the trouble with fractions. the four big ideas of fractions the parts are of equal size there are a...
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The Trouble with FractionsThe Trouble with Fractions
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The Four Big Ideas of The Four Big Ideas of FractionsFractions
The parts are of equal size
The parts are of equal size
There are a specific number of parts
There are a specific number of parts
The whole is dividedThe whole is divided
The parts equal the whole
The parts equal the whole
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Fractions: EquivalenceFractions: Equivalence
Fractions with different numbers can be equalFractions with different numbers can be equal
one wholeone whole
1/81/8
1/21/2
1/41/4 1/41/4
= 4/4= 4/4= 2/2= 2/2 = 8/8= 8/8
1/21/2
1/41/41/41/4
1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8
1/11/1
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Fractions: Non-EquivalenceFractions: Non-Equivalence
If the whole = 8If the whole = 8
1/2 of 2 = 1
1/2 of 2 = 1
1/2 of 8 = 4
1/2 of 8 = 4
If the whole = 2
If the whole = 2
Does 1/2 equal 1/2? One-half of what?Does 1/2 equal 1/2? One-half of what?
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Fractions: Symmetry of AreaFractions: Symmetry of Area
Recognizing whether shapes have same size (i.e. equal parts)Recognizing whether shapes have same size (i.e. equal parts)
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Fractions: Confusing NamesFractions: Confusing Names
The larger the number (denominator), the smaller the quantity.The larger the number (denominator), the smaller the quantity.
1/121/12
1/21/2
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Fractions: Visual ConfusionFractions: Visual Confusion
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Fractions Strategy: Fractions Strategy: MemorizationMemorization
The complexity of fractions makes it more likely that students will forget that fractions represent quantities.
This leads to memorization without understanding:
•“Find the common denominator, then add”
•“Flip it and multiply”
•“The bigger the denominator the smaller the fraction”
The complexity of fractions makes it more likely that students will forget that fractions represent quantities.
This leads to memorization without understanding:
•“Find the common denominator, then add”
•“Flip it and multiply”
•“The bigger the denominator the smaller the fraction”
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Fractions: Prerequisite Fractions: Prerequisite UnderstandingUnderstanding
Solid understanding of foundational numeracy
•Quantity
•Part-whole relationships
•Equal groupings
•Reversibility
Solid understanding of foundational numeracy
•Quantity
•Part-whole relationships
•Equal groupings
•Reversibility
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Source: OECD PISA 2006 databaseSource: OECD PISA 2006 database
Hong Kong-ChinaHong Kong-China
FinlandFinland
KoreaKorea
NetherlandsNetherlands
LiechtensteinLiechtenstein
JapanJapan
CanadaCanada
BelgiumBelgium
Macao-ChinaMacao-China
SwitzerlandSwitzerland
AustraliaAustralia
New ZealandNew Zealand
Czech RepublicCzech Republic
IcelandIceland
DenmarkDenmark
FranceFrance
SwedenSweden
AustriaAustria
GermanyGermany
IrelandIreland
SloveniaSloveniaUnited KingdomUnited Kingdom
PolandPoland
Chinese Taipei
Chinese Taipei EstoniaEstonia
Macao-ChinaMacao-China
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-- National Center for Educational Statistics, 2007-- National Center for Educational Statistics, 2007
The Nation’s Report CardThe Nation’s Report Card
US Students Proficient in MathUS Students Proficient in Math
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Center for Research in Math & Science Education, Michigan State UniversityCenter for Research in Math & Science Education, Michigan State University
Top Achieving Countries
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Center for Research in Math & Science Education, Michigan State UniversityCenter for Research in Math & Science Education, Michigan State University
1989 NCTM Topics by Grade1989 NCTM Topics by Grade
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Number of TopicsNumber of Topics
GradeGrade
US vs Top Achieving US vs Top Achieving CountriesCountries
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4th Grade
“There are 600 balls in a box, and 1/3 of the balls are red.
How many red balls are in the box?”
International Test Item
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“Teachers face long lists of learning expectations to address at each grade
level, with many topics repeating from year to year. Lacking clear,
consistent priorities and focus, teachers stretch to find the time to present important mathematical topics effectively and in depth.”
“Teachers face long lists of learning expectations to address at each grade
level, with many topics repeating from year to year. Lacking clear,
consistent priorities and focus, teachers stretch to find the time to present important mathematical topics effectively and in depth.”
-- NCTM Curriculum Focal Points-- NCTM Curriculum Focal Points
Changing CourseChanging Course
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‣Focus on developing problem solving, reasoning, and critical thinking skills.
‣Develop deep understanding, mathematical fluency, and an ability to generalize.
‣Focus on developing problem solving, reasoning, and critical thinking skills.
‣Develop deep understanding, mathematical fluency, and an ability to generalize.
‣Instruction should devote “the vast majority of attention” to the most significant mathematical concepts.
‣Instruction should devote “the vast majority of attention” to the most significant mathematical concepts.
NCTM RecommendsNCTM Recommends
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Math curricula should:
‣Be "streamlined and should emphasize a well-defined set of the most critical topics in the early grades."
‣Emphasize "the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts."
‣Teach with "adequate depth."
‣Have an "effective, logical progression from earlier, less sophisticated topics into later, more sophisticated ones."
‣Have teachers regularly use formative assessment.
Math curricula should:
‣Be "streamlined and should emphasize a well-defined set of the most critical topics in the early grades."
‣Emphasize "the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts."
‣Teach with "adequate depth."
‣Have an "effective, logical progression from earlier, less sophisticated topics into later, more sophisticated ones."
‣Have teachers regularly use formative assessment.
The manner in which math is taught in the U.S. is "broken and must be fixed."
The manner in which math is taught in the U.S. is "broken and must be fixed."
National Math Panel ReportNational Math Panel Report
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“A major goal for K-8 mathematics education should be proficiency with fractions, for such proficiency is foundational for algebra and, at the present time, seems to be severely underdeveloped.”
“A major goal for K-8 mathematics education should be proficiency with fractions, for such proficiency is foundational for algebra and, at the present time, seems to be severely underdeveloped.”
National Math Panel ReportNational Math Panel Report
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