In middle school, says this self-effacing tenth grader, "I didn't have much of a work ethic." By this, she means she threw herself into competition math and got serious about the flute, which was easier to practice during her family's two years in New Jersey. She had played the piano since she was four, but the flute started in third grade when her father took her to a wind-quintet concert. Ilaria said she wanted to play the piccolo. "It was a joke, but the next day, my mom brought home a flute." As the more social of the two instruments, it became her favorite. Since moving to Boston, she has played in New England Conservatory's preparatory orchestras.
Ilaria says her life changed when she came to Commonwealth in that, rather than "really needing to find things to do" because middle-school homework took so little time, she had to start making hard choices among passionate interests. To begin with, "my eighth-grade self would never have believed I would love homework so much." She placed into Calculus Advanced as a ninth grader but visited geometry classes for fun. This year, she audited Creative Writing, reluctantly dropping it because "I had to find ten hours a week" when she was accepted to a research program called PRIMES. This entails working on original scholarship in a small group with a graduate-student mentor. Ilaria apologizes for not getting very far in conveying her group's topic, enumerative combinatorics, to a curious English teacher. "It took me three weeks to understand it," she explains.
But when she talks about PRIMES, Ilaria crisply defines a pair of categories: competition math and research math. Quoting one of her early math mentors, she describes competition math as "racing," research math as "exploring." To train for math competitions, students practice solving problems fast, reviewing concepts so they can apply them on the fly and drilling to build up speed. Exploring enumerative combinatorics means "you're trying to solve a problem over a whole year, which means you're stuck most of the time." You have to fight through a lot of frustration and sometimes just stop and go for a walk.
The racing/exploring boundary is permeable, though. Ilaria has noticed that Commonwealth kids who loved competition math in middle school "because they were bored in math class" eagerly embrace the "exploring" our proof-based courses entail. As they progress through the curriculum, their hunger for "racing" tends to fade. Members of the math team sometimes resist using the meetings to train for competitions. Ilaria herself now takes exams like the AMC and AIME without any preparation. At the same time, she wants to keep qualifying for the next level of competition. Her goal is the Math Olympiad, with its nine hours of...proof-based problems.