Should Students Wait until College to Take Calculus?

By David Bressoud

You can now follow me on Twitter @dbressoud

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.

Figure 2. Relationship between grade earned in college calculus and course grade or SAT/ACT score. The symbol area is proportional to the number of students in each group. The dotted line represents the mean grade (80.7) Source: Sadler and Sonnert, 2018, page 312.

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.

Calculus as a Modeling Course at Macalester College

By David Bressoud

You can now follow me on Twitter @dbressoud

When I talk with individuals who are wrestling with improving their calculus program, I often describe calculus at Macalester. For over 15 years, we have approached the first calculus course as a modeling course, drawing inspiration from many of the early calculus reform efforts. This month’s column will look at how we came to revise Calculus I in this way, a sample of the curriculum, and thoughts on implementation.

Old Main lawn. Macalester College

Origins

The revision of Calculus I began when Professor Kaplan, then a faculty member whose research was in mathematical models of biological phenomena, looked at transcripts of students who had passed through Calculus I and II. He discovered that, although this is framed as a full-year course, few students took it as such. As was true then and still holds true, the bulk of Calculus I enrollments come from Biology and Economics majors for whom only Calculus I is required and usually only Calculus I is taken. But the traditional Calculus I does not make sense as a stand-alone course. Most of these students were learning how to find derivatives with little sense of why they were doing it. Calculus II enrollments were predominantly prospective mathematics, physics, and chemistry majors as well as the strongest economics majors. Even fifteen years ago, almost all of these students arrived at Macalester having already earned credit for Calculus I. Rather than a course that picked up two-thirds of the way through a course they had already completed, what they needed was a more intensive understanding of both differential and integral calculus.

With financial support from the administration, Kaplan began to shape the introductory courses that our biology majors most needed, a Calculus I with a focus on modeling that could stand on its own, to be followed by a statistics course that emphasized statistical modeling. The sequence that resulted has been described in "The First Year of Calculus and Statistics at Macalester College" (Flath et al, 2013) in the MAA Notes volume that I reviewed in Mathematics for the Biological Sciences (February, 2014).

We are a small college and cannot afford to offer more than one flavor of calculus. Kaplan arranged for the funding to include team-teaching these courses during the first two developmental years. This involved a large fraction of our departmental faculty in shaping these courses, ensuring both a great deal of useful feedback and a strong buy-in to Kaplan’s vision. Major efforts of outreach and explanation with the partner disciplines that required calculus eventually brought them all on board, either enthusiastically as in the case of biology and economics, or reluctantly as with physics. When the time came to decide whether we would embrace this as our only Calculus I course, the department unanimously supported it.

Curriculum 

I last taught Calculus I as a modeling course in fall, 2015. Over the years, this course has been subject to continual monitoring and adjustment. What I describe here is simply a snapshot of one moment in an evolving process, but the goals and essential elements of the course have not changed. We want students to finish the course with an appreciation for calculus as a tool for modeling dynamical systems, which means an emphasis throughout on differential equations. In addition, the most interesting and instructive dynamical systems are multi-dimensional, including SIR and predator-prey models. The course employs functions of several variables from the start. Finally, the emphasis is on numerical and qualitative analysis of these models. The procedures of differentiation and integration get less attention that in a traditional course.

No existing textbook fits the course we have built, but we used Hughes-Hallett et al. Applied Calculus (HH). In 2015, there were seven major sections to the course, described below, with indications of the relevant sections of the 5th edition. To anyone who has access to Moodle and wishes the full syllabus and supplementary materials, I can send the Moodle backup for this course.

