"The first game we're going to play is where you see the squares and you have to tell me which side has more of them, okay?" says Anna Matejko.
She is talking to six-year-old Olivia Hall, whose head is so deep in the maw of an fMRI machine that only her feet can be seen sticking out from the giant magnetic imaging device.
It's an odd way to spend a Sunday morning, but the Grade 1 student has come to University Hospital in London, Ont., so that Ms. Matejko can study the fleeting neural fireworks going off in Olivia's brain as she responds to a series of simple counting and number problems.
A doctoral student at the University of Western Ontario, Ms. Matejko is trying to understand how kids Olivia's age – typically transitioning from an intuitive sense of things such as quantity and order to a more abstract way of thinking about numbers that is the basis for later math learning – differ in both brain activation and performance.
If Ms. Matejko can detect those differences reliably, she may also be able to predict which children will struggle with math in later grades – and help teachers intervene before years of frustration and embarrassment take their toll and leave a child firmly convinced that he or she is "bad at math."
The work is part of a broader effort now under way at Western's Numerical Cognition Laboratory to understand how the human brain learns about numbers – until recently an under-studied domain of cognitive science.
"As a society we have paid more attention to reading in the past few decades," says Daniel Ansari, Canada Research Chair in Developmental Cognitive Neuroscience and head of the lab. "But more and more I think people are recognizing the importance of math skills … and I think with the research we're really catching up."
Their research has particular resonance now, as anxiety grows across North America about students' math abilities.
In Ontario, provincial test scores have fallen five years in a row, renewing debate about whether the curriculum should move away from concept-based learning and return to old-school arithmetic drills in early grades. Some parents are so concerned about gaps in math education they are hiring private tutors for their children, creating what some fear is a two-tier system; one math-tutoring company, Kumon Canada, says its enrolment in math programs has jumped 23 per cent in three years.
Few doubt that the consequences of poor math skills extend beyond the classroom. A U.S.-Swiss study published in May found that, during the subprime mortgage crisis of 2008, the ability of homeowners to perform basic mathematical calculations was a predictor of whether or not they were likely to default on their mortgage. More broadly, the Obama administration has declared improving math and science education a high priority for economic competitiveness.
Ready or not
Like all learning, exposure to new ideas rewires kids' brains so that they can handle new tasks. But the outcome is not the same for all children.
For example, David Geary, who leads a long-running research program in math cognition at the University of Missouri in Columbia, has been comparing the math performance of 180 13-year-olds with their numerical knowledge in kindergarten. The results show that differences in mathematical performance persist.
The same group of kids are now 16, and Prof. Geary says the most recent data, as yet unpublished, show the same trend: "If you start school low in math achievement, independent of other things, you tend to stay low."
More troubling: Slow starters in math are unlikely to catch up, in large part because math involves an ascending progression of concepts where the ability to grasp each new idea depends on a firm understanding of what came before.
"It's easier to catch up in reading.," says Prof. Geary. "But in math, the curriculum proceeds whether or not you're ready for it to proceed."
Clearly, many factors can affect math performance. Even though studies suggest that gender is not a big factor in determining number skills at an early age, girls remain underrepresented in higher-level math education, presumably due to social or cultural hurdles. Socioeconomic status can also play a role in how well children are prepared to learn math.
Yet beyond these external factors, and independent of overall intelligence and language skills, there appears to be a separate mental faculty for dealing with numbers that helps determine how well a child grasps math. Those with serious impairment to this faculty may account for the 5- to 7-per-cent of people who suffer from a condition called developmental dyscalculia, analogous to dyslexia in reading.
When it comes to the direct link between the brain and performance, the challenge for researchers is to pinpoint precisely which fundamental concepts are essential for children to be able to keep up in class. Just as many children with reading problems can be helped by the right teaching strategies at the right time, the hope is that those who struggle with numerical concepts can be similarly supported if identified soon enough.
Drawing from his own work, Prof. Geary says those fundamental concepts include a clear understanding of numbers and the quantities they represent, seeing how those quantities relate to one another, as on a number line, and being able to understand that larger numbers can be broken down into smaller ones, the way 7 can become 3 and 4, for example.
"We're beginning to understand what kids need to know at the beginning of First Grade in order to be really ready to learn formal math," he says.
Wired for numbers
The growing scientific interest in numerical ability is also motivated by a broader interest in how the human brain overlays a complex skill on top of what appears to be preexisting biological ability.
