Education Week - November 12, 2014 - Special Report - S6
November 12, 2014
Making Sense of the Math: The Common Core in Practice > www.edweek.org/go/math-report
Approach to Fractions
Seen as Key Shift in Standards
Number line emphasized
By Liana Heitin
or many elementary teachers, fractions have traditionally sprung to mind lessons involving pizzas,
pies, and chocolate bars, among other varieties of
"wholes" that can be shared. But in what many experts are calling one of the biggest shifts associated
with the Common Core State Standards for mathematics, more teachers are now being asked to emphasize
fractions as points on a number line, rather than just parts of
a whole, to underscore their relationships to integers.
As common-core proponents see it, the new standards do a
much better job of putting fractions into context, which will
help students make connections across other math concepts.
"The ultimate underlying principle is you want kids to
understand that fractions are numbers," said William G.
McCallum, a mathematics-education professor at the University of Arizona, in Tucson, and one of the lead writers of
the common standards. "They're new, but they're not in a
"I should not have to change what I know about numbers
to learn fractions," said Zachary Champagne, an assistant
in research at the Florida Center for Research in Science,
Technology, Engineering, and Mathematics at Florida State
University, in Tallahassee.
Fractions instruction in schools has long been seen as a
problem area. In 2010, the U.S. Department of Education's
Institute of Education Sciences released a report on effective
K-8 fractions instruction as part of its What Works Clearinghouse. The report noted that half of 8th graders could not
place three fractions in order from least to greatest on the
2004 National Assessment of Educational Progress in math.
It also found that fewer than 30 percent of 17-year-olds could
convert 0.029 into a fraction.
The recommendations from that report, as well as those of
a 2008 federal study by the National Mathematics Advisory
Panel that called difficulty with fractions "pervasive," are
reflected in the common core.
In addition to emphasizing fractions as points on the number line, the common standards differ from previous standards in other important ways: They delay arithmetic with
fractions until students have a thorough understanding of
what a fraction is; they eliminate explicit instruction on lowest common denominators; they do not differentiate between
proper and improper fractions; and they place an early emphasis on decimal equivalents.
Under the common core, 1st and 2nd graders learn some
basic vocabulary on fractions, including describing parts of
shapes as halves and quarters, and 6th graders learn division of fractions. But the bulk of fractions instruction goes on
between grades 3 and 5.
The 3rd grade rollout of fractions is intended to be slow and
steady. The standards require students to view fractions as
divided wholes and as numbers on a number line, as well as
to reason about a fraction's size. There's no arithmetic with
fractions that year.
"We're allowing time for students to explore and delve
deeply into the meaning of fractions," said Denise M.
Walston, the director of mathematics for the Washingtonbased Council of the Great City Schools.
In the past, teachers and textbooks have rushed into operations before students really understood the basics of fractions, said Jonathan A. Wray, the instructional facilitator for
BY GRADE LEVEL
Under the common-core standards,
fractions instruction extends across
grades 3 through 6, with operations
being taught after students have a
firm understanding of where fractions
lie on a number line. The standards,
shown here by grade, have been
edited for length and clarity.
secondary mathematics curricular programs in Maryland's
Howard County public schools. Mr. Wray, who also helped
write the 2010 ies report, said he has seen students make
it to middle school still thinking of the numerator and denominator as separate numbers. "They're using algorithms to
come up with equivalent fractions, they're using cross-multiplication-that was getting them by, but they didn't understand
anything behind it," he said.
Teachers need help with
this. This certainly isn't
the way most of us
JONATHAN A. WRAY
fractions as numbers
* nderstand 1/b as whole
divided into b parts
* nderstand a fraction as a
number on the number line
* xplain equivalence of
fractions and compare
fractions by reasoning
about their size
Putting fractions on a number line early can also help solidify students' understanding that fractions can and should
be compared to whole numbers, experts say.
Circular and rectangular representations can still be useful, said Mr. McCallum, especially in the context of learning
about quarters and halves. But "it's not so easy to divide a
pizza into five equal pieces," he said.
In addition, the number line helps ensure students use
consistent units. Mr. Wray pointed out that students who
are trying to compare fractions with a circular model may
end up drawing two circles of significantly different sizes. If
they shade one-half of the larger circle and three-fourths of
the smaller circle, they could make the argument that onehalf is greater than three-fourths, he said.
"The fact that [number lines] are mentioned explicitly as a
teaching and learning tool in the common core, I think that
has changed the landscape a little bit," said Mr. Wray.
Of course, many teachers have been putting fractions on
number lines for years, said Ms. Briars, but the common core
makes this "much more of a central representation."
Howard County (Md.) Public Schools
Students first need a good grasp of what a unit fraction is-
that is, a fraction with a one in the numerator, the basic unit of
measurement for larger fractions-before moving on to operations, according to the standards' authors.
"This emphasis on units and unit fractions, and how nonunit fractions are built from unit fractions, they did this
very purposefully to help students use the knowledge they
have from how whole numbers work," said Diane J. Briars,
the president of the Reston, Va.-based National Council of
Teachers of Mathematics. "It helps teachers hammer home
the point that a denominator is just the label that tells you
the size of the partition," she said.
A notable absence from the standards, meanwhile, is any
mention of "finding the lowest common denominator."
Students have traditionally spent large amounts of time
practicing reducing fractions to their lowest form, and in
many classrooms, an answer was marked wrong if it was not
"The question is, 'Why?' " said Mr. McCallum. "It's not
Students do need to compare equivalent fractions, which
means they will need to simplify at times. "But it's not
an overriding concern," said Mr. McCallum. "And there
are situations where it positively is getting in the way of
Mr. Champagne, of Florida State University, offered an exam-
Education Week - November 12, 2014 - Special Report
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