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

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Solve this… 3 ÷ 1/7

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Solve this… 3 ÷ 1/7 Tell someone near you how you solved it.

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

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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.

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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?

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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.

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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?

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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

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Background of the CCSSM Minnesota adopted the CCSS in ELA/literacy only 45 States + DC have adopted the Common Core State Standards

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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).

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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). So what do these standards look like anyway?

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CCSSM Content Standards Wordle for Grades 3-5

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Content 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

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Content Standards 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

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Content Standards Number & Operations in Base Ten4.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.

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Content Standards Number & Operations in Base Ten4.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

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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).

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

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

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

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

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Standards for Mathematical Practice Wordle

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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

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

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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)

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Draft Assessment Claims for Mathematics* SMARTER Balanced Assessment Consortium 1Concepts and Procedures 2Problem Solving 3Communicating Reasoning 4Data Analysis and Modeling * See Draft Item Spec released January 26, 2012

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Content Standards 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

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Solve this…

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What did you do?

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Perspective… What do you think fourth grade students would do? How might they solve 4 x 7 x 25?

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Perspective… Are you observing this sort of mathematics talk in classrooms? Is this sort of math talk important?

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

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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

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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

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Perspective… In an effort to simplify students learning pathways and minimize barriers (stigler, et. al., 1999), teachers often provide students with efficient procedures too early. When we do this – we minimize students opportunities to engage in these practices.

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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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Content Standards 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...

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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 What strategies can we use? What does it mean to use strategies to multiply?

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Consider 6 x 7 What strategies can we use? How can using strategies to multiply these factors help students look for and make use of structure? (SMP7) What does it mean to use strategies to multiply?

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

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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

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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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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

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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

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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

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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

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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

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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

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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

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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

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

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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).

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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?

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

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Consider this 5 th grade class.

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What was the misconception?

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

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The 8 Standards for Mathematical Practice: 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 With which practice were the fifth grade students engaged?

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The 8 Standards for Mathematical Practice: 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 With which practice were the fifth grade students engaged?

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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

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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?

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Describing the Standards … 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).

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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.

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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.

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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.

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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.

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

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