(Photo credit: Brittney Oswald, Center for Inspired Teaching)
At age 16, I invented functions in math class. Functions didn’t look like f(x) to me; I didn’t even know the term “function.” Instead, “functions” looked like fountains. On the first day of class, my teacher walked in, his Einstein-like grey hair pointing in all directions. As students trickled in, he already had a slideshow of various fountains playing on the board and told us to observe them. Over the next four months, we examined how fountains work. We began as biologists, closely examining the cycle of water flow. Next we were architects as we decided how our fountains would look and how the water would flow. Finally, we were engineers as we practiced virtually designing what angle to position the faucet and how to scale a miniature model of our fountains.
I built and rebuilt models out of old shoeboxes and rulers. I discovered the relationship between the angle of spout and the arc of the water. I wrote an equation that would determine the angle of water for all fountains. I eagerly presented my equation to the class, convinced I would blow everyone away. Quickly, I learned the whole class had developed a similar equation as we all shared our reasoning. Our teacher smiled and pointed to our equations on the board. “What you have here,” he said, “is a function.” Maybe I wasn’t the original inventor of functions, but I felt the joy of discovering the equation like I was the first person who ever did. My confidence and excitement as a learner, which my inspired teachers cultivated at school, drew me to Center for Inspired Teaching.
The past summer I watched Inspired Teaching Fellows rediscover what it feels like to truly learn something on their own. As a part of the Inspired Teacher Certification Program, Fellows experience the kind of inquiry-based learning that they implement in their classrooms — the kind of learning that encourages students to ask why, to explore and discover.
Fellows tackled a problem they all know how to solve: 1/4 x 1/5. In unison, they called out the answer: 1/20. Fellows are then challenged even further: Why? The goal is not that Fellows, or their students, simply know the right answer; the goal is that Fellows will know how to find the right answer. In small groups and clusters, the pre-service teachers huddled around markers, graph paper, LEGO, blocks, and other materials on the carpet. They began to explore why 1/4 x 1/5 = 1/20. Fellows drew, discussed, built models, and created stories. One group even wrote a recipe to demonstrate and unpack the problem. After a while of grappling with their reasoning and asking, in a combination of curiosity and frustration, “Why does this work?,” Fellows came back together as a whole group to present their work.
One group said they decided to represent the problem in a way they all knew: food. “Imagine,” one Fellow explained as she stood by the whiteboard, “I was at a dinner party with one leftover pizza and I brought 1/4 home.” She drew a large pizza on the board and shaded in the eaten pieces. “When I got home,” she continued, “my family all wanted some, so I separated my portion of the pizza, one fourth, into 5 separate pieces.“ She continued to draw. “After my family members each took one fifth of the fourth of the pizza I had, in the end, I only had one twentieth of the whole pizza to eat for myself.” The cohort applauded her work and the next group presented their process. They had persevered through their curiosity and frustration, and wound up satisfied by their learning. The room, which was once filled with tension, was now full of joy. Instead of computing equations, they created explanations. Whether they used blocks, pizzas, stories, or graphs, each group discovered why and how the solution works.
When Fellows fully grasped and communicated the concepts underlying the multiplication of fractions – moving beyond mere memorization of the formula, perhaps for the first time –their faces lit up like mine did when I constructed an equation for my fountain. My time with Inspired Teaching has reinforced my belief that there is a profound difference between surface-level learning and learning through discovery. As I work with Inspired Teaching Fellows and hear stories from their classrooms, I am reminded of the joy of learning like a biologist, an architect, an engineer. Inspired Teachers bring joy and discovery into their classrooms every day. I’m hopeful for a world where every student gets to learn this way.