Many students in elementary grades struggle to grasp the perennially vexing concept of fractions. If those struggles persist into middle school, it can undermine their ability to master more complex math.
That’s because the ability to build upon foundational fractions skills and apply them to lessons on rates, ratios, and proportional reasoning serves as a gateway to higher-level math classes, like algebra, said Melinda Griffin, a senior researcher at the American Institutes for Research who studies math instruction.
“Kids see fractions as things that sit outside the number system, instead of seeing them as just another number that we use when we do math,” said Griffin, who recently developed professional development tools to improve 6th grade teachers’ ability to cover fractions.
Data show the challenge of teaching fractions remains significant for teachers as students advance into upper-elementary and middle school grades.
In an online survey of 729 educators and administrators who teach or supervise the teaching of middle and high school math classes conducted between Jan. 28 and March 5, the EdWeek Research Center asked which foundational math skills most of their students lack and act as impediments to them making progress in the subject.
Nine in 10 respondents said they viewed fractions as a foundational shortcoming for their students.
More educators identified fractions as a problem than areas such as number sense, spatial reasoning, and basic geometry.
Answers to fraction problems confuse students and break “rules”
Academic researchers and educators have long flagged students’ inadequate understanding of fractions as a top concern.
“Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra,” said an influential 2008 report from the National Mathematics Advisory Panel.
The panel, created by an executive order by President George W. Bush, called fractions “the most important foundational skill not presently developed” and said students need to both understand fractions conceptually and have enough procedural fluency to identify strategies and solve problems.
For older elementary and middle-grades students, it can be difficult to balance conceptual and procedural understanding when lessons advance beyond adding fractions to multiplying and dividing them, Griffin said.
Answers to those problems seem to break “rules” established when students learned whole numbers, such as the idea that multiplying two numbers always leads to a larger sum, she said.
Concepts like cross-multiplying and dividing “tend to be taught in a very formula-based way: ‘Just follow the steps and you’ll get an answer,’” Griffin said. “But do you understand what that answer means?”
Equipping teachers with new strategies to teach fractions
Teachers of older elementary students, who often teach all core subjects, can struggle to build students’ comfort with fractions because they aren’t confident in their own abilities, said Yinmei Wan, a senior researcher at AIR. Middle school math teachers, by contrast, may be drawn to math because of their own proficiency in the subject, and as a result it can be difficult for them to identify the precise gaps in students’ understanding.
Wan and Griffin worked together to develop the Teaching Fractions Toolkit, a collection of free professional development resources, on behalf of the Regional Education Lab Midwest, a federally funded organization that works with educators to design and pilot research-based education solutions.
The tool kit—designed for 6th grade teachers—includes PD modules, which teachers can complete online or in person; asynchronous work for participants to complete between group sessions; and online apps students can use to visualize concepts like placing fractions on a number line or comparing ratios.
It also includes separate guides for facilitators and a resource for administrators to improve instruction in fractions.
Working together with a panel of math experts and teachers from throughout Illinois, the researchers identified and piloted strategies to help students understand and apply fractions in everyday problem-solving.
For example, students may use tape diagrams to map out how many servings are in a pitcher of orange juice, assuming the pitcher holds five glasses and each serving is 3/5th of a glass.
First, a student would divide the tape into five equal portions to represent glasses, then they would divide each of those portions into five, dividing the total number by three to determine the answer. Teachers can then help the students understand how the visual representation relates to the underlying computational strategy.
In another lesson, students may use stacked number lines to compare rates of time and distance in order to solve problems like: “If a driver is going four miles per hour in a traffic jam, how long will it take them to reach the next exit, which is six miles away?”
Sixth grade is a critical time for mastering fractions, Wan said.
“It’s a transitional age grade and it’s almost the last opportunity for kids to catch on,” she said. “If they fall behind, it will be a serious challenge moving forward.”