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

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

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

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

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

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

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

K : Halves 2 in 352 pages G1 : Equal parts, One half, One third and one fourth 2 in 352 pages G2 : Unit fractions, wholes and parts, comparing fractions, fraction of a group 12 in 592 pages G3 : Fractions and Decimals 44 in 595 pages, one of the 12 chapters G4 : Fractions and Decimals (addition & subtraction) 92 in 603 pages, 2 of the 12 chapters G5 : Fractions and Decimals (multiplication & division) 100 in 603 pages, 2 of the 12 chapters G6 : Operations with fractions / 100 in 591 pages 1 of the 12 chapters US G4 : Fractions and Decimals 60 in 180 pages, 3 of the 9 chapters * G5 : Fractions and Decimals 131 in 227 pages, 5 of the 9 chapters G6 : Fractions and percents 101 in 184 pages, 4 of the 8 chapters China

Time allocated on learning fractions and decimals G4 G5 G6 China G6 G5 G4 G3 G2 G1K US

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

Exposure to fractions Meaning and features of decimals Addition and subtraction with decimals Multiplication and division with decimals Computations mixed with four basic operations with decimals Late G4 Early G5 G4 G5 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 The divisibility of numbers The meaning and features of fractions Addition and subtraction of fractions ( including computation mixed with fractions and decimals) Late G5 Early G6 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.) 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

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

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

Forms to represent fractions G4 (China) G5 (China) ÷ 4 = A B C D

Fraction as a way of presenting division To express the quotient of the following divisions with a fraction: 4 ÷ 52 ÷ 97 ÷ ÷ ÷ 83 2 ÷ 75 ÷ 823 ÷ ÷ ÷ 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?

Forms to represent fractions

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

Knowledge foundation for learning fractions Starting from kindergarten:

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

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

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

A history view of the evolution of school arithmetic School Arithmetic

Teachable School Arithmetic A history view of the evolution of school arithmetic

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

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

The evolution of the juxtaposed-strands organization Number of strands

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

2000 NCTM Principles and Standards 1.Number and Operations 2.Algebra 3.Geometry 4.Measurement 5.Data Analysis and Probability 6.Problem Solving 7.Reasoning and Proof 8.Communication 9.Connections 10.Representation Juxtaposed strands 1.Number and Operations 2.Geometry 3.Measurement 4.Applications of mathematics 5.Functions and graphs 6.Sets 7.Mathematical sentence 8.Logic 1962 California Strands Report 1. Number and Operations Arithmetic as the core subject School Arithmetic Primary Geometry Measurement Simple Equations Primary Statistics

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