By Eileen Toh
Researchers at the Stanford Graduate School of Education (GSE) have discovered new, more successful classroom methods to teach negative numbers to children using symmetry.
Their study focused on the brain’s use of visual symmetry, “a primitive ability that humans have,” said Daniel Schwartz, professor at the GSE and an author of the study. Using symmetry, students can visualize negative and positive numbers as reflections of each other, which enhances their grasp of abstract numbers.
Over the past 15 years, neuroscientists have asserted that there is a region of the brain that compares physical sizes by assigning each of them a symbolic, numeric magnitude.
From this information, in 2012, the researchers hypothesized that the brain repurposes many capabilities to solve math problems. In their five-year study, the researchers focused on the brain’s use of visual symmetry to represent math problems.
The researchers conducted behavioral laboratory research on adults completing simple mathematical computations, such as determining the larger of two numbers.
According to Jessica Tsang Ph.D. ’12, the lead author of the study, when adults had to identify the midpoint between numbers, they were quicker when the numbers were closer to zero and more symmetric about the number line at zero. For example, people can more easily compute the midpoint between -7 and 7 than 12 and 20. The researchers also found that for more symmetric number pairs, the brain was more active.
The researchers then proposed that students would learn more quickly if they applied visual symmetry to the concept of abstract numbers.
“Negative and positive numbers are reflections of each other at zero, [but] students often don’t see that,” Schwartz said. “If students see that the numbers are reflections of each other, they can learn negative numbers better.”
At elementary schools in the Bay Area, the authors developed and tested tools that implemented visual symmetry on fourth graders to teach math. Schwartz said that when the researchers first created their materials, students did learn symmetry better, but no improvement in learning numbers was observed. Despite correctly distinguishing between negative and positive numbers, students would still put them in the wrong order.
“When the kids would draw a number line, some of them would put negative numbers on the right side of the number line and positive numbers on the left,” Schwartz said. “They understood symmetry, but didn’t get the concept of bigger or smaller numbers.”
The researchers then went through a process of finding mistakes with their tools, fixing and tweaking. They also compared their students’ results to those of students who were taught with typical approaches that do not incorporate visual symmetry.
One tool they developed was a magnetic number line for adding and subtracting integers. Given the problem 6 + (-3), students would attach six blocks to the right of zero and three to the left. They could then fold the number line at zero, and the number of leftover blocks due to symmetry would give the answer: three.
After three weeks, the researchers found that symmetry became an “instantaneous and reflexive” part of students’ thinking that allowed them to do better on topics not yet covered in the classroom. According to Tsang, the children were able to apply the ideas of symmetry on their own to solve problems that were not learned before.
The researchers hope that, with this new strategy, teachers can use symmetry to better teach abstract numbers.
“It’s rare when research and neuroscience generate a new way to teach,” Schwartz said. “With this research, we examined why different types of instruction are effective or not and helped children enhance their future learning.”
Contact Eileen Toh at eileen.toh ‘at’ saratogafalcon.org.