The Odd Math on Calculus Prerequisites

There’s a conundrum in college mathematics policies: Taking prerequisite math courses in college doesn’t appear to increase most students’ chances of completing calculus, yet most higher ed institutions continue to require them.

Early studies even suggest that prerequisite courses—such as precalculus and college algebra—may actually interfere with many students’ eventual completion of calculus.

Large proportions of students who succeed in precalculus don’t continue to calculus and leave STEM majors, according to Eric Hsu and David Bressoud, citing studies at public universities across four states. Another large study found that precalculus students who did continue to calculus appeared to fare worse than similarly prepared students who started in calculus. 

Is the conventional wisdom about the efficacy of prerequisites for calculus wrong, or is faculty confidence in prerequisites merited? More research in multiple contexts is needed to say for sure, as noted in a 2023 Just Equations report. 

“An intended remedy for perceived underpreparedness could instead be a structural barrier to improving Calculus completion,” I wrote with co-author Marcelo Almora Rios. “But unless colleges begin to monitor sequence success rates in addition to course success rates, this issue will go unaddressed.” 

A new law in California is bringing more attention—and research—to this issue. The law, Assembly Bill 1705, places limits on calculus prerequisites across the state’s community college system. The related research focuses on how community colleges should implement the law’s requirements. Its findings are exposing a divide across the higher education landscape about how to improve calculus completion and access to STEM majors, and even drawing official fire from university professors. 

The new research—conducted by community college institutional researchers—finds that attrition associated with prerequisite courses reduces students’ chances of ever taking or passing calculus, echoing prior studies. Some math departments in the system have demonstrated success with eliminating prerequisites: For example, when Cuyamaca College allowed STEM-bound students to start their math sequences in a calculus class with additional support, success went up, particularly for underserved students.

But many math faculty in community colleges as well as at CSU have responded with disbelief. They worry that bypassing prerequisites will eliminate opportunities for students with weak algebra skills, particularly students who missed out on critical preparation during the pandemic. Since a shaky math foundation could hamper progress into STEM fields, they say students shouldn’t be deprived of the opportunity to take prerequisite courses. 

At CSU, calculus prerequisites are the norm. Conflicting policy directions between the two systems could produce an incongruous result: Students at open-access community colleges could be taking fewer prerequisites than CSU freshmen, who were in the top third of graduating seniors. 

Discord over new law 

California’s AB 1705 builds on prior legislation that phased out many community college remedial math courses but left college-level prerequisites intact. Like its predecessor, AB 1705 doesn’t merely mandate shortening sequences: It requires offering additional support for students to succeed in their gateway math course—calculus, in this case. 

The law doesn’t take effect until the fall of 2025, starting with an initial two-year experimentation period. Hopefully, that will be enough time to develop consensus about how to move forward. 

For now, though, discord over AB 1705 is amplifying discontent over implementation of the prior law, which focused on increasing students’ chances of passing their college-level math courses by placing students directly into those courses, rather than into remedial courses. 

For community college students majoring in fields with a single math requirement—often statistics—the prior law led to shorter math sequences and increased completion of college-level math. There has been less agreement, however, about the math trajectories of students in STEM fields requiring calculus. Such students often face multiple college-level prerequisites—including some that were added after remedial courses were phased out, because faculty felt students needed the additional preparation.  

STEM students at the CSU face a similar situation: the CSU completely eliminated stand-alone remedial courses six years ago. But college-level prerequisites for calculus range from 3 to 17 units and take one to four semesters, according to our 2023 report Staying the Course.

At the majority of campuses, the least prepared students face two semesters of prerequisites. At one—Chico State—students can take as many as four semesters of prerequisites before entering a calculus course. 

Shock over longitudinal findings 

In interviews for Staying the Course, most CSU math faculty expressed confidence in their use of prerequisite sequences. Yet we found no longitudinal studies looking at rates of calculus completion for different prerequisite patterns.  

