The Mathematical Mind Offers Neuroscientists a Master Class in Concentration
Expertise bulks up the brain’s ability to think deeply, a skill that may generalize across tasks
By Hanna Poikonen
Think of the last time you concentrated deeply to solve a challenging problem. To solve a math puzzle or determine a chess move, for example, you might have had to screen through multiple strategies and approaches. But little by little, the conundrum would have come into focus. Numbers and symbols may have fallen into place. It might have even felt, at some point, like your problem effortlessly resolved itself on the blackboard of your mind.
In recent research, my colleagues and I set out to investigate the neural mechanisms underlying these experiences. Specifically, we wanted to understand what happens in the brain while a person engages in abstract and demanding thought—so we designed a study involving math expertise.
Mathematics relies on an ancient brain network located in the parietal regions at the top and center of the brain’s outer folded cortex. That network helps us process space, time and numbers. Past studies on neurocognition in mathematics have focused on brain activity while considering problems that take a few seconds to solve. These studies have helped illuminate brain activity that supports focused attention and a special form of recall called working memory, which helps people keep numbers and other details top of mind in the short term.
But our study used longer, more complex math challenges that involve multiple steps to solve. These problems are more akin to the tricky puzzles that mathematicians must tackle regularly. We found that people with more experience in mathematics enter a special state of deep concentration when thinking about challenging math problems. Understanding that state could help scientists to someday understand the power of concentration more broadly, as well as the possible trade-offs of off-loading our problem-solving to our devices.
For our experiment, we recruited 22 university students—at both the graduate and undergraduate level—who were in math and math-related programs, such as physics or engineering, along with 22 fellow students in disciplines with minimal to no quantitative emphasis, such as physiotherapy and arts. We determined each student’s verbal, spatial and numerical intelligence quotient (IQ), as well as their level of math anxiety.
We asked the students to watch step-by-step presentations that explained how to solve several challenging math problems—such as proving a Fibonacci identity. Throughout this demonstration, students wore a cap covered with electrodes so that we could noninvasively track electrical activity in their brain. After each presentation, they had to report whether they thought they had understood the demonstrations and how engaged they felt during this experience. We also encouraged the participants to watch the demos carefully by telling them that they would have to explain the problem afterward.
We found that the students with greater math expertise showed markedly different brain activity than those with less. For example, the students whose coursework involved little mathematics showed more signs of complex activity in the prefrontal cortex, an area just behind the forehead that is engaged in all kinds of cognitive efforts. This finding may reflect how hard they were working to understand the various steps of the complex math demonstrations.
But things really got interesting when we turned to students who engaged in quantitative thinking regularly. We noted significant activity that appeared to link the frontal and parietal regions of their brain. More specifically, these areas exhibited a pattern of activity that neuroscientists describe as delta waves. These are very slow waves of electrical activity that are typically associated with states such as deep sleep. Of course, these students were wide awake and deeply engaged—so what was going on?
Some recent research suggests that these “sleepy” slower delta waves may play a crucial role in the cognitive processing that supports deep internal concentration and information transfer between distant brain regions. For example, recent studies show that large-scale delta oscillation emerges among experienced meditators when they enter meditative states. One reason that meditation, mathematical problem-solving and sleep resemble one another might be that, in each case, the brain needs to suppress irrelevant external information and unneeded thoughts to really focus and concentrate on the task at hand. (Indeed, even sleep can be a busy time for the brain. Sleep research has revealed deep sleep’s irreplaceable role in memory consolidation; slow-wave sleep retracts the neural patterns that were previously activated during a learning task.)
In fact, we suspect that the long-distance delta oscillation we observed may play a central role whenever people are immersed in contextual and complex problem-solving. For instance, we have found that dancers and musicians show similar delta waves when watching dance or listening to music. This suggests engaging brain networks in this way could be useful for many tasks involving concentration. It’s likely that when people who have extensive experience in a task are deeply engaged in that effort, these same slow delta waves are involved, even as the specific brain networks vary. It’s also possible—though we’ll need to investigate further to be sure—that this state of deep concentration is generalizable: develop this way of thinking in one domain, whether it’s tackling trigonometry or playing the violin, and it could help you in others.
Though our experiments involved students and not, say, champion mathematicians or Nobel laureates, the differences in brain activity that we observed are still a testament to the power of practice in expertise. Our student participants did not significantly differ in their IQ or level of math anxiety, for example. Rather repetition and deliberate or intentional study helped some of these graduate and undergraduate students become more efficient masters of quantitative thinking.
By the same logic, these findings hint at a trade-off that people should keep in mind—particularly as artificial intelligence and other tools offer tantalizing shortcuts for various forms of problem-solving. Each time we off-load a problem to a calculator or ask ChatGPT to summarize an essay, we are losing an opportunity to improve our own skills and practice deep concentration for ourselves. To be clear, technologies can boost our efficiency in important ways, but the seemingly “inefficient” hard work we do can be powerful, too.
When I consider how frenetically we switch between tasks and how eagerly we externalize creativity and complex problem-solving to artificial intelligence in our high-speed society, I personally am left with a question: What happens to our human ability to solve complex problems in the future if we teach ourselves not to use deep concentration? After all, we may need that mode of thought more than ever today to solve increasingly complex technological, environmental and political problems.
