the trouble with fractions. the four big ideas of fractions the parts are of equal size there are a...

The Trouble with FractionsThe Trouble with Fractions

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

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

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?

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)

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

Fractions: Visual ConfusionFractions: Visual Confusion

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”

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

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

-- 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

Center for Research in Math & Science Education, Michigan State UniversityCenter for Research in Math & Science Education, Michigan State University

Top Achieving Countries

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

Number of TopicsNumber of Topics

GradeGrade

US vs Top Achieving US vs Top Achieving CountriesCountries

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

“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

‣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

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

“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