By Jessica Tomer

"It's definitely fun to do math for seventy-five minutes," says Kyle ’22, reflecting on taking the American Mathematics Competition exam. And he means it. 

Now a senior, Kyle came to Commonwealth, in part, because he heard a rumor that the school offered a Category Theory class for a single student. (Status: confirmed.) "I don't know that a lot of schools would have been able to offer something like that, which is really nice," he says. "I found that totally interesting and compelling." So Kyle applied, was accepted, and ended up placing into Calculus 1 Advanced as a freshman. There he got his first taste of mathematical proofs—though his teacher, Rob Sherry, assured him: this was just the beginning. Theoretical Calculus was where the real fun began. 

Kyle, with his long-standing interest in math (his parents happen to be actuaries) and accelerated progress through Commonwealth's math curriculum, may not be an average math student—but he's not too many standard deviations from it. Most Commonwealth students progress through a math curriculum of geometry or algebra to precalculus to calculus, with the option of branching into full- and half-credit electives such Abstract Algebra, Statistics, Topology, and Axiomatic Set Theory. The latter two courses, currently offered to cohorts of four and eight students, respectively, are among the most advanced math electives Commonwealth offers—and they exist because students like Kyle, having already seized upon other advanced courses like Theoretical Calculus, asked for them.

"That they're willing to create new classes to help us pursue our interests is really cool," Kyle says. "I had seen myself going in some very applied math direction. But I think now, going into pure math and research math would be really, really enjoyable, something I would really love to be able to do. And I wouldn't have known about that as an opportunity if I hadn't taken the classes at Commonwealth." 

Keep reading to learn more about what it's like to take and offer advanced math courses in high school, from curriculum to community, and what success in these classes really looks like...

A College-Level Curriculum

"Whatever mathematics courses a student is ready for and interested in, we can offer. We have that ability," says Al Letarte, a ten-year veteran of Commonwealth's math faculty. And we have "that ability," in no small part, due to his great relish for developing those courses and shepherding students through them. (Axiomatic Set Theory was a jaunty summer project for him.)

"I was really impressed because we had two advanced math electives...just because a couple of juniors and seniors requested it," says Linda ’23. Now taking Topology and Axiomatic Set Theory as a junior, Linda's question is where to go from here? She's intrigued by Complex Analysis, a class that Commonwealth doesn't offer—yet. "I think it's really extraordinary that I can take Topology as a junior," Linda says. "It feels like this school will never run out of high-level math courses."

"The promise that we make—to take kids as far as they want to go—is one we can actually keep. And we do," Mr. Letarte says. "These courses are not offered in any other high school. They're not. And I've looked everywhere." He teaches the advanced math electives "exactly," he says, the way he learned them as a third-year undergraduate math major. "I'm using the same curriculum, I'm using the same textbooks, and I have the same expectations of the students," he says. "We're talking about full-scale college-level courses that would normally be available only to math majors." A fact many graduates corroborate, he notes with mirth, when they come back to visit Commonwealth. 

"I think the depth at which we learn to think about math is really unique to Commonwealth," says Ilaria, another senior and student in Topology and Axiomatic Set Theory. "My Commonwealth classes taught me not just how to apply the Fundamental Theorem of Calculus, but how to prove it from the axioms of the real numbers; I learned not just how to choose the right formulas, but to understand the structure underneath each concept or proof." 

Not only are the courses technically rigorous, but they respect historical contributions to math, Mr. Letarte says, asking students to consider such breakthroughs as trisection of the angle (Wantzel, 1836), proof of Euclid's fifth postulate (Beltrami, 1868), and quadrature of the circle (Lindemann/Weierstrass, 1882)—to say nothing of the foundations of math dating back to ancient Mesopotamia. To even have such conversations requires a thorough understanding of the mathematical concepts at play. "Even as early as geometry, we are giving kids an appreciation for the development of research math," he says. 

Mr. Sherry agrees. "We have different levels of classes and different ways of teaching for different students," he says. "We're trying to be able to meet students where they are. And for the more advanced students who can do abstract work, I don't know of any [high] school that has the offerings that we have." 

The students in these classes already have "mathematical maturity," Mr. Sherry says. "I didn't start thinking about math in that way until I was a junior [studying math at Yale]." Abstract Algebra, a year-long course at Commonwealth, is comparable to a semester-long undergrad college course, he says. Topology is closer to a graduate-level topic.

Commonwealth courses like Mathematical Logic and Axiomatic Set Theory even brush against Mr. Letarte's dissertation research ("Covering Properties on the Hyperfinite Timeline"), meaning he can discuss concepts with his students—like using model theory to build a non-standard universe—that are beyond their years and thrilling to teach. "It's amazing," Mr. Letarte says. "I mean, this is what math really is."

