teaching fractions: the differences caused by two kinds of curriculum organization
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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. - PowerPoint PPT PresentationTRANSCRIPT
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
Arithm
etic
Primary Geometry
Measurement
What might they be ?
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 subtractionNumbers 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 tablesNumbers up to 10,000 , notation, addition and subtraction
Multiplication with multiplier as a one-digit numberDivision with divisor as a one-digit number
Many-digit numbers, notation, addition and subtractionMultiplication with multiplier as a two-digit number
Division with divisor as a two-digit numberMultiplication with multiplier as a three-digit number
Division with divisor as a three-digit numberFractions – the basic concepts
Decimals – meaning and featuresDecimals – addition and subtraction
Decimals – multiplication and divisionDivisibility ( factors, multipliers, prime number, factorization, GCD, LCM)
Fractions – meaning and featuresFractions – addition and subtraction
Fractions – multiplicationFractions – division
PercentsRatio and proportion
Area of triangles & trapezoids;Prism and cubic(volume) Simple equation
Area of rectanglesAngles & 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 subtractionNumbers 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 tablesNumbers up to 10,000 , notation, addition and subtraction
Multiplication with multiplier as a one-digit numberDivision with divisor as a one-digit number
Many-digit numbers, notation, addition and subtractionMultiplication with multiplier as a two-digit number
Division with divisor as a two-digit numberMultiplication with multiplier as a three-digit number
Division with divisor as a three-digit numberFractions – the basic concepts
Decimals – meaning and featuresDecimals – addition and subtraction
Decimals – multiplication and divisionDivisibility ( factors, multipliers, prime number, factorization, GCD, LCM)
Fractions – meaning and featuresFractions – addition and subtraction
Fractions – multiplicationFractions – division
PercentsRatio and proportion
Area of triangles & trapezoids;Prism and cubic(volume) Simple equation
Area of rectanglesAngles & 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 subtractionNumbers 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 tablesNumbers up to 10,000 , notation, addition and subtraction
Multiplication with multiplier as a one-digit numberDivision with divisor as a one-digit number
Many-digit numbers, notation, addition and subtractionMultiplication with multiplier as a two-digit number
Division with divisor as a two-digit numberMultiplication with multiplier as a three-digit number
Division with divisor as a three-digit numberFractions – the basic concepts
Decimals – meaning and featuresDecimals – addition and subtraction
Decimals – multiplication and divisionDivisibility ( factors, multipliers, prime number, factorization, GCD, LCM)
Fractions – meaning and featuresFractions – addition and subtraction
Fractions – multiplicationFractions – division
PercentsRatio and proportion
Area of triangles & trapezoids;Prism and cubic(volume) Simple equation
Area of rectanglesAngles & 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 subtractionNumbers 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 tablesNumbers up to 10,000 , notation, addition and subtraction
Multiplication with multiplier as a one-digit numberDivision with divisor as a one-digit number
Many-digit numbers, notation, addition and subtractionMultiplication with multiplier as a two-digit number
Division with divisor as a two-digit numberMultiplication with multiplier as a three-digit number
Division with divisor as a three-digit numberFractions – the basic concepts
Decimals – meaning and featuresDecimals – addition and subtraction
Decimals – multiplication and divisionDivisibility ( factors, multipliers, prime number, factorization, GCD, LCM)
Fractions – meaning and featuresFractions – addition and subtraction
Fractions – multiplicationFractions – division
PercentsRatio and proportion
Area of triangles & trapezoids;Prism and cubic(volume) Simple equation
Area of rectanglesAngles & 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)
6.03
1
6.55
1
13.52
1
49.3
1
118.4
1
176
1
5.91
1
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
3
11.71
1
1.82
1China
Time allocated on learning fractions and decimals
G4
G5
G6
ChinaG6G5G4
G3G2
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
KShow manipulatives
Good? Bad? No good, no bad?
Forms to represent fractionsG4 (China)
G5 (China)1
2 3
3 ÷ 4 =
A
B
C
D
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 ÷ 832 ÷ 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?
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
Why24 + 3 = 27instead of
24 + 3 = 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 for2/3 + 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
Arithm
etic
Teachable School Arithmetic
A history view of the evolution of school arithmetic
1850 1900 1950 2000
Teachable School Arithmetic
1850 1900 1950 2000
1852 the compulsory school attendance laws of Massachusetts
1902 The Child and the Curriculum by John Dewey
Practical Arithmetic
OrRule
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
1850 1900 1950 2000
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
ArithmeticNew Math
---------Back to Basics
NCTM
Practical Arithmetic
OrRule
Arithmetic
School Arithmetic
The evolution of the juxtaposed-strands organization
1962 1968 1974 1985 1989 1992 1999 2000
7 9 7 7 13 8 5 10Number of strands
The evolution of core-subject organization
School
Arithm
etic
Primary Geometry
Measurement
The structure of Chinese Curriculum
School
Arithm
etic
Primary Geometry
Measurement
Simple Equations
PrimaryStatistics
2000 NCTM Principles and Standards
1. Number and Operations2. Algebra3. Geometry4. Measurement 5. Data Analysis and Probability 6. Problem Solving 7. Reasoning and Proof8. Communication 9. Connections10. Representation
Juxtaposed strands
1. Number and Operations2. Geometry3. Measurement 4. Applications of mathematics 5. Functions and graphs 6. Sets7. Mathematical sentence 8. Logic
1962 California Strands Report
1. Number and Operations
Arithmetic as the core subject
School
Arithm
etic
Primary Geometry
Measurement
Simple Equations
Primary Statistics
The End