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

Mathematical Mathematical PracticePractice

Juli 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…

Perspective…Perspective…

What do you think fourth grade students would do?

How might they solve 4 x 7 x 25?

Perspective…Perspective…

Are you observing this sort of mathematics talk in classrooms?

Is this sort of math talk important?

Perspective…Perspective…

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

Background of the Background of the CCSSMCCSSM

“… 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

Background of the Background of the CCSSMCCSSM

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:

Making Sense of the Making Sense of the Mathematical Mathematical PracticesPractices

• Problem Solving

• Reasoning and Proof

• Communication

• Representation

• Connections

NRC Strands of Mathematical Proficiency:

Making Sense of the Making Sense of the Mathematical Mathematical PracticesPractices

• Adaptive Reasoning

• Strategic Competence

• Conceptual Understanding

• Procedural Fluency

• Productive Disposition

Standards of Standards of Mathematical Practice Mathematical Practice WordleWordle

Perspective…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.

The 8 Standards for Mathematical Practice:

Making Sense of the Making Sense of the Mathematical Mathematical PracticesPractices

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

The 8 Standards for Mathematical Practice:

We will only address 4 We will only address 4 todaytoday

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.

Impact on Depth… Impact on Depth… (CCSS)(CCSS)Grade 4 Cluster:Grade 4 Cluster: Use place value understanding Use place value understanding

and properties of operations to perform multi-and properties of operations to perform multi-digit arithmetic.digit arithmetic.

4.NBT.5:4.NBT.5: Multiply multi-digit numbers using Multiply multi-digit numbers using strategies based on place value and the strategies based on place value and the properties of operations. Illustrate and explain properties of operations. Illustrate and explain the calculations by using equations, rectangular the calculations by using equations, rectangular arrays, and/or area models.arrays, and/or area models.

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)

Grade 3 Cluster:Grade 3 Cluster: Understand properties of Understand properties of multiplication…multiplication…

3.OA.5:3.OA.5: Apply properties of operations as Apply properties of operations as strategies to multiply and divide.strategies to multiply and divide.

Grade 3 Cluster:Grade 3 Cluster: Multiply and divide within 100 Multiply and divide within 100

3.OA.7:3.OA.7: Fluently multiply within 100, using Fluently multiply within 100, using strategies such as the relationship between strategies such as the relationship between multiplication and division or properties of multiplication and division or properties of operations.operations.

Impact on Depth… Impact on Depth… (CCSS)(CCSS)

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 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?

What does it mean to use strategies to multiply?

Now solve 4 x 7 x 25…

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

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.

We need to help students We need to help students make sensemake sense of of them. Not just solve them.them. Not just solve them.

Empowering Young Empowering Young LearnersLearners

Consider the following problems:Consider the following problems:

Jessica has 7 key chains. Calvin has 8 key Jessica has 7 key chains. Calvin has 8 key chains. How many key chains do they have all chains. How many key chains do they have all together?together?

Jessica has 7 key chains. Alex has 15 key Jessica has 7 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?

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

Empowering Young Empowering Young LearnersLearners

Now consider this problem:Now consider this problem:

Jessica has 7 key chains. How many Jessica has 7 key chains. How many more key chains does she need to have more key chains does she need to have 15 key chains all together?15 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?

Empowering Young Empowering Young LearnersLearners

Empowering Young Empowering Young LearnersLearnersWe need students to make sense of We need students to make sense of problem situations problem situations as well asas well as each each other’s thinking.other’s thinking.

Consider these students as they reason Consider these students as they reason about division.about division.

Empowering Young Empowering Young LearnersLearnersWe need students to make sense of We need students to make sense of problem situations problem situations as well asas well as each each other’s thinking.other’s thinking.

Consider these students as they reason Consider these students as they reason about division.about division.

Notice how the teacher’s questions Notice how the teacher’s questions focus on making sense of the problem. focus on making sense of the problem.

Empowering Young Empowering Young LearnersLearnersWe need students to make sense of We need students to make sense of problem situations problem situations as well asas well as each other’s each other’s thinking.thinking.

Consider these students as they reason Consider these students as they reason about remainders.about remainders.

Notice how they need support to construct Notice how they need support to construct viable arguments and critique the viable arguments and critique the reasoning of others. reasoning of others.

The 8 Standards for Mathematical Practice:

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

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

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.