Evidence Is Mounting That Calculus Should Be Changed. Will Instructors Heed It?
Calculus is a critical on-ramp to careers in science, technology, engineering and mathematics (STEM). But getting to those careers means surviving the academic journey.
While there’s been progress of late, it’s been “uneven” and Black, Hispanic and women workers are still underrepresented in some STEM fields. Traditional methods of calculus instruction may be knocking students off the path to these vital occupations, which is why advocates warn that getting diverse students into these careers may require instructional models more responsive to students. Meanwhile, the country is struggling to fill vacancies in related fields like semiconductor manufacturing, despite sizable investments — a feat that may require stabilizing the pipeline.
Good news: There’s mounting evidence that changing calculus instruction works for the groups usually pushed out of STEM. At least, that’s according to a randomized study recently published in the peer-reviewed journal Science.
The study — which involved 811 undergraduate students at Florida International University, a large public university in Miami — is perhaps the biggest randomized study of active learning methods in calculus, says Laird Kramer, a physicist at the university and one of the study’s authors. Researchers tapped alternative models of calculus teaching that have shown evidence that they work, according to Kramer.
The study, which occurred over three semesters, randomly assigned students to either learning through lectures, the old-school way, or through “active” calculus instruction that emphasizes student engagement. Those active methods limited the amount of lecture time, instead focusing on small groups and using “learning assistants,” other undergraduates who were on the teaching team. Instead of sitting through lectures and working through procedural rules, students in the experimental groups were expected to focus on calculus concepts such as derivatives. Outside of class, they worked on problems on their own, while during class, they thought like mathematicians by reasoning out problems with limited guidance.
Its conclusion? That the traditional lecture method of teaching calculus isn’t as effective as active models. Those who learned from active methods did significantly better across race, gender and major, according to the study. (Students majoring in biology saw the biggest bump.) Over each of the three semesters of the experiment, there was a “medium/large effect size.”
It’s common for students who are used to learning math from lectures to be reluctant to think critically at first, learning assistants from the study say. But eventually, they get it. “[The students] move away from that algorithmic knowledge of mathematics, just following steps and just working like a little robot,” says Daniela Zamora Zuniga, a former economics student who was a learning assistant from 2019 through 2022.
Zuniga, now graduated, learned calculus through the active learning model, and it led her to pursue math courses outside of the degree requirements, she says.
That’s similar to something she noticed in other students who took the course. The students she’s kept up with, Zuniga says, report carrying an understanding of calculus forward into other STEM courses. That can relieve the pressure they feel around advanced math, freeing up mental space to devote to science, Zuniga adds.
Sometimes, in these classrooms, students who are apprehensive of calculus because they might have weak background knowledge can end up being the best students, says Juan Sanchez Quintana, a senior at Florida International University who was a learning assistant during the study. Quintana assisted the experimental classrooms, and says that his participation has fueled his desire to teach college math after he graduates. Quintana, a math education major, estimates that he’s been a learning assistant for about 120 class periods. In the end, he came away as a proponent of the model, because “I’ve seen it work.”
That these newer methods of teaching impart more learning isn’t surprising to the study authors. But, Kramer says, the research does serve a purpose by adding to the store of evidence that these methods work. He and his co-authors hope that bringing scientific rigor to the studies of these methods of teaching calculus might sway skeptical colleagues to change how they teach.
Widening the Gateway
As a gateway course to STEM, calculus can be seen as a make it or break it moment for students, especially ones who are typically excluded from these careers. “If you’re struggling, it’s a barrier for you,” Kramer says.
In conducting the study — funded by the National Science Foundation — researchers wanted to let students experience what it’s like to be a mathematician.
The researchers figured that Florida International, one of the largest public research universities in the country, had a unique chance to help students who are underrepresented in STEM disciplines better connect with the subject matter. The university has a lot of Hispanic and women students, two underrepresented groups, the study notes. Whether many of those students pass calculus varies: In the six semesters leading up to the study, the pass rates for introductory calculus — which included classes taught using some limited active learning methods — spanned from 13 to 88 percent. Failure could mean potential biologists, mathematicians or engineers being pushed out of the field.
Kramer and others have been experimenting with active teaching methods for a number of years, and wanted to break the notion that some students are born with natural abilities in calculus and that teachers are supposed to identify the gifted few. “Our study shows that [any] student can grow” under the right circumstances, Kramer says. “And that’s really our responsibility as faculty, is to put students in environments where they can succeed, and [where] they are going to be able to achieve things that they might not have thought possible.”
Kramer projects certainty that these models are effective. These ways of teaching can be a lot more fun, too, Kramer says. But they break the preconceived notion of calculus as a weed-out course, he says, which can raise the hackles of professors skeptical of education research, and that increases the need for strong evidence.
Will this latest study be enough to convince colleagues to wander away from traditional lecturing methods?
“It should be very compelling evidence to anybody who looks at the study,” Kramer says. But people are messy. “My suspicion is that people will even be skeptical over this, even though it has a strong effect size, we’ve taken care of all the potential biases, as best as humanly possible, and it is published in Science, which is known to be an extremely rigorous process,” Kramer adds.
Instructors may still cling to lecture models, Kramer says, perhaps because “it helps their ego that they get to be the sage in front of a bunch of students professing how awesome they personally are.”
Nevertheless, there were possible limitations to the experiment that bear mentioning.
While the researchers say it was impossible to randomize the teachers, since the instruction relies on specialized knowledge, students were randomly assigned to either traditional classes or active learning classes. Randomizing the teachers could have raised more problems than it solved by introducing potential biases around active learning, Kramer argues.
But for some observers, this is a notable limitation. Jon Baron, a former chair of the National Board for Education Sciences and former vice president of evidence-based policy for Arnold Ventures, has called the study “encouraging but less than definitive” since it failed to randomly assign teachers.
A learning assistant noted another potential hindrance: These models don’t inspire as much enthusiasm when taught online.
When Quintana, the learning assistant, took calculus during the pandemic campus closures, the active learning methods were already in place, he says. But, Quintana notes, because students like himself were so fatigued by virtual learning, it didn’t really have as big of an effect. They didn’t interact in the breakout sessions as much, and didn’t really want to be there.
Still, to Quintana, it beat suffering through lectures.
“I can’t even think how long it would have been for me to take calculus without any type of active learning, like, no learning assistance at all,” Quintana says.