It takes a mathematical village
Toronto's Fields Institute uses a team-based approach to encourage collaboration on real-life problems
For Claire Heffernan, a third-year student at McGill University, there has always been something satisfying about mathematics that other subjects can't quite deliver. With history or literature, students may grapple with matters that often come down to a point of view. But when a math problem is posed, she says, "there is an answer." In high school, it was her passion for the irrefutable that first drew Ms. Heffernan toward the sciences. But math, she says, "math has the most of that."
Yet, for the past few months, Ms. Heffernan has found herself looking at math in an utterly different way: not as a tool for delivering answers but as terrain to be explored and that requires teamwork to negotiate.
The change in perspective was triggered by a summer at Toronto's Fields Institute, a powerhouse for quantitative reasoning and one of Canada's main portals to the wider mathematical world. The institute is named after John Charles Fields, a prominent University of Toronto mathematician from a century ago who is best known for creating the prestigious Fields Medal, widely regarded as the Nobel Prize of mathematics. But while it is located on the University of Toronto campus, it is an independent institute supported by a number of universities and funding agencies and populated by a rotating population of multifaceted international researchers who come to pursue their own research programs.
This year, more than 300 undergraduate math students from around the globe competed to attend the Institute's innovative summer research school – the most in the program's eight-year history. Ms. Heffernan was among the 35 applicants who were chosen. Her prize: a chance to experience firsthand what it's like to try to discover original mathematical knowledge.
Peter Gibson, a professor at York University and one of a dozen supervisors who posed math-related problems for the program this year, helped guide the students in their research. "I promised them that they would be working on pure research. I wasn't going to fake it and have an answer in my pocket," he says.
Dr. Gibson's group consisted of four students, including Ms. Heffernan, who bonded around a problem drawn from the science of acoustics. In a nutshell, the problem involves using how sound waves bounce off a wall or other material to reconstruct what the material is like beneath the surface. Geologists have been doing something similar for decades, measuring echoes from explosive detonations to map out potential oil deposits. But that method involves making approximations that overlook many of the subtle characteristics and complexities of reflected sound. A more precise alternative could allow engineers to devise materials that are better suited to absorbing unwanted sounds, Dr. Gibson says.
"Coming in, I wasn't sure what they would be capable of," he says of the students who chose to work on the problem. But in the end, the group exceeded his expectations by coming up with a successful approach to one aspect of the problem that he says he would likely not have tried. "If we're lucky, this could be part of a research paper."
The outcome is further evidence that the institute has found a formula for engaging promising students with professional mathematical research in a meaningful way – a goal that is easier said than done.
"Everyone wants to see undergraduates more involved in research," Dr. Gibson says. "The fact is that in math, that's just wildly unrealistic."
He compares the challenge to asking a group of "super fit" people to climb a mountain that no one has attempted before. Without some experience, the climber's physical ability may not be enough to succeed.
Where the Fields program has excelled is in matching elite students with one another and with the right mentors and research problems. The result provides an early taste of mathematics at a professional level that few non-mathematicians ever get to see or know. It may seem an unexpected approach in a discipline that is often portrayed as the domain of lone geniuses scribbling away at their equations in isolation. In fact, much of mathematics is collaborative and aimed at opening doors for other areas of research or industry.
"We've asked people in industry about what kind of skills they value and they say teamwork and communications," says Huaxiong Huang, the institute's deputy director. "Our motivation is to build up those skills."
It is one example of how the institute, which recently celebrated its 25th anniversary, has managed to put itself at the intersection point of mathematics and society. Its philosophy could better equip all students, not just the most mathematically gifted, to deal with an increasingly analytical and data-driven world.
Dr. Salisbury notes that the institute has in the past offered input on how to improve Ontario elementary school teacher preparation in mathematics, currently a front-burner issue after a disastrous set of provincewide test scores were released earlier this month. The institute's 2005 submission to the Ontario College of Teachers, much of which is still relevant, was largely ignored, he says.
He adds that while the students who participate in the undergraduate summer program represent some of the best young mathematical talent in the world, the approach the institute takes is not unlike what any good school math curriculum is trying to achieve.
"Teaching the math, and then letting students struggle with applying it to problems that come from outside the classroom, is what produces students who can use what they've learned," he says.
But while the classroom setting is typically set up to teach and then test students on their individual competence, true research often thrives where there is interaction.
In that respect, "it's a very different environment from school," Dr. Gibson says. "Here, nobody really knows what's involved. You're finding new routes."
Students agree that the environment can take some getting used to, but it can quickly lead to a collective effort as they learn to leverage each other's talents and experiences to find ways around obstacles. That's what William Hart, an Oxford University student who was also in Dr. Gibson's group, says he was looking for when he decided to apply to the Fields program after having once tried an individual research project.
"I thought it would be a better experience to do something different," says Mr. Hart, who adds that an unexpected side effect of the group work was all the social interaction that took place over the summer, from evening sessions of watching Game of Thrones at the U of T residence where the students are housed to dinners out where the conversation might turn to matters and interests beyond mathematics.
Equally important is the chance for students to watch and interact with the mathematical superstars who regularly pass through the institute's doors. The building's airy floor plan helps, featuring a central staircase, open central atrium and plenty of natural meeting points with blackboards at the ready where mathematical conversations can spontaneously spring up.
"There's kind of a buzz around the place. It's not distracting but it makes you feel like you're part of a larger enterprise," says Barbara Keyfitz, a former director of the Fields Institute and a professor of mathematics at Ohio State University.
This summer, the buzz was infectious as the largest cohort of summer students yet took up their problems with relish and raced to make progress in order to present their findings at an end-of-summer symposium.
Tyler Wilson, an industry-liaison officer with the institute, was also a group supervisor this year. An avid baseball fan and former semi-pro player, his problem involved finding a way to simulate the trajectory of a spinning baseball in flight.
"Their level of enthusiasm for the problem has been quite tremendous," he says. "At night, over weekends, in hours that I was asleep, they were coming up with new things."
Another change this year was the proportion of female students in the summer program – roughly one third, which is the most the program has seen so far. With many researchers voicing concerns over equity in the academic community and an ongoing perception that technical fields can be unwelcoming to women and minorities, Dr. Huang said that the program was consciously making an effort to improve the ratio.
"When we reviewed the application files, we were delighted to see a lot of excellent female applicants this year, which made our job easier," he says, adding that the program also strives for a balance of regional and international representation.
For Ms. Heffernan, the mix clearly worked. Not only has she made friends she expects to keep in touch with in the coming year, "it's made me like research," she says. "And it's made me want to do more."