In spring 2015 the MAA’s Progress through Calculus (PtC) grant (NSF#1430540) surveyed all U.S. Departments of Mathematics that offer a graduate degree in Mathematics to learn about departmental practices, priorities, and concerns with respect to their mainstream courses in precalculus through single variable calculus. I have reported on some of the results from this study in November, 2015. This month’s column describes a variety of data relative to mainstream Calculus I that were collected in that survey. The full report can be found under PtC Reports (link from maa.org/cspcc).
The survey was sent to the chairs of all departments of mathematics in the United States that offer a graduate degree in Mathematics (PhD or Master’s). We received responses from 134 of the 178 PhD-granting universities (75%) and 89 of the 152 Master’s-granting universities (59%).
Given how ineffective the standard precalculus course is known to be (see my Launchings column from October, 2014), we were particularly interested in efforts to teach precalculus topics concurrently with calculus. Accomplishing this through a stretched-out Calculus I is now fairly common (20 of 222 respondents use this approach to incorporate precalculus topics into Calculus I). Eleven universities have courses or options with extra hours to allow time on precalculus, and three offer precalculus courses designed to be taken concurrently with Calculus I. We also found 14 universities with an accelerated calculus specifically designed to meet the needs of students entering with AP® Calculus credit. Three universities have special lower credit courses that enable students who begin in a non-mainstream Calculus I to transition to mainstream calculus.
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| Table 1: Number of surveyed universities that reported using each of the listed variations in single variable calculus classes. |
Every five years, CBMS surveys departments of mathematics in the U.S. to get enrollment numbers, but those are only gathered for the fall term. In this survey, we were particularly interested in how these numbers vary over the full year, both academic and summer terms. While we only have results for a sample of universities, and no undergraduate colleges, the numbers are large enough, 150,000 in Precalculus, 200,000 in mainstream Calculus I, and 160,000 in subsequent mainstream single variable classes, to get a good idea of how these enrollments distribute over the year. For Precalculus, 57% of the enrollment occurs in the fall term. Fall term accounts for 60% of the Calculus I students. Not surprisingly, Calculus II is predominantly a second-term course (47%), but 40% of the students who take Calculus II do so in the fall. The distribution among the terms is complicated by the fact that some universities are on a quarter system, others on semesters. What I have labeled 2nd Term, is either spring semester or winter quarter. The 3rd Term refers to the spring quarter for those on a quarter system. Summer aggregates all summer terms. Figure 1 shows actual numbers from the universities that responded to give an idea of how enrollments drop off. For the purposes of the survey, “Precalculus” was defined as the last course before mainstream Calculus I. It is variously called Precalculus, College Algebra, College Algebra with Trigonometry, or Preparation for Calculus. Calculus II includes all mainstream single variable calculus courses that follow Calculus I. On a semester system, there is usually just one. On a quarter system, there usually are two such courses.
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| Figure 1: Distribution of enrollments by term among the 205 universities that respond to this question. 2nd term = spring semester or winter quarter. 3rd term = spring quarter. Calculus II includes all mainstream single variable calculus classes that follow Calculus I. |
The number of contact hours (including recitation sections) in Calculus I averaged 4.17 (SD = 0.77) at PhD-granting universities and 4.25 (SD = 0.64) at Masters-granting universities. The DFW rate in mainstream Calculus I was 21% (SD =12.2), at PhD-granting universities and 25% (SD = 13.7) at Masters-granting universities.
The next table (Table 2) reports the fraction of universities in which Calculus I is frequently taught by each type of instructor. For each category of instructor, the options were “Never,” “Rarely,” or “Frequently.”
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| Table 2: Percentage of universities for which each category of instructor frequently teaches mainstream Calculus I. |
Recitation sections were far more common at PhD-granting universities. All classes have recitation sections for 49% of the institutions, some classes at 6%, and there are no recitation sections at 45% of the universities. For Masters-granting universities, the percentages were 18% for all classes, 6% for some classes, and 76% for no classes.
We also found that active learning was much more common at Masters-granting universities than PhD-granting universities. Figures 2 and 3 record primary instructional format for mainstream Calculus I. “Some active learning” includes techniques such as use of clickers or think-pair-share. “Minimal lecture” includes Inquiry Based Learning and flipped classes. “Other” usually means too much variation to be able to identify a primary instructional format. We did find that 35% of the PhD-granting universities did report having at least some sections that were using active learning approaches.
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| Figure 2. Primary instructional format for regular classes (not recitation sections) at 214 PhD-granting universities. |
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| Figure 3. Primary instructional format for regular classes (not recitation sections) at 109 Masters-granting universities. |
At 73% of the PhD-granting universities and 74% of the Masters-granting universities that offer recitation sections, they are simply homework help, Q&A, and review. Recitation sections are built around active learning approaches 21% of the time at PhD-granting universities, 4% of the time at Masters-granting universities.
Table 3 reports which elements of mainstream Calculus I are common across all sections. We see much more uniformity at PhD-granting universities. In view of our findings from the earlier Characteristics of Successful Programs in College Calculus that coordination of course elements was one of the significant factors of successful calculus programs (see my Launchings column from January 2014), the results of this study suggest a great deal of room for improvement.
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| Table 3: Percentage of reporting universities that have these elements across all sections of mainstream Calculus I. |
Another aspect of coordination that was characteristic of the most successful programs was the practice of regular meetings of the course instructors. As shown in Table 4, there is also a great deal of room for improvement here.
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| Table 4: Response to "When several instructors are teaching in the same term, how often do they typically meet as a group to discuss the course?" |
The situation at PhD-granting universities is disappointing. The primary means of instruction is still large lecture with few or no structured opportunities for students to reflect on what is being presented to them, supplemented by recitation sections in which graduate students simply go over homework and answer student questions. At the Masters-granting universities, where classes are smaller and there is more emphasis on teaching, there is little coordination, often resulting in highly variable instruction. But there is room for hope. While there is no previous study with comparable data, there appears to be good deal of experimentation. My own experience in visiting these predominantly large public universities is that they are aware that what they are doing is not working, and they are looking for ways to improve what happens in this critical sequence.
