CAMBRIDGE, Mass. — Math professor Martin Weissman is rethinking how his university teaches calculus.
Over the summer, the professor from the University of California at Santa Cruz spent a week at Harvard to learn how to redesign some of the math courses his institution offers related to life sciences. Right now, they are part of a “leaky pipeline,” Weissman said. Thousands of students go through these courses, he adds, but a lot of them don’t graduate with degrees in those fields.
Falling off that path can lock students out of science, technology, engineering and math (STEM) careers. And despite some “uneven” progress in recent years, STEM fields are just not as diverse as industry leaders would like. Some educators place a share of the blame on calculus courses, which can push out otherwise interested students.
That’s a phenomenon Weissman noticed at his university. “There are math requirements for those majors. And students slowly seep off and change majors because they have difficulty with the math,” he says.
UC Santa Cruz sees a lot of underrepresented students disproportionately drip out of that leaky pipeline, Weissman says. That includes a number of Black, American Indian, Alaska Native and Hispanic students. Biologists at the school look at the math taught in traditional calculus courses, he adds, and wonder why it’s even being taught, because the math isn’t practically useful for the field. Meanwhile, the calculus instruction has to be slowed down enough that it’s not as effective for math people as it could be.
“I think we're in an uncomfortable zone, where a lot of calculus classes are serving no one,” Weissman concludes.
Around the country, “math wars” are raging over attempts to increase equity by playing down calculus from the curriculum in favor of statistics or computer science, or by delaying when students take algebra. But there’s also a quieter revolution taking place that applies a different strategy to achieve the same principles. Its aim is not to abandon calculus, but rather to yank calculus instruction into the 21st century, by teaching students through the use of real-world problems. Changing the way calculus is taught, proponents argue, helps more students find math approachable and relevant, making them therefore more likely to succeed while studying it.
This is the more responsive approach that Weissman studied in July at Harvard, where he joined two dozen other college educators from around the country, tucked inside the air-conditioned, blackboard-walled rooms of Harvard’s Science Center. The week-long training ran from mornings into afternoons, with chummy lunch breaks in the faculty lounge, or the buzzing cafe in the Science Center lobby. The educators sat through lectures on pedagogy, the finer points of math and how to apply it to actual biological problems.
Sessions were prone to explanations such as how “physics-based simulations” became the buzzword in Hollywood, leading animation teams to use modeling techniques for hits like “Frozen,” “Brave” and “Toy Story,” which include life-like representations of walking through snow and bouncing curly hair. These digressions were placed alongside technical explanations of “cardiac defibrillation,” the rippling of electrical pulses as they move through the heart, as a way to show how to connect complicated math to the world outside of the classroom.
The training also had the educators plan, observe and teach classes based on these principles to eager high schoolers enrolled in a summer camp on campus.
The teaching experts who sponsored the training hope it will prepare college instructors to become “advocates,” empowering them to explain and defend the rigor of this way of teaching calculus to skeptical scientists from other departments. They expect it to be only the opening shot in an academic revolution.
But if calculus instruction is going to change, it may take some persuasion.
A Silent Revolution
The trouble with calculus is widely understood. The solution? Less so.
As these Harvard training sessions took place, the California State Board of Education finally approved a new framework that sets out to make math more culturally responsive and inquiry-based. It’s an attempt to respond to some of the pressures Weissman identified by kindling students’ math interest.
But it’s been controversial, causing “knock-down, drag-out math wars” that have included parent protests, threats and academic-on-academic social media spats about whether calculus should be reworked. That’s in part because the framework prioritized alternatives to calculus and also recommended delaying Algebra I, an onramp course to high school math and a critical juncture in the race to calculus, until ninth grade for most students. Critics have alleged the framework rejected rigor for “wokeism.”
In fact, many of recent attempts to keep calculus from being an obstacle to a STEM career focus on deemphasizing calculus, instead directing students to take other math courses like statistics or computer science.
The idea for the Harvard sessions came from a quieter attempt to revolutionize math instruction, relying on similar ideas, emanating from the University of California, Los Angeles.
Over the past decade, UCLA revamped its calculus for life sciences courses, focusing them more strictly on math concepts and real-world biological questions, rather than on procedural rules for derivatives and integrals — which its advocates describe as a paradigm shift for calculus instruction.
This idea is what drew instructors to sweaty Cambridge in July. UCLA’s model caught the attention of the Harvard math department, which decided to host a training over the summer for college instructors looking to refashion their own calculus courses. The session was meant to catalyze change, encouraging those instructors to open their own revised courses modeled on the ones being taught at UCLA.
As part of that, the college instructors observed and taught lessons to teenagers participating in a summer program being hosted at the same time at Harvard. It was meant to allow the educators to see these new methods in action, and to try them out personally.
In an early morning class, bleary-eyed and still vibrating from coffee, the instructors met with high school-aged students. The students had previously “warmed up” by grappling with datasets on COVID-19 mortality rates, trying to figure out what that data meant for policy.
“What’s your morning process?” the instructor asked.
The students, broken up into groups around tables, considered the question. “Brushing teeth” was the most common response.
The students then learned to map out the likely impact of teeth scrubbing on plaque growth, before pivoting to other possible applications of advanced concepts like vector spaces and differential equations.
During classes like this, instructors for the program studiously referred to these methods as “change equations,” a non-threatening phrase substituted to prevent the high schoolers from shying away from intimidating language like “differential calculus.” It’s connected to the claim that these classes can capture the rigor of advanced math, only without the anxiety it usually brings.
