Just outside a cafe on Duke campus, mathematician Jonathan Mattingly sits down with his colleagues, pulls a legal pad and pen from his bag and quickly draws a stick-figure form of a picket fence.
The drawing seems simple. But it represents the deep theoretical underpinnings of democracy and also the introduction of a half dozen North Carolina School of Science and Math (NCSSM) students to the life of a mathematician.
"There's nothing in standard schoolwork that really resembles the work of a mathematician," says NCSSM math professor Dan Teague, who watches as Mattingly, his former student and now his collaborator, works on the drawing. "In a chemistry lab, you do things that are similar to what a chemist might do, but you can't often do that with mathematics."
To change that, Teague and Mattingly have developed a mathematical research course for NCSSM students. This fall marks the fourth year Duke professors will work with the students to give them first-hand experience doing math research.
"How do you get students to go into mathematics? It's not by teaching them one more math course," Teague says. "It's by letting them experience the energy that comes by solving real-world problems mathematically."
Finding the right problems, however, is a challenge. The students, Teague says, have the passion and energy to do the work, but they don't yet have the mathematical tool kit to begin working on the most challenging questions in the field.
"At first [math research] can be daunting because you might begin with very limited knowledge or intuition of the problem at hand," says Christy Vaughn, a former NCSSM student who participated in the research course and is now a junior majoring in math at Duke. Still, she says, "there are all kinds of problems just waiting to be studied, and it is exciting to tackle a new one."
The most important feature of the research course is the students' awakening to the process of discovery and independent thought. "At this age, they are used to being right. They are at the point where they would rather do nothing than be wrong," Teague says. In the course, however, there's no way to find out if the students' approach to a problem is going to work until they've done it. "That process really helps them think clearly about whatever it is they are thinking about, whether it is mathematical or non-mathematical," he says.
It's true, Mattingly says, thinking of his own work with Teague, who introduced him to the process of discovery for the first time. "The experience ended up pushing me in the direction I took in college and ultimately in my career," he says, adding that "[The School of] Science and Math was a really important thing in my life in every way, especially socially and intellectually, and the self-confidence that the research experience, and Dan's mentorship and modeling class, gave me was a large part of that."
He feels so strongly about the school's influence on his life that being able to give back to it and his mentors actually factored into his choice to accept a faculty position at Duke in 2002. At the time, he was weighing the pros and cons of taking a position at two schools. "One of the things I listed as a plus for Duke was being back here, closer to Science and Math and having the opportunity to get involved there," he says.
His opportunity to "pay off childhood debts" came about five years ago when Steve Warshaw, NCSSM's Vice Chancellor for Academic Programs and also one of Mattingly's former research advisors, called to ask if he could mentor a student. Teague heard about it and called Mattingly too. After a few conversations, the two decided to expand the mentorship experience into a research course open to a group of students who would work as a team to solve open-ended math problems.
Last year the students looked at the mathematical foundations of voting schemes. Pointing to the drawing, Mattingly says the vertical lines represent where candidates fall based on political ideology. Students can calculate how votes are distributed to each candidate and what happens to those votes when a contender leaves the race. Students can also see where to align themselves politically if they want to win a race, Teague says.
"At a fundamental level, we are investigating what is democracy," says Duke mathematician Hubert Bray as he watches Mattingly work on the drawing. "You can look at all these ways of determining the winners of an election. You can look at their mathematical properties. At the end of the day, though, you then have to judge those mathematical properties and decide which voting scheme looks more like democracy."
Vaughn says looking at these problems, which professional mathematicians don’t quite understand yet, really expanded her interest in math research. Being able to work on campus with Duke faculty also reinforced her decision to attend the university, she adds.
Though he now mentors Vaughn, Mattingly says he tries to resist the urge to draw all the high school students into math. "I want to make sure that there are more people who go out and work as lawyers, economists and doctors, who do other things, but still see that the process of discovery is no different in math than in any other intellectual pursuit," he says, folding his drawing and sticking in his bag.
As the three mathematicians get up to leave the cafe, they talk about mentoring math high school teachers in leading their students in research. Teague has done this already, and now Mattingly wants to bring the teachers to campus too to get them involved solving more open-ended math problems.
Teague also reminds Mattingly about the coming year's research course. Immediately they start throwing around topics -- maybe knot theory or graphs related to evolutionary biology. A final decision isn't made that afternoon, but it's clear that next year another group of NCSSM students will learn what it's like to be a mathematician, or at least to think like one.