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“This class reminded me of how I used to think about math before it became unenjoyable for me. Now that I actually enjoy it again, I am easily motivated to use mathematics in both my college career and in life.” – Anonymous Student Evaluation
One afternoon in my office, a veteran of military operations in Iraq and former police officer showed me a math question and said, "I saw that problem, and it freaked me out." Here was a citizen, trained in mortal combat, stunned by a simple math question. Days later, at jury duty, I met a building inspector who thought he could never learn mathematics, even though he needed it to advance his career. Then, back in the jury room, a lady studying social leadership noticed I was reading a math book and asked, "Are you a teacher or something? Because I don't see normal people reading stuff like that." Indeed, enjoying mathematics or being good at it is generally seen as strange or abnormal in our society. Math has become something scary and unapproachable, even more frightening perhaps than deadly conflict.
And in friendships, it’s generally not cool to talk about math. One day I wrapped a fun picture book called "Really Big Numbers" and handed it to a friend as a gift. She said she would take it "as long as it’s not a math book!" But it was a math book, given with the hope that a fun book for "kids" of all ages might dispel some of her fear of math. Research into human psychology finds mounting evidence that fears and anxieties pent up in people day by day have a debilitating effect on their personal happiness, physical health, performance at work, and interactions with other people. These are the people we find in our classrooms, and here, we as teachers enter one of the most charged and rewarding opportunities for changing people’s lives for the better.
I found myself on a sunny Tuesday afternoon teaching a complex analysis class of mostly college seniors, half physics majors and half math majors. The topic was to see how complex numbers help in determining the motion of a spring from a corresponding differential equation. But how do we know the equation that the motion satisfies? Only by experiment with actual springs. So I asked if anyone had studied how springs move in their physics labs. "Oh, yes," said a physics major, "we had a lab about springs, I remember." But no one could tell me what they discovered, simple as the outcome was. After visiting an undergraduate physics lab that semester, I knew why. There was a lab manual with detailed instructions on how to set up an apparatus and what numbers to look for to fill in the blanks. The students read the instructions carefully, wrote down the numbers they were supposed to find, and went to the next question. It was teamwork. One read the instructions as another operated the apparatus accordingly, and the next person recorded the answers. So the lab proceeded as the hours passed.
But nobody could tell me the basic principles of how a spring behaved, and I was not just going to tell them, to have it forgotten again (besides, it was too beautiful outside to waste such a wonderful afternoon!), so I backed up. What would happen if we pulled a spring twice as far? Would the force double, or more than double? Does friction on the table affect the force with which the spring pulls back? Does it affect the speed of the spring? If so, how? Together, we re-created the experiments, appealing to experience and intuition and simple child’s play. We found the differential equation that the spring-mass system satisfies, and the class could explain to me what each part meant. It wasn’t so weird anymore. The mathematics read like a language that they could understand and use to communicate what they wanted to say.
We were now ready to play with the mathematics to find the resulting motion. Surprises came up, and the class had to figure out for me what the strange scenarios meant. Can a spring just slide back slowly into its original position after being stretched and let go? That’s what the math was telling us, but aren’t springs supposed to oscillate? Perhaps there was a mistake in the math, but we could not find it. The class became very curious. Momentarily, someone realized that the damping may be so strong as to wipe out all oscillations, for example, as in a spring moving in honey. The next surprise came when the formula showed up again, in complex space and time, to give solutions that oscillate in real space and time.
At the end of the course, a student wrote that "this area of math (and the parts of physics that utilize it) seem far less terrifying and much more accessible." Another described the passion of the interactions as "incredibly engaging and fun to be in." Many of these students have now graduated, and I hope they will take with them the renewed joy for math and science that they gained…leaving behind any terror they may have had when the class began.
