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

The 8 Standards for Mathematical Practice:Making Sense of the Mathematical Practices 1 Make sense of problems and persevere in solving them 2 Reason abstractly and quantitatively 3 Cons

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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@ucf.edu

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Solve this…

3 ÷ 1/7

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A student said this…

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

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“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.”

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“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?

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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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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 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

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Minnesota adopted the CCSS in ELA/literacy only

45 States + DC have adopted the Common Core State Standards

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“… 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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“… 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?

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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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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.

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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.

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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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The Standards for Mathematical Practice are based on:

Making Sense of the

Mathematical Practices

• 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).

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NRC Strands of Mathematical Proficiency:

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NRC Strands of Mathematical Proficiency:

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Standards for Mathematical Practice Wordle

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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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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)

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The 8 Standards for Mathematical Practice:

Mathematical Practices

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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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

Impact on Depth… (CCSS)

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Solve this…

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What did you do?

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Are you observing this sort of

mathematics talk in classrooms?

Is this sort of math talk important?

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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?

of others

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of others

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What does it mean to use strategies to multiply?

When do students begin to develop these strategies?

Impact on Depth…

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Grade 3 Big Idea 1: Develop understanding of

multiplication and division and strategies for

basic multiplication facts and related division

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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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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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Consider 6 x 7

What does it mean to use strategies to multiply?

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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?

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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?

What does it mean to use

strategies to multiply?

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Making Sense of Multiplication

Consider 6 x 7 How about 4 x 27?

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of others

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Reason abstractly and

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.

2

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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?

2

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quantitatively

Key words seem helpful

2

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quantitatively

Key words seem helpful, or are they….

2

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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?

2

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quantitatively

How would a child who has been conditioned

to use key words solve it?

2

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quantitatively

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?

2

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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…

2

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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…

2

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quantitatively

8 + = 13

(How might making a ten help?)

2

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Which Practices Have We

Addressed?

of others

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Which Practices Have We

Addressed?

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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

 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

5

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Use appropriate tools strategically

Consider this Kindergarten class

5

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Use appropriate tools strategically

Consider this Kindergarten class

What did you notice?

5

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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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Consider this…

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Consider this…

Solving this wouldn’t require much

perseverance… but what if we said…

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Consider this…

cookie should each friend get? – Give

each person the biggest unbroken piece

of cookie possible to start

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Consider this…

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Consider this…

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Consider this…

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Consider this…

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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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