Fractions are the basis for most higher-level mathematics. Students need to master the numerical values in earlier grades to tackle topics like algebra later. There’s only one hitch: Fractions can strike fear.
Even if students get the basic concepts right—that fractions don’t behave the same way that whole numbers do—the rules of how to operate on fractions can stump learners. Their abstract nature also makes fractions notoriously hard to teach.
Only 15 percent of math teachers said it’s “not at all challenging” for them to teach fractions, decimals, or percentages, according to a nationally representative EdWeek Research Center survey of educators conducted this spring. Forty-five percent said these concepts are “slightly” or “somewhat” challenging to teach, and 16 percent said they’re “very challenging.” (Another 24 percent said they don’t teach these concepts.)
In the early grades, students may come in with an innate sense of fractions because they’ve learned to split a candy bar into halves or fourths. But teachers can get stuck moving students from this organic “parts-of-a-whole” understanding to thinking about fractions as unique numbers, with their own set of operational rules.
The EdWeek Research Center survey asked math teachers of all grade levels which mathematical operations on fractions their students could do. Addition and subtraction were the most common responses, with more than half of math teachers indicating their students can perform those operations. Roughly 3 in 10 said their students had not yet been taught these concepts.
Just 38 percent said their students can divide fractions. This operation continues to stump students well into high school—69 percent of high school math teachers said their students can divide fractions, compared to the 84 percent who said their students can add fractions.
How do students understand fractions?
Students often treat fractions like whole numbers well into the upper-elementary grades.
A common mistake is to confuse the size of fractions based on which has the larger denominator: They may think that 1/6 is a bigger number than 1/4, for instance.
Another common struggle: the parts-of-a-whole problem. Fractions are introduced through the “pizza model,” but students’ thinking often gets stuck there. Taking two slices from an eight-slice pizza is easy to comprehend, but taking two slices out of two different pizzas—one with six slices and another with seven—is difficult to grasp.
If students believe that a fraction is always less than 1, they will have a hard time understanding ratios or proportions written in fraction form.
Another sticking point for many students is adding fractions with different denominators—for example, coming up with 11/12 when asked to add 3/5 and 8/7. This conceptual problem can crop up in later grades, too, if it isn’t tackled earlier.
Teachers need to focus on getting students’ basic knowledge of fractions right. But they don’t have to start from scratch. Experts suggest that teachers should tap into the information students already have.
Here are three research-aligned strategies to help students master fractions.
1. Build on informal knowledge.
Students come to school with an informal understanding of proportionality—they are familiar with the idea of sharing a candy bar or a cookie in equal proportions.
Teachers can use this basic knowledge as a ramp to introduce fractions by asking students to divide one object into equal parts or several objects equally among themselves. Some students may draw pictures, while others may use manipulatives to make separate piles.
Regardless of how they reach the answer, experts say students should be encouraged to estimate and experiment on their own.
With each iteration of this activity, teachers can increase the level of complexity and eventually introduce the required math vocabulary.
Working on early proportional reasoning is useful for students to grasp the next set of fraction concepts: equivalence and proportional thinking.
2. Trust the number line.
The number line is a math teacher’s most trusted friend when it comes to fractions and can be used well beyond elementary classrooms. It forms the conceptual basis for students to relate fractions to each other and to whole numbers and to grasp that there are an infinite number of fractions between any two numbers.
Measurement is a good place to start. The use of fraction strips—manipulatives that provide a visual representation—can help young students measure and compare different items’ lengths.
Teachers can use these strips to introduce the concept of improper fractions (11/2; 6/4; 23/16) and show that fractions can be more than one whole (1 and 1/2). These strips can be stacked up vertically to see how 1/2=2/4=4/8 or 3/6 (equivalence) or how close or far a fraction is to zero compared with other fractions.
In higher grades, experts suggest that students mark these notations on the number line to understand their relationship.
Research suggests “fading” from concrete representations, like pies or pizzas, to fraction strips, and finally to abstract notations of x/y on a number line. This means it’s not a strict start-and-stop between concepts but more of an overlap.
Experts also suggest using representations together—like the pizza model and the number line in one lesson—so that students can compare the different ways of expressing a fraction.
3. Putting it into practice.
The jump from learning about fractions to adding and then multiplying them happens fast. At every stage, students need to understand the operation conceptually and pair that with learning how to solve. Otherwise, students could make common mistakes like incorrectly adding the numerators and denominators.
Here, using a visual aid or a manipulative can help. Students may initially think 1/2+1/2 =2/4, but if they saw a picture, they would know 1/2 and 1/2 make 1. Before students begin adding fractions, teachers can stress the example of sharing or dividing candy bars equally, so they know to compare like denominators.
Let students guess the answers and explain their thinking before they solve a fraction problem. Teachers can use the student’s own thinking to explain the solution. For instance, a student may estimate that 1/2+1/5 is greater than 1/2 but smaller than 3/4 but still calculate the answer at 5/7. Now, the teacher can show why that’s wrong: 5/7 is smaller than 1/2 in size.
Research indicates that when teachers start with fraction arithmetic, they should use smaller values to reduce the cognitive load of adding or subtracting big numbers. Also, teachers should use the same unit—like a chocolate bar—when comparing and adding fractions initially.
Another potential pitfall that teachers encounter when teaching fractions is an over-reliance on cross-multiplication to identify equivalent fractions or solve for an unknown.
Once students have learned it as a trick, the specter of cross-multiplication is difficult to exorcise. Students tend to try it on every fraction problem, even when it’s not appropriate, which can leave them with incorrect answers. This is where real-world applications come in handy. Teachers can get students to think about ratio and proportions in terms of adjusting recipes or calculating a car’s mileage.
They can also boost multiplicative reasoning by using the “build up” method— “If two dishes are needed for five people, how many dishes do we need for 15?” Students will add on both sides of 2:5 ratio to get the answer of six dishes. Teachers can use this strategy as a precursor to teaching cross-multiplication.
Data analysis for this article was provided by the EdWeek Research Center. Learn more about the center’s work.
