The pennies on the wall outside Tim Whiteford's office on the third floor of Saint Edmund's Hall illustrate math concepts - Pi, angle degrees, Fibonacci sequences, fractals - but as he sees it, they're just as much about art and poetry.
"I spend my life understanding how children learn math, and the idea I want to convey with this penny thing is that we have to captivate kids in math with an aesthetic component," says Whiteford, a Saint Michael's College education professor who for years has been looking for a vehicle "to push the idea that math is 'the science of pattern and the art of making sense.'"
Inspired by a project created by his student, Lydia Koch '14, he's lately been adding a new clause to that mantra: "Now I also put, '…and the art of using cents!'" says Whiteford, a white-haired and bearded Englishman with a passion for Celtic music besides his love of math education. "When you see key concepts aesthetically presented, and not just with numbers, it changes everything. I think all faculty will agree that we learn so much from our students, and every so often it’s something really tangible," he says. "That's what happened here with Lydia."
Koch is a senior education and English major from Essex Junction who long has been fascinated by pennies, particularly after she started using them as a three-dimensional medium to create art several years ago. "Basically I'm obsessed by them," Koch says. "I want to make a kitchen floor out of them in the future when I get a house! It's just a decorative art interest, and it's the only 3-D art medium I've ever worked with."
Her personal, ever-growing collection of thousands of pennies is divided by decade minted. She's fascinated by old "wheat-back" pennies (pre-1958), and her oldest is from 1909 - the first year Lincoln pennies were minted. "The color and shape and demarcation on them are different based on the era, and the art potential with different shades based on history, I love that idea," she says.
When Professor Whiteford assigned students in a class that she took last spring - Elementary Math and Science 325 - to find and explain 12 examples of math found in some aspect of the everyday world, her choice was clear. "I knew pennies had a lot of geometry because of the shape, and also data analysis with so many ways to classify them," she says.
The goal of the project, Whiteford says, "is to help them see math is relevant to life." He was impressed enough by Koch's PowerPoint for the class about the math of pennies that he put up a printed paper copy of it on the wall near his office. In order to draw more eyes to it and get conversations going, he had the idea to create a few of the basic patterns from her project out of real pennies, which he stuck with Elmer's Tack to the painted cinderblock walls.
"That's where it started," Whiteford says of his subsequent explorations of recent months, which have taken Koch's penny math to a whole new level. "I thought, 'This is neat showing all these relationships with pennies because it had never occurred to me before. I realized how simple and quick and easy it was to show fractals, for instance, since it's all based on the idea of three." He pours out some pennies on a table and makes fast triangles out of three of them pushed together to illustrate. Larger subsequent triangles built with ever more pennies get to the heart of the fractal concept, he demonstrates.
One fractal-based figure that he built on his wall from Koch's presentation was something called the Sierpinski Triangle, based on a Polish mathematician’s work. From there, his imagination just took over until the walls near his office became covered with math concepts explained with pennies: There's a huge penny circle that people can stand against and show angle degrees with their hands and body; or the pi ratio, illustrated by 22 pennies making a circle with a diameter of 7 pennies; a 44-penny circle has a 14-penny diameter.
Both Whiteford and Koch love the coincidence that there is a well-known math phenomenon called the "Koch Snowflake Fractal" that's named for a famous old-time Swedish mathematician, Helge von Koch, who shares Lydia's family name. Naturally, a Koch Snowflake illustrated with pennies now is on Whiteford's wall too.
"These concepts are not beyond what elementary students learn – after all, they learn algebra from kindergarten with a problem like, '3 + ? = 9,' for instance," Whiteford says. "Everything is a pattern. If you can see the pattern in math, you can understand it, you can predict, and once you predict, you know you'll understand." He's determined to challenge a "fixed mindset" that so many people have, and often express to him: that they "can't do math."
"I tell them, 'it's not your fault' - this this notion usually is based on a two-week experience in third grade where they had a problem with math!" he says. Whiteford preaches the imperative of "knocking out" fixed mindsets in human lives, whether with kids or politicians. "It's easy to get into a fixed mindset, and once you do, that's the death of learning,” he says. "I see pennies as a way of helping our students, and maybe some faculty, open up about math."
Koch says she was "blown away" when she returned from summer vacation and saw what Whiteford had done with her initial idea. While she continues to collect pennies, she's handed off the teaching aspect of them to her professor. "If I need a stress reliever, I'll still lay them out in geometric patterns, and all my friends still bring me their pennies - I still collect them!" she says. Whiteford says students, faculty, staff and even parents on tours will walk by his pennies-on-the-wall displays and engage the interactive displays, or seek him out for extended conversations.
"I started calling it the penny arcade," he says.