By David Bressoud

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I have often cited data from the Sadler and Sonnert FICSMath study (Factors Influencing College Success in Mathematics, sponsored by NSF grant #0813702), a large-scale study of 10,437 students in mainstream Calculus I in the fall of 2009 at a stratified random sample of 134 U.S. colleges and universities. Sadler and Sonnert have just published their insights from this study into the following question: Are the students who will enroll in Calculus I in college well-served by studying it first in high school?

Figure 1. Phil Sadler (left) and Gerhard Sonnert. |

To allay the suspense, their answer is a qualified “yes.” Sadler and Sonnert demonstrate that, for most students, having taken any kind of calculus in high school raises college calculus performance by about half a grade. However, they also found that the level of mastery of the high school mathematics considered preparatory for calculus varies widely. It is a far more powerful predictor of how well students will do than whether or not they have seen calculus before.

The FICSMath study had a very simple design. Questionnaires were answered in class, exploring a wide range of variables that might influence student performance in Calculus I. These included race and gender, year in which Algebra I was taken, year in college, college precalculus (if taken), career interest, parental education, high school calculus (if taken), preparation for calculus including courses taken, grades received, and SAT or ACT scores. The single dependent variable was the grade received for the course. The authors employed a hierarchical linear model. They found that about 18% of the variation in grades could be explained at the institution or instructor level. Their model enabled them to focus just on the student effect.

By far the biggest effect at the student level came from preparation for calculus. Figure 2 shows the relationship between grades earned in college calculus and grades earned in high school mathematics courses or on SAT or ACT quantitative exams. The average grade across the entire study was 80.7%, a low B–. We see that less than an A on any high school math course and less than 600 on the SAT or 26 on the ACT suggests a grade of C or less, on average, in college Calculus I. While C is a passing grade, it is a strong signal that there is considerable risk in continuing the pursuit of calculus.

The six variables indicating various aspects of mathematics preparation were combined into a “Calculus Preparation Composite Score” that was very highly correlated with the probability of taking calculus in high school (Figure 3).

Figure 3. Relationship between calculus preparation composite and probability of taking high school calculus. Source: Sadler and Sonnert 2018, page 313. |

This demonstrates the difficulty of untangling preparation for calculus from whether a student took calculus in high school. With the calculus preparation composite normalized to a mean of 0 and a standard deviation of 1, the authors found that at every level of preparation, taking calculus in high school led to an improvement in the college calculus grade (Figure 4). For students in their first year of college with an average level of preparation, the boost is 5 points, or half a grade. Intriguingly, the benefit is greatest for the students with the weakest preparation. The benefit is less for students who enroll in Calculus I after their first year in college.

Figure 4. Relationship between college calculus performance, high school preparation, taking high school calculus, and year taking calculus in college. |

In the introduction to their paper, the authors discuss how the debate over the place of calculus in high school echoes a much older and more fundamental disagreement over the extent to which mathematics is hierarchical. Does every mathematical topic have a set of prerequisites that must be mastered before any progress can be made, or can students benefit from a spiraling effect, introducing new concepts while revisiting the mathematics on which they rest?

From my experience, most mathematicians and mathematics educators recognize that spiraling is an essential part of learning. It is commonplace to assert that one never learns a subject until one has moved on to the course that builds upon it. At the same time, they acknowledge that students whose foundational knowledge is too weak will struggle as they move forward. The familiar adage is that a student does not fail calculus because they do not understand the calculus but because they have not mastered precalculus.

To the college instructor who sees students missing exam questions because of mistakes at the level of precalculus or earlier, the rapid expansion of calculus into our high schools seems a misplaced allocation of resources. And yet, requirements of prerequisite knowledge before admission to calculus that are too strict can limit access to mathematically intensive careers, especially for first generation students and those from under-resourced schools. This is compounded by the fact that, generally speaking, we do a miserable job of remediation. I documented this in “First, Do No Harm.” In this paper, Sadler and Sonnert reveal that—with other variables controlled—taking precalculus in college lowered the Calculus I grade by a small but statistically significant amount, an observation described in greater detail in Sonnert & Sadler, 2014.

We must expect that students will enter Calculus I with deficiencies that will need to be recognized and addressed within the context of the new material in this course. The rapid expansion of courses that offer expanded labs, stretched out curricula, or co-curricular offerings designed to address these deficiencies speak to the growing recognition that this is the case. What we can and should expect by way of preparation for college calculus will need to be institutionally specific, dependent on the goals of the course, the implemented curriculum, the nature of the student body, and a continuing data-based appraisal of how well current support structures and curricula are serving our students.

**References**

Sadler,
P. & Sonnert, G. (2018). The path to college calculus: the impact of high
school mathematics coursework.

*Journal for Research in Mathematics Education*. 49(3), 292–329.
Sonnert,
G. & Sadler, P.M. (2014). The impact of taking a college pre-calculus
course on students’ college calculus performance.

*International Journal of Mathematical Education in Science and Technology*, 45(8), 1188–1207.