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Empowering Learners through the Standards for Mathematical Practice of the Common Core

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Empowering Learners through the Standards for Mathematical Practice of the Common Core

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  1. Empowering Learners through the Standards for Mathematical Practice of the Common Core Juli K. Dixon, Ph.D. University of Central Florida juli.dixon@ucf.edu

  2. Solve this… 3 ÷ 1/7

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

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

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

  6. Perspective… When asked to justify the solution to 3 ÷ 1/7 Another student said 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.”

  7. Perspective… When asked to justify the solution to 3 ÷ 1/7 Another student said 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 what does it have to do with the CCSSM?

  8. Background of the CCSSM • 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

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

  10. 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).

  11. 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). • We’ve already met this challenge in Florida. How can we use our momentum to take us further and deeper?

  12. NGSSS Content Standards Wordle

  13. CCSSM Content Standards Wordle

  14. 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

  15. Content Standards Measurement and Data K.MD Describe and compare measurable attributes. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. 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. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

  16. Content Standards Domain Measurement and Data K.MD Describe and compare measurable attributes. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. 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. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. Cluster Standard Standard Cluster Standard

  17. 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).

  18. Making Sense of the Mathematical Practices The Standards for Mathematical Practice are based on: • 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).

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

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

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

  22. Standards for Mathematical Practice Wordle

  23. 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.

  24. Perspective… Lead with Mathematical Practices Implement CCSS beginning with mathematical practices, Revise current materials and assessments to connect to practices, and Develop an observational scheme for principals that supports developing mathematical practices. (CSMC, 2010)

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

  26. 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.

  27. Impact on Depth… (CCSS) Domain Number & Operations in Base Ten 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. Cluster Standard

  28. Solve this…

  29. Solve this…

  30. What did you do?

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

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

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

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

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

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

  37. Impact on Depth… (NGSSS) 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.

  38. Impact on Depth… (CCSS) Operations & Algebraic Thinking 3.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 100. 7. Fluently multiply within 100, using strategies such as the relationship between multiplication and division or properties of operations...

  39. Impact on Depth… (CCSS) Operations & Algebraic Thinking 3.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 100. 7. Fluently multiply within 100, using strategies such as the relationship between multiplication and division or properties of operations...

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

  41. 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?

  42. 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?

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

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

  45. Reason abstractly and quantitatively 2 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.

  46. Reason abstractly and quantitatively 2 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?

  47. Reason abstractly and quantitatively 2 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