Beauty in the Beast

Beauty in the Beast: Falling 4 Math

by Krista Collins, chair of mathematics I hated math. As a student in middle and high school, I found that the only numbers that interested me were those on the clock, and the only operation I eagerly performed was subtraction: How many minutes are left in this awful class? I had learned early on that my mistakes would result in my being sent alone to the board at the front of the room and that my questions about “why” could unleash an embarrassing rebuke about how I was trying to divert attention from my weak algebra skills. Finally, in my senior year, I had a teacher who answered my question about right triangles, in an algebra class, no less. She devoted an entire class to different proofs of the Pythagorean Theorem—one with pictures, some with variables, and even one originally done by President Garfield. For a student like me, planning to major in political science, the story of Garfield was a revelation. Math was not just for a select group of math nerds; anyone could participate. I fell in love with math! Today, as a math teacher, rarely do I find a math topic that fails to inspire a story, a real-world connection or a monologue about how beautiful a concept is just because it is mathematical. While I love to work with talented honorslevel kids, I have found that my greatest successes come when I help those who have been completely paralyzed by math discover its relevance and beauty. Joe was one of the most traumatized math students I have ever encountered. He was a very good math student, but he was completely intimidated by his brusque, my-way-or-the-highway algebra teacher. By the end of eighth grade, Joe had dropped two letter grades and been humiliated out of the honors track. When he entered my classroom the following September, he was no longer interested in math. He would come in, quickly sit in the back corner of the room and avoid eye contact. He would not volunteer an answer and would stammer with embarrassment whenever he was called upon. However, as the weeks went on, I noticed that Joe was beginning to listen a little more closely. One day, I shared the story of Fermat’s Last Theorem (zn = xn + yn). He claimed to have found a proof for it in the late 1600s and that the margin of his book was too small for him to write it out. The theorem was finally proved in 1993 by Andrew Wiles—and it took him over 200 pages! Joe was getting hooked. He began to realize that even great mathematicians struggle with problems. He was learning that it was OK not to know the answer immediately, that wrestling with a problem was normal. Joe watched closely as I encouraged other students to try to solve problems at the board. Early in the term, I could see him wince as they made mistakes, as if waiting for the verbal lashing that he expected them to receive. During our study of exponent rules, a student incorrectly evaluated x0 = 1. Her confusion about how to work the problem led to a 35-minute group discussion about why exponent rules work and how to prove that x0 = 1. Joe began to learn that mistakes lead to interesting discussions. And he learned that it isn’t just students who make mistakes. I make them all the time. When I do, I admit it and ask students to help analyze where I went wrong. Not only do they love to “out-math” the teacher, but they also learn what every great mathematician learns at some point: Mistakes will happen, but you can learn so much from them. Joe saw how other students in the class were getting excited about math, too. During a discussion of Pascal’s Triangle, an ordered arrangement of the counting numbers, Meg began to see the many connections between this grouping of counting numbers and higher level math, as well as to simple problems. She was truly awed by the simplicity and the elegance of math that has been around for centuries. She begged me to stop teaching for a moment so that she could sit and look at the triangle and appreciate it—and she was not kidding. Mary, a less able student, became willing to give math a chance when she saw how it could be applied to “real life.” During a unit of graph theory, she saw how these “complicated” math algorithms could be applied to simple problems like delivering mail or plowing snow from streets. While she could not completely understand the entire math involved, she was more willing to try when she could see its connection to her life. The experiences of students like these affected Joe, allowing him to think about new dimensions of math. Math wasn’t just about learning mechanical operations to find correct answers. As the weeks turned into months, Joe emerged from his “math coma.” He slowly moved toward the front of the class. He volunteered to work certain problems at the board, and he began to offer suggestions to his classmates. By the end of the year, Joe was coming into class with math stories he had found on the Internet, and he was eager to talk about higher math problems that interested him. Most important, he started entering class with a smile on his face. The key to getting students interested in math is to teach them to think and work like mathematicians. They need to see math as a living, breathing discipline filled with stories about real people, a subject worthy of passion and intensity. As Joe illustrates, once students have gained an appreciation of the beauty, elegance, and relevance of the mathematics, they become open to new mathematical ideas. Once they understand the importance of mistakes and feel safe enough to make them, fear is replaced by curiosity and confidence. The beast vanishes. It is this transformation that inspires me and that I strive to share with my students.