Why Is Doing Arithmetic With Fractions So Difficult?
Teaching children math they will use
Posted Sep 30, 2017
On his website andertoons.com, illustrator Mark Anderson has a cartoon in which a boy is handing his teacher an assignment. “To show how well I understand fractions,” he says, “I only did half of my homework.”
Certainly this is a sentiment that many of us share, as we all struggled with fractions in grade school. A recent study found that U.S. eighth-graders in an advanced math class performed at chance on a multiple-choice test of fractions. Adult students at a community fared no better. And while grade school math teachers did understand how to add or divide fractions properly, most couldn’t explain why you need to find the lowest common denominator in addition or invert the second fraction in division. Small wonder they had difficulty explaining these concepts to their students.
But why do so many people struggle with fractions? In a recent article, psychologists Robert Siegler and Hugues Lortie-Forgues outline the reasons why rational number arithmetic is so difficult. By this term, the researchers include not only fractions expressed in the form a/b but also in decimal and percent formats. Although fractions of the a/b form present the most difficulties for children and adults, decimal notation and percentages pose significant problems of their own. In fact, expressions such as “She gave 110% on her performance” belie the fact that many people don’t completely understand that percentages are fractions.
The researchers looked at performance on rational number arithmetic among students and teachers in North America and East Asia. They identified a number of reasons why rational number arithmetic is so difficult, but these reasons can be divided into two classes—inherent and cultural sources of difficulty.
Rational numbers are more complex than whole numbers, and so they’re more difficult to understand and use. Inherent sources of difficulty are universal, and even students in East Asia, who famously outperform their peers throughout the world in math, still have difficulty with fractions. The authors point out two inherent sources of difficulty:
- It’s difficult to understand what rational numbers mean. Each whole number is represented by a single symbol (1, 2, 3, 37, 996, and so on). However, rational numbers can be expressed as fractions, decimals, or percentages. It’s not at all obvious why 1/2, .5, and 50% all refer to the same quantity. Even more confusing, any rational number can be represented by an infinite number of different fractional expressions. The numbers in the series 1/2, 2/4, 3/6, 4/8, and 5/10 appear to be getting larger. After all, both the numerators and the denominators are increasing. And yet they all represent the same quantity.
- Arithmetic operations with rational numbers are far more complex than they are with whole numbers. The methods that students learned for adding, subtracting, multiplying, and dividing are fairly straightforward to perform. Furthermore, the reasons why they work are easy to demonstrate with objects—such as blocks or buttons—that children can manipulate and count. But the methods for fractional arithmetic are complex, and the reasons for them are opaque. Why do you need to find the lowest common denominator when adding or subtracting fractions but not when multiplying them? And why do you have to invert the second fraction and multiply when dividing? Dunno, just the way it’s done. With such a shallow understanding of rational number arithmetic, no wonder so many people have difficulty with it.
Although these inherent complexities plague all of us, other sources of difficulty with rational number arithmetic are cultural in original. It’s these “culturally contingent sources of difficulty” that account for the better performance among East Asian students compared with their North American counterparts. Specifically, these include:
- Teacher knowledge. Grade school math teachers in China, Japan, and Korea receive better instruction in rational number arithmetic than do teachers in the U.S. or Canada. When asked what is meant by the expression 7/4 ÷ 1/2, most U.S. teachers could not provide an explanation, but most Chinese teachers could. It’s hard to provide quality instruction to your students when you only half understand the concepts yourself.
- Textbook quality. A comparison of grade-school math textbooks shows that Korean textbooks devote far more space to rational number arithmetic and provide more practice problems than U.S. textbooks. To the extent that practice makes perfect, Korean schoolchildren have a decided advantage over their American peers.
- Language. The Chinese number system is much easier to learn than European systems because it’s strictly base ten. There are no irregular number names, such as “eleven,” “twelve,” or “twenty.” These are rendered instead as “ten-one,” “ten-two,” “two-ten,” and so on. As a result, Chinese children learn to count at a much earlier age than North American or European children. Fractional expressions are also more straightforward. For example, the fraction 1/3 is expressed as “one third” in English but as “one of three parts” in Chinese. In other words, the meaning of the fraction is explicit. Both Japan and Korea adopted the Chinese number system many centuries ago, and so they also reap the benefits of this highly logical system.
One reason why so many people experience math phobia is because they don’t see the relevance of mathematics in their daily lives. Traditionally, algebra has been the standard math course in high school—and it’s often repeated at the college level. But many people protest—and I think rightly so—that they never use algebra in their daily lives.
In higher education, there’s an ongoing debate about whether algebra or elementary statistics is a more appropriate math course for non-STEM majors. After all, students in the liberal arts will never need to solve a quadratic equation, but they will need to deal with statistical information, such as polls, surveys, census data, and economic reports. And these kinds of data are frequently presented as rational numbers—fractions, decimals, and percentages.
A recent survey found that 82% of white-collar workers, 70% of blue-collar workers, and 40% of service workers used fractions in their work. (And you just used fractions as you interpreted these statistics.) Clearly, rational number arithmetic does have relevance in people’s lives, and so we need to do a better job teaching it.
However, improvements are going to have to start at the very top, with better education for our teachers. We can also benefit by looking at how rational number arithmetic is taught in East Asia. After all, such an exercise can provide valuable lessons in how to improve our own math instruction here in North America.
Siegler, R. S., & Lortie-Forgues, H. (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science, 26, 346-351.