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Teaching Fractions: The differences caused by two kinds of curriculum organization

Teaching Fractions: The differences caused by two kinds of curriculum organization. Liping Ma. W ×. W ÷. W +. W –. Two kinds of curriculum organization. With a c ore subject. Juxtaposed strands. What might they be ?. Primary Statistics. Primary Geometry. School Arithmetic.

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Teaching Fractions: The differences caused by two kinds of curriculum organization

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  1. Teaching Fractions: The differences caused by two kinds of curriculum organization Liping Ma

  2. W × W ÷ W + W – Two kinds of curriculum organization With a core subject Juxtaposed strands What might they be ? Primary Statistics Primary Geometry School Arithmetic Simple Equations Measurement

  3. Circle (perimeter & area); cylinder & cone (area & volume) Primary Statistics Area of triangles & trapezoids; Prism and cubic (volume) Simple equation Area of rectangles Angles & lines Perimeter of rectangles Length / Weight Length Time / Weight Money Division with divisor as a three-digit number Numbers up to 100 , addition and subtraction(with concept of regrouping) Multiplication and division with multiplication tables Numbers up to 10,000 , notation, addition and subtraction Multiplication with multiplier as a two-digit number Division with divisor as a one-digit number Many-digit numbers, notation, addition and subtraction Division with divisor as a two-digit number Multiplication with multiplier as a one-digit number Numbers 11 to 20 , addition and subtraction(with concept of regrouping) Multiplication with multiplier as a three-digit number Fractions – division Ratio and proportion Percents Fractions – multiplication Fractions – addition and subtraction Fractions – meaning and features Fractions – the basic concepts Decimals – multiplication and division Decimals – addition and subtraction Decimals – meaning and features Divisibility ( factors, multipliers, prime number, factorization, GCD, LCM) Numbers 0 to 10 , addition and subtraction Organizing the topics (the tightest chain and breakups) G6 G5 G4 G3 G2 G1

  4. Circle (perimeter & area); cylinder & cone (area & volume) Primary Statistics Area of triangles & trapezoids; Prism and cubic (volume) Simple equation Area of rectangles Angles & lines Perimeter of rectangles Length / Weight Length Time / Weight Money Division with divisor as a three-digit number Numbers up to 100 , addition and subtraction(with concept of regrouping) Multiplication and division with multiplication tables Numbers up to 10,000 , notation, addition and subtraction Multiplication with multiplier as a two-digit number Division with divisor as a one-digit number Many-digit numbers, notation, addition and subtraction Division with divisor as a two-digit number Multiplication with multiplier as a one-digit number Numbers 11 to 20 , addition and subtraction(with concept of regrouping) Multiplication with multiplier as a three-digit number Fractions – division Ratio and proportion Percents Fractions – multiplication Fractions – addition and subtraction Fractions – meaning and features Fractions – the basic concepts Decimals – multiplication and division Decimals – addition and subtraction Decimals – meaning and features Divisibility ( factors, multipliers, prime number, factorization, GCD, LCM) Numbers 0 to 10 , addition and subtraction Organizing the topics (the tightest chain and breakups) G6 G5 G4 G3 G2 G1

  5. Circle (perimeter & area); cylinder & cone (area & volume) Primary Statistics Area of triangles & trapezoids; Prism and cubic (volume) Simple equation Area of rectangles Angles & lines Perimeter of rectangles Length / Weight Length Time / Weight Money Division with divisor as a three-digit number Numbers up to 100 , addition and subtraction(with concept of regrouping) Multiplication and division with multiplication tables Numbers up to 10,000 , notation, addition and subtraction Multiplication with multiplier as a two-digit number Division with divisor as a one-digit number Many-digit numbers, notation, addition and subtraction Division with divisor as a two-digit number Multiplication with multiplier as a one-digit number Numbers 11 to 20 , addition and subtraction(with concept of regrouping) Multiplication with multiplier as a three-digit number Fractions – division Ratio and proportion Percents Fractions – multiplication Fractions – addition and subtraction Fractions – meaning and features Fractions – the basic concepts Decimals – multiplication and division Decimals – addition and subtraction Decimals – meaning and features Divisibility ( factors, multipliers, prime number, factorization, GCD, LCM) Numbers 0 to 10 , addition and subtraction Organizing the topics (the tightest chain and breakups) G6 G5 G4 G3 G2 G1