Finding a Math Community

"When I came to Commonwealth, everything changed," says Thomas ’24, who also placed into Calculus 1 Advanced as a freshman. "All of my math classes were engaging, and I was able to ask questions and receive direct feedback on my work." He says he appreciated Mr. Sherry's "lighthearted nature and teaching style" in those classes, particularly his willingness to adjust the textbook curriculum to meet his students' needs. Mr. Sherry, a fixture in Commonwealth' math department since 2000, also helped guide Thomas's progress through the math curriculum. 

Thomas decided to take Theoretical Calculus because he wanted the challenge of a proof-based class, joining the upper-level course as one of two tenth graders. "It felt intimidating at first," he says, "but I quickly got to know my classmates better. No one treats you as inferior just because you are in a younger grade, and it isn't hard to start up a conversation, especially when it comes to math!" 

Linda, now in her junior year, also came to Commonwealth ready to take Calculus as a ninth grader—despite only having taken algebra I and geometry at her middle school. But she taught herself the intervening math (precalculus and algebra II) and "escaped to calculus," given her performance on Commonwealth's math placement test for incoming students. "I really appreciated the opportunity to do that," she says. "I didn't have experience with advanced math. I had a tiny bit of competition background. But [Mr. Sherry and Mr. Letarte] explained things really well." 

Linda excelled and found her place in Commonwealth's "math community." From there it was sophomore year with Differential Equations and auditing Abstract Algebra—and COVID-19. Despite the challenges of hybrid learning, the class "went through the entire book and most of the material for Differential Equations," she says. "Even though the material is hard, I think the students inside the class all tried really hard. There was a good environment." 

Linda is also a member of Commonwealth's Math Team, which is actually co-captained by Ilaria and Kyle, with fellow senior Daria ’22. They prepare a new lecture for the group each week on topics from Young Tableaux to Farey Series to The Math Behind SET (the card game), and they help members prepare for competitions like the Massachusetts Math Olympiad, New England Math League, and the aforementioned American Mathematics Competition, an annual opportunity for students to test their skills on a national level. All Commonwealth students are automatically registered for the AMC exam, though they are not required to take it, to ensure no one is left behind who might want to compete.

Math Beyond the Classes

"Doing more math makes you think about the world more quantitatively and abstractly," Linda says. "In the Axiomatic Set Theory class, we went back to the foundations of math, like constructing sets and all that. It's very cool to see how such easy notions of sets can have very abstract constructions."

Calculus 1 Advanced and Differential Equations gave her a "solid background" for her advanced science classes, like Physics, which features some multivariable calculus. And friends in Commonwealth's third-level Computer Science class, The Theory of Computer Science, have found their experience with set theory and abstract math helpful as well. "Problem solving is an important life skill that I am teaching through my math classes," says CS3 teacher Meena Boppana. "The skill of breaking down a novel problem is more important than any specific content. I try to incorporate problem solving into all my classes, but it especially comes across in Computer Science 3."

Mirai ’24 sees those connections, too. "I now notice more how math is involved in different things in the world," he says, from the shape of buildings to robots using derivatives to the computer code he enjoys writing to the harmonies he plays on the cello. "There's so much in math, so many different branches," Mirai says. "They've helped me appreciate math and how much there is to it."

Before coming to Commonwealth, Mirai also had relatively advanced math experience, studying algebra, precalculus, geometry, and number theory in middle school. That robust foundation made joining Calculus 1 Advanced a natural fit for his freshman year. ("I wanted to challenge myself," he says. "It ended up going really well.") Now, as a sophomore, he's taking Theoretical Calculus alongside Thomas. Both have a predilection for proofs, despite their challenging nature. "It's often difficult to understand the proof," Mirai says, "but I can see myself getting better at it, which is nice." 

Mirai echoes his classmates on the applicability of math in his science classes. He recognizes the formulas in chemistry and anticipates next year's physics class will be that much easier given his familiarity with calculus. "I think it really helps to know more advanced math," he says.

Proof-based courses in particular help students cultivate logic skills and the ability to quickly form cohesive, rational arguments, even outside of math. Ilaria says she can "follow a logical proof in [her] mind, even when it leads to a startling conclusion in the physical world," particularly in her Theory of Relativity class. "It's been really exciting to try to wrap my mind around these counterintuitive ideas. I see special relativity as a sort of 'axiomatic' version of physics, because we spend a lot of time deriving laws consistent with two original postulates. This perspective has made the material easier to think about."

"Learning about mathematical proofs made me appreciate the logic inherent in a lot of philosophy," Kyle says. Taking Axiomatic Set Theory, for example, has led to some heady questions. Among them, "What is truth?" Advanced math helps ask and try to answer that kind of question, he says. 

