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

Solve this… 3 ÷ 1/7

Perspective… A student said this… When asked to justify the solution to 3 ÷ 1/7

Perspective… 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.

Perspective… 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 justification?

Perspective… Another student said this… When asked to justify the solution to 3 ÷ 1/7 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… Another student said this… When asked to justify the solution to 3 ÷ 1/7 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 Minnesota adopted the CCSS in ELA/literacy only 45 States + DC have adopted the Common Core State Standards

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). Weve already met this challenge in Florida. How can we use our momentum to take us further and deeper?

NGSSS Content Standards Wordle

CCSSM Content Standards Wordle

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

Content Standards 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.

Content Standards 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. Domain Cluster Standard Cluster

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).

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 Councils (NRC) Adding It Up (NRC, 2001).

NCTM Process Standards: Making Sense of the Mathematical Practices Problem Solving Reasoning and Proof Communication Representation Connections

NRC Strands of Mathematical Proficiency: Making Sense of the Mathematical Practices Adaptive Reasoning Strategic Competence Conceptual Understanding Procedural Fluency Productive Disposition

NRC Strands of Mathematical Proficiency: Making Sense of the Mathematical Practices Adaptive Reasoning Strategic Competence Conceptual Understanding Procedural Fluency Productive Disposition

Standards for Mathematical Practice Wordle

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.

Perspective… 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)

The 8 Standards for Mathematical Practice: Making Sense of the Mathematical Practices 1Make sense of problems and persevere in solving them 2Reason abstractly and quantitatively 3Construct viable arguments and critique the reasoning of others 4Model with mathematics 5Use appropriate tools strategically 6Attend to precision 7Look for and make use of structure 8Look for and express regularity in repeated reasoning

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.

Number & Operations in Base TenNBT 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)

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?

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)?

The 8 Standards for Mathematical Practice: With which practices were the fourth grade students engaged? 1Make sense of problems and persevere in solving them 2Reason abstractly and quantitatively 3Construct viable arguments and critique the reasoning of others 4Model with mathematics 5Use appropriate tools strategically 6Attend to precision 7Look for and make use of structure 8Look for and express regularity in repeated reasoning

The 8 Standards for Mathematical Practice: With which practices were the fourth grade students engaged? 1Make sense of problems and persevere in solving them 2Reason abstractly and quantitatively 3Construct viable arguments and critique the reasoning of others 4Model with mathematics 5Use appropriate tools strategically 6Attend to precision 7Look for and make use of structure 8Look for and express regularity in repeated reasoning

What does it mean to use strategies to multiply? When do students begin to develop these strategies? Impact on Depth…

Grade 3 Big Idea 1: Develop understanding of multiplication and division and strategies for basic multiplication facts and related division facts. MA.3.A.1.2: Solve multiplication and division fact problems by using strategies that result form applying number properties. Impact on Depth… (NGSSS)

Impact on Depth… (CCSS) Operations & Algebraic Thinking3.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 Fluently multiply within 100, using strategies such as the relationship between multiplication and division or properties of operations...

Impact on Depth… (CCSS) Operations & Algebraic Thinking3.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 Fluently multiply within 100, using strategies such as the relationship between multiplication and division or properties of operations...

Consider 6 x 7 What does it mean to use strategies to multiply?

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?

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?

Making Sense of Multiplication Consider 6 x 7 How about 4 x 27?

The 8 Standards for Mathematical Practice: With which practices were the fourth grade students engaged? 1Make sense of problems and persevere in solving them 2Reason abstractly and quantitatively 3Construct viable arguments and critique the reasoning of others 4Model with mathematics 5Use appropriate tools strategically 6Attend to precision 7Look for and make use of structure 8Look for and express regularity in repeated reasoning

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

Reason abstractly and 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

Reason abstractly and 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? Key words seem helpful 2

Reason abstractly and 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? Key words seem helpful, or are they…. 2

Reason abstractly and 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

Reason abstractly and 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? How would a child who has been conditioned to use key words solve it? 2

Reason abstractly and 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? 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

Reason abstractly and 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 = 16, so… 2

Reason abstractly and 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 = 16, so… 2

Reason abstractly and 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? 8 + __ = 13 (How might making a ten help?) 2

The 8 Standards for Mathematical Practice: Which Practices Have We Addressed? 1Make sense of problems and persevere in solving them 2Reason abstractly and quantitatively 3Construct viable arguments and critique the reasoning of others 4Model with mathematics 5Use appropriate tools strategically 6Attend to precision 7Look for and make use of structure 8Look for and express regularity in repeated reasoning

The 8 Standards for Mathematical Practice: Which Practices Have We Addressed?

Use appropriate tools strategically This practice will be very difficult to capture in textbook-driven instruction.

Use appropriate tools strategically This practice supports hands-on learning This practice supports hands-on learning Tools must include technology Tools must include technology Tools manipulatives, number lines, and paper and pencil Tools manipulatives, number lines, and paper and pencil Mathematically proficient students know which tool to use for a given task. Mathematically proficient students know which tool to use for a given task. 5

Use appropriate tools strategically Consider this Kindergarten class. 5

Use appropriate tools strategically Consider this Kindergarten class. What did you notice? 5

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

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?

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 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? Solving this wouldnt require much perseverance… but what if we said… 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? – Give each person the biggest unbroken piece of cookie possible to start. 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? – Give each person the biggest unbroken piece of cookie possible to start. 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? – Give each person the biggest unbroken piece of cookie possible to start. 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? – Give each person the biggest unbroken piece of cookie possible to start. 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? – Give each person the biggest unbroken piece of cookie possible to start. Engaging Students in Reasoning and Sense Making

Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

Consider this… So how much of a cookie would person A get? - How much is this all together? Engaging Students in Reasoning and Sense Making

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… Engaging Students in Reasoning and Sense Making

How do we support this empowerment? … a lack of understanding [of mathematical content] effectively prevents a student from engaging in the mathematical practices… 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… 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?

Engaging Students in Reasoning and Sense Making We need to question students when they are wrong and when they are right. 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 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. We need to look for opportunities for students to reason about and make sense of mathematics.

Consider this 5 th grade class.

What was the misconception?

What was the misconception? With which practices were the students engaged?

The 8 Standards for Mathematical Practice: How might you change your practice to address these now? 1Make sense of problems and persevere in solving them 2Reason abstractly and quantitatively 3Construct viable arguments and critique the reasoning of others 4Model with mathematics 5Use appropriate tools strategically 6Attend to precision 7Look for and make use of structure 8Look for and express regularity in repeated reasoning

Where do we start?

How do we support this empowerment? What needs to occur at the administrative level? What needs to occur at the administrative level? What needs to occur to support teachers? What needs to occur to support teachers? What needs to occur to support students? What needs to occur to support students?

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. 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. Drill addition/multiplication facts only after your child explores strategies. Help your child become more proficient in using mathematics at home. 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. 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. 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. 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. Social norms need to be established in classroom and professional development settings to address misconceptions in respectful ways.

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