By contrast, community college institutional researchers at the Research and Planning Group for California Community Colleges (RP Group) have recently conducted three studies examining how students’ math prerequisite paths relate to their success in calculus. The studies focus on business majors, STEM majors, and biology majors. The findings, which cast doubt on traditional assumptions about prerequisites, have implications for how the system implements AB 1705. 

For many, those findings came as a shock. CSU math faculty even passed an official resolution challenging the RP analysis. All three reports concluded that no group of calculus-bound students would be better off taking a prerequisite course than starting in a calculus course. Remarkably, even students who had not taken Algebra II in high school are—on average—better served by skipping over prerequisites, according to the reports. 

That last finding has been understandably hard for many CCC and CSU faculty to swallow. After all, many have been teaching the prerequisites in question for many years and have countless examples of students who prevailed through their calculus sequences and benefited as a result

Furthermore, research has consistently shown that reforms such as AB 705 and AB 1705 involve a trade-off: Increases in overall sequence success are accompanied by decreases in success rates in individual courses. This 2014 study, for example, found declines of 10 to 15 percent in course pass rates. It makes sense that allowing more students to take a gatekeeper course could lower the pass rate for the course, even while raising the overall numbers of students who complete the course. That’s because of inevitable attrition in longer sequences.  

But faculty tend to be far more aware of—and, thus, more concerned about—course pass rates than they are of attrition across a course sequence. Students who don’t return are typically off their radar. Faculty understandably worry—and morale can suffer—when pass rates noticeably decline. 

The resolution from CSU math chairs conveys the concern that the new policies will “negatively impact the quality of calculus instruction and outcomes, diminishing the average academic preparation of students majoring in STEM fields who transfer to the UC and the CSU.” They even note fears of losing accreditation in some fields, such as engineering. 

Chief among their critiques, the professors assert that the RP study “disregards local community college placement criteria that differentiated the preparation of students initially placed into calculus from those initially placed into calculus prerequisites.” 

This claim is head-scratching. The resolution doesn’t explain how placement criteria would change the results—and it doesn’t seem that they would. The research describes students’ fates in the courses they eventually took, regardless of how they landed there. (That’s what is possible since the state’s data system does not include information on placement, though it would have been helpful for RP to mention this fact.) 

The better claim—and perhaps what the profs were getting at—is that we don’t know how a group of students without Algebra II on their transcripts were allowed into Calculus in the first place. Could there be something unique or special about these students that isn’t apparent? Might there be experience not reflected on their transcripts? 

Those are good questions, since this is an observational study. But even if such students are unique, the fact remains that only a quarter of students without Algebra II had a chance to take a calculus course over two years if they were not given direct access to the course. That is no ringing endorsement of the status quo, assuming the goal is to increase access to STEM fields.

Another assertion by the CSU professors is that the study’s analysis of prepandemic data may not apply to today’s students. This is a fair point. Current students may well need more math fundamentals than students in 2019 did. That doesn’t change the fact that prior policies were not successful in providing the needed foundation for the majority of students. It’s hard to imagine they would be more successful with less prepared students. But it is a question worth studying. 

To address these deficiencies, the CSU faculty call for multiple reviews of the RP report—a peer review, plus reviews by the CSU and UC faculty senates. 

I am always a fan of more research, but three reviews sounds like overkill, especially since the RP appendices provide so much data that it wouldn’t be a stretch for a group of math faculty to replicate their analysis. The CSU faculty don’t say whether they have done so. 

Fortunately, David Bressoud, one of the nation’s foremost calculus education experts, has done just this—quietly and over the summer break. In Bressoud’s latest Mathematics Association of America blog, he concludes, with some caveats, that “the surest route to success in Calculus I is to enroll directly in it.”

The caveats include the fact that only 230 of the 35,663 students in the study enrolled directly in calculus without having taken Algebra II. They constitute only 8 percent of students without Algebra II on their transcripts. 