  6. Circle (perimeter & area); cylinder & cone (area & volume) Primary Statistics Area of triangles & trapezoids; Prism and cubic (volume) Simple equation Area of rectangles Angles & lines Perimeter of rectangles Length / Weight Length Time / Weight Money Division with divisor as a three-digit number Numbers up to 100 , addition and subtraction(with concept of regrouping) Multiplication and division with multiplication tables Numbers up to 10,000 , notation, addition and subtraction Multiplication with multiplier as a two-digit number Division with divisor as a one-digit number Many-digit numbers, notation, addition and subtraction Division with divisor as a two-digit number Multiplication with multiplier as a one-digit number Numbers 11 to 20 , addition and subtraction(with concept of regrouping) Multiplication with multiplier as a three-digit number Fractions – division Ratio and proportion Percents Fractions – multiplication Fractions – addition and subtraction Fractions – meaning and features Fractions – the basic concepts Decimals – multiplication and division Decimals – addition and subtraction Decimals – meaning and features Divisibility ( factors, multipliers, prime number, factorization, GCD, LCM) Numbers 0 to 10 , addition and subtraction Organizing the topics (the tightest chain and breakups) G6 G5 G4 G3 G2 G1

  7. The time allocated for learning fractions Forms to represent/express fractions Students’ prior knowledge (cognitive foundation for learning fractions) Three differences

  8. US G6 : Fractions and percents 101 in 184 pages, 4 of the 8 chapters 1 1 G5 : Fractions and Decimals 131 in 227 pages, 5 of the 9 chapters 6.03 1.71 G4 : Fractions and Decimals60 in 180 pages, 3 of the 9 chapters* 1 13.52 G6 : Operations with fractions / 100 in 591 pages 1 of the 12 chapters G5 : Fractions and Decimals (multiplication & division) 100 in 603 pages, 2 of the 12 chapters 1 118.4 G4 : Fractions and Decimals (addition & subtraction) 92 in 603 pages, 2 of the 12 chapters G3 : Fractions and Decimals44 in 595 pages, one of the 12 chapters China G2 : Unit fractions, wholes and parts, comparing fractions, fraction of a group 12 in 592 pages 1 1 1 G1 : Equal parts, One half, One third and one fourth 2 in 352 pages 176 3 1.82 1 1 1 K : Halves 2 in 352 pages 5.91 49.3 6.55

  9. G1 G4 G5 G4 G6 G3 G5 K G2 G6 US China Time allocated on learning fractions and decimals

  10. K G6 G3 G4 G1 G5 G2 G6 G4 G5 China U. S.

  11. G6 G4 G5 Late G4 • Exposure to fractions • Meaning and features of decimals • Addition and subtraction with decimals • The Divisibility of numbers • Divisors and Multipliers • The numbers divisible by 2, 5, 3 • Prime numbers, composite numbers, factoring prime factors • Greatest common divisor (G. C. D.) • Least common multiple (L. C. M.) • Multiplication and division with decimals • Computations mixed with four basic operations with decimals Early G5 • The divisibility of numbers • The meaning and features of fractions • Addition and subtraction of fractions ( including computation mixed with fractions and decimals) Late G5 • Meaning and features of fractions • Meaning of fractions • Proper fraction, improper fraction, mixed numbers • The basic feature of fraction • Reduction of a fraction / “cross reduce” • Reduction to common denominator • Reduction between fractions and decimals Early G6 • Multiplication of fractions • Division of fractions • Computation with the four operations mixed with fractions and decimals • Percents (including computation mixed with fractions and percents

  12. The time allocated for learning fractions Forms to represent/express fractions Students’ prior knowledge (cognitive foundation for learning fractions) Three differences

  13. K G6 G3 G4 G1 G5 G2 Forms to represent fractions Good? Bad? No good, no bad? Show manipulatives

  14. 3 2 1 C 3 ÷ 4 = D A B Forms to represent fractions G4 (China) G5 (China)

  15. Fraction as a way of presenting division To express the quotient of the following divisions with a fraction: 4 ÷ 5 2 ÷ 9 7 ÷ 12 16 ÷ 49 33 ÷ 83 2 ÷ 7 5 ÷ 8 23 ÷ 24 37 ÷ 50 47 ÷ 100 • To cut a cord of 5 meters into 6 pieces of same length, how long each piece will be? • Lily is reading a story book of 48 pages. She has already read 31 pages. What fraction of the book has she finished for now? • A farmer has a piece of land of 3 acers. He evenly divided it into 7 pieces and use 1 piece to plant pepper. What fraction of the land is used to plant pepper ? How big is this piece?

