Approach to Fractions Seen as Key Shift in Common Standards
For 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 different galaxy."
"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 Washington-based 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 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.
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.
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 one-half 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."
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 simplified.
"The question is, 'Why?' " said Mr. McCallum. "It's not mathematically important."
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 understanding."
Mr. Champagne, of Florida State University, offered an example. "Fifty-two one-hundredths is more than half," he said. "If you put that in lower terms, it becomes 26/50. If you go further, it's 13/25. But that's way harder to picture than 52/100."
Asking students to reduce a fraction or write it in its lowest terms can give them the impression that a fraction is getting smaller, said Ms. Briars: "I think that not emphasizing that makes a lot of sense. The number you want in the denominator depends on the situation you're in. If I'm working with time, I'm thinking about 60ths, so it makes a lot more sense to work with that than thirds and halves."
And often, students calculate a correct answer but then make a mistake when trying to reduce the fraction. "There's nothing mathematically that says you have to do that," said Mr. Champagne. "Fourteen-sixteenths [can be] a correct answer."
The authors of the common core also chose not to differentiate between proper and improper fractions. "It's part of the same mantra of trying to get people to see fractions as numbers," said Mr. McCallum. "You can start counting forward on the number line by thirds, and there's no law that says you have to stop at 1."
In fact, "a lot of people think a fraction means less than one," he said, when that is far from true.
Instead of spending time having students practice the steps for changing improper fractions into mixed numbers, the common standards presume that students will figure out how to do that on their own through better conceptual grasp.
"The meaning develops out of the work with the number line," said Ms. Walston of the CGCS. "They will see, 'Oh, a quick way I can do it is to divide the numerator by the denominator.' "
Fraction-decimal equivalents can also be demonstrated more easily through the number-line approach, some educators say.
With more traditional methods of instruction, "students end up thinking there are all different types of numbers—fractions, mixed numbers, and decimals—and they're all different animals," said Mr. McCallum. But the number-line approach makes it clear that these can all be the same number: "You're simply writing the number in a different way."
Consistency with the number-line approach across the grades can also help develop conceptual understanding, said Justin Minkel, an elementary teacher in Springdale, Ark., and the 2007 Arkansas Teacher of the Year. "If you use colorful, shaded circles [for fractions] and then jump to money [for decimals], I don't think kids are learning that connection," he said.
All the work with the number line in 3rd grade is meant to lead to a more thorough understanding of what it means to add, subtract, multiply, and divide with fractions. The common standards repeatedly ask students to "apply and extend previous understandings" when working with fractions—that is, if students are able to show addition on a number line, they should also be able to show addition with fractions there.
Challenges for Teachers
What has typically happened in classrooms is "you learn how to add whole numbers, you learn algorithms for adding multidigit whole numbers, and then you learn addition with fractions, and it's completely different," said Mr. McCallum. The number line makes addition "much more like what addition was before. You're just using fifths instead of ones as the unit where you measure things out."
For teachers who've taught fractions as "a different animal" from whole numbers until now, assimilating the new approach will take some time and effort.
"We know the rule and how to get the answer really quickly," said Mr. Champagne, "but that's not going to be enough." For instance, when dividing fractions, teachers can no longer just tell students to "invert and multiply." They also need to explain why the rule works and use it to solve real-world problems, he said. "If you ask any K-12 teacher to write a word problem for dividing 1/2 by 3/4, it's really tough."
In some cases, teachers will need to relearn, or at least get a refresher on, the concepts themselves. "Especially for elementary teachers who are generalists and teach multiple subjects, they may not have had the opportunity to develop their understanding of the mathematics concepts as deeply as the common core is calling for," said Ms. Briars of the NCTM. At her group's annual conference last spring, more than 35 sessions focused on fractions, including a well-attended keynote address.
"Teachers need help with this," said Mr. Wray. "This certainly isn't the way most of us learned fractions."
Video: Teaching Fractions Under the Common Core
Vol. 34, Issue 12, Pages s6,s7,s8