Union professor’s calculus textbook aims to change how students learn by including infinitesimals

JACKSON, Tenn. — During the spring semester in 2004, Bryan Dawson was teaching his calculus students about limits when one student began asking some questions about infinitesimals.

“I didn’t have good answers,” Dawson recalled. “And so, that sat on the back of my mind for a while.”

That initial student encounter eventually led to Dawson, university professor of mathematics at Union, tackling the topic of infinitesimals in textbook treatments years later. Following the publication of his first textbook with Oxford University Press in 2021, Dawson has now completed his second textbook, “Multivariable Calculus Set Free: Infinitesimals Ride Again,” also published by Oxford.

The book’s title is drawn from Psalm 51:12 in the First Nations version: “Set me free and make me whole, and once again my heart will dance for joy.” Its publication marks the completion of Dawson’s capstone scholarly contribution to the field of mathematics.

“I’m convinced that eventually the mathematical community will make the decision that this is the way to teach beginning calculus,” Dawson said. “I really just praise God for the gift that he gave me through this. But I’m convinced that this contribution to the mathematical community will last centuries.”

Those student questions in 2004 stuck with Dawson over the years. He was busy with a family and helping to home school his children, so his priority at the time was classroom teaching and not additional scholarly activity. But after his children had grown, he began thinking about what kind of scholarship he might pursue.

He began poking around with infinitesimals, or infinitely small numbers. He examined textbooks that had been published in the late 1960s and early 1970s, concluding that the methods they taught made the burden on students worse. He was convinced there was a better way than what the mathematical community had promoted for decades.

After more research and some trial and error, Dawson landed on a system that he thought would be easier for students. In the spring 2014 semester, he began testing out his methods in the classroom.

“There was an immediate improvement in student scores and an increase in the sophistication of problems they could do,” he said. “And so, I thought, ‘This is good. You’re onto something.’”

He invested more heavily in his classroom instruction through the fall of 2014 and 2015. He was granted a research leave to work on his first textbook, which covered Calculus 1 and Calculus 2. A second research leave in 2024 gave Dawson the opportunity to make major progress on the new Calculus 3 textbook.

“I have often prayed that this textbook will be a blessing to all readers,” Dawson writes in the preface. “May it always be so.”

The electronic version of the book is available now. The print version will be available later this fall.