Empowering Learners through the Common Core State Standards in Grades 3-5 Juli K. Dixon, Ph.D....

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Empowering Empowering Learners through Learners through the Common Core the Common Core State StandardsState Standards

in Grades 3-5in Grades 3-5Juli K. Dixon, Ph.D.Juli K. Dixon, Ph.D.

University of Central University of Central FloridaFlorida

juli.dixon@ucf.edujuli.dixon@ucf.edu

Solve this…Solve this…

3 ÷ 1/7

Tell someone near you how you solved it.

Perspective…Perspective…

A student said this…A student said this…

When asked to justify the solution to 3 ÷ 1/7

“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 Is this an acceptable justification?justification?

Another student said Another student said this…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.”

Another student said Another student said this…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 How is this justification different and what does it have to do with the what does it have to do with the CCSSM?CCSSM?

Background of the Background of the CCSSMCCSSM

• 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

Minnesota adopted the CCSS in ELA/literacy only

45 States + DC have adopted the Common Core State Standards

“… 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 CCSSM Content Standards Wordle for Standards Wordle for Grades 3-5Grades 3-5

Content StandardsContent 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

• 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

Number & Operations in Base Ten4.NBTUse 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.

Number & Operations in Base Ten4.NBTUse place value understanding and properties of operations to perform multi-digit arithmetic

Domain

Cluster

Standard

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).

The Standards for Mathematical Practice are based on:

Making Sense of the Making Sense of the Mathematical Mathematical PracticesPractices

• 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).

NCTM Process Standards:

• Problem Solving

• Reasoning and Proof

• Communication

• Representation

• Connections

NRC Strands of Mathematical Proficiency:

• Adaptive Reasoning

• Strategic Competence

• Conceptual Understanding

• Procedural Fluency

• Productive Disposition

NRC Strands of Mathematical Proficiency:

• Adaptive Reasoning

• Strategic Competence

• Conceptual Understanding

• Procedural Fluency

• Productive Disposition

Standards for Standards for Mathematical Practice Mathematical Practice WordleWordle

The 8 Standards for Mathematical Practice:

1 Make sense of problems and persevere in solving them

2 Reason abstractly and quantitatively3 Construct viable arguments and critique the

reasoning of others4 Model with mathematics5 Use appropriate tools strategically6 Attend to precision7 Look for and make use of structure8 Look for and express regularity in repeated

reasoning

According to a recommendation from the Center for the Study of Mathematics Curriculum (CSMC, 2010), we should lead with the Mathematical Practices.

Lead with Mathematical Practices1Implement CCSS beginning with mathematical practices,2Revise current materials and assessments to connect to practices, and3Develop an observational scheme for principals that supports developing mathematical practices.

(CSMC, 2010)

Draft Assessment Claims for Mathematics*

SMARTER Balanced SMARTER Balanced Assessment Assessment ConsortiumConsortium

1 Concepts and Procedures

2 Problem Solving

3 Communicating Reasoning

4 Data Analysis and Modeling

* See Draft Item Spec released January 26, 2012

Number & Operations in Base TenNBTUse place value understanding and properties of operations to perform multi-digit arithmetic

Domain

Cluster

Standard

What did you do?What did you do?

What do you think fourth grade students would do?

How might they solve 4 x 7 x 25?

Are you observing this sort of mathematics talk in classrooms?

Is this sort of math talk important?

What does this have to do with the Common Core State Standards for Mathematics (CCSSM)?

With which practices were With which practices were the fourth grade students the fourth grade students engaged?engaged?

reasoning

With which practices were With which practices were the fourth grade students the fourth grade students engaged?engaged?

reasoning

In an effort to simplify studentIn an effort to simplify student’’s learning s learning pathways and minimize barriers (stigler, et. pathways and minimize barriers (stigler, et. al., 1999), teachers often provide students al., 1999), teachers often provide students with efficient procedures too early.with efficient procedures too early.

When we do this – we minimize studentsWhen we do this – we minimize students’’ opportunities to engage in these practices. opportunities to engage in these practices.

What does it mean to use strategies to multiply?

When do students begin to develop these strategies?

Impact on Depth…Impact on Depth…

Operations & Algebraic Thinking3.OAUnderstand 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...

Consider 6 x 7Consider 6 x 7

What strategies can we use?What strategies can we use?

How can using strategies to multiply these How can using strategies to multiply these factors help students look for and make factors help students look for and make use of structure? (SMP7)use of structure? (SMP7)

The Standards for Mathematical Practice help us to focus on processes, not just products.

reasoning

Reason abstractly and Reason abstractly and quantitativelyquantitatively

Reasoning abstractly and quantitatively Reasoning abstractly and quantitatively often involves making sense of often involves making sense of mathematics in real-world contexts.mathematics in real-world contexts.

Word problems can provide examples of Word problems can provide examples of mathematics in real-world contexts.mathematics in real-world contexts.

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

Consider the following problems:Consider the following problems:

Jessica has 8 key chains. Calvin has 9 key Jessica has 8 key chains. Calvin has 9 key chains. How many key chains do they have chains. How many key chains do they have all together?all together?

