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Core Content State Standards … for Learning … as Learning Math in Grades 3-12

Core Content State Standards … for Learning … as Learning Math in Grades 3-12. Daniel J. Heisey, Ph.D. Mathematics Coordinator NJ Department of Education daniel.heisey@doe.state.nj.us. Fear and Angst. Presentation Roadmap. NAEP – gatekeeper for NCLB

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Core Content State Standards … for Learning … as Learning Math in Grades 3-12

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  1. Core Content State Standards… for Learning… as LearningMath in Grades 3-12 Daniel J. Heisey, Ph.D. Mathematics Coordinator NJ Department of Education daniel.heisey@doe.state.nj.us NJDOE 2011

  2. Fear and Angst NJDOE 2011

  3. NJDOE 2011

  4. Presentation Roadmap • NAEP – gatekeeper for NCLB • Conceptual understanding of fractions • Core Content State Standards [CCSS] • Grade 3 understand fractions as numbers and build fractions from unit fractions • Grade 4 add and subtract fractions with like denominators and generate decimal equivalents • Grade 5 multiply fractions (including mixed numbers) and divide fractions in special cases • Diagnostics and learning progressions NJDOE 2011

  5. NAEP Data Reading and Math Gap (ETS summary, 2010) African American, Latino, and poor of all races vs. their wealthier, mostly white peers • By 4th grade, they are 2 yearsbehind • By 8th grade, they are 3 yearsbehind • By 12th grade, they are 4 yearsbehind High School Dropouts (75% in jail are school dropouts) 68% of six graders score basic (or worse) in math Only 17% of H.S. seniors are proficient in math NJDOE 2011

  6. NAEP Released Item What fraction of the figure is shaded? Answer: _______________ Did you us a calculator on this question? ______ Grade 4 NAEP 2007 22% incorrect NJDOE 2011

  7. NAEP Released Item Luis is making a game spinner. He wants the chance of landing on red (R) to be twice the chance of landing on blue (B). Show how he could label his spinner. Number of blues ____ Number of reds ____ Grade 4 NAEP 2007 59% incorrect NJDOE 2011

  8. Gene Wilhoit Gene Wilhoit is the executive director of the Council of Chief State School Officers [CCSSO]. This video was made in November 2010. It is a presentation to the U. S. Congress about the federal role in the state-initiated Common Core initiative … a historical perspective. NJDOE 2011

  9. Gene WilHoit in Congress NJDOE 2011

  10. Fraction beginnings: Part vs. WholeGrade 2: CCSS 2.G.3Partition circles and rectangles into two or four equal shares and describe shares using words halves, thirds, half of, third of, etc., and describe the whole as two halves, three thirds, four fourths.Recognize equal shares of identical wholes need not have the same shape.Numerals for fractions (1/2, 1/3, 1/4, etc.) are not used at grade 2.Two fractions are compared only if they refer to the same whole.Grade 3: NJDOE 2011

  11. . Note: The above is not a complete map (e.g. proportional relationships). NJDOE 2011

  12. NJDOE 2011

  13. Critical areas in grades 3-5 -- plus grade 6 ….. HANDOUT #1 NJDOE 2011

  14. The first mention of multiplying or dividing fractions in NJCCCS was “implied” in this single standard in grade 6 >>>>>>>>>>> There is no lead up, and no progression of ideas. The grade 5 NJCCCS is not very helpful. • In Grade 5: • 4.1.5 B. Numerical Operations • 2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with: • Pencil-and-paper • Mental math • Calculator • In Grade 6: • 4.1.6 B. Numerical Operations • 2. Construct, use, and explain procedures for performing calculations with fractions and decimals with: • Pencil-and-paper • Mental math • Calculator NJCCCS vs. CCSS 4.NF.3 & 4.NF.4 …………... HANDOUTs #2 & #3 NJDOE 2011

  15. Common Core demands we revamp the mile-wide, inch-deep approach in curriculum and textbooks. Key moments in the curriculum (like 4.NF.4) demand that we slow down and devote more time to allow for reasoning / thinking / discussing as well as the necessary hard work and practice. • CCSS gives three years to the division of fractions thread: • In grade 4, we multiply a fraction by a whole number. • In grade 5, we multiply a fraction by a fraction and we divide a unit fraction by a whole number or a whole number by a unit fraction. • Finally, in grade 6, we divide a fraction by a fraction. NJDOE 2011

