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Moving Ahead with the Common Core Learning Standards for Mathematics

Moving Ahead with the Common Core Learning Standards for Mathematics. CFN 609 Professional Development | February 9, 2012 RONALD SCHWARZ Math Specialist, America’s Choice,| Pearson School Achievement Services. Block Stack.

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Moving Ahead with the Common Core Learning Standards for Mathematics

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  1. Moving Ahead with the Common Core Learning Standards for Mathematics CFN 609Professional Development | February 9, 2012 RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services

  2. Block Stack 25 layers of blocks are stacked; the top four layers are shown. Each layer has two fewer blocks than the layer below it. How many blocks are in all 25 layers? Math Olympiad for Elementary and Middle Schools

  3. AGENDA • Standards for Math Content: Conceptual Shifts • Standards for Math Practice • What’s Different • Impact on Instruction • Math Performance Tasks • Resources

  4. What are Standards? • Standards define what students should understand and be able to do. • The US has been a jumble of 50 different state standards. Race to the bottom or the top? • Any country’s standards are subject to periodic revision. • But math is more than a list of topics.

  5. DESPITE GAINS, ONLY 39% OF NYC 4TH GRADERS AND 26% OF 8TH GRADERS ARE PROFICIENT ON NATIONAL MATH TESTS NAEP & NY STATE TEST RESULTS NYC MATH PERFORMANCE PERCENT AT OR ABOVE PROFICIENT 4th Grade 8th Grade 2003 2009 2003 2009 2003 2009 2003 2009 NAEP NY State Test NAEP NY State Test

  6. What Does “Higher Standards” Mean? • More Topics? But the U.S. curriculum is already cluttered with too many topics. • Earlier grades? But this does not follow from the evidence. In Singapore, division of fractions: grade 6 whereas in the U.S.: grade 5 (or 4)

  7. Lessons Learned • TIMSS: math performance is being compromised by a lack of focus and coherence in the “mile wide. Inch deep” curriculum • Hong Kong students outscore US students in the grade 4 TIMSS, even though Hong Kong only teaches about half the tested topics. US covers over 80% of the tested topics. • High-performing countries spend more time on mathematically central concepts: greater depth and coherence. Singapore: “Teach less, learn more.”

  8. Common Core State Standards Evidence Base • English language arts • Australia • New South Wales • Victoria • Canada • Alberta • British Columbia • Ontario • England • Finland • Hong Kong • Ireland • Singapore Mathematics Belgium (Flemish) Canada (Alberta) China Chinese Taipei England Finland Hong Kong India Ireland Japan Korea Singapore For example: Standards from individual high-performing countries and provinces were used to inform content, structure, and language.

  9. Why do students have to do math problems? • To get answers because Homeland Security needs them, pronto • I had to, why shouldn’t they? • So they will listen in class • To learn mathematics

  10. Answer Getting vs. Learning Mathematics United States How can I teach my kids to get the answer to this problem? Use mathematics they already know. Easy, reliable, works with bottom half, good for classroom management. Japan How can I use this problem to teach mathematics they don’t already know?

  11. Three Responses to a Math Problem • Answer getting • Making sense of the problem situation • Making sense of the mathematics you can learn from working on the problem

  12. Answer Getting Getting the answer one way or another and then stopping Learning a specific method for solving a specific kind of problem (100 kinds a year)

  13. Butterfly method

  14. Use butterflies on this TIMSS item 1/2 + 1/3 +1/4 =

  15. Answers are a black hole:hard to escape the pull • Answer getting short circuits mathematics, especially making mathematical sense • Very habituated in US teachers versus Japanese teachers • High-achieving countries devise methods for slowing down, postponing answer getting

  16. Posing the problem • Whole class: pose problem, make sure students understand the language, no hints at solution • Focus students on the problem situation, not the question/answer game. Hide question and ask them to formulate questions that make situation into a word problem • Ask 3-6 questions about the same problem situation; ramp questions up toward key mathematics that transfers to other problems

  17. What problem to use? • Problems that draw thinking toward the mathematics you want to teach. NOT too routine, right after learning how to solve • Ask about a chapter: what is the most important mathematics students should take with them? Find problems that draw attention to this math • Begin chapter with this problem (from lesson 5 thru 10, or chapter test). This has diagnostic power. Also shows you where time has to go. • Near end of chapter, external problems needed, e.g. Shell Centre

  18. What do we mean by conceptual coherence? Apply one important concept in 100 situations rather than memorizing 100 procedures that do not transfer to other situations: • Typical practice is to opt for short-term efficiencies, rather than teach for general application throughout mathematics. • Result: typical students can get 80s on chapter tests, but don’t remember what they ‘learned’ later when they need to learn more mathematics • Use basic “rules of arithmetic” (same as algebra) instead of clutter of specific named methods • Curriculum is a ‘mile deep’ instead of a ‘mile wide’

  19. Teaching against the test 3 + 5 = [ ] 3 + [ ] = 8 [ ] + 5 = 8 8 - 3 = 5 8 - 5 = 3

  20. Write a word problem that could be modeled by a + b = c • Result or total unknown; e.g. 5 + 3 = ? – Mike has 5 pennies. Sam gives him 3 more. How many does Mike have now? • Change or part unknown; e.g., 5 + ? = 8 – Mike has 5 pennies. Sam gives him some more. Now he has 8. How many did he get from Sam? • Start unknown; e.g., ? + 3 = 8 – Mike has some pennies. He gets 3 more. Now he has 8. How many did he have at the beginning?

