Modeling the real world

There’s a story that Dan Teague, who has taught in NCSSM’s math department since 1982, loves to tell about NCSSM’s unexpected arrival on the international mathematical modeling scene. 

It began in the summer of 1986 at Princeton University, where Teague was teaching in a workshop. There as a speaker was Sol Garfunkel, the director of the Consortium for Mathematics and Its Applications (COMAP), which hosted an annual international competition for university math students. Teague asked the director if NCSSM could participate in the event.

Garfunkel agreed, almost as if in passing, Teague recalls. 

And so, with Garfunkel’s off-handed blessing, NCSSM entered the COMAP Mathematical Contest in Modeling the following year and squared off against universities from around the world. 

The format was simple: over the course of a long weekend, participating teams chose one of two problems to solve using mathematical modeling, then submitted their solution in a paper to the competition’s judges for a blind review. 

When the winners were revealed (deemed “Outstanding” solutions), the judges turned to one another in confusion. None of them had ever heard of NCSSM.

Unsure what to do, they turned to a directory of North Carolina universities. No Science and Mathematics university there. 

Eventually, someone with knowledge of the school appeared. NCSSM, they said, was a high school.

A high school? A competition for undergraduate math students could not name a high school as one of its winners, Teague recalls hearing in a phone call from Garfunkel. Apparently, in his brief conversation with Teague the year before at Princeton, Garfunkel had missed the fact that NCSSM was a high school.

Dan Teague

“I reminded him that I had asked him if we could do it and he had said yes,” Teague says. “He said, ‘Yeah, sorry about that, but the rules are that this competition is for undergraduate students.’”

“You can’t be more undergraduate than high school,” Teague responded.

There was a long pause, Teague says, before Garfunkel finally spoke again. “I think that might be it,” he said.

It was. NCSSM has been participating in the competition ever since, winning recognition alongside some of the most prestigious universities in the world.

Real-world scenarios

The Society for Industrial and Applied Mathematics, the world’s largest professional association devoted to applied mathematics, provides perhaps the most easily-understood definition of mathematical modeling. It is, the Society says, “the process of creating a mathematical representation of a real-world scenario to make a prediction or provide insight.”

Here’s how it works: a problem is presented, modelers assemble a multitude of relative numerical facts specific to that problem, algorithms are created (if none already exist) through coding to run the numbers, it’s all entered into a computer programmed with a set of parameters, and the computer does the rest. If the results are inaccurate or unreliable, the known inputs and/or parameters are refined, and the whole thing is run again.

The “real-world scenario” in the Society’s description of modeling is what modeling is all about, says Taylor Gibson, NCSSM’s dean of mathematics. He points to a project from a few years ago in which NCSSM students set out to reconfigure the school’s exam schedule to eliminate a significant number of scheduling conflicts vexing students and teachers alike. Initial solutions eliminated nearly all exam conflicts, but were unrealistic in their suggestions.

Taylor Gibson

So the modeling students returned to the problem, refining their work over and over until finally they arrived at a solution that resolved about half of the 400-plus scheduling conflicts in a way that was doable. NCSSM adopted the new schedule.

Moments like that, Gibson says, when a student realizes that a real problem seemingly unrelated to math can, in fact, be solved using math modeling, are quite often “transformational.” 

That is exactly how Graham Pash ’15 describes his experience with mathematical modeling at NCSSM. Prior to arriving at NCSSM from Terry Sanford High School in Fayetteville, Pash had never even heard of math modeling until he enrolled in a modeling class the first semester of his senior year. 

“That class really changed a lot of the way that I think about math and approach problems in general,” Pash says. “It personally changed my trajectory because I had kind of always figured that I was going to do engineering in college … and I did do that, but it really sent me down a path of becoming more mathematically inclined, of feeling more that I had a place in math.”

It led Pash to double major in applied mathematics and mechanical engineering at North Carolina State University. Today he is a Ph.D. student in the Computational Science, Engineering, and Mathematics program in the Oden Institute at University of Texas-Austin.

“Problems that matter”

There’s more to modeling than seeking mathematical solutions to practical problems in a classroom, Gibson says. It can also teach students how to problem solve nearly anything, in and out of the classroom.

“It’s a lot like life,” he says. “There are often a lot of different things you could do, and they all have their pros and cons, and you need to be able to decide how to pick one. Mathematical modeling trains students to evaluate multiple solutions under different contexts and choose the one they think is best for the given situation.” 

That open-ended nature of modeling is a shock for many students upon first encounter. “It blows their mind to think about math as a field where maybe there isn’t one right answer,” Gibson says. “Sometimes there are many right answers, or no right answer but some good answers. Lots of times you’re measuring not whether you got one right answer or not, but measuring how close you came to an objective goal.”

For Pash, it was the first time he had truly felt in control of a problem. “In a lot of math classes you have to learn this concept and that concept and you do a bunch of homework problems on those concepts and there’s not that much flexibility for allowing you to approach problems in different ways,” he says. “But the modeling class [at NCSSM] is incredibly open-ended. Here’s a problem statement: Go! I just really enjoyed having the opportunity to be creative in the way that you attack problems and how it helps your mind sort of learn new ways to come at things the more you do it.” 

Christine Belledin, who has taught modeling alongside Teague for a number of years, mirrors Pash’s point. “What’s so much fun about [modeling at NCSSM] is that it’s a very creative class,” she says. “Sometimes the most creative solutions come from the least advanced mathematical methods. That’s really empowering for students because they can see how the math that they know can actually be applied to solve problems that matter.”