  16. Build on prior work of multiplying whole numbers. • Build fractions from unit fractions: • 5/4 means 5 x ¼. It is also ¼ + ¼ + ¼ + ¼ + ¼, which builds from unit fractions using additive reasoning. (see 4.NF.3) • (3) We achieve the more complex case 5 x ¾ by saying, 5 times 3/4equals (5 times 3) fourths equals 15 fourths. This stresses properties of operations, making arithmetic a rehearsal for algebra. Source: http://www.p12.nysed.gov/ciai/mst/math/standards/revisedg4.html NJDOE 2011

  17. Fraction progressions … • Grade 3: Develop an understanding of fractions as numbers. • Grade 4: Understand fraction equivalence and ordering. Build fractions from unit fractions and apply and extend previous understandings of operations on whole numbers. Use decimal notation for fractions and compare fractions. • Grade 5: Use equivalent fractions to add and subtract fractions (including mixed numbers w unlike denominators). Apply and extend previous understandings of multiplication and division to multiply and divide fractions. NJDOE 2011

  18. Thinking about 3/4 … NJDOE 2011

  19. Unit Fractions … 1/b … as building blocks Grade 3: Students name the “numeral” for each unit fraction as 1/2, 1/3, 1/4, etc. Fractions are built from unit fractions. 3/4 is 3 copies of 1/4 … 3/4 = 1/4 + 1/4 + 1/4 4/5 is 4 copies of 1/5 … 4/5 = 1/5 + 1/5 + 1/5 + 1/5 Note: No need to introduce proper or improper fractions. The quantity 5/3 is the sum of 5 parts of a whole divided into 3 equal parts. Students can easily find 5/3 on the number line. Grade 3: Students identify “equivalent” fractions as having the same size or the same position on the number line. NJDOE 2011

  20. Domain: Number – FractionsCluster: Develop understanding of fractions as numbers. 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts … each part has size 1/b and the endpoint of that part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. NJDOE 2011

  21. Domain: Number – FractionsCluster: Develop understanding of fractions as numbers. 3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or at the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6, = 2/3 … explain using visual fraction models. c. Express whole numbers as fractions, & recognize fractions equivalent to whole numbers. Examples: Express 3 as 3/1 & recognize 6/1 = 6; locate 4/4 and 1 at the same point on a number line. d.Compare two fractions with the same numerator or denominator by reasoning about their size. Comparisons need to refer to the same whole. Use symbols >, =, < for comparisons … explain using visual fraction models. NJDOE 2011

  22. Conceptual Understanding ¾ What happens to the value of a fraction if: -- the numerators is increased by 1 ? -- the denominator is decreased by 1 ? -- the denominator is increased by 1 ? -- the value of the numerator and denominator are doubled ? Which comparison is true? … Explain WHY using a number line. ½ < ¾ versus ¾ < ½ NJDOE 2011

  23. Grade 4 Fractions: Equivalent Fractions Grade 4: CCSS 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) using fraction models, with attention to how the number and size of the parts differ even though the two fractions are the same size. Use this principle to recognize and generate equivalent fractions. This fundamental property of equivalent fractions impacts much of the computation and procedural work in grade 4 … Comparing … adding … subtracting … finite decimals … NJDOE 2011

  24. Grade 4 : Fraction Questions Create an area model for 2/3 = (4 x 2)/(4 x 3). How can 2/5 + 1/3 be represented as a length? Part of a whole vs. Part of a part. Convert mixed numbers to improper fractions. Multiply fractions by whole numbers. What visual fraction model shows 0.2 > 0.17 ? NJDOE 2011

  25. Area models multiply whole numbers 14 x 6 Each block shown represents 1/6 multiply unit fractions 1/2 x 1/3 1/3 of 1/2 = 1/6 1/2 of 1/3 = 1/6 NJDOE 2011

  26. Multiply Fractions (Area Model) 2/3 x 4/5 = 8/15 (2-dimensional arrays) 2/3 of 15/15 = 10/15 4/5 of 15/15 = 12/15 either8/15 is 4/5 of 10/15 …or 8/15 is 2/3 of 12/15 NJDOE 2011

