Friday, August 31, 2018

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.


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.

Wednesday, August 1, 2018

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

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.


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.

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.

Saturday, June 30, 2018

Departmental Turnaround: The Case of San Diego State University

By David Bressoud

You can now follow me on Twitter @dbressoud

Paul Zorn and I have just published a special issue of PRIMUS on Improving the Teaching and Learning of Calculus (Bressoud & Zorn, 2018) . It contains eight articles that should be of interest to anyone who is discontented with the current state of calculus instruction at their institution. Four of these articles present case studies of universities that have made significant changes within the past few years: San Diego State University (SDSU), the University of Illinois-Chicago, Colorado State University, and the University of Hartford. The most extensive revamping occurred at San Diego State University, which is where I am focusing below.

MAA’s national study of calculus instruction, Characteristics of Successful Programs in College Calculus (CSPCC) , identified seven practices (Bressoud & Rasmussen, 2015; see the Appendix for descriptions) that we observed in the most effective programs. A few years ago, San Diego State University, facing unacceptably high failure rates and low persistence rates in its Precalculus through Calculus II sequence, decided to work on all seven areas. The result has been a dramatic improvement in these courses. Naneh Apkarian, who was a doctoral student in mathematics education within the mathematics department during this process, is the lead author on this account (Apkarian et al., 2018) .

Figure 1: The landmark Hepner Hall at San Diego State University.

With roughly 30,000 undergraduates, San Diego State University is a large public university, part of the California State University System, and chronically underfunded. It is a Hispanic-Serving Institution where 84% of students are on some form of financial aid. Science, technology, engineering, and mathematics (STEM) majors account for 10% of bachelor’s degrees. The mainstream precalculus and single variable calculus courses enroll about 1,500 students each fall. The Department of Mathematics and Statistics consists of 17 faculty in pure and applied mathematics, seven in statistics, and six in mathematics education.

Michael O’Sullivan was appointed chair of the department in 2014. He made it his mission to revamp lower-division mathematics instruction. The effort began that fall with the creation of a Calculus Task Force charged with proposing a system for coordinating the courses in the Precalculus to Calculus II sequence (P2C2). As Rasmussen and Ellis (2015) have documented, one of the most important characteristics of successful P2C2 programs is coordination of the essential elements of each course including policies, learning objectives, and exams and their scoring rubrics. Coordination also involves regular communication among those teaching different sections. At San Diego State University, total autonomy—to the point where different instructors were using different textbooks, homework systems, and even course content—had been the rule.

As the department expanded its data collection beyond simple pass rates, they discovered that only 17% of those who began with Precalculus successfully completed Calculus II, only 10% within the standard three semesters. This made mathematics faculty aware that something was seriously wrong and needed to change.

Because the discontinuation of large lectures was not financially feasible, the implementation of active learning to address this completion rate was concentrated in breakout sections led by Graduate Teaching Assistants (GTAs). The chair successfully lobbied to increase breakout sections from one to two hours per week and managed to reduce the size of most of these sections.

The chair also tied into a university initiative, Building on Excellence, to fund a new Mathematics Learning Center within the library building, directed by the office of the Dean of Science—ensuring its continued funding—but led by the department. The static 40-question placement exam was replaced by ALEKS Placement, Preparation, and Learning, with the license paid by the California State University System and student payments of $20 per proctored exam.

While these contributions were serendipitous, I have found that—particularly in situations of tightly constrained budgets—deans and provosts are keen to direct resources toward strategic initiatives with the potential for high impact. I have frequently encountered deans who asserted that if only the department would come forward with a well-thought-out and cost-effective plan for improving student outcomes, the money could be found to fund it.

As the authors reported, the effort at revision was successful because of the attention paid to opening and maintaining communication channels with stakeholders in this process (see Figure 2).

Figure 2: Significant communication channels between the mathematics department and various administrative programs as they relate to the seven targeted program features. Source: Apkarian et al. 2018, p. 540.

The result is a calculus program of which the department is justly proud, as reflected in this video. Students find the new Math Learning Center particularly helpful because its work is tightly connected to what is happening in all sections of each course.

The Department of Mathematics and Statistics at San Diego State University is a good example of how a program can be transformed. Its story illustrates the role of leadership from the department chair, buy-in and effort from a core of committed faculty, and strong two-way communication with all of the stakeholders.