"While there is a general notion that math is linear—there is only one answer to each problem—as you get more advanced, there becomes more than one way of doing a problem or proving a statement," Thomas says. "It's our job as mathematicians to find these methods."

For Ilaria, the relationship between math and other classes is more symbiotic. "Honestly, I think that my other classes have more significantly impacted the way that I think about math," she says, unable to choose a favorite course this year between Topology, English, Relativity, Bible History, and Life Drawing. "Even though I love math and I want to pursue it in the future, I've learned to enjoy every other subject as well, and I'm grateful to all of my teachers who share their enthusiasm every day."

The Path to Success

So, do all students come to Commonwealth fully formed math rock stars? No. And even those who do arrive with the passion and experience of Yale undergrad math majors still find the work challenging. 

"They think that the kids in our advanced math sections are there because they can put together high-level stuff with no effort. That's not true," Mr. Letarte says, drawing a parallel between writing and solving math problems. The notion that "good writers" can sit down and bang out brilliance with no effort "is completely false." And the same is true of mathematicians. "The people who excel in writing excel because they understand that if they labored over something for an hour and have to crumple it up and throw it away, that doesn't mean that they're bad writers. Part of the process of turning out amazing writing is understanding that you have to bang your head against the wall. And that is what people misunderstand about math. They think somehow that if you bang your head against the wall, it means you're a bad mathematician. The people who are amazing mathematicians are amazing because they know that that's a normal part of the creative process that goes into doing high-level math." 

"One of my favorite things about Commonwealth math classes is that it's easy to ask questions," Ilaria says. "The math teachers create a friendly atmosphere, and even the hardest classes I've taken are filled with excitement. I'm never afraid to take a guess or make a mistake."

All agree that homework is the secret to success. And the students are quick to note that it's not "busy work." "For me, doing math homework often takes a while, but I think it's really worth it because the homework helps me understand the topics better," Mirai says. "There is a lot of homework, but it pays off when you realize how much you actually learn!" Thomas adds. Ms. Boppana agrees: "For me it boils down to: No 'drill and kill.' Every homework problem is different. You will have to think about it rather than just applying a formula."

Mr. Letarte urges his students to lay out their mathematical arguments with elegance and brevity. But as any mathematician (or writer) will tell you, elegance and brevity take time. "Math at the high level is nothing but creativity," he says. "And there's a kind of beauty in a mathematical argument that is exactly like the feeling you experience when you listen to incredible music, or you read a literary piece that's beautifully written with not a single unneeded word in it. And people don't realize that." 

Advice for Students

Pursuing advanced math courses at any grade level can be intimidating, but these students recommend taking the leap. "Just go for it!" says Thomas. "Advanced math classes are definitely very challenging and time consuming, but it's well worth the effort. The teachers here make sure you understand the material, and your deepened understanding of math is the result!"

In addition to attentive teachers, other students can be just as helpful. Ilaria came to Commonwealth largely due to the math offerings, and because she met some female students who were taking advanced STEM courses in particular. "I didn't know many other girls interested in math at the time; that made me really excited about Commonwealth," she says. "Those students were inspirations to me throughout my time at Commonwealth, and I hope that, now that I'm a senior, I've become the same kind of role model for one or two of the younger students."

Diving into the rigorous Calculus Advanced proofs as a first-year student prepared Kyle for his upper-level math ambitions, and he suggests other students with similar interests do the same. "Trying to prove [mathematical] 'truth' isn't as grand as it sounds," he says. "Don't be as afraid of it."

If you don't rack up the prerequisites to take classes as advanced as, say, Topology, you can experience the same sort of "pure math" during courses like Theoretical Calculus, which is totally attainable by senior year, says Mr. Sherry. "It's proof based on axiomatic systems," he says. "It's the same approach." That exposure to pure math actually starts as early as Geometry at Commonwealth, he notes.

For younger students, including those considering Commonwealth, Mr. Sherry stresses patience. "You don't have to be intimidated by people who just have had more exposure [to math] than you have," he says. "Give yourself time. Don't feel you have to decide now. Just be open. And who knows what you'll actually find exciting." If a love of advanced math blossoms later in high school, you can always pursue it in college. "What we want is for you to be challenged by math, for you to grow as a math thinker, and to learn whatever you learn from it well," Mr. Sherry says. "That's the most important thing, not how many courses you get through." 

It takes guts to push yourself academically, Linda says. But don't let those advanced classes intimidate you. "Follow your heart," she says, "and join the Math Team!" 

See Math Offerings At Commonwealth

Jessica Tomer is the Director of Communications at Commonwealth School. This article was published in the Winter 2022 issue of Commonwealth Magazine.