An ‘Exploratory’ Way to Learn Math
J
ane Doherty ’19 is really enjoying college, but she has a confession—she misses math at Colorado Academy. A Freshman at Georgetown University majoring in neurobiology, Doherty will have plenty of opportunity to practice the math she learned at CA in collegiate statistics and calculus courses. But she feels prepared for any future challenge because of the years she spent at CA in what she calls a “numerical playground,” where she gained “a deep love for math.” “The way we learned math reflected the fact that the subject is not just a collection of equations to be memorized,” she says. “Rather, CA allowed me to appreciate math for the awesome thing that it is—an entire system that leverages small known patterns to solve for large unknowns.” Doherty describes CA’s approach to teaching math as “exploratory” and “investigative.” And that approach, she readily admits, occasionally led to moments of frustration. “When new material was introduced in class, we weren’t just handed equations, which would’ve required much less time and brain power,” she says. “Instead, we were expected to manipulate mathematical patterns we already knew to solve unfamiliar problems, which takes much longer and is a lot harder.” What Doherty may not have fully realized during her years at CA was that the frustration she felt and the occasional struggles she had with math were not accidental. They were a deliberative part of the curriculum for Upper School students, where Chair of the Mathematics
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Colorado Academy Journal
Department Peter Horsch says teachers look for a balance between struggle and success as students develop their math skills. “I like hearing students say, ‘I’m not sure, but here’s what I am thinking,’” Horsch says. “We want students to take on problems they are not sure they know how to do, perhaps struggle, perhaps not get to a final result, but they persevere. And ultimately they see even the struggle is rewarding.”
Math in Ninth Grade Step into Horsch’s Math 1 class, a combination of algebra and geometry, and you may feel as though you have taken a wrong turn. The students are seated in small groups of four, talking quietly to each other, as if they might be discussing the themes of a classic novel. Instead, they are talking math. Horsch floats from group to group offering feedback. You never hear him say the words “that’s right” or “that’s wrong.” Instead, you hear this: “What mathematical evidence supports your conclusion?” “Your answer is good, but how do we know it’s right?” “What was your process here?” Students take new problems and migrate with their small groups to whiteboard walls, where they can fill the perimeter of the classroom with numbers, arrows, fractions, and different approaches to solving a problem. At this point, the class becomes an aerobic exercise for Horsch as he bounces on cross-trainers around the room from group to group, coaching
their mathematical thinking, smiling all the time. “I love watching and listening to students doing math,” he says. “I know that it’s a tool that will allow them to understand their world, whether they go into business, engineering, design, medicine, scientific research, art, or even teaching.” Some groups of students join others to view their thinking. No one gives up. Horsch moves around the room, reviewing their whiteboard work and asking questions. “I’m not convinced, are you?” “That looks like a hard strategy. I wonder if there is another approach?” “That’s excellent. But does it really work?” Suddenly, Ninth Grader Dori Beck is jumping up and down at the whiteboard. “Five works! Five works! I can tell you why! Can we have more problems?” It is a classic “Aha!” moment. Beck explains what prompted the celebration. “There are different ways to approach a problem, and Mr. Horsch is interested in how we as a group worked together, using different viewpoints, to get to an answer,” she says. “You start to learn why things don’t work; there is a lot of trial and error involved, and when you get it, it’s rewarding.” For Ninth Grader Hudson Parks, new to CA in his Freshman year, this approach to learning math is a departure from middle school. “I was used to having a lecture, taking notes, practicing the problem for homework, then taking the test,” he says. Parks describes the approach at CA as more “interactive,” with communication >>>