  16. Forms to represent fractions

  17. The time allocated for learning fractions Forms to represent/express fractions Students’ prior knowledge (cognitive foundation for learning fractions) Three differences

  18. Knowledge foundation for learning fractions Starting from kindergarten:

  19. Starting with a solid foundation of the basic operations with whole numbers: A group of fractional units as a unit Multiplication and division with fractions Notation of fractions Addition and subtraction with fractions Fractional Unit One group of objects as a Unit (II) – Any number of objects considered as “1” Multiplication and division with whole numbers Whole number notation Addition and subtraction beyond 10 One group of objects as a Unit (I) – 10 and power of 10 considered as “1” One object as a Unit Addition and subtraction within 10 Unit 1

  20. Starting with a solid foundation of the basic operations with whole numbers: A group of fractional units as a unit Multiplication and division with fractions Notation of fractions Addition and subtraction with fractions Fractional Unit Who starts teaching fractions earlier, US or China? One group of objects as a Unit (II) – Any number of objects considered as “1” Multiplication and division with whole numbers Whole number notation Addition and subtraction beyond 10 One group of objects as a Unit (I) – 10 and power of 10 considered as “1” One object as a Unit Addition and subtraction within 10 Unit 1

  21. Why a foundation starting built when working with whole numbers? When working with whole numbers students learn: When being introduced to fractions students learn: The idea of “fractional unit” The idea of “unit” When fractions being added, they have to be of a common denominator Only like numbers (the numbers with same kind of unit) can be added What is the fractional unit of 2/3? of 4/7? of 5/11? Why? Why do we need to find a common denominator for 2/3 + 4/7 = ? Why 24 + 3 = 27 instead of 24 + 3 = 53 ?

  22. School Arithmetic A history view of the evolution of school arithmetic

  23. 1850 1900 1950 2000 A history view of the evolution of school arithmetic Teachable School Arithmetic

  24. 1852 the compulsory school attendance laws of Massachusetts 1902 The Child and the Curriculum by John Dewey 1904 China adopted Western school system 1906 Calvin W. Mateer, an American missionary wrote first school arithmetic textbook for China 1850 1900 1950 2000 1881 Tuskegee Normal School for Colored Teachers was established Teachable School Arithmetic Practical Arithmetic Or Rule Arithmetic School Arithmetic

  25. 1852 the compulsory school attendance laws of Massachusetts 1902 The Child and the Curriculum by John Dewey 1957 Sputnik 1962First Strands Report 1989 NCTM St. 1850 1900 1950 2000 Teachable School Arithmetic Practical Arithmetic Or Rule Arithmetic Progressive Practical Arithmetic School Arithmetic New Math --------- Back to Basics NCTM

  26. The evolution of the juxtaposed-strands organization 1962 1968 1974 1985 1989 1992 1999 2000 Number of strands 7 9 7 7 13 8 5 10

  27. Primary Statistics Primary Geometry School Arithmetic Simple Equations Measurement The evolution of core-subject organization Primary Geometry School Arithmetic Measurement The structure of Chinese Curriculum

  28. Primary Statistics Juxtaposed strands Primary Geometry School Arithmetic • Number and Operations • Algebra • Geometry • Measurement • Data Analysis and Probability • Problem Solving • Reasoning and Proof • Communication • Connections • Representation • Number and Operations • Geometry • Measurement • Applications of mathematics • Functions and graphs • Sets • Mathematical sentence • Logic 1. Number and Operations Simple Equations Measurement 1962 California Strands Report Arithmetic as the core subject 2000 NCTM Principles and Standards

  29. The End

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