Rethinking Calculus for Future Engineers

For Siddhant Bohra, math had always felt relatively straightforward. But as he pursued robotics engineering at the University of Michigan, the 20-year-old found that the clean, theoretical problems he encountered in traditional calculus courses didn’t always resemble the challenges he wanted to tackle.

That changed last year when Bohra, then a sophomore, enrolled in ROB 201: Calculus for the Modern Engineer, a course that combines elements of Calculus I and II and differential equations with programming and robotics applications.

“I learn the best when I’m interested in something, and the only way I’m interested in something is knowing why or how this is going to help me,” Bohra said. “Sometimes you have to do things to learn, and not just read and figure out your interests from that. ROB 201 is that class that made me open my eyes to so many things.”

Redesigning calculus: First offered in 2024, ROB 201 was designed to more closely connect calculus with the work students would encounter as engineers, teaching them mathematics through computation, modeling and engineering projects.

Jessy Grizzle, a professor of robotics at the University of Michigan who developed the course, said the idea grew in part from years of hearing students describe a disconnect between the math they were learning and how they expected to use it as engineers.

“I kept hearing from students that they felt beat down in the math department and unmotivated by just learning abstract concepts without them being tied to applications,” Grizzle said. “People become engineers because they want to create devices that we all hope improve the world.”

“So when the math seems completely divorced from that, there’s a segment of students who are flat out turned off by that,” he added. “They really need it tied to something more substantial, more concrete.”

Grizzle structured the course around increasingly complex robotics problems. Rather than beginning with differentiation—as traditional calculus courses typically do—students start with integration, using real drone data to calculate a drone’s trajectory from its acceleration.

“I took the subject apart and rearranged it so that we started with integration, because it’s much easier than differentiation,” Grizzle said.

The course also pairs traditional pencil-and-paper calculus exercises that have exact solutions with messier, real-world problems students use computer programming to solve. Students learn to code in Julia, allowing them to use computation when a neat solution isn’t possible. Grizzle said the approach exposes students earlier to the kinds of challenges they will encounter as engineers, where data can be noisy and solutions less straightforward.

“We need to understand the hard [problems] because then we have confidence in how we can modify the procedure … [and] apply it in a more complicated situation that they’re facing on a student team, in an internship and eventually in a career,” Grizzle said. “We’re doing math with a purpose.”

ROB 201 isn’t the robotics program’s only departure from the traditional math sequence. In 2020, the program moved linear algebra, traditionally taken later in the math sequence, into students’ first year. Grizzle said the change gives students earlier exposure to mathematics foundational to robotics and machine learning while putting those who took calculus in high school and those who didn’t on more equal footing.

The change has also allowed students to reach a benchmark machine-learning course about a year earlier than those following the conventional math sequence, while performing similarly in the course, he said. The machine-learning course now enrolls about 300 students each year.

Refining the model: Student feedback also revealed a potential trade-off in ROB 201. Grizzle found that the course’s emphasis on coding and computational tools left some students with less practice performing calculations by hand, making them slower than their peers in subsequent engineering courses. Take-home assignments and readily available AI also made it easier for students to become overly dependent on computational tools.

As a result, Grizzle is adjusting the course this fall without abandoning the computational approach at its core. Students will take five low-stakes quizzes and an in-class final, which together will emphasize the hand calculations and foundational concepts they will need in later engineering courses. During those assessments, students can use limited handwritten notes, but not AI or software.

Homework and projects, meanwhile, will continue to emphasize coding and modern computational tools, allowing students to practice solving the kinds of complex problems they are likely to encounter as engineers. Grizzle said that includes teaching students how to use AI effectively rather than avoiding it altogether.

“Because AI is very important for a practicing engineer, I want them to learn how to do clean programming themselves,” Grizzle said. “And also how to get help from AI to do programming, and how to get help from AI as a tutor.”

Challenging the status quo: For Grizzle, being able to put what students learn in ROB 201 into practice is part of the goal. As part of a University of Michigan student project team, Bohra used concepts from ROB 201 to help program a navigation system for a mock Mars rover that competed in the international University Rover Challenge.

In ROB 201, Bohra worked on a project that used calculus to estimate and control the position of a drone. He said the experience became especially relevant when he later joined the rover team and encountered the same underlying mathematical concepts.

“When I was reading through the textbooks that my [project] lead gave me, all of the fundamental mathematics theory is what that ROB 201 project gave me,” Bohra said.

Isaac Mekaru/University of Michigan

Bohra’s experience reflects what Grizzle sees as a larger question about how to define student success in a course like ROB 201. Passing calculus is one measure, but he also wants students to leave the course engaged, confident in their understanding of the fundamentals and prepared to apply them in later engineering courses.

Rethinking student success, Grizzle said, also means questioning long-standing assumptions about how calculus should be taught to engineering students.

“The course I created is much more helpful for most students, but the broader opinion of faculty across the College of Engineering is fear of changing from the old ways,” Grizzle said. “All faculty went through that sequence, even the faculty in engineering, and they’re afraid to move away from that because they’re afraid their students will be less prepared than they are.”

“Faculty want everything to be just like it was when they were students,” he added. “So we have a lot of work to do to get people to open their minds to the fact that novelty can be good.”

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