If you’ve never thought of math as a creative art, you’re probably not alone. But three Letters and Science professors will disagree with you. Gabriella Pinter, Chris Hruska, and Boris Okun, all professors within the Department of Mathematical Sciences, teach middle and high schoolers how to find creative solutions to unconventional math problems through the UWM Math Circle.
At each of Math Circle’s weekly sessions, the professors use an entire hour to pose one or two problems to the youth. They do not suggest a method; they do not even promise that there is an answer. They do, however, guide the students in brainstorming ideas, working together as a class, and recognizing the underlying patterns and principles involved in the problem. What’s more, this circle has a point.
“It’s giving them a feel of what it is like to actually do real mathematics,” said Hruska. “Or any real science,” added Okun.
A different kind of classroom
Real-world problems of any kind, they said, are rarely as straightforward as the ones you find in a textbook. The Math Circle complements classroom experiences by posing open-ended, frequently abstract questions that encourage young students to draw on a variety of skills, including their own ingenuity, to approach the problem.
One problem they discussed involved dominoes and a chessboard. They took a chessboard – an eight by eight grid – and removed two opposite corner squares. They then asked the kids to cover the board with dominos so that every domino covered exactly two squares. The problem is admittedly contrived, but it’s also more difficult than it first appears.
Pinter, Hruska, and Okun make a point to distinguish the experience of the Math Circle from that of a traditional classroom. Aside from the unusual problems, they also deliberately remove common classroom elements. “There’s no homework, and there’s no commitments that you have to do something by a certain date,” said Hruska. “They come if they are interested, and if they continue to be interested they continue coming. And if they are not interested, they just don’t come back.”
The professors further distinguish their program from a math class by focusing the teens on learning to articulate the root of a problem – why an approach does or doesn’t work – rather than only coming up with an answer. For example, it turns out the chessboard problem is impossible, and most students reach that conclusion after much trial and error. But explaining why it’s impossible means understanding the fundamental predicament behind it, which the professors try to draw out of the students through discussion.
The UWM Math Circle is the only math circle in Milwaukee, though the idea is not entirely novel. It happens to be a common extracurricular activity in eastern European countries, and immigration from those countries since the 1990s has brought the tradition to the United States. Today, math circles are scattered throughout the country, usually attached to a university and a bit more prevalent on the East and West Coasts than in the Midwest. Both Pinter and Okun, who grew up in Hungary and Russia, respectively, attended math circles as kids and said that their own positive experiences were major factors in initiating the program here. They’ve patterned this math circle after ones they enjoyed but have also incorporated ideas from successful math circles they’ve researched as adults. The goal in the UWM Math Circle has always been to help youth develop critical thinking skills and confidence in approaching situations they haven’t encountered before.
“If you’re afraid that ‘Oh, I don’t want to do this problem because I don’t know how to do that type of problem,’ then you won’t be very successful in your adult life no matter what career you’re in,” said Hruska.
“From my experience,” said Okun, “most of my classmates in math circles didn’t go into math careers, but they were successful in other areas. The point is to develop some sort of analytical thinking which is important in many careers.”
In other cities, math circles sometimes operate differently to meet different needs. One based in Madison, for example, functions to generate interest in math careers by presenting students unusual but true problems in math history as a lecture series similar to the UWM Science Bag. Other circles are gathering places for educators interested in brainstorming new ways to teach traditional subjects in their classes.
When asked how she would like to see the UWM Math Circle grow in the future, Pinter responded that she would like to extend the program to elementary school students. She believes in the importance of showing students early in their education that math relies on creativity and diverse thinking. And though funneling kids into math careers has never been the purpose of the program, she feels that more students would choose to pursue those fields if they recognized this.
“These students who are typically very creative and who can put things together in interesting ways – they cannot use their creativity in the math classroom, typically, so they turn to different things like music, for example, where they can freely express their creativity. I think you can do that in math also, but they don’t see it,” Pinter said.
Creativity is important to math because not all problems are solved with an equation. For instance, one way to explain the chessboard problem is to take a Sharpie and color one half of a domino black and leave the other half white. You do this to show that no matter where you place the domino on the board, it’s going to cover one black square and one white square. You realize, then, that opposite corners of a chessboard are always the same color. This means when you cut two corners out, you’re stuck trying to cover a board that has more squares of one color than the other with dominoes that can only cover an equal number of each. It’s impossible, but it takes a little creativity to explain why, and that’s the real point of the problem.
UWM Math Circle runs throughout the school year on Wednesday nights at 5:30 p.m. in room E424A of the Engineering and Mathematical Sciences (EMS) building. Anyone interested is welcome to attend; however, problems are selected to be most appropriate for students in grades 7-12. More information can be obtained by contacting Gabriella Pinter at firstname.lastname@example.org.