Associate Professor of Mathematics
“I feel my job is to challenge my students. If that makes me the hardest professor in the department, I’m willing to live with that because I feel it’s part of my job to make my students think,” said UMF Associate Professor of Mathematics Nic Koban.
Mathematics students enjoy his Abstract Algebra and Geometry courses, but Koban also understands that some students are intimidated by math and he works step-by-step with them to explain algebra and abstract math concepts.
“Typically, I try to break it down so students can understand it from a simple point and we can build up again,” said Koban.
“People have told me they used to be good at math until x was thrown in — but that x is where math really begins with unknown variables,” Koban said.
Math for Everyone
“The Mathematics department at the University of Maine at Farmington serves a broad spectrum of students. We have so many different types of students — we try to take of all their needs. We have our general classes for non-Math major students. We have Education students who don’t have to be theoretical mathematicians, but need to be strong, competent mathematicians who go out to teach kids. Then we have those students majoring in Mathematics, who go on to grad school,” Koban said.
“I just had a student in here and we were working on a mathematics problem and she said if we replaced that number 10 with the letter D, she’d be totally thrown off. I explained to her that whatever you can do to the number 10 — you can do to D — you can square it, divide it by two, and take the square root” said Koban.
“It’s really not hard, but I realize it’s uncomfortable. It’s okay to express it as D2 = 8 or D2 ÷ 2,” Koban said.
By generalizing, you don’t have to go through these steps all the time. If you tell me D=10 or 55 I can plug it in without having to go through all the algebra,” said Koban.
“It’s conversations or blackboard lectures like that where I get students to understand that algebra is not evil or rocket science. Sure, it can be difficult, but maybe it’s not as bad as they thought,” Koban said.
Koban can give individualized attention to his Mathematics students because there are usually no more that 25 student in each class. UMF’s small size allows him to know all his students, he said.
Teaching Future Math Teachers
“I want our Education students to be strong mathematicians who are also good at teaching because if there’s a problem in education, the way to solve it is to send out well prepared teachers who really understand their field,” Koban said.
Koban is teaching Mathematics Education students using a modified Moore method where students are self-directed, with the professor there to guide them. Students work on the problems and present solutions to the entire class.
“Students roll up their sleeves and dig through the material they’re trying to understand. This class is reading a textbook written by UMF Associate Professor Mathematics Paul Gies. At every class, they tell me which of the problems they think they can do. I divvy them up and they present their problem and solution. If there are questions or something goes wrong, then there’s a discussion,” said Koban.
“These students are going to be teaching grades K-8. They’re going to have to go through textbooks, understand the problems, organize their thoughts, and present the material in front of a class so that every pupil in the room understands. I thought this was the perfect time for them to get first-hand experience at doing all that. I may not be the most popular person in their world, but it’s good practice,” Koban said with a chuckle. “In the UMF Mathematics department, we see a broad spectrum of student needs and abilities I think we do a good job of taking care of our students’ needs and that’s what we should be doing,” Koban said.
Geometric Group Theory
“A group is set of objects that you can do algebra on. Most people think algebra is about numbers, and certainly, that’s true. You can do algebra on other abstract objects that aren’t necessarily numbers. My favorite examples are braids. If you think about attaching three pieces of rope to the ceiling and as you lower them down to the floor, you braid them together and attach the ends of the rope to the floor. Those objects can actually be considered an algebraic object,” Koban said.
“If you want to put two of these together, you go from ceiling to floor with one of the braids, and then make the floor of the first braid the ceiling of the next braid and go down again. Then, remove the middle piece and you have another braid. So in a sense, you can add two braids together,” said Koban.
“You can have a braid that acts like the number zero,” Koban said. “There are also ideas of negation that this braid minus another braid would give you the identity braid. So there’s the idea of three and negative three added together gives you zero.”
“And so these objects can have algebraic qualities. If I want my set to have these algebraic qualities, then those things are called groups. But I don’t use algebra to study groups. I find information about groups by looking a geometric spaces that talk about the group. Knowing the group, I sit and build a geometric space. I study that geometric space, take a look at its properties, and from that I can tell something about the group. That’s geometric group theory,” Koban said.
Avoiding Robot Crashes and Mid-Air Collisions
“The example with the braids may seem kind of strange. Who cares about lowering ropes or braiding them up? But if you think about them as objects moving through time and space, then there are real-world applications,” Koban said.
“If you had three robots moving around that you didn’t want crashing into each other, that’s the same as three ropes in the braids not intersecting each other. If you want the robots to perform a series of moves and another series of moves and then undo those moves or do nothing, that’s the algebraic structure being demonstrated in the braid group,” said Koban.
“The same could be true of planes landing and taking off. We certainly don’t want planes running into each other. This idea of objects not hitting one another is called the fundamental group of a configuration space. Many groups can give you that information and other groups have other real-world applications,” Koban said.
Michael D. Wilson Research Fellow / Scholar Program
The Michael D. Wilson Research Scholars and Fellows Program has sponsored undergraduate research at UMF since 2006. The program, named after a UMF student who died soon after he graduated, provides students with funding to conduct their own high-level research, the kind of research often found only at the graduate school level. UMF faculty select 12-18 Wilson Scholars each semester and two Wilson Fellows are selected each year to pursue more extensive research projects. Koban has sponsored two Wilson Fellows and several Wilson Scholars.
“Every year, I hire a research assistant to help me with my research. They work on samples and come back with results. We try to generalize those results and make theorems and prove theorems, all sorts of good stuff. Each time, this process has led my assistant to become inspired to discover their own little project, and each time I’ve recommended that they try for a Wilson Scholar award,” Koban said.
“I hire my assistant in the summer and we spend those three months in the summer training in the graduate level math they need to know for Geometric Group Theory. I get them ready to work on research projects with me. It’s challenging, but so far so good,” said Koban.
Koban’s research assistants are among the select number of Mathematics students who have gone on to graduate studies in schools such as Tufts University, North Carolina State University and Binghamton University.
The Family who Teaches Together
Koban met his wife and fellow UMF Mathematics faculty member, Associate Professor of Mathematics Lori Koban, when they were both in a graduate student study group at Binghamton University in New York.
“Lori is a wonderful teacher and fabulous mathematician. I have learned tons from her,” Koban said.
After completing their doctorates, the couple both got teaching positions at Western Carolina University. Although they enjoyed their time in North Carolina, they couldn’t resist the opportunity to return to Lori’s home state of Maine when two Mathematics positions opened at UMF.
Now, in addition to teaching, the Kobans spend much of their free time attending their son’s and daughter’s baseball games, karate lessons, dance recitals, and gymnastic meets.