“With this group of students, I am leery of a recommendation to start with calculus,” Bressoud notes, echoing the concerns of the CSU faculty. “That 8 percent is very small. They certainly represent very highly motivated students, and probably students who found other means outside the high school classroom to obtain the knowledge they would need to succeed.” 

Then he adds a caveat to that caveat, focused on the 75 percent of non–Algebra II takers who never make it to calculus if placed in a prerequisite course:

I am worried by that incredibly high attrition rate. These are students who wanted to major in a STEM field and almost certainly were told to begin with College Algebra. It is not just the ability to pass the prerequisite courses. We know that a lack of persistence to the next course is at least as big a barrier as failure to pass.

‍Overplacing vs. overcorrecting 

Eric Hsu, Bressoud’s co-author on the precalculus placement study, has a stake in this debate as math chair at San Francisco State. He shares Bressoud’s ambivalence. Hsu acknowledges that colleges currently may be overplacing students into prerequisite courses such as precalculus. Most students who took precalculus in high school shouldn’t have to repeat it, he notes. 

At the same time, he worries about the outcome of AB 1705. Getting rid of precalculus entirely would be overcorrecting in ways that could hurt some students. “Precalculus needs a big overhaul,” he told me, “but it would be weird to eliminate any possibility of students taking precalculus in college.” 

Indeed, when RP examined students who took prerequisites and did persist to calculus, they concluded that the prerequisite courses appeared to offer a modest boost in calculus success. This seems to validate faculty intuition about the students they actually see in their classes: They could be benefiting from the longer sequence (even though none of the 12 variables RP examined identified this group), though we don’t know how that group would have fared if they’d gone straight to calculus.

Perhaps the problem perhaps is not the courses themselves, but the invisible cost of requiring them: Two-thirds of STEM majors do not complete Calculus I within two years of starting their math sequence. Disproportionately, these students are Black and Latinx, groups that are already underrepresented in STEM. 

In the end, it’s fair to ask whether the goal is ensuring effective algebra preparation for students entering STEM or ensuring that more students have the chance to pursue STEM at all. Ideally, we shouldn’t have to choose between the two. But, based on the evidence to date, it’s clear that we don’t yet have a strategy to address both. 

The CSU math faculty care about student success. They engage with K–12 and community college math through research and other partnerships far more than their counterparts in math departments at UC. So it is encouraging to see them engaging with these questions. 

If they don’t find Bressoud’s review sufficient, hopefully the CSU profs will study the RP appendices and share their conclusions. A look at Cuyamaca’s findings would also illuminate the stunning results that campus has reported: an 83 percent increase in the number of students taking Calculus I as their first math course, a 67 percent increase in sections of Calculus II, and an anticipated doubling of Calculus III sections. If such outcomes are replicable, it would be game-changing. 

Since good placement practices are necessary but not sufficient, it would also be important to examine Cuyamaca’s approach to precalculus skill development, active learning, and faculty development to understand what is behind its apparent success.

The CSU faculty would also contribute a great deal to these questions by conducting a similar analysis of math prerequisites within their own system. This would shed light on which of the array of strategies in place at CSU are most effective for students. 

If there are prerequisite strategies within the CSU that are boosting students’ likelihood of succeeding in STEM, while also bolstering their algebra foundation, that evidence needs to be shared widely. The same goes for the prospects of Cuyamaca’s approach at CSU campuses as well. 

At the same time, RP could benefit from writing its reports for a wider audience, rather than just for insiders who are familiar with community college policies and placement rules. That includes highlighting the limitations of its study, such as the absence of placement information. 

RP could also analyze some of the implications. For example, what does it mean that a subset of directly placed students take two or three tries to pass Calculus? For another, how do students in various math-preparation paths eventually perform in STEM disciplines, the ultimate outcome under debate? Incorporating these broader contexts would help make research more transparent and meaningful for insiders as well as external partners, such as CSU faculty. 

The stronger the collaboration, the better. The upcoming two-year experimentation period provides a critical window for research and innovation to uncover the best possible placement policies to maximize opportunities for the next generation of California community college students.