Content Standards
Measurement and Data
K.MD
Describe and compare measurable attributes.
1.Describe measurable attributes of objects, such as length or weight.
Describe several measurable attributes of a single object.
2.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.
3.Classify objects into given categories; count the numbers of objects in each
category and sort the categories by count.
Content Standards
Domain
Measurement and Data
K.MD
Cluster
Describe and compare measurable attributes.
Standard
1.Describe measurable attributes of objects, such as length or weight.
Describe several measurable attributes of a single object.
Standard
2.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.
Cluster
Classify objects and count the number of objects in each category.
Standard
3.Classify objects into given categories; count the numbers of objects in each
category and sort the categories by count.
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
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
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.
Perspective…
Lead with Mathematical Practices
1Implement CCSS beginning with mathematical
practices,
2Revise current materials and assessments to
connect to practices, and
3Develop an observational scheme for principals
that supports developing mathematical practices.
(CSMC, 2010)
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… (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)
Domain
Number & Operations in Base Ten
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… (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)
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
How can using strategies to multiply these
factors help students look for and make use of
structure? (SMP7)
What strategies can we use?
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?
Making Sense of
Multiplication
Consider 6 x 7
How about 4 x 27?
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
Reason
abstractly
and
2
quantitatively
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.
Reason
abstractly
and
2
quantitatively
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?
Reason
abstractly
and
2
quantitatively
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
Reason
abstractly
and
2
quantitatively
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….
Reason
abstractly
and
2
quantitatively
Now consider this problem:
Jessica has 8 key chains. How many more key
chains does she need to have 13 key chains
all together?
Reason
abstractly
and
2
quantitatively
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?
Reason
abstractly
and
2
quantitatively
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?
Reason
abstractly
and
2
quantitatively
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…
Reason
abstractly
and
2
quantitatively
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…
Reason
abstractly
and
2
quantitatively
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?)
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
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
7 Look for and make use of structure
Use appropriate tools
strategically
This practice will be very difficult to capture in
textbook-driven instruction.
Use appropriate tools
5
strategically




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.
Use appropriate tools
5
strategically
Consider this Kindergarten class.
Use appropriate tools
5
strategically
Consider this Kindergarten class.
What did you notice?
The exploration of fractions
provide excellent opportunities
for student engagement with
the Standards for
Mathematical Practice.
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?
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?
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 wouldn’t require much
perseverance… but what if we said…
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.
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.
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.
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.
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.
Engaging Students in
Reasoning and Sense Making
Consider this…
So how much of a cookie would person A
get?
Engaging Students in
Reasoning and Sense Making
Consider this…
So how much of a cookie would person A
get?
Engaging Students in
Reasoning and Sense Making
Consider this…
So how much of a cookie would person A
get?
Engaging Students in
Reasoning and Sense Making
Consider this…
So how much of a cookie would person A
get?
Engaging Students in
Reasoning and Sense Making
Consider this…
So how much of a cookie would person A
get?
Engaging Students in
Reasoning and Sense Making
Consider this…
So how much of a cookie would person A
get?
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?
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…
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).
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?
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.
Consider this 5th grade class.
What was the misconception?
What was the misconception?
With which practices were the
students engaged?
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
Where do we start?
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?
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.
How do we support this
empowerment?

What we know best might be the most
difficult to change.
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.
Empowering Learners
through the Standards
for Mathematical
Practice of the
Common Core
Juli K. Dixon, Ph.D.
University of Central Florida
juli.dixon@ucf.edu