Over the past year, since I took over the common-core math beat, I’ve been thinking a lot about fractions.
As I wrote in November, the Common Core State Standards for mathematics emphasize fractions as points on a number line, rather than just parts of a whole. Now, more teachers are pinning numbers to clotheslines to demonstrate fractions rather than divvying pizzas and fruit pies. Many experts have called this the biggest shift from previous state standards in math.
In a phone call last week, Hung-Hsi Wu, a professor emeritus of mathematics at the University of California, Berkeley, who helped write the common standards in math, gave an impassioned explanation of why instruction on fractions, which he calls “the backbone of school mathematics,” needed to change.
Fractions are “really an abstract concept” Wu said. “For whole numbers, you have your fingers and your toes [to count on]. But for fractions, where in the world do you see 11/13?” Students are given a flood of terms to memorize when they start learning fractions—numerator, denominator, proper, improper—that are often not well-explained conceptually.
Wu claims that the beginnings of math anxiety in students can often be traced to “the day they go to school and learn about fractions.”
Some research has indeed linked math anxiety to early exposure to negative math experiences. (Studies have also pointed to adults’ discomfort with math and genetic factors as potential causes.)
Wu went a step further, pinpointing the heart of what he calls “fraction phobia": fear of addition of fractions.
“By the time students get to fractions, there’s ingrained knowledge in their minds that adding things is combining things—which is a correct image,” he said. “So a teacher says, ‘OK kids, we’re going to learn how to add 2/3 and 1/5. Great, [the students think], let’s put things together. Unfortunately, the next thing the teacher tells them is, ‘First, look for the least common denominator.’”
Similarities to Whole Numbers
That’s when students begin to panic, he said. They think, “I’ve been learning to put things together, but instead of putting things together, you want me to monkey with the numbers? What am I supposed to do?”
Instead of changing the rules, teachers can show 2/3 on a number line, and 1/5 on a number line. Students see that 2/3 + 1/5 is the same as the length of those line segments next to each other. Adding fractions on a number line works the same as adding whole numbers.
But how can students convert that line segment back into a fraction?
“They have rulers, so you ask them, ‘I have 2 centimeters and 1 inch. How do I put them together?’” he said. Students will relate to the need for a common unit.
Teachers also need to emphasize the concept of equivalent fractions, which he calls “the most important thing about fractions.”
Wu has been lobbying to move away from pizzas and teach fractions on the number line for nearly 20 years. The common-core standards incorporate many of the ideas on fractions from a 2008 federal study by the National Mathematics Panel, of which Wu was a member.
For more on the differences between the common core and most previous state standards with regard to fractions, see the video below and this explanation on fractions division.