  • Apkarian, N., Bowers, J., O’Sullivan, M., and Rasmussen, C. (2018). A Case study of change in the teaching and learning of Precalculus to Calculus 2: what we are doing with what we have. PRIMUS. 28:6, 528-549, DOI: 10.1080/10511970.2017.1388319
  • Bressoud, D., and Rasmussen, C. (2015). Seven characteristics of successful calculus programs. AMS Notices. 62:2, 144–146.
  • Bressoud, D. and Zorn, P. (2018). Improving the Teaching and Learning of Calculus. PRIMUS vol. 28.
  • Rasmussen, C., and Ellis, J. (2015). Calculus coordination at PhD-granting universities: more than just using the same syllabus, textbook, and final exam. In Bressoud, Mesa, and Rasmussen (Eds.), Insight and Recommendations from the MAA National Study of College Calculus. MAA Notes #84. Washington, DC: MAA Press. 

Appendix: Seven Characteristics of Successful Programs in College Calculus
  1. Local Data. Regular collection and use of local data to guide program modifications as part of continual improvement efforts.
  2. Placement. Effective procedures for placing students appropriately into their first Precalculus to Calculus II (P2C2) course (both initial placement and re-placing students after the term begins).
  3. Coordination System. A coordination system for instruction that (i) makes use of a uniform textbook and assessments and (ii) goes beyond uniform curricular elements to include regular P2C2 instructor meetings in development of de facto communities of practice.
  4. Course Content. Course content that challenges and engages students with mathematics.
  5. Active Pedagogy. The use and support of student-centered pedagogies, including active learning strategies.
  6. GTA Preparation & Development. Robust teaching development programs for teaching assistants.
  7. Student Support Service. Proactive student support services (e.g., tutoring centers, services for first-generation students) that foster students’ academic and social integration

Friday, June 1, 2018

Explosive Growth of Advanced Undergraduate Statistics

By David Bressoud

You can now follow me on Twitter @dbressoud

The 2015 CBMS Survey is now available. Last month I reported on Trends in Mathematics Majors. This month I am looking at what has happened to enrollments in particular mathematics courses. The column has three section: Enrollments by Category, where we see that the fastest growing category is Advanced Undergraduate Statistics; Calculus Enrollments, noting that the growth here is almost exclusively within the research universities where it is tied to the strong growth in engineering enrollments; and Dual Enrollment, where the story is about the dramatic increase in four-year institutions now offering dual enrollment courses.

Enrollments by Category
The first graph (Figure 1) shows strong growth in course enrollments in 4-year undergraduate programs, exceeding 2.5 million for the first time. This is certainly tied to the rampant growth in the number of prospective STEM majors (Figure 2). The number of prospective engineering majors grew from 108,000 in 2005 to 156,000 in 2010, peaking at 194,000 in 2015. Over the same period, prospective physical science majors grew from 30,000 to 40,000. Students entering with the intention of majoring in the mathematical sciences grew from 10,000 to 16,000.
Figure 1: Undergraduate enrollments by course category in mathematics and statistics departments at 4-year institutions.
Intro Level includes College Algebra and Precalculus; Calculus Level includes sophomore courses in linear algebra and differential equations.

Figure 2: Number of entering full-time first-year students at 4-year institutions intending to major in five core STEM disciplines.
Data from The American Freshman, published by the Higher Education Research Institute.

The most remarkable growth among categories of courses was for Advanced Statistics, any course beyond a first college-level statistics course, almost doubling from 60,000 in 2010 to 110,000 in 2015. This is in line with the growth in the number of Bachelor’s degrees awarded in Statistics, from 858 in 2010 to 1509 in 2015. Figure 3 shows that this growth has occurred primarily within departments of statistics, although there has also been strong growth at Bachelor’s level colleges and a remarkable turnaround in Master’s granting universities.

Figure 3: Enrollments in Advanced Undergraduate Statistics by type of department.
Departments of mathematics are characterized by the highest degree offered by the department.

Calculus Enrollments 

Calculus enrollments have also seen strong growth, driven by increases in prospective STEM majors (Figure 4). The MAA Progress through Calculus study found that for mainstream Calculus I, fall enrollments account for about 60% of all mainstream Calculus I enrollments throughout the year, while fall Calculus II enrollments account for about 40% of all Calculus II enrollments. Thus, about 550,000 students study Calculus I each year at a post-secondary institution. This compares with roughly 800,000 students who study calculus in high school each year (NCES data).

