Where Creativity Meets Logic: Lawrentians Dive into Number Theory and Differential Equations

For Lawrenceville mathematics educator Anton Fleissner, “Introduction to Number Theory and Differential Equations” is more than a typical high school math class. This college-level seminar, available to students who have completed or are enrolled in other 500-level (advanced) math courses, offers a unique glimpse into the abstract world of mathematics while linking its lessons to real-world applications.

Fleissner explained that the course blends two distinct mathematical realms: number theory and differential equations. He described number theory as "the study of numbers from the perspective of divisibility and primes." To him, the subject captures both the ancient wonder of mathematics and its modern relevance. "If you are interested in why there are infinitely many Pythagorean triples or why the square root of two is irrational, those are statements in number theory known long ago," he said. "It’s mystical, but at the same time, it has deep real-world applications.”

For example, Fleissner highlighted how its principles now underlie critical technology. He noted that check digits, which verify the accuracy of numbers tracking numbers for an assortment of travel tickets, as well as book publishing identification numbers (ISBNs), are grounded in number-theoretic properties. This, he pointed out, demonstrates the impressive ability of number theory to ensure security and catch human errors in important systems.

“Number theory is foundational to modern cryptography and blockchain, which rely on the concepts of prime numbers and divisibility to keep our data safe and our digital world in order. That would be the application of number theory in the strictest sense,” said Sofia Liu ’25. “But, if we were to broaden number theory to include the boolean algebra which we’ve spent half the term learning, the applications will be extended to the construction of circuits and modern computers.”

It’s this application that drew Liu to the class. “I have always been interested in cryptography and blockchain, fascinated by how numbers protect ‘truths’ and thereby build trusts in our world,’’ she said. “So, I see number theory as math’s answer and solution to the philosophical questions and real-world problems of ‘how do you build trust and defend authenticity?’”

Nitza Kahlon ’25 is attracted by the mix of creativity and the need to generate logical arguments to back up claims. “It adds so much in other areas of life,” she said. “Creativity and problem solving are key skills that go from school, to sports, to projects, and more.”

Number theory has always been her favorite area of mathematics, having been exposed to it through math competitions. “It begins with the simplest of things that most high schoolers already know: whole numbers (integers), primes, divisibility, etc. But what’s special about Number Theory is the way it takes these simple elements and builds upon them, creating complexity that can be very interesting and beautiful,” she said. “‘Beautiful’ math is kind of an oxymoron, but the way these deep concepts can be represented simply and eloquently is the ‘beauty’ of the subject – building to the complex from the simple. Coming to a concise and clear proof or an elegant solution has always been my favorite part of mathematics.”

Later in the year, the class will delve into differential equations, which offer a more direct connection to everyday life. "Differential equations are really just about equation solving where the equations themselves involve derivatives," Fleissner explained. Students will model dynamic systems, such as how the temperature of an object changes when placed in a different environment or how a spring behaves. "It’s almost easier to answer what the application is for differential equations," he said, with examples ranging from thermodynamics to aerodynamics.

Despite the complexity of the subject matter, Fleissner's goal is not to overwhelm students with difficult concepts, but rather to teach them to think like mathematicians. His Harkness-style teaching encourages collaboration and questioning.

Ava Martoma ’25 appreciates the collaborative atmosphere. “Every week or so, Mr. Fleissner creates a new, anagram-based seating chart, so we are exposed to working with different groups of people,” she explained. “We collaborate in small groups over different problems before presenting our work to the larger class. As we present, other class members ask questions and challenge our work, and we defend it, giving us a solid understanding of what we are doing.”

Martoma said she enjoys the classroom dynamic because everyone is “so interested in learning higher level mathematics, we each put a lot of effort into contributing positively to discussions in the work we do.”

Kevin Xu ’26 said, “As the coursework becomes more challenging, support from peers becomes more invaluable as everyone has their own unique experiences and perspectives for each problem.”

Fleissner strives to replicate the kind of intellectual discovery that has shaped human understanding for centuries. "What we’re making is kind of a simulacrum of what humankind has once done,” he said.

One of the most fulfilling aspects of teaching for Fleissner is seeing students engage deeply with the material. Because the class isn’t teaching a set curriculum, in preparation for an AP exam, for example,

“We can work at our desired pace and spend more time on things that we find interesting,” said Xu.

Fleissner shares a moment of pride when students ask questions that anticipate the next topic. "Almost every day of class, there’s a student asking a question to which I can respond, 'That is exactly what we were about to talk about next,'" he said. This, for him, is a sign of their genuine interest and growth.

Although the course is challenging, Fleissner believes students rise to the occasion. “I often worry, ‘Is this too hard? Or too easy?’” he admitted. But he’s gratified when students thank him for a thought-provoking test. “They want to be challenged,” he said, noting that the balance between difficulty and accessibility is crucial.

Martoma said it’s been challenging to understand every concept “because they are so different from other kinds of math I’ve learned, but it’s also very satisfying because it takes time to get there.... I love the work we are doing and the pace we are going at because we are able to truly think deeply about things. Mr. Fleissner does a great job helping you get there.”

Mila Cooper ’26 said she would “absolutely recommend” the course to other Lawrentians. “I would stress that the class emphasizes a lot of thinking outside of the box, a lot of trial and error, and especially a lot of collaboration.” If you like math, she noted, this is the course for you.”

“It may seem intimidating, but it is actually one of my most fun and chill classes, so I would definitely recommend it," said Xu.

For Fleissner, teaching this advanced mathematics course goes beyond simply transmitting knowledge. His goal is to cultivate curiosity, analytical thinking, and intellectual hunger. As he puts it, "I hope that this course continues to develop into a full meal for that hunger.” With his dedication and expertise, he’s helping students not only understand complex mathematical theories but also see the broader beauty and utility of the subject.

For additional information on all Lawrenceville School news, please contact Lisa M. Gillard H’17, director of public relations, at lgillard@lawrenceville.org.