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Empowering Learners through the Common Core State Standards 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 State Standards?

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Background of the CCSSM (Common Core State Standards) 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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CCSSM Content Standards Wordle

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Content Standards Critical Areas – much like our big ideas Domains – group related clusters Clusters – group related standards Standards – define what students should know and be able to do

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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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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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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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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 Impact on Depth… (CCSS)

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

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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...

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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...

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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 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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The 8 Standards for Mathematical Practice: With which practice were the third 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 practice were the third 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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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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Where do we start? ….in Kindergarten.

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Making Sense of Numbers to 20 How do we help students to truly make sense of number patterns?

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Making Sense of Numbers to 20 How do we help students to truly make sense of number patterns? Consider number pairs for 8.

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K.OA.3: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = and 5 = 4 + 1). Describing the Standards

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Watch the following video of a Kindergarten class as they explore Number Pairs for 8.

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Describing the Standards Watch the following video of a Kindergarten class as they explore Number Pairs for 8. Look for how patterns are addressed.

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The 8 Standards for Mathematical Practice: Which practice was of focus here? 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 practice was of focus here?

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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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The 8 Standards for Mathematical Practice: What is your biggest take away regarding the practices and your role? 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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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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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. We need to be aware of the changes so that we can prepare ourselves to meet the needs of our students. We need to be aware of the changes so that we can prepare ourselves to meet the needs of our students.

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

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