Calculus as Preparation for Calculus

By David Bressoud @dbressoud

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How should we place students who need mainstream calculus for their intended major? This was one of the questions faced by the California Community Colleges (CCC) as their Chancellor’s office began work on their Multiple Measures Assessment Project for placement into English and mathematics courses. Of course, one cannot expect a single answer. It must depend on a student’s prior experience before matriculating at one of their colleges. The surprising answer they found was that the greatest chance for ultimate success in Calculus was direct placement into Calculus, regardless of whether a student’s highest high school mathematics course was calculus, precalculus, trigonometry, statistics, or algebra 2, or the student had not even reached algebra 2.

Their findings are summarized in Figure 1. Regardless of highest high school mathematics course, all students who simply began in Calculus I have had at least a 62% success rate in passing this course with a C or better within two years, with far better rates of success than those who started at any other transfer-level college course along the calculus pathway: College Algebra, Trigonometry, or Precalculus.

Figure 1: Preparatory Pathways and STEM Calculus Completion, page 6

Fortunately, the report containing this result, Preparatory Pathways and STEM Calculus Completion: Implication of the AB 1705 Standards, has appendices with extensive breakdowns of the data that were collected, enabling me to do my own analysis. While that revealed some caveats and subtleties, I have come away with the conclusion that the surest route to success in Calculus I is to enroll directly in it.

The data are based on the records of CCC students who on matriculation between 2012–13 and 2019–20 had declared an intention to pursue a STEM major for which mainstream Calculus I is required and whose first mathematics course was in the transfer-level calculus pathway (College Algebra, Trigonometry, Precalculus, or STEM Calculus I). They restricted the study to students for which they had complete high school transcripts and either a record of passing calculus at the community college within two years or two full years of records in the CCC system. Students were not required to pass calculus on their first attempt. A few (a total of three) took at least five attempts before they passed, but they had to accomplish this within two years of the start of their first transfer-level course. The data are based on a cohort of 35,663 students who qualified for the study.

My first question was about those students who had not even made it to Algebra 2 in high school, much less trigonometry or precalculus, before enrolling in calculus and succeeded. Not surprisingly, their numbers are small. A total of 230 students enrolled directly in calculus without having seen Algebra 2. This is only 8% of the 2868 who came to college without Algebra 2. These did have reasonable success: 103 passed on their first attempt, an additional 28 on the second, and 11 more on the 3rd or 4th, yielding a success rate of 142/230 = 62%.

Most of the loss of students who did not start with calculus was the result of attrition along the way. Just over three-quarters (76%) of those who entered without Algebra 2 never got to the point of enrolling in calculus. Those who stuck with it all the way through did better when they got to calculus. Of those without Algebra 2 who did not start in calculus but eventually enrolled in it, 79% would pass it at some point during the two years.

With this group of students, I am leery of a recommendation to start with calculus. That 8% 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. But 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 pre-requisite courses. We know that a lack of persistence to the next course is at least as big a barrier as failure to pass.

Attrition on the road to calculus is not just a problem for students without Algebra 2 on their high school transcript. Table 1 shows the percentage of those who did not go directly to calculus and never enrolled in calculus as well as the passing rate of those who did not go directly to calculus but eventually did enroll in it.

Table 1: For students who intended to major in a STEM discipline that required Calculus I, percentage who never enrolled in Calculus I and percentage who passed Calculus I as a fraction of those who enrolled in Calculus I after starting in College Algebra, Trigonometry, or Precalculus

It is amazing that of those who passed trigonometry or precalculus in high school and who needed calculus for their intended major, half never enrolled in calculus. Also, there is very little difference in the pass rates of those who went directly to calculus and those took a preliminary course. Even more surprising, looking at Table 2 and the data for students who studied calculus in high school, there is a higher pass rate for those who went directly to calculus then for those who eventually got into calculus.

Table 2: For students who intended to major in a STEM discipline that required Calculus I, percentage that started with Calculus I and percentage of these students who passed Calculus I.

Conclusions

There seems little point in asking someone who passed calculus in high school to start with a course below calculus when they get to college, even if it was not a fully college-equivalent course and a placement exam reveals gaps or weaknesses in their preparation. This is especially true if they are serious about aiming for a STEM major that requires calculus. I believe that this is also true for those whose highest high school course was trigonometry or precalculus. It is very possible that they are not fully prepared to succeed in calculus, but they will have more opportunities to identify and address any deficiencies within calculus than in precalculus or its precursors. This probably will require supplemental supports. In addition, there should be a recognition that many students will need to take Calculus I more than once, but at least they can come out of a failed first attempt with a better understanding of what they need to remedy.

For students who have not taken trigonometry or precalculus in high school, the numbers who have gone directly into calculus and succeeded are so small that I am hesitant to draw any conclusions beyond two obvious ones.

  1. You still want to keep the path to calculus as short as possible. The longer the path, the more students you will lose.

  2. Students who appear to lack the necessary prerequisites but go directly to calculus and succeed are interesting students who are worth studying. What was it they did that enabled them to succeed despite their leap into far more demanding mathematics?

Finally, these data reveal that the traditional sequence of courses designed to prepare students for calculus—College Algebra/Trigonometry/Precalculus—is failing the majority of community college students with aspirations of a STEM major. These courses need to be deconstructed and re-imagined using what we have learned about what motivates students and truly prepares them for calculus.

Reference

California Community Colleges in partnership with The RP Group. (2024). Preparatory Pathways and STEM Calculus Completion: Implication of the AB 1705 Standards https://rpgroup.org/Portals/0/Documents/Projects/MultipleMeasures/AB705_Workshops/PreparatoryPathwaysSTEMCalcCompletion_February2024.pdf

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