Jessica has 8 key chains. Alex has 15 key Jessica has 8 key chains. Alex has 15 key chains. How many more key chains does Alex chains. How many more key chains does Alex have than Jessica?have than Jessica?

Jessica has 8 key chains. Calvin has 9 key Jessica has 8 key chains. Calvin has 9 key chains. How many key chains do they have chains. How many key chains do they have all together?all together?

Key words seem helpfulKey words seem helpful

Jessica has 8 key chains. Calvin has 9 key Jessica has 8 key chains. Calvin has 9 key chains. How many key chains do they have all chains. How many key chains do they have all together?together?

Key words seem helpful, or are they….Key words seem helpful, or are they….

Now consider this problem:Now consider this problem:

Jessica has 8 key chains. How many Jessica has 8 key chains. How many more key chains does she need to have more key chains does she need to have 13 key chains all together?13 key chains all together?

How would a child who has been How would a child who has been conditioned to use key words solve it?conditioned to use key words solve it?

How might a child reason abstractly and How might a child reason abstractly and quantitatively to solve these problems?quantitatively to solve these problems?

Which Practices Have We Which Practices Have We Addressed?Addressed?

reasoning

Which Practices Have We Which Practices Have We Addressed?Addressed?

reasoning

The exploration of fractions The exploration of fractions provide excellent opportunities provide excellent opportunities for student engagement with for student engagement with the Standards for Mathematical the Standards for Mathematical Practice.Practice.

How do we support How do we support this empowerment?this empowerment?““… … a lack of understanding [of a lack of understanding [of mathematical content] effectively mathematical content] effectively prevents a student from engaging in the prevents a student from engaging in the mathematical practicesmathematical practices”” (CCSS, 2010, p. 8).(CCSS, 2010, p. 8).

When and how do we develop this When and how do we develop this understanding?understanding?

We must anticipate student misconceptions and use them as spring boards to learning.

Consider this 5th grade class.

What was the What was the misconception?misconception?

With which practice were the With which practice were the students engaged?students engaged?

The 8 Standards for Mathematical Practice:1 Make sense of problems and persevere in

solving them2 Reason abstractly and quantitatively3 Construct viable arguments and critique the

reasoning

With which practice were the With which practice were the fifth grade students fifth grade students engaged?engaged?

The 8 Standards for Mathematical Practice:1 Make sense of problems and persevere in

solving them2 Reason abstractly and quantitatively3 Construct viable arguments and critique the

reasoning

With which practice were the With which practice were the fifth grade students fifth grade students engaged?engaged?

How might you change your How might you change your practice to address these practice to address these now?now?

reasoning

Where do we start?Where do we start?

There are at least three ways to think about this:

1.Where do we start as teachers and administrators?

2.Where do we start as users of mathematics? Thinking mathematically.

3.Where do we start with respect to grade level?

Describing the Describing the StandardsStandards

““… … a lack of understanding [of a lack of understanding [of mathematical content] mathematical content] effectively prevents a student effectively prevents a student from engaging in the from engaging in the mathematical practicesmathematical practices”” (CCSS, 2010, p. 8).(CCSS, 2010, p. 8).

Engaging Students in Engaging Students in Reasoning and Sense Reasoning and Sense MakingMaking We need to question students when We need to question students when

they are wrong they are wrong and and when they are when they are right.right.

We need to create an environment We need to create an environment where students are expected to share where students are expected to share their thinking.their thinking.

We need to look for opportunities for We need to look for opportunities for students to reason about and make students to reason about and make sense of mathematics.sense of mathematics.

Advice to help parents Advice to help parents support their children:support their children: Teach procedures only after they are Teach procedures only after they are

introduced in school. Ask your child to introduced in school. Ask your child to explain his or her thinking to you. explain his or her thinking to you. Discuss this with your teacher.Discuss this with your teacher.

Drill addition/multiplication facts only Drill addition/multiplication facts only after your child explores strategies.after your child explores strategies.

Help your child become more proficient Help your child become more proficient in using mathematics at home.in using mathematics at home.

How do we support How do we support this empowerment?this empowerment? What we know best might be the What we know best might be the

most difficult to change.most difficult to change.

How do we support How do we support this empowerment?this empowerment? Teachers need content knowledge for Teachers need content knowledge for

teaching mathematics to know the tasks teaching mathematics to know the tasks to provide, the questions to ask, and to provide, the questions to ask, and how to assess for understanding.how to assess for understanding.

Math Talk needs to be supported in the Math Talk needs to be supported in the classroom.classroom.

Social norms need to be established in Social norms need to be established in classroom classroom andand professional professional development settings to address development settings to address misconceptions in respectful ways.misconceptions in respectful ways.

Empowering Empowering Learners through Learners through the Common Core the Common Core State StandardsState Standards

in Grades 3-5in Grades 3-5Juli K. Dixon, Ph.D.Juli K. Dixon, Ph.D.

University of Central University of Central FloridaFlorida

juli.dixon@ucf.edujuli.dixon@ucf.edu

Empowering Learners through the Common Core State Standards in Grades 3-5 Juli K. Dixon, Ph.D....

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