  27. Multiply mixed numbers (area model) A typical grade 5 problem 4 ½ x 7 ¼ = ?? A composite model showing 4 areas 7 ¼ 4 ½ 4 ½ x 7 ¼ = (4 x 7) + (½ x 7) + (4 x ¼) + (½ x ¼) 32 ⅝ = 28 + 3 ½ + 1 + ⅛ NJDOE 2011

  28. Length model :: Unit Fractions How can 2/5 + 1/3 be represented as a length? Draw a model of 2/5 plus 1/3 using the number line. 01 Reference: Lamon, Susan. Teaching Fractions and Ratios for Understanding, 2nd Edition, New York, NY: Routledge, 2008. NJDOE 2011

  29. Length model :: Unit Fractions What fraction is located at the point “X”? 0X2/3 Reference: Lamon, Susan. Teaching Fractions and Ratios for Understanding, 2nd Edition, New York, NY: Routledge, 2008. NJDOE 2011

  30. Part of a whole vs. Part of a part Part of a part: Bill had 2/3 of a cup of juice. He drank 1/2 of his juice. How much juice did Bill have left? This problem cannot be solved by subtracting 2/3 - 1/2 because the 2/3 refers to a cup … but the 1/2 refers to the amount of juice Bill had and not to a cup of juice. A similar problem: If ¼ of a garden is planted with daffodils, 1/3 with tulips and the rest with vegetables, what fraction of the garden is planted with flowers? NJDOE 2011

  31. Mixed and Improper Fractions 4 2/5= 4 + 2/5 = 1 + 1 + 1 + 1 + 2/5 Tricks are not helpful to students with memory problems, think about the concept and purpose to the calculations “Four and two – fifths” … “and” says to add … 4 + 2/5 = 5/5 + 5/5 + 5/5 + 5/5 + 2/5 = 20/5 + 2/5 = 22/5 Trick: The whole number times the fraction’s denominator plus the numerator equals the new numerator. The number in the denominator does not change. NJDOE 2011

  32. Multiplication of fractions by a whole number 4 2/5 is not the same as 4 x 2/5 Tricks are not always helpful … “Four times two fifths” … “times” means add 2/5 four times 4 x 2/5 … 2/5 + 2/5 + 2/5 + 2/5 = 8/5 TRICK:Multiply the whole number by the numerator and that product becomes the new numerator in the answer. NJDOE 2011

  33. Visual models for decimal fractions The “tenth” scale vs. the “hundredths” scale … 0.2 < 0.17 is not true 2 is “less than” 17 but … 0.2 = 0.20 which is 0.03 greater than 0.17 Compare all fractions by referencing the same part ... 0 0.1 0.2 1/4 0.3 0 0.10 0.20 0.25 0.30 NJDOE 2011

  34. How do Fraction Models Differ Across Grades Concrete Manipulatives: Grades K-2 Representational Drawings: Grades 2-3 Abstract Number Procedures: Grades 3-4 1. There were 16 apples. Rhonda ate 1/4 of them. How many are left? 2. Mom used one-third of 12 eggs. How many are left? 3. Tom ate 4 hazelnuts, which was 1/8 of the nuts. How many were there in the first place? 4. Lisa used $5 to buy a gift which took 1/3 of her savings. How much did she have in the beginning? How much does she have now? NJDOE 2011

  35. The Khan Academy – a resource The web-access to this video library is free and individual videos can be purchased for 99¢. The videos show only a blackboard and chalk writing. The invisible “teacher” explains the math by audio. Not the lessons are not aligned to the CCSS skills. For example, the video is multiplying fractions (a grade 5 CCSS skill) with integer math included (a grade 6 CCSS skill). NJDOE 2011

  36. COMMON CORE MATH STANDARDSClearer, Fewer, Higher No more … … content a mile wide and an inch deep … standards that require “un-packing” … skills re-taught (spiraled) in the next grades NJDOE 2011

  37. Addition of Whole Numbers NJDOE 2011

  38. NJCCCS CCSS % of Standards/ CPI’s Grade Levels Grade Levels NJDOE 2011

  39. The CCSS Format of Standards Domain (Topic) Grade Level Standard Cluster Algebra Symbol NJDOE 2011

  40. CCSS Developer Commentary http://www.americaschoice.org/uploads/Common_Core_Standards_Resources/PhilDaro_MathStandards/PhilDaro_MathStandards.html In this video Phil Daro applauds Common Core developers as being mindful of common sense skill levels selection “less is more” idea. http://successatthecore.com/teacher_development_featured_video.aspx?v=44 NJDOE 2011

