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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 602Professional Development | February 17, 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 • What’s Different • Math Performance Tasks • Formative assessment • 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. Foil FOIL • (a + b)(c +d) = ac + bc + ad + bd • Use the distributive property • This IS the distributive property when a is a sum: a(x + y) = ax + ay • Sum of products = product of sums • It works for trinomials and polynomials in general

  16. Answers are a black hole:hard to escape the pull • Answer getting short circuits mathematics, especially making mathematical sense • High-achieving countries devise methods for slowing down, postponing answer getting

  17. A dragonfly can fly 50 meters in 2 seconds. What question can we ask?

  18. Rate × Time = Distance

  19. 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 the 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

  20. Bob, Jim and Cathy each have some money. The sum of Bob's and Jim's money is $18.00. The sum of Jim's and Cathy's money is $21.00. The sum of Bob's and Cathy's money is $23.00.

  21. 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 • Near end of chapter, external problems needed, e.g. Shell Centre

  22. 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 do OK on unit 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

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

  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 apples + 5 apples = 8 apples • 3 ones + 5 ones = 8 ones • 3 tens + 5 tens = 8 tens • 3 inches + 5 inches = 8 inches • 3 tenths + 5 tenths = 8 tenths • 3(¼) + 5(¼) = 8(¼) • 3(x + 1) + 5(x+1) = 8(x+1) Daro, 2010

  33. There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

  34. A Student’s Response There are 125 sheep and 5 dogs in a flock. How old is the shepherd? 125 x 5 = 625 extremely big 125 + 5 = 130 too big 125 - 5 = 120 still big 125  5 = 25 That works!

  35. 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)

  36. 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.

  37. (Page 39)

  38. Critical Areas: Grade 6 Ratio and Rate: • Connecting to whole number multiplication and division. • Equivalent ratios derive from, and extend, pairs of rows in the multiplication table. Number: • Dividing fractions in general. • Extending rational number system to negative integers (order, absolute value).

  39. Critical Areas: Grade 6 Expression and Equations: • Use of variables, equivalent expressions. • Solve simple one-step equations. Statistics: • Different ways to measure center of data. Geometry: • Find areas of shapes by decomposing, rearranging or removing pieces, and relating shapes to rectangles.

  40. (Page 46)

  41. Critical Areas: Grade 7 Proportional Relationships: • Use to solve variety of percent problems. • Graph and understand unit rate as the steepness of the line, or slope. Unified Understanding of Number: • Fraction, decimal and percent are different representations of rational numbers. • Same properties and operations apply to negative numbers. • Use to formulate equations, solve problems

  42. Critical Areas: Grade 7 Geometry: • Area and circumference of a circle. • Surface area of solids. • Scale drawing and informal constructions. • Relationships among plane figures Data: • Compare two data distributions, to see differences between populations. • Informal work with random sampling.

  43. (Page 52)

  44. Critical Areas: Grade 8 Proportional Reasoning: • Equations for proportions as special linear equations: y = mx or y/x = m • Constant of proportionality is the slope, graphs are lines through the origin. Functions: • Function as a rule that assigns to each input exactly one output. • Functions describe situations where one quantity determines another.

  45. Critical Areas: Grade 8 Geometry: Ideas about distance and angles, how they behave under translations, rotations, reflections and dilations and ideas about congruence and similarity

  46. 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)

  47. A Coherent Curriculum Is organized around the big ideas of mathematics Clearly shows how standards are connected within each grade Builds concepts through logical progressions across grades that reflect the discipline itself.