Preparation for Calculus: A Research Agenda
By: David Bressoud @dbressoud
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences
Just Equations has just published their latest report: Staying the Course: Examining College Students’ Paths to Calculus, authored by Marcelo Almora Rios and Pamela Burdman. It illuminates the difficulties of an important and increasingly common question:
For students who aspire to an academic major that requires calculus but who arrive at college with weaknesses in their mathematical preparation, how can we best support them so that they can succeed?
This has always been a difficult question to answer, made all the more pressing by the significant gaps in mathematical preparation created by the inadequacies of and inequities created by remote learning during the COVID pandemic. Staying the Course both presents the known research on this question and describes the extremely varied responses of the 23 campuses of the California State University system (CSU). These campuses provide particularly informative case studies because of the 2017 CSU decision to eliminate all non-credit-bearing developmental courses. This has opened opportunities for students who will not need calculus, but it leaves the question of what it takes to facilitate success in calculus. The situation is complicated by the fact that mathematical preparation of CSU students runs the gamut from those with good scores on AP Calculus to those who have not gone much beyond high school Algebra II.
The first issue is how departments should assess readiness for calculus. One does not want to push students into a course in which they are certain to fail, but undervaluing their readiness can be damaging in two respects. The first is what this does to their own sense of mathematical ability. For women and students in other traditionally underrepresented groups, the self-confidence needed to pursue a mathematically challenging major can be irreparably harmed by under-placement. Second, it is well documented that as the string of prerequisites lengthens, there is a compounding of the attrition that occurs both within courses and when faced with the decision whether or not to continue to the next course. Persistence rates are often as low as or lower than passing rates.
The 23 CSU campuses exhibit the full range of options for placement from home-grown placement tests to commercial offerings such as ALEKS to placement that relies primarily on the high school transcript to self-placement. All have their weaknesses. An interesting point raised in this report is that students from underrepresented groups are more likely to self-place lower than equally well-prepared students from well-represented groups (see Kosiewicz & Ngo, 2019; Cuellar Mejiaet al., 2020). As Eric Hsu—current Chair of the math department at San Francisco State—and I have reported in “Placement and student performance in Calculus I,” multiple measures that heavily weight high school math course-taking and GPA are usually the most reliable.
Across the CSU system, there is a tremendous variety in the structure of calculus prerequisites, reaching its extreme at Cal State Chico where a student might be directed through a sequence of four prerequisites before being admitted to calculus. The intent is well-meant. The authors quote Michael O’Sullivan, until recently Chair of the math department at San Diego State University,
The last thing I want is for someone to get into a class and fail because they don’t have prerequisite knowledge, because then they’re stuck. […] If a student retakes a class, the chances that they fail are very high.
In response, the authors point out that we do not have rigorous research demonstrating that a longer prerequisite sequence improves chances of success in calculus for certain at-risk students, and, if so, whether it is preferable to other possible strategies.
Eric Hsu has been a skeptic of any changes that lengthen the route to calculus. As he is quoted in this report,
I came into the whole business very concerned about Precalculus, whether it’s an effective class in preparing students for Calculus. If I had been emperor, I probably would have never created stretch Precalculus. I came into it feeling that underplacement is automatically bad.
Yet, as he goes on to reflect,
These are adults with choices, and now we’ve gone through a pandemic, and students come in, “You know, I didn’t really learn a lot online and I want to take it easy. I’ve got plenty of things to stress about, and I want to take a two-semester version of this.” That’s where my thinking is evolving.
Many campuses are content with their placement procedures based on success rates in calculus, but that is not the whole story. One must consider whether there are students who could succeed in calculus but never get the chance because of the obstacles that are placed in their path or the lack of supports that would enable them to be successful. Relying on filters to ensure acceptable passing rates is almost certain to foster inequities. As Eric commented,
Let’s say ALEKS is the best test ever, and if it says you’re not good enough to pass Calculus, it works for 90 percent of the students. Some people would think that’s a really accurate test and we saved 90 percent of students from making a horrible mistake. I feel like, that’s a lot of students who could have passed Calculus that you kept out of it. Shouldn’t we give people the choice?
The takeaway is that the CSU system, like most of American higher education, is in the midst of a vast, uncontrolled experiment in preparing students for and enabling them to succeed in calculus. This report has no answers. What should be clear is that there are no simple answers. But that does not negate the fact that there is much that we can and need to learn from the careful study of the effects of the various structures and procedures that are being employed.
References
Almora Rios, M., and Burdman, P. (2023). Staying the Course: Examining College Students’ Paths to Calculus. Just Equations. https://justequations.org/resource/staying-the-course-examining-college-students-paths-to-calculus
Cuellar Mejia, M., Rodriguez, O., & Johnson, H. (2020). A new era of student access at California’s community colleges. Public Policy Institute of California. https://www.ppic.org/wp-content/uploads/a-new-era-of-student-access-at-californias-community-colleges-november-2020.pdf
Hsu, E., & Bressoud, D. (2015). Placement and student performance in Calculus I. In D. Bressoud, V. Mesa, & C. Rasmussen (Eds.), Insights and recommendations from the MAA national study of college calculus (pp. 59–68). MAA Press. https://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf
Kosiewicz, H. and Ngo, F. (2019). Giving community college students choice: The impact of self-placement in math courses. American Educational Research Journal 57(3), 1358–1391. https://journals.sagepub.com/doi/abs/10.3102/0002831219872500
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