Empowering Learners through the Standards for Mathematical Practice of the Common Core Juli K. Dixon, Ph.D. University of Central Florida [email protected]

Empowering Empowering Learners through Learners through the Standards for the Standards for

Mathematical Mathematical Practice of the Practice of the Common Core Common Core

Juli K. Dixon, Ph.D.Juli K. Dixon, Ph.D.

University of Central University of Central FloridaFlorida

[email protected]@ucf.edu

Solve this…Solve this…

3 ÷ 1/7

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.”




“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).


“… 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 NGSSS Content Standards WordleStandards Wordle

CCSSM Content CCSSM Content Standards WordleStandards Wordle

Content StandardsContent 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


Measurement and Data K.MDDescribe 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.


Measurement and Data K.MDDescribe 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.

Domain

Cluster

Standard

Standard

Standard

Cluster


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


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.


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)

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

Impact on Depth… Impact on Depth… (NGSSS)(NGSSS)Grade 4 Big Idea 1:Grade 4 Big Idea 1: Develop quick recall of Develop quick recall of

multiplication facts and related division facts multiplication facts and related division facts and fluency with whole number multiplication.and fluency with whole number multiplication.

MA.4.A.1.2:MA.4.A.1.2: Multiply multi-digit whole numbers Multiply multi-digit whole numbers through four digits fluently, demonstrating through four digits fluently, demonstrating understanding of the standard algorithm, and understanding of the standard algorithm, and checking for reasonableness of results, checking for reasonableness of results, including solving real-world problems.including solving real-world problems.

Number & Operations in Base TenNBTUse 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… Impact on Depth… (CCSS)(CCSS)



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






reasoning

What does it mean to use strategies to multiply?

When do students begin to develop these strategies?

Impact on Depth…Impact on Depth…

Grade 3 Big Idea 1:Grade 3 Big Idea 1: Develop understanding of Develop understanding of multiplication and division and strategies for multiplication and division and strategies for basic multiplication facts and related division basic multiplication facts and related division facts.facts.

MA.3.A.1.2:MA.3.A.1.2: Solve multiplication and division fact Solve multiplication and division fact problems by using strategies that result form problems by using strategies that result form applying number properties.applying number properties.

Impact on Depth… Impact on Depth… (NGSSS)(NGSSS)


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...


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



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)

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)

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

How might this sort of thinking influence How might this sort of thinking influence the order in which facts are introduced in the order in which facts are introduced in grade 3?grade 3?


Making Sense of Making Sense of MultiplicationMultiplication


How about 4 x 27?How about 4 x 27?






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.

2


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?

2



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

2



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….

2


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?

2




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?

2




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?

2


Consider this problem:Consider this problem:

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

I know that 8 + 8 = 16, so…I know that 8 + 8 = 16, so…

2


Consider this problem:Consider this problem:

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

I know that 8 + 8 = 16, so…I know that 8 + 8 = 16, so…

2




8 + __ = 138 + __ = 13

(How might making a ten help?)(How might making a ten help?)

2


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




reasoning


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

Use appropriate tools Use appropriate tools strategicallystrategically

This practice will be very difficult to This practice will be very difficult to capture in textbook-driven instruction.capture in textbook-driven instruction.


This practice supports hands-on This practice supports hands-on learninglearning

Tools must include technologyTools must include technology

Tools manipulatives, number lines, and Tools manipulatives, number lines, and paper and pencilpaper and pencil

Mathematically proficient students Mathematically proficient students know which tool to use for a given know which tool to use for a given task.task.

5


Consider this Kindergarten class.Consider this Kindergarten class.

5


Consider this Kindergarten class.Consider this Kindergarten class.

What did you notice?What did you notice?

5

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

Engaging Students in Engaging Students in Reasoning and Sense Reasoning and Sense MakingMakingConsider this…Consider this…

A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get?

Consider this…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 Engaging Students in Reasoning and Sense Reasoning and Sense MakingMaking

Consider this…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…


Consider this…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.










Consider this…Consider this…So how much of a cookie would person A get?













- How much is this all together?


Consider this…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 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).

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?

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 right.when they are 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.

Consider this 5th grade class.

What was the What was the misconception?misconception?

What was the What was the misconception?misconception?

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

How do we support How do we support this empowerment?this empowerment? What needs to occur at the What needs to occur at the

administrative level?administrative level?

What needs to occur to support What needs to occur to support teachers?teachers?

What needs to occur to support What needs to occur to support students?students?

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 Standards for the Standards for

Mathematical Mathematical Practice of the Practice of the Common Core Common Core

Juli K. Dixon, Ph.D.Juli K. Dixon, Ph.D.

University of Central University of Central FloridaFlorida

[email protected]@ucf.edu

Documents

Empowering Learners through the Standards for Mathematical Practice of the Common Core Juli K. Dixon, Ph.D. University of Central Florida [email protected]