This idea that a rudimentary number sense is hardwired into our brains at birth has a long history, supported by decades of work with animals, and by some unusual instances of brain damage in humans.
In the 1990s, Stanislas Dehaene, a neuroscientist based at College de France in Paris and a pioneer in the study of numerical ability, documented a subject he dubbed the "Approximate Man." The patient had lost much of his mathematical reasoning ability, among other cognitive skills, as the result of ruptured blood vessels in his brain.
Curiously, though, he was still able to approximate quantities. He could estimate that a year was about 350 days long and that a quarter of an hour lasted about 10 minutes. It was the exactness of calculation that eluded him. His case seems to suggest that humans possess an underlying ability to deal with quantity in an approximate form, just as animals have been shown to, and that this innate ability can still be glimpsed when higher order skills are cut off.
The question for researchers today is whether this innate sense, known as non-symbolic number processing, is relevant to the acquisition of more formal math skills. A study published this week by Elizabeth Brannon, a cognitive neuroscientist at Duke University, suggest there is a connection.
Prof. Brannon and her team had a group of six-month-old infants watch and react to a stream of changing dot patterns, noting how readily they spotted changes in the number of dots, in contrast to changes in dot size and arrangement. "There's a lot of variability," Prof. Brannon says. "Some babies show a really strong effect and some babies don't seem to notice at all."
Three years later, the same children, now preschoolers, were given standardized tests to probe their math ability. Those who were better at discriminating a change in the number of dots when they were babies also tended to score higher at age three-and-a-half.
"The possibility is that infants with good approximate number sense are finding it easier [three years later] to map number symbols onto these pre-verbal number concepts," says Prof. Brannon.
Nevertheless, the role of non-symbolic number processing has lately become a subject of scientific debate.
Dr. Ansari argues that while it likely plays a part in the development of number skills, its relationship to those skills is not clear. This ambiguity could simply be a matter of variations in the way experiments have been run by different groups – or it could mean that the initial construction of a child's ability to work with numbers involves something more than mapping symbols onto what the brain is born with.
"The story is not as straightforward as it once seemed," Dr. Ansari says.
Yet, despite these complexities, Dr. Ansari and other say cognitive research can inform the way math is taught at school.
"Teachers are changing the brains of students in their classrooms." he says. "There should be a connection between neuroscience and education."
Part of the challenge is winning over the education establishment, from board administrators to faculties of education, which has not always been welcoming to cognitive science in the classroom.
More recently, there are signs that the tenor of the relationship is changing, though, thanks in part to demonstrated improvements to reading instruction over the past several years based on brain research. Researchers are also working more directly with schools to find useful tools.
For instance, Dr. Ansari's lab has developed a two-minute "paper-and-pencil" test of numerical ability that could, he says, be administered to kindergartners to help spot differences that could predict future math trouble.
Steve Killip, manager of research and assessment for the Thames Valley District School Board collaborates with Dr. Ansari on projects running at London-area schools. He considers the cognitive research at an "early stage" in terms what it may ultimately mean for classroom practices. But he adds that "there's more of a sense of partnership and collaboration" compared to an earlier generation of scientists working with the school system to conduct cognitive research.
The school system may also appreciate what the increasing body of research in math cognition says about classroom methodology – including the fact that some of the most well-worn debates over math education may well be moot.
"The bottom line is that dichotomies such as 'creative' vs. 'rote' math are false dichotomies," Dr. Ansari says. "Math learning involves both procedures … so biasing the curriculum in any one direction will likely do more harm than good."
Nancy Jordan, a professor in the School of Education at the University of Delaware, has spent the last five years developing and testing ways to help kindergarten-age children who lag behind in their number skills. She says she has observed a huge shift among the school systems she has worked with, which not so long ago offered no extra help at all for the younger students because math difficulties before Grade 3 are easily overlooked.
"It was kind of a wait-to-fail model," she says. "Since that time there's been a lot more in terms of screening and intervention. But we still need to provide instruction that will lead to sustained gains."
Back in the fMRI machine, six-year-old Olivia is watching a cartoon to give her brain a break after responding to another round of cognitive tasks. Ms. Matejko will meet with her again later in the school year to see how her responses have changed and how that compares with other students in her sample.
"I think this work will help us understand where individual differences come from," she says. "And perhaps, later on … create change."