As part of a four-hour curriculum on integers, fourth-grade students played games using a special number line that could be folded in half at the zero point, allowing the symmetry between positive and negative quantities to stand out. (Image credit: Courtesy AAALab@Stanford)

Building on new discoveries about how the brain grapples with abstract mathematics, researchers at Stanford Graduate School of Education have developed a classroom strategy for teaching children the often baffling concepts surrounding negative numbers.

The new strategy recruits the brain’s use of visual symmetry to make sense of the physical world, and it could have profound implications for the way elementary schools teach math.

Using symmetry appears to have helped not just in teaching children about negative numbers but in improving their ability to solve higher-level math problems they haven’t seen before.

“Learning about negative numbers is one of the first times that kids learn about abstract numbers – it’s a gateway to more abstract learning,” said Jessica Tsang, a Stanford researcher and lead author of a new study with Daniel Schwartz and Kristen Blair of Stanford and Laura Bofferding of Purdue University.

The study marks the latest and most concrete result of a five-year project that bridges the gap between new insights from neuroscience and the testing of new classroom teaching tools for fourth-graders in the San Francisco Bay Area. It is published in the current issue of the journal Cognition & Instruction.

Researchers have long thought that the brain harnesses and adapts “perceptuo-motor” capacities to make sense of abstract ideas.

For the purposes of making sense of math, the most widely studied of these capacities has been the ability to compare physical magnitudes such as size. Over the past 15 years, neuroscientists have confirmed that the same region of the brain that assesses physical magnitudes is also key to comparing the magnitudes represented symbolically by numerals.

Three years ago, the Stanford researchers proposed that the brain actually re-purposes several additional capabilities, such as sequencing and ordering, to solve math problems. In this project, the researchers focused on the brain’s ability to process visual symmetry.

In 2012 the researchers found that most adults identified the midpoint between a negative number and a positive number more easily if the integers were more symmetric about zero.

For example, people were faster computing the mid-point of 6 and -4 than 8 and -2, even though both pairs of digits were displayed in the same locations on a video screen. Moreover, using functional magnetic resonance imaging, the authors found that visual symmetry-coding regions of the brain became more active for the more symmetric pairs.

Given that the adult brain recruits symmetry circuits for conceptual math problems, the researchers hypothesized that helping young students engage this native ability would improve their mathematics learning.

For the new study, the authors created a guide and tools that incorporate ideas about using symmetry to teach negative numbers. As intuitive as it might seem for people to rely on symmetry as a way to make sense of the world, few curricular materials now used for teaching math make explicit use of it.

The researchers then tested the tools with students and compared the results to those of other students who were taught with the existing approaches.

One of the new classroom teaching tools was a specially designed hands-on manipulative device. Students worked with a magnetic plastic strip that was numbered. To solve the problem 3 + -2, students attached three magnetized blocks to the right of zero and two blocks to the left of zero. The manipulative further included a hinge at zero, the point of integer symmetry. Students folded the two sides together, and the number of extra blocks on either side gave the answer, in this case +1. The hinge at zero helped students recruit their native abilities with symmetry, and the numbers on the little platform helped them coordinate the sense of symmetry with the symbolic digits.

After four hours of instruction spread over three weeks, the researchers documented encouraging results.

First, the children quickly applied the hands-on lessons with symmetry to build new strategies for solving abstract problems involving symbolic digits. When the symmetry students were asked to add up a string of negative and positive numbers, for example, many used a balancing strategy to simplify the problem.

Some paired up negative and positive numbers that would add up to zero and cancel each other out. Others clustered the positive and negative integers on separate sides, which made the balance between the positives and negatives more apparent.

Second, students were likely to incorporate symmetry as an almost automatic part of their thinking. That’s important, said Tsang, because many skills – such as decoding words in reading – are more effective when they become instantaneous and reflexive.

But the biggest surprise was on what educational researchers call “generativity” – the tendency of students to apply the ideas of symmetry on their own to problems they haven’t encountered before.

As it turned out, students who learned to rely on symmetry didn’t simply do better than other students on the material they had just been taught. They also did better on topics that they hadn’t yet studied, such as making sense of negative fractions and solving pre-algebraic problems.

“The big difference was that the symmetry instruction enabled students to solve novel problems and to continue learning without explicit instruction,” said Schwartz, who holds the Nomellini & Olivier Professorship in Educational Technology at Stanford. “By untangling how the brain comes to know mathematics, we helped with a major goal of education – putting children on an upward trajectory of future learning.”

This research was funded by the National Science Foundation and the Marcus and Marianne Wallenberg Foundation.