  21. Some Addition and Subtraction Situations

  22. Some More Addition and Subtraction Situations

  23. Some Multiplication and Division Situations

  24. Anna bought 3 bags of red gumballs and 5 bags of white gumballs. Each bag of gumballs had 7 pieces in it. Which expression could Anna use to find the total number of gumballs she bought? A. (7 × 3) + 5 = B. (7 × 5) + 3 = C. 7 × (5 + 3) = D. 7 + (5 × 3) =

  25. Math Standards Mathematical Practice: varieties of expertise that math educators should seek to develop in their students. Mathematical Content: Mathematical Performance: what kids should be able to do. Mathematical Understanding: what kids need to understand.

  26. Standards for Mathematical Content Organization by Grade Bands and Domains (Common Core State Standards Initiative 2010)

  27. Progressions within and across Domains Daro, 2010

  28. Math Content Greater focus – in elementary school, on whole number operations and the quantities they measure, specifically: Grades K-2 Addition and subtraction Grades 3-5 Multiplication and division and manipulation and understanding of fractions (best predictor algebraic performance) Grades 6-8 Proportional reasoning, geometric measurement and introducing expressions, equations, linear algebra

  29. Why begin with unit fractions?

  30. Unit Fractions

  31. Units are things that you count • Objects • Groups of objects • 1 • 10 • 100 • ¼ unit fractions • Numbers represented as expressions • Daro, 2010

  32. Units add up • 3 pennies + 5 pennies = 8 pennies • 3 ones + 5 ones = 8 ones • 3 tens + 5 tens = 8 tens • 3 inches + 5 inches = 8 inches • 3 ¼ inches + 5 ¼ inches = 8 ¼ inches • 3(1/4) + 5(1/4) = 8(1/4) • 3(x + 1) + 5(x+1) = 8(x+1) Daro, 2010

  33. How CCLS support change The new standards support improved curriculum and instruction due to increased: FOCUS, via critical areas at each grade level COHERENCE, through carefully developed connections within and across grades RIGOR, including a focus on College and Career Readiness and Standards for Mathematical Practice throughout Pre-K through 12 (Massachusetts State Education Department)

  34. Critical Areas • There are two to four critical areas for instruction in the introduction for each grade level, model course or integrated pathway. • They bring focus to the standards at each grade by providing the big ideas that educators can use to build their curriculum and to guide instruction.

  35. Critical Areas: Kindergarten • Whole numbers: comparing, joining, separating, counting objects that remain or in combined sets • Shapes: shape, orientation, spatial relationships • Two-dimensional: square, triangle, circle, rectangle, hexagon • Three-dimensional: cube, cone, cylinder, sphere

  36. Critical Areas: Grade 1 • Addition and subtraction fluency to 20: model add-on, take-from, put-together, take-apart, compare • Place value: beginning of grouping by 10s and 1s • Measurement of length: concept of equal-sized units, transitivity • Figures: compose and decompose plane and solid shapes, understand part-whole relationships

  37. Critical Areas: Grade 2 • Base ten: place value of 1000s, 100s, 10s and 1s; counting by 5s, 10s, 100s • Addition and subtraction fluency within 100; problems within 1000 • Rulers: recognize and use inches and centimeters • Shapes: describe and analyze by sides and angles; begin foundation for later area, volume, congruence, similarity, symmetry

  38. Critical Areas: Grade 3 • Multiplication and division: using equal-sized groups, arrays, area models; finding unknown number of groups or unknown group size; solving problems involving single-digit factors • Fractions: built out of unit fractions, use to represent part of a whole, is relative to size of the whole, use to represent numbers equal to greater than, less than one

  39. Critical Areas: Grade 3 • Area of a shape: square units; since rectangular arrays can be decomposed into identical rows or columns, area is connected to multiplication • Shapes: classified by sides and angles; area of part of a shape expressed as a unit fraction of the whole

  40. Critical Areas: Grade 4 • Base ten: place value to 1,000,000, distributive property, multi-digit multiplication, estimate or mentally calculate products, quotients with multi-digit dividends, interpret remainders based on context • Fractions: equivalence, addition and subtraction with like denominators, multiply fraction by whole number • Geometric figures analyzed and classified by properties: parallel or perpendicular sides, angle measures, symmetry

  41. Critical Areas: Grade 5 • Fractions: fluency in addition and subtraction with unlike denominators, multiplication, division in special cases (unit fractions by whole numbers and vice-versa) • Base ten: two-digit divisors

  42. Critical Areas: Grade 5 • Decimals: place value, operations to hundredths with estimation, relationship between decimals and whole numbers (e.g., when multiplied by a power of ten can become a whole number) • Volume: use cubic units, estimating and measuring, finding volume of right rectangular prism

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