That’s a key part of the sales pitch for the course. “Our class has no prerequisites. Period,” says Alan Garfinkel, one of the UCLA professors who designed it, when asked by a teacher about talking to students about prerequisites in calculus.
That’s not typical. When this subject is usually taught, it’s done procedurally. Students are given a set of rules for solving these equations and then drilled on them, with the “why are we learning this?” question answered afterward. But students in these classrooms confronted the problems they wanted to solve first, only getting the equations after the curiosity had set in.
It left an impression. “Today I got to be a teacher again! Euler’s method to 20 amazing High School students with varying levels of mathematics background! Loved honoring that mathematics is a web of ideas as opposed to a linear trajectory filled with pre-requisites,” one instructor posted on social media.
Many of the educators at the event said they were attracted by the desire to increase student engagement and to make math more relevant to students’ lives.
But the impact the educators hope for reaches beyond the classroom, too. If high school and higher education can get more students to reason mathematically, it will make them productive thinkers, says Lindsey Henderson, a secondary math specialist at the Utah State Board of Education, who attended the training. That’s what the businesses in Utah’s Silicon Slopes, the state’s burgeoning tech sector, say they want, she adds.
For Weissman, of UC Santa Cruz, the fact that this course is being taught at a large institution already is important. When it comes to math instruction, he says, “There are always people promising revolutions.” But UCLA’s method does seem to work for large institutions, according to Weissman. The University of Arizona now offers a version of the class. A study of the course published by its creators suggests it’s been successful in engaging underrepresented students.
And Weissman doesn’t foresee much of a fight in implementing it: “I'm not beholden to a traditional textbook, so I don't have to make sure that I cover certain methods that I really do think just don't need to be taught anymore.”
Change Equations
At the same time the week-long workshop for instructors took place, Harvard also ran a two-week program for high school students based on the idea that high schoolers can be taught to solve problems using principles of advanced calculus.
The teacher workshop included designing and teaching classes to that class of 36 high schoolers, something the attendees weren’t warned about more than a day or two before.
“We wanted a way to have workshop participants see what's possible in the classroom,” says Brendan Kelly, the director of introductory math at Harvard and one of the event organizers. If you haven’t seen students thinking through the problems, it can be hard to vividly imagine what it might look like in your classroom, Kelly says.
The traditional sequence for math in middle school and high school is algebra I, geometry, algebra II/trigonometry and then pre-calculus, with advanced students making it to calculus. Increasingly, calculus is seen as a necessary bolster for competitive college applications.
For the high school summer program at Harvard, though, only algebra II was required. Students at the program had mostly taken AP Calculus, though not all of them had. One student said she had only taken pre-calculus before entering the course.
Student campers gave high marks to the experience.
“For me, like, I've honestly never considered a major in math,” says Judy Yen, a rising junior from the private Taipei American School in Taiwan. Yen wants to enter the medical profession, and the course left her considering a math double major or minor in her future, she adds.
For others, the lesson was that math can lead to benefits beyond school. “I can actually get jobs rather than just studying. Really, I can actually, maybe get a job that's related to math,” says Charles Sciarrino, a rising senior from Staten Island Academy, a college preparatory day school in New York. “And I just find that really cool,” he adds.
Still, it was a Harvard summer school class, implying that most students who participated not on scholarship had access to the funds to afford summer school in Massachusetts — which cost $5,300 for tuition and room and board for the two-week program — not to mention a prior interest in math. Will it translate to other schools and contexts?
There’s some privilege there, Kelly admits. But he firmly believes that the learning that happened there is possible anywhere: “I think it's a real deficit mindset to think that first-generation or low-income students wouldn’t have that same enthusiasm and curiosity. I just fundamentally disagree with that. Young people are curious about the world. And when you put compelling questions in front of them, they respond with excitement and engagement.”
The 28 educators at the workshop training seemed positive. “The course helps gain access for a broader range of student populations, for us to get students excited about math and cross-pollinate to all the other divisions as well,” says Steven LeMay, an associate professor at Curry College, a private college in Massachusetts, who attended the training.
LeMay was tasked with figuring out whether the revamped calculus will work for Curry, and he seemed generally optimistic. Curry College likely won’t have the fight that UCLA reported in attempting to transition its students, LeMay predicts. The college doesn’t have a standalone math major, and there’s been a push from LeMay’s colleagues to freshen the school’s technology use, LeMays says.
Other higher ed instructors, however, expressed concern over whether it would translate into their less resource-rich colleges. Their institutions, they say, were worried about whether their students would get transfer credits at other colleges for these courses, and they were skittish about possibly disrupting their own institutions’ math departments by keeping students from more traditional calculus classes.
In the end, Kelly of Harvard says, the dream is to have similar courses that integrate calculus concepts in life sciences, economics, social sciences, physical sciences and engineering taught at colleges and high schools. (Kelly has taught a similar modeling course for economics and social sciences for the last few years.)
But it’s hardly a foregone conclusion. One major challenge to spreading this method of math instruction more broadly? Money. The Harvard summer training was popular with potential teachers, but it was hard to get funding to support the program, Kelly says. He reports that he was unsuccessful twice in applying for a grant from the National Science Foundation — which Kelly attributes to a general lack of enthusiasm for attempts to alter calculus and a belief that it wasn’t a proper training course — but it was funded by a gift from an anonymous Harvard alumnus to the math department. Continuing the work will mean securing sustainable funding, he adds. That may be easier now that the first session has wrapped up, Kelly predicts.
But it’s still early days, Kelly says: “I think across the country, we are barely getting off the ground.”
