Transcript The Trouble with Fractions

The Trouble with Fractions
The Four Big Ideas of Fractions
The parts
are of
equal size
The whole is
divided
The parts
equal the
whole
There are a
specific number
of parts
Fractions: Equivalence
1/1 = 2/2 = 4/4 = 8/8
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
1/4
1/4
1/4
1/2
1/4
1/2
one whole
Fractions with different numbers can be equal
Fractions: Non-Equivalence
Does 1/2 equal 1/2? One-half of what?
If the whole = 2
1/2 of 2 = 1
If the whole = 8
1/2 of 8 = 4
Fractions: Symmetry of Area
Recognizing whether shapes have same size (i.e.
equal parts)
Fractions: Confusing Names
1/12
1/2
The larger the number (denominator), the smaller
the quantity.
Fractions: Visual Confusion
Fractions Strategy: Memorization
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 Understanding
Solid understanding of foundational numeracy
• Quantity
• Part-whole relationships
• Equal groupings
• Reversibility
Chinese Taipei
Netherlands
Canada
Macao-China
Japan
France
Germany
Korea
United Kingdom
Macao-China
Ireland
Poland
Austria
Finland
Australia
Slovenia
Hong Kong-China
Belgium
Iceland
Denmark
New Zealand
Liechtenstein
Czech Republic
Estonia
Sweden
Switzerland
Source: OECD PISA 2006 database
The Nation’s Report Card
US Students Proficient in Math
-- National Center for Educational Statistics, 2007
Top Achieving Countries
Center for Research in Math & Science Education, Michigan State University
1989 NCTM Topics by Grade
Center for Research in Math & Science Education, Michigan State University
US vs Top Achieving Countries
Number of Topics
Grade
International Test Item
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?”
Changing Course
“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 Recommends
‣ Instruction should devote
“the vast majority of
attention” to the most
significant mathematical
concepts.
‣ Focus on developing problem solving, reasoning, and
critical thinking skills.
‣ Develop deep understanding, mathematical fluency, and
an ability to generalize.
National Math Panel Report
Math curricula should:
‣Be "streamlined and should emphasize a well-defined
set of the most critical topics
in the
early grades."
The
manner
in
‣Emphasize "the mutuallywhich
reinforcing
benefits
math
isof
conceptual understanding, procedural fluency, and
taught
in
the
U.S.
automatic recall of facts."
is
"broken
and
‣Teach with "adequate depth."
must
be
fixed."
‣Have an "effective, logical progression from earlier, less
sophisticated topics into later, more sophisticated ones."
‣Have teachers regularly use formative assessment.
National 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.”