Christine Belledin

The creativity inherent in modeling allowed Pash to learn from others by investigating their methodology and logic. Often he saw in others’ work things he wished he had done in his own. “It’s cool to see how somebody else approached it and came to a similar conclusion and sometimes you’re like ‘Oh, I agree with them, I wish I had taken a little bit of what they did and applied it to how I did it.’”

There are varied approaches to teaching modeling as well. For starters, Teague and Belledin both eschew formal tests, daily quizzes, or a running calculation of a grade (though there are grades offered for completed projects). That can be a bit disconcerting for kids focused on up-to-the-minute calculations of their class grade. 

Two, neither spends much time up front teaching students how to model. Doing so, Teague says, would burn half of a semester before students ever got to model. Instead, students are led directly into a project, then taught on the go through teacher feedback on what went right and what went wrong.

Finally, neither Teague nor Belledin have in mind any particular way they want a student to approach a challenge. Rather, they just want them to do something. “Often a student will ask me, ‘What do I do now?’ Teague says, “and I always tell them ‘you decide.’” 

It takes a bit of adjustment, but in time, many students fully embrace that style of learning.

“The whole course is figuring out how to do what you don’t know how to do,” Teague says. “Modeling is a big thing. There are lots of pieces, and I tell my students not to expect to master it right away. You are going to struggle with it a bit. So, what you need to do is jump in, be bad at it to begin with, and live with that. And each time you do it, you get better and more comfortable.”

Because modeling is often such an open-ended immersion into the unknown, it can be challenging for the teachers as well, allowing them to grow and learn alongside their students. “Sometimes I’ll get a question and I’ll have to say, ‘You know what? Let me go think about this,’” Belledin says. “It’s definitely a different kind of course.”

For Pash, math modeling at NCSSM finally made clear to him the true meaning of a cliched but true saying in mathematics. “I think what that class and what that program instills in you is that math is a language,” he says. “I used to hear that before I took the class and I always thought they meant, ‘oh, it means a number stands for a letter or a word.’ But it’s more than that. It’s about how you use those numbers to communicate, to think about a problem and all your assumptions, and everything that’s baked into those equations that you write down. It means that someone can come in and read them and know exactly what you’re talking about.”

Secrets to success

NCSSM has had unprecedented success in mathematical modeling competitions, even when competing against elite universities. By Teague’s estimation, NCSSM has had more solutions deemed “Outstanding” in the last 30-plus years of the COMAP competitions than all but one other school. Even more than Duke University. In one recent Mathematical Contest in Modeling competition, NCSSM was the only school from the United States to earn an “Outstanding” distinction. And in the 2017 International Mathematical Modeling Challenge, another COMAP competition which usually names more than one winner, NCSSM’s solution was so unique and impressive that the judges felt there needed to be some distinction. So, they named NCSSM the sole winner.

So how is it that a high school does so well? Part of it, Gibson says, is that NCSSM has the curricular freedom to offer courses in mathematical modeling. “It’s not a course that’s easily introduced in more typical high schools because it doesn’t easily align with high school graduation requirements or state curriculum requirements,” even though, he says, he has seen evidence that students in more traditional high schools do well in the course if it’s made available to them. 

Too, NCSSM tries very hard to incorporate the idea of modeling open-ended problems using math students already know into more than one course. In particular, Gibson notes NCSSM’s Precalculus and Modeling course as well as Modeling with Differential Equations.

Engaging modeling in other courses and exploring the range of issues that can be addressed through modeling tremendously widens students’ perspectives on what is possible, Belledin says. “It seems such a waste to learn pure mathematics without learning also the power of what it can do.”

But a lot of NCSSM’s success, Gibson says, is due directly to Dan Teague, even though Teague always deflects any praise offered his way. Teague seems to have a way with students that no one else – and Gibson includes himself here – can. “He gets results out of the kids, and he does it with a smile,” Gibson says. “He does such a good job of training them to be thinkers and doers and unafraid to try and fail. And that’s really what you need to be a successful math modeler. If we could bottle up that secret sauce that Dan has – but we just don’t know how he does it. It’s incredible.”

Belledin agrees. “Dan is really the heart and soul of the program,” she says. “He really built it from nothing.”

Unsurprisingly, Teague gives credit to the students. “Our students do so well because, most often, their solutions are different from others,” he says. “They don’t know what they don’t know and so they approach things in truly novel and creative ways,” building their own models from the inside out instead of relying on standard formulas and approaches heavily used by math modeling students. 

But the biggest factor contributing to the math modeling students’ successes, Teague says, is the talent of his colleagues in all disciplines at NCSSM. To successfully model a real-world scenario, he explains, you have to have a well-rounded and deeply informed approach.

“Most of the problems we win on come from the social sciences,” he says. “You have to know a lot of science and you have to know a lot of social science. You have to know how to write code and you have to be able to explain very complex ideas simply and clearly. This is stuff they don’t learn in math class at NCSSM. They learn all this in our other courses. So, with the awards our students win, we in the math department will take a little bit of credit for it, but their English teachers and their computer science teachers and their social science and natural science teachers all play a huge role in what they are able to accomplish. It really is more of a statement about the school’s curriculum and the school’s faculty than it is a statement about the math curriculum and the mathematics faculty.”