To Repeat or Not to Repeat

By David Bressoud @dbressoud

As of 2024, new Launchings columns appear on the third Tuesday of the month.

When should a student repeat in college a course that they have already taken and passed in high school? In preparation for College Algebra/Precalculus or Calculus, the evidence is now very clear that requiring students to repeat a course they passed in high school is not an effective way of addressing deficiencies. They are more likely to succeed in the course to which they aspire either through directed self-study—one example being the use of ALEKS—or through supplemental instruction that might be provided via a second supporting course or by expanding the desired course through extra hours or an additional term. When the question is whether to repeat a calculus course taken in high school, the College Board has run several studies that demonstrate that repeating this course provides no advantage for the subsequent course (see “AP Calculus: What we Know,” Launchings, June 2009 [Bressoud, 2009]).

But the first term in college is stressful. Many students who could take credit for one or more terms of calculus choose not to, making it easier to begin their college study of mathematics with a good grade and less pressure. With this in mind, three researchers at Stanford University have recently published a paper on “Should I start at MATH 101? Content repetition as an academic strategy in elective curriculums” [Harrison et al, 2022].

To study this question, they conducted interviews with a group of students at a highly selective research university in the western US that enrolls approximately 7000 undergraduates, 60% of whom earn a degree in a STEM field. A mathematics placement exam is administered, but placement is not enforced. While they started with 85 students, full data were collected from about 50. They looked at factors that influenced the decision to either repeat their last high school class or proceed directly to the next class. They also looked at how this affected their first mathematics grade.

Not surprisingly, students who repeated a course earned higher grades (mean grade A/A–) than those who did not (mean grade A–/B+). The interesting result is that students with economic advantage, upper-middle and middle class, were more likely to repeat than those from lower-middle- or working-class families. Half of middle and upper-middle class students chose to repeat. Only 27% of lower middle- or working-class students repeated, and only 17% of first-generation college students repeated. 

The authors speculate that awareness of the nature of college demands and of the options that are available favors students with economic and generational advantage. Quotes from some of the less advantaged students supported this. One student assumed that the placement test should be followed, and another reported that their academic advisor recommended taking the most demanding class for which they were qualified. An additional piece of evidence that privilege favors retaking the last high school course came from the 42 non-international students for whom they had full data. Twelve of these had attended private schools, and all but two repeated their first course. Of the 14 who attended public high schools but had taken a course in college while in high school, half chose to repeat. Of the remaining 16 US students, only three chose to repeat.

The evidence that privilege favors retaking the last high school course leads the authors to three conclusions:

1. Curved grading puts students who are repeating an introductory course in direct competition with those who are not, thus penalizing students who are studying this material for the first time.

2. Mastery-based grading with its focus on whether specific skills and knowledge have been obtained is more equitable.

3. Institutional advising might encourage incoming students, especially first-generation college students, to consider repeating the last course they took in high school.

It is worth noting that MIT deals with some of these problems by not assigning grades in the first semester (see https://registrar.mit.edu/classes-grades-evaluations/grades/grading-policies/first-year-grading).

I agree with the authors’ conclusions and acknowledge that economic and generational advantage probably factor into student decisions on whether to repeat a course, but I have difficulty with their evidence that these advantages shape the decision. The problem is that at this university a year of single variable calculus is followed by a term spent on multivariable calculus and linear algebra, and the decision to repeat was most common for students facing multivariable calculus/linear algebra. Fifteen of the fifty students had studied multivariable calculus and/or linear algebra while in high school. Only one chose to move directly on to the next course. Of the 33 students facing the choice of whether to repeat a single variable calculus course, ten chose to repeat it (see Figure 1).

Figure 1. Relationship between highest math course before matriculation and content repetition in first math course. Novices are those who chose not to repeat the course. Source: Harrison et al, 2022, Figure 4, p. 140.

It is clear from student comments that multivariable calculus/linear algebra is a demanding course of which many students are wary. There seems good reason not to skip it. Furthermore, just having access to such a course while in high school is a sign of privilege. This paper by itself does not present convincing evidence that repetition of introductory mathematics courses is a result of student economic and generational advantages. At least at the university under study, the choice of whether to repeat a mathematics course is highly dependent on which course would be repeated.  

What would be most illuminating is a focus on calculus. The College Board studies on the effects of repeating a calculus course are more than a quarter century old, and they say nothing about the effects of combining students who have and those who have not studied this material before. The national survey conducted by the Mathematical Association of America in 2010 [Bressoud, 2015] revealed that 67% of Calculus I student at PhD-granting universities were repeating the course they had taken in high school. Even at regional public universities, the figure was 40%. I expect that these percentages are substantially higher now. At the same time, disadvantaged students are far less likely to take calculus in high school. Calculus in high school is taken by less than 10% of students at schools with over 50% of the student body on free or reduced lunch. At schools with less than 25% of students on free or reduced lunch, over 25% have studied calculus [NCES, 2022]. In colleges and universities, those learning calculus for the first time are almost always in competition with those for whom it is not new. It is in calculus that we have the strongest argument against grading on a curve.

References 

Bressoud, D. (2009). AP Calculus: What we know. Launchings, June 2009. https://docs.google.com/document/d/12SICsMuFtT_rMx8wSoH-0G4QuG2xQxA02kXp2jy3rj8/edit

Bressoud, D. (2015). The Calculus Students. Pages 1–16 in Insights and Recommendations from the MAA National Study of College Calculus. Bressoud, Mesa and Rasmussen eds. MAA Notes #84. Washington, DC: MAA Press. https://maa.org/ptc.

Harrison, M. H., Hernandez, P. A., & Stevens, M. L. (2022). Should I Start at MATH 101? Content Repetition as an Academic Strategy in Elective Curriculums. Sociology of Education, 95(2), 133-152. https://doi.org/10.1177/00380407221076490

NCES (2022). Percentage of public and private high school graduates who completed selected mathematics courses in high school, by selected student and school characteristics, Table 225.40, https://nces.ed.gov/programs/digest/d22/tables/dt22_225.40.asp?current=yes

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