There are some basic properties of numbers any 3rd grader can tell you: Each number is represented by a single symbol, and followed by a single successor. Multiplication makes a number bigger; division makes it smaller.
The problem is, none of those qualities鈥攖rue of whole numbers鈥攊s true when it comes to fractions, one of the most chronically troublesome basic mathematics areas for children and adults alike. Now, as the Common Core State Standards push for earlier and deeper understanding of fractions, researchers and teachers are exploring ways to ensure students learn more than a sliver of the fractions pie.
鈥淒evelopmental research shows even very young children have a fundamental grasp of fractions that can be built on through instruction,鈥� said Nancy C. Jordan, a professor of education in the University of Delaware鈥檚 School of Education.
But, she added, 鈥淚f children are taught math in a way that鈥檚 very rote, where they memorize procedures 鈥� it really doesn鈥檛 help you much.鈥�
鈥榃hole New World鈥�
The traditional approach to teaching fractions can make it more likely for students to show superficial progress without real understanding, some researchers and educators argue.
鈥淲e鈥檝e had a tendency in our traditional scope and sequence of math that you teach all this whole number stuff鈥� and then all of a sudden you get to fractions and it鈥檚 a whole new world of what to do鈥攅verything they learned in whole numbers has nothing to do with how you do fractions,鈥� said Linda M. Gojak, the president of the National Council of Teachers of Mathematics, in Reston, Va. 鈥淚t鈥檚 one of the hardest things for kids to get their heads around.鈥�
Cynthia Hacker, the education director for Sylvan Learning of Irmo/Lexington in Columbia, S.C., sees that confusion a lot. For more than a dozen years, the center has run a weeklong 鈥淔raction Action鈥� summer camp, at which students play games using shapes and number lines to compare fractions of different sizes and practice multiplying and dividing mixed and improper fractions.
While the camp focuses on elementary students, Ms. Hacker said one rising 12th grader who came to the center to boost his performance in high school math has ended up using the fractions games, too. 鈥淗e鈥檚 a bright student; he just somewhere along the way missed the foundations of fractions, so now he鈥檚 having a lot of trouble with algebra.鈥�
He鈥檚 far from alone. Students鈥� lack of fractions understanding has been cited as second only to word-problem difficulty as the top handicap for students learning algebra, in a survey of a representative sample of 1,000 algebra teachers conducted by the National Mathematics Advisory Panel鈥攁 federal task force created under former President George W. Bush to evaluate evidence on math teaching and learning.
The typical American approach to teaching fractions can overemphasize procedures at the expense of understanding the relationships among numbers, which is needed for higher math, according to Lynn Fuchs, a professor of special education at Vanderbilt University in Nashville.
In the United States, curricula overwhelmingly focus on understanding fractions as parts of a whole, using area models and pie charts, and teaching students the procedures for adding or multiplying, for example.
Some Asian countries, by contrast, focus on what Ms. Fuchs calls a 鈥渕easurement interpretation鈥� of fractions: how they fall on a number line, the relationships between numbers that are represented by a fraction.
Leah Hurtubise, the director of the Mathnasium Forest Hill center in the Canadian city of Toronto, who operates a summer 鈥淔raction Frenzy Boot Camp,鈥� often sees middle school students who had seemed to understand fractions in elementary school, but do not understand the relationships among numbers well enough to apply what they have learned to decimals or percentages in 7th or 8th grades.
鈥淚f you ask what is 7/10, they can tell you .70, but if you ask to convert 7/9 they鈥檙e completely lost,鈥� Ms. Hurtubise said. 鈥淭hey鈥檙e not grasping that they are dividing the top number by the bottom number.鈥�
That鈥檚 soon to become an even bigger hurdle, as the common-core math standards push more of the early work on fractions into 3rd grade, as opposed to 4th and 5th grades. It also calls for teachers to focus less on the procedures for specific fraction problems and more on getting students to understand the relationships between numbers that underlie a fraction problem.
鈥淚t鈥檚 actually a very different way to teach fractions,鈥� Ms. Gojak said. 鈥淚t may have been implicit, but it鈥檚 not something that has been clear to teachers, and more than likely it鈥檚 not the way they learned it.鈥�
鈥淚 want kids to understand in the long run that the meaning of the operation hasn鈥檛 changed鈥� in a fraction-division problem, Ms. Gojak said. 鈥淚鈥檓 not just flipping and multiplying; the numerator is telling us how many things we have, and the denominator is describing the size of the piece.鈥�
In a 2012 essay, Robert Siegler, a professor of cognitive psychology at Carnegie Mellon University, in Pittsburgh, and his colleagues call fractions a 鈥�new frontier鈥� for understanding students鈥� numeracy development.
Building Better Fractions
The National Center on Improving the Learning of Fractions, based at the University of Delaware a multi-university project supported by the Institute of Education Sciences, is trying to help students and teachers become more adroit with fractions.
Researchers Ms. Jordan, Mr. Siegler, and Ms. Fuchs are working with Nashville and other public school districts to explore more effective ways to teach these concepts.
Beyond simply being able to count, Mr. Siegler said, fraction knowledge in 5th grade 鈥渦niquely predicts鈥� a student鈥檚 10th grade math achievement, above and beyond the student鈥檚 IQ, family background, or even knowledge of other parts of mathematics.
Ms. Jordan鈥檚 research has found that a student鈥檚 ability to understand and estimate where fractions would fall on a number line and explain magnitude鈥攖hat a number represents a set of items which can be changed or compared to other sets鈥攚ill predict how well they perform in math over the long term. She is developing a screening tool for 4th-6th grade students that would identify children who are having trouble learning fractions and what areas of instruction might need to be emphasized. It is expected to go to field tests next year.
Mr. Siegler and his team at Carnegie Mellon are developing a board game intended to help early elementary students understand and compare the magnitude of different fractions. The computer-based game, 鈥淐atch the Monster,鈥� asks students to find a monster hiding along a number line by estimating the point closest to a location using a fraction-related hint. For example, a student given the fraction 9/4 might guess the monster was hiding close to 2, between 2 and 3 on the number line. If a student estimates correctly, the cartoon monster 鈥渄ies a dramatic death,鈥� Mr. Siegler said.
Similarly, in one of Ms. Fuch鈥檚 studies, students in 53 4th grade classes in 13 schools were randomly assigned to participate in either their school鈥檚 regular fractions instruction or a 12-week intervention focused on helping them understand underlying relationships in fractions. Students who participated in the intervention performed significantly better than peers who had not participated on a test of fractions problems culled from the National Assessment of Educational Progress. Moreover, the achievement gaps between students considered 鈥渁t risk鈥� in math鈥攖hose initially in the lowest 35th percentile on a standardized math test鈥攁nd those not at risk closed significantly, but only for those who participated in the program.
鈥淲e are teaching children to think of fractions in terms of quantities, how different-sized fractions compare to one another,鈥� Ms. Fuch said. 鈥淲e鈥檙e trying to teach them a more sophisticated understanding of fractions and help them do well.鈥�
