Wednesday, October 1, 2014

The Pitfalls of Precalculus

Many of the students who aspire to study science or engineering never manage to get through calculus. For many if not most of them, the stumbling block is not calculus as such; it is an inadequate grasp of the mathematics of precalculus. This is why most colleges and universities administer a placement exam and offer a course called Precalculus. This is a high school course, but most colleges and universities offer it for students who want to take calculus but are believed to lack the mathematical foundation needed to succeed. Does precalculus in college actually work? Do the college students who take precalculus go on to succeed in calculus? Does a college course in precalculus even help student performance in calculus? Recent research by Sonnert and Sadler suggests that the benefits of college precalculus are marginal at best. At worst, it can be damaging.

Precalculus, as the name suggests, should be a course that students take to prepare for calculus. In fact, the University of Illinois at Urbana-Champaign has changed the name of this course to Preparation for Calculus, emphasizing the fact that this is not and should not be taken as a course to satisfy a mathematics requirement, but only as preparation for success in calculus. Unfortunately, precalculus often proves to be terminal, even for students who do well. A variety of local studies have shown that large numbers of students who are successful in precalculus choose not to continue on to calculus. A Texas Tech study [Jarrett 2000] found that a third of their students who earned a B or higher in their precalculus course failed to enroll in Calculus I. There are similar data from Arizona State University [Thompson et al 2007]: Among declared Engineering majors who earned a C or higher in Precalculus, 38% failed to enroll in Calculus I. It was worse for other majors: 55% of Physical Science majors, 56% of Mathematical Science majors, and 65% of Life Science majors who earned a C or higher in Precalculus failed to enroll in Calculus I. Herriott and Dunbar [2009] reported comparable data from the University of Nebraska-Lincoln and a collection of colleges in Illinois.
What about the students who do go on to Calculus I? Has a semester of college-level precalculus helped them? Gerhard Sonnert and Philip Sadler have just reported on the first large-scale study to address this question [Sonnert & Sadler 2014]. What they found is that for students whose high school mathematical preparation lies between the mean and one standard deviation below the mean (roughly the 15th to 50th percentile) of all Calculus I students, taking precalculus in college produced a small and not statistically significant improvement in their expected grade: around 1 point on a 100-point scale. More than one standard deviation below the mean, their study showed worse grades in Calculus I if they also took precalculus, but here the numbers are so small that the results are problematic. For students whose high school preparation was 0.3 standard deviations or more above the mean, taking precalculus in college was associated with a reduction in their Calculus I grades by 6 or more points, a result that was statistically significant at p < 0.05 for students at 0.3 standard deviations and numerous higher cut-offs.
The Sonnert-Sadler analysis relies on data collected in their FICS-Math study (Factors Influencing College Success in Mathematics, NSF #0813702) that collected both extensive background information and final grades of approximately 10,500 students enrolled in Calculus I in fall 2009. From their own analysis and that of others, once you control for variables such as gender and socio-economic status, high school preparation is the best predictor of success in college calculus. Sonnert and Sadler used a hierarchical logistic regression to identify six high school variables that were highly correlated with success and from which they could build a score to describe readiness for calculus. The variables they identified are
  1. SAT/ACT math scores
  2. Took precalculus in high school
  3. Grade in high school precalculus
  4. Took non-AP calculus in high school
  5. Took AP Calculus in high school
  6. Grade on AP Calculus exam
Because of the large sample with which they were working, they were able to find substantial numbers of students with comparable readiness scores, many of whom had taken precalculus in college, many of whom had not. At each level of preparation, from 1.8 standard deviations below the mean to 1.0 standard deviations above, in increments of 0.1 standard deviations, they were able to simulate a discontinuity regression, comparable to the one described in my column from January 2012, First, Do No Harm. This demonstrated the results reported above.

The exceedingly modest gains for students below the mean are not very surprising. Across all subjects, effective remediation is tough to pull off. There are some precalculus programs that seem to be helping. Identification and investigation of these will be part of MAA’s next study of the precalculus/calculus sequence, Progress through Calculus (NSF #1420389). Some of the more promising directions include courses that weave precalculus review into the introductory calculus course. The University of Illinois has also had success with its use of ALEKS in their Preparation for Calculus. Unfortunately, the vast majority of college precalculus is still taught as recapitulation of the course students took and failed to master in high school, just coming at them a lot faster.

The harm that appears to be done by putting good students into precalculus is a more intriguing result, perhaps a reflection of the damage done to their self-confidence.


Herriott, S.R. & Dunbar, S.R. 2009. Who takes college algebra? Primus. 19:1, 74–87.

Jarrett, E. 2000. Evaluating Persistence and Performance of ‘Successful’ Precalculus Students in Subsequent Mathematics Courses, M.S. Thesis, Texas Tech University.

Sonnert, G. & Sadler, P. 2014. The impact of taking a college pre-calculus course on students’ college calculus performance. International Journal of Mathematical Education in Science and Technology. DOI: 10.1080/0020739X.2014.920532

Thompson, P. et al. 2007. Failing the future: Problems of persistence and retention in Science, Technology, Engineering, and Mathematics (STEM) majors at Arizona State University. Report prepared & submitted by the Provost’s Freshman STEM Improvement Committee.