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Empowering Learners through the Common Core State

Standards in Grades 3-5

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

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 the CCSSM?

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

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). - So what do these standards look like anyway?

CCSSM Content Standards Wordle for Grades 3-5

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

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

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.

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

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 Councils (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

Making Sense of the Mathematical Practices

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

Perspective

According to a recommendation from the Center for

the Study of Mathematics Curriculum (CSMC, 2010),

we should lead with the Mathematical Practices.

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)

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

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

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?

(No Transcript)

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

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

With which practices were the fourth grade

students engaged?

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

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.

Impact on Depth

What does it mean to use strategies to

multiply? When do students begin to develop

these strategies?

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...

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 can using strategies to multiply these

factors help students look for and make use of

structure? (SMP7)

(No Transcript)

The Standards for Mathematical Practice help us

to focus on processes, not just products.

Making Sense of the Mathematical Practices

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

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.

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?

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

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.

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?

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?

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?

Which Practices Have We Addressed?

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

Which Practices Have We Addressed?

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

The exploration of fractions provide excellent

opportunities for student engagement with the

Standards for Mathematical Practice.

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?

We must anticipate student misconceptions and use

them as spring boards to learning.

Consider this 5th grade class.

(No Transcript)

What was the misconception?

What was the misconception? With which practice

were the students engaged?

With which practice were the fifth grade students

engaged?

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

With which practice were the fifth grade students

engaged?

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

How might you change your practice to address

these now?

The 8 Standards for Mathematical Practice

- Make sense of problems and persevere in solving

them - Reason abstractly and quantitatively
- Construct viable arguments and critique the

reasoning of others - Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated

reasoning

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?

Describing the Standards

a lack of understanding of mathematical

content effectively prevents a student from

engaging in the mathematical practices (CCSS,

2010, p. 8).

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.

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 Common Core State

Standards in Grades 3-5

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