1. Functions as Models. (6 days, HH 1.1–1.3, 1.5–1.7, 1.9–1.10, 8.1–8.2, and supplemental materials). In one sense this was a review of the functions that students should be familiar with from high school: linear, power, exponential, logarithmic, and trigonometric functions, as well as functions of two variables. But the emphasis was on the phenomena that are modeled by each of these types of functions. For exponential and logarithmic functions, attention was paid to the relationship with doubling times. For trigonometric functions, we focused on how to translate knowledge of the range and period of a periodic phenomenon into the formulation of the corresponding sine or cosine. This is also when we introduced students to the software they would be using, in our case R-Studio (chosen so that they could use the same software for the statistical modeling course).
2. Units, Dimensions, and Estimation. (3 days, supplemental materials) This is a unit that focuses on key quantitative skills that all college graduates, especially those in quantitative fields, should possess, but are never explicitly taught: understanding scale, the effect of powers of ten, how dimension affects scale, dimensional analysis as a short-cut to finding and remembering formulas, and the kind of estimation found in Fermi problems.
3. Concepts of Derivatives.  (4 days, HH 2.1–2.3, 8.3, and supplemental materials) We avoid a formal definition of the derivative in terms of limits and instead focus on what is happening to the average rate of change as the time intervals get shorter. As soon as we have explained the concept of the derivative, we extend it to partial and directional derivatives of functions of two variables.
4. Symbolic Differentiation. (5 days, HH 3.1–3.5, 8.3–8.4, and supplemental materials) This is a fairly traditional treatment of derivatives. Topics include derivatives of polynomials as well as exponential, logarithmic, and trigonometric functions, and the product, quotient, and chain rules. We spend one of these days fitting data to various kinds of models.
5. Optimization. (5 days, HH 4.1–4.3, 8.5–8.6, and supplemental materials) This section starts with traditional optimization techniques and problems, but then moves on to optimizing functions of two variables and constrained optimization problems for functions of two variables, including a very geometric explanation of Lagrange multipliers.
6. Integration and Accumulation. (7 days, HH 5.1–5.5, 6.1, 6.3, and supplemental materials) This starts with integration as accumulation, leading up to the Fundamental Theorem of Integral Calculus, 2 days of antidifferentiation as a tool for evaluating definite integrals, followed by a one-day introduction to integrals of functions of two variables.
7. Models of Change. (7 days, HH 10.1–10.7 and supplemental materials) This proceeds from a basic introduction to differential equations, through slope fields as means of visualizing solutions, exponential growth and decay, the SIR model, and predator-prey models, ending with a discussion of stability and equilibria.

Thoughts on Implementation

This variation on Calculus I will not work everywhere. It is difficult because there is no textbook that is a good fit, and we have found that faculty teaching it for the first time need a good deal of support. It also does not articulate well with the standard calculus curriculum. At Macalester, with very few students transferring in or out, this is not a problem, but it would be at public universities.

The change in Calculus I also forced major changes to Calculus II. Eventually, Macalester redesigned the entire Calculus I through III sequence to fit this image of calculus as a modeling course with single variable and multivariable functions handled simultaneously. We now call this sequence Applied Multivariable Calculus I, II, and III. This is scary for the student who thinks of multivariable calculus as the course that follows two semesters of single variable calculus, but the title provides an accurate description.

The sequence works very well for us. Learning why calculus is useful has attracted many students into further courses. It has also led to beefing up our upper division applied mathematics and statistics options. This past spring, we graduated 54 majors in mathematics or applied mathematics and statistics out of a graduating class of about 500. Next year, we expect at least 60 majors in mathematics or applied mathematics and statistics. It definitely is working for us.

Nothing communicates what is valued in a course better than how student success is assessed. For that reason, I am concluding this article with links to the exams I administered in 2015. Midterms 1 and 2 were given in class. The final exam was a take-home. In addition, students were graded on WeBWorK problems, more challenging weekly problems that required careful write-up, and Reading Reflections submitted the night before each class to ensure that students had read the relevant material before class.

• Midterm I, based on sections 1, 2, and 3
• Midterm II, based on sections 4, 5, and 6
• Final Exam, comprehensive, but emphasis on section 7

Reference

Flath, D., Halverson, T., Kaplan, D. and Saxe, K. 2013. The first year of calculus and statistics at Macalester College. pp. 39–44 in Undergraduate Mathematics for the Life Sciences: Models, Processes, and Direction. Ledder, Carpenter, and Comar, eds. MAA Notes #81. Washington, DC: Mathematical Association of America. www.maa.org/publications/ebooks/undergraduate-mathematics-for-the-life-sciences