Figure 4: Fall term mainstream calculus enrollments (meaning that they lead to the usual upper division mathematical sciences courses), combined from all 2- and 4-year institutions.

Supporting the claim that most of the growth in calculus enrollments can be attributed to the growth in prospective engineering majors, Figures 5–7 show that the increase in calculus enrollments has occurred at the universities that also offer a PhD in mathematics, predominantly the large research universities.

Figure 5: Fall enrollments in mainstream Calculus I, by type of institution.
Figure 6: Fall enrollments in mainstream Calculus II, by type of institution.
Figure 7: Fall enrollments in mainstream Calculus III&IV, by type of institution.

The connection to engineering is reinforced by an interesting though not surprising observation. In 2005, I plotted the number of prospective engineering majors against the total number of students enrolled in all mainstream calculus classes (single and multi-variable) in PhD-granting departments (Figure 8). The correlation, at slightly over two students enrolled in the fall for each engineering major is remarkably tight, with a Pearson r=0.99.

Figure 8: Number of entering freshman intending to major in Engineering against total fall enrollment in all mainstream calculus (single and multi-variable).
Pearson’s r = 0.99.

The 2010 and 2015 data do not come close to fitting this line. It overestimates calculus enrollments by about 35%. Fitting a line to the data from 1995 to 2015 yields the graph in Figure 9. The multiplier effect of each prospective engineer has dropped to a little over 1, evidence that whereas an engineering major would, in the past, study single or multi-variable calculus in two fall terms, they now usually take calculus in only one fall term.

Figure 9: Number of entering freshman intending to major in Engineering against total fall enrollment in all mainstream calculus (single and multi-variable).
Pearson’s r = 0.97.

Dual Enrollment

CBMS began tracking dual enrollment in 2005, courses offered by a 2- or 4-year college, taught in a high school by a high school teacher, but carrying both high school and college credit. In 2005, 50% of 2-year departments, but only 14% of 4-year departments offered dual enrollment courses in mathematics. By 2015, these percentages had climbed to 63% at 2-year institutions and 26% at 4-year institutions. We conclude this column with Figures 10 and 11, showing the number of fall enrollments in the four most common dual enrollment courses: College Algebra, Precalculus, Calculus I, and Statistics.

Figure 10: Fall term dual enrollment at 2-year colleges.
Figure 11: Fall term dual enrollment at 4-year institutions.

Tuesday, May 1, 2018

Trends in Mathematics Majors

By David Bressoud 

You can now follow me on Twitter @dbressoud

By the time this column appears, the full CBMS 2015 survey of math departments should be available at I reported some of the data on faculty demographics in my October and November Launchings columns. This month I want to report on what is happening to undergraduate mathematics majors.

From 2010 to 2015, the number of bachelor’s degrees in the mathematical sciences grew by just over 3,000, from 19,242 to 22,265, almost a 16% increase (Figure 1). However, most of the growth was in Actuarial Science (from 849 to 2354), Statistics (from 858 to 1509), joint majors (e.g. biomath, the total rising from 1222 to 1821), and “other” (including Operations Research, from 231 to 907). Degrees in Mathematics Education fell from 3,614 to 2,880. Traditional mathematics and applied mathematics degrees only rose by 326, from 12,468 to 12,794.

Figure 1. Bachelor’s degrees awarded by departments of Mathematics or Statistics.
Source: CBMS Surveys.

For comparison, the total number of Bachelor’s degrees over the years 2010 to 2015 increased by 15%, and the number of degrees in STEM fields (specifically bioscience, computer science, engineering, mathematical sciences, or physical sciences) rose by 34%, from 238,000 to 319,000.

The period 2010 to 2015 saw a decrease in the percentage of Bachelor’s degrees in Mathematics or Statistics earned by women, dropping from 42.4% to 40.8% (Figure 2). This does not include degrees in Mathematics Education awarded by Math departments. If we include them, then women earned 43.3% of the Bachelor’s degrees in 2015.

 Figure 2. Women as % of Mathematics or Statistics Bachelor’s degrees, organized by highest degree offered by the mathematics department. Source: CBMS Surveys.

Figure 3 shows the representation of African-Americans, Hispanic-American, Asian-Americans (including Pacific Islanders), and nonresident aliens. Here we are drawing on data from the