  41. CCSS Assessments coming 2014-15 In 2014-15, assessments for the Common Core will be administered via the internet. The Partnership for Assessment of Readiness for College and Careers [PARCC] is an alliance of states working together to develop common assessments serving nearly 25 million students. PARCC’s work is funded through a four-year 185 million dollar grant from the U.S. DOE. PARCC’s partners include 200 higher education institutions and is led by its member states and managed by Achieve. PARCC’s goal is to ensure that all students graduate from high school college and career ready. (See www.parcconline.org ) NJDOE 2011

  42. Kinds of Assessments • Readiness • Benchmarks • Diagnostics • Common Core is a new paradigm. Teachers will teach to mastery so that all students can demonstrate mastery. • Diagnostic assessments will be needed to manage the intervention events for struggling math students. NJDOE 2011

  43. ASSESSMENT NOTE RE: Student Progress “It is impossible for a norm-referenced test to align with standards. Norm-reference tests tell you how well students are doing compared to each other … Standards mean that student progress must be compared to the standard, not to how well or poorly others do.” • Quoted from … Workshop: Teaching to Academic Standards NJDOE 2011

  44. Readiness Test Items 1. A batch of muffins requires 2/3 cups of sugar. A loaf of zucchini bread requires 3/4 cups of sugar. How many cups of sugar are needed to make one batch of each recipe? a. 5/7 b. 1 5/12 c. 1 1/2 d. 1 7/12 2.  What is 4 divided by one-half? a. 1/2 b. 1/8 c. 2 d. 8 3. If AX = B and A, B, and X are not equal to zero, then ______ ? a.X = A divided by B b.X = A minus B c.X = B divided by A d.X = B times A 4. Which number is largest ?  a. 0.72 b. 0.080 c. 8/9 d. 6/7 NJDOE 2011

  45. Diagnostic Test Items Find the equivalent common fraction. 1. 1/4 a) 11/14 b) 3/5 c) 2/8 d) 4/1 2. 5/6 a) 6/5 b) 50/60 c) 15/16 d) 5/11 3. 1 3/4 a) 7/4 b) 13/4 c) 7/14 d) 3/14 4. 3 2/5 a) 32/5 b) 17/5 c) 6/5 d) 2/32 NJDOE 2011

  46. Diagnostic Testing Diagnostic tests assess fewer skills – “less is more” Skills are ordered as a learning progression -- Four test items assess each skill -- Each Item has similar difficulty -- 3 or 4 items correct demonstrates mastery Guessing is NOT a factor: guess 1 out of 4 25% guess 2 out of 4 6.25% guess 3 out of 4 1.6% guess 4 out of 4 0.4% … very few false positives ! NJDOE 2011

  47. Student scores and number correct per skill (left tally matrix) • Student mastery indicators (right OK matrix) NJDOE 2011

  48. Classroom Learning Progressions A learning progression is a carefully sequenced set of building blocks that students must master en route to mastering a more distant curricular aim. -- James Popham, ASCD, 2007 Classroom learning progressions are: – often “customized”; – not etched in stone; – often referred to as learning trajectories; and, – needed to find where students are at. NJDOE 2011

  49. A scoop holds ¾ of a cup. How many scoops of bird seed are needed to fill a bird feeder that holds 3 cups? Show how to use pictures to solve this problem. … or explain your solution in words. Fractions and Word Problems Number line l__ı__ı__ı__l__ı__ı__ı__l__ı__ı__ı__l 1 1 1 2 2 2 3 3 3 4 4 4 Arrays 1 2 3 12 3 1 2 3 1 2 3 NJDOE 2011

  50. Length model … explained … Draw a model of 2/5 plus 1/3 using the number line. 0 ← 2/5 → ↑ ← 1/3 → ↑1 11/15 count unit fractions … 1/5 = 3/15 … 1/3 = 5/15 therefore ... 2/5 + 1/3 = 6/15 + 5/15 = 11/15 NJDOE 2011

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