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Buchholz High School student discovers and publishes new calculus technique

Glenn Bruda, a 17-year-old high school student, is credited with the discovery of a novel calculus formula that he has named the Maclaurin Integration technique. (Photo courtesy of Jennifer Bruda)

Glenn Bruda woke up in the middle of the night to fetch a glass of water. It was late, past 1 a.m., and he had school in a few hours. 

Before falling asleep, he had completed his nightly ritual: running mathematical techniques in his trusty notebook. Bruda is a senior at Buchholz High School in Gainesville. He also competes with the Santa Fe College math club. 

He haphazardly shuffled to his bathroom sink. The 17-year-old carried the cup back to his bedroom, expectant of immediate sleep. 

That’s when it hit him – out of the blue – the solution to a formula that had stumped him for months.

He frantically searched for a pen and notebook. He spilled his water. His hands seemed to move on their own as they jotted down the solution to a calculus proof in pitch black. Bruda tucked it away and slept for a few more hours.

That Monday, on May 24, 2021, he revisited his notes. He worked the formula again and again in his head, double- and triple-checking to make sure it equated to more than sleep-induced nonsense. It checked out. He felt satisfied. Then he went to school.

What Bruda had concocted turned out to be an entirely new calculus technique to solve integral equations. He cross checked it with mathematics professors at Santa Fe College and the University of Florida, who later encouraged him to submit his discovery for publication. On Jan. 30, 2022, it waspublished to Cornell University's “arXiv”, an open-access archive for scholarly scientific articles. 

“Mathematics is a creative discipline that requires tenacity and experimentation, and the technique Glenn developed is an example of that,” said Dr. Kevin Knudson, chair of the math department at the University of Florida. Knudson worked closely with Bruda to confirm his technique as legitimate. 

In calculus, an integral represents the area underneath a curve. The process in calculating the value of an integral is called integration. Outside of mathematics, integral equations can be applied to everything from complex construction projects to electrical engineering. 

“Mathematicians, since calculus was invented some 250 years ago, have developed what we call integration techniques to find that area underneath the curve,” Bruda said. “We have a lot of them. And they range from relatively easy, which is what is taught in mainstream calculus classes, to extremely difficult, which only experts and top researchers use to tackle really difficult problems.”

Bruda titled his integration technique the “Maclaurin Integration” because it is derived from a formula named after Colin Maclaurin, a famous 18th century Scottish mathematician. He is now in the process of writing a second mathematical paper that will propose another original integration technique.

The Maclaurin Integration technique Bruda developed is designed to be “one of the most versatile integration techniques” because, with it, “there is essentially zero human labor involved in calculating integrals.” 

Where most integration techniques can only be applied to around 10% to 40% of integral problems, Bruda’s technique applies to approximately 73% of integrals. That means it is almost two times more effective than most mainstream techniques.

Integration techniques are most common in Calculus II courses, which – despite not having graduated high school – Bruda claims as his favorite mathematics course to date. Bruda is a dual-enrolled student at Santa Fe College, where he is currently enrolled in courses like Differential Equations and Physics I with Calculus.

“This is an extraordinary accomplishment for anyone at any level, to come up with an approach to solve an age-old problem,” said Katey Arnold, chair of the math department at Santa Fe College. “He is an excellent example of the high school dual enrollment program connecting students with ways to expand their opportunities. He’s taken it to the next level.”

Part of the inspiration behind Bruda’s love for mathematics is his family. His father Michael Bruda teaches Geometry, Algebra I Honors and supports the instruction of Financial Algebra at Buchholz. His mother, Jennifer Bruda, and 12-year-old sister, Maggie Bruda, are English enthusiasts, but relentless supporters nonetheless.

“To me, as a layperson, it’s hard to look at and understand the scope of what he’s accomplished because it literally looks like a foreign language,” Jennifer Bruda said. “Glenn even told me the math language is unique at this point with math itself. It’s not shared with any other entity in the world. And even if I don’t understand all of it, I’m very proud.”

He’s now applying for the Davidson Fellows Scholarship, a prestigious scholarship for gifted high school students with rewards of $10,000, $25,000 and $50,000. The scholarship is awarded to students who create a “significant piece of work” at the graduate level. The Davidson Institute defines significant work as work “that experts in the field recognize as meaningful and has the potential to make a positive contribution to society.”

Bruda graduates high school soon. He is a Gainesville native who hopes to study mathematics at the University of Florida. He even has dreams of teaching math, of sharing his enthusiasm for the mathematical language with others. One day he might even instruct students on his own techniques and theories. Until then, he maintains his nightly ritual of solving math equations to calm his mind before bed. 

He still has the notebook where he jotted down his first original integration technique. He safeguards it.

“I kind of use it as motivation sometimes,” Bruda said. 

When he finds himself wading through mounds of schoolwork and grows frustrated, he turns to that note he wrote in the early hours of the morning in May 2021. 

“I'll just look at that, and I'll be like, ‘Yeah, just keep going. You'll be fine.’”

Natalia is a reporter for WUFT News who can be reached by calling 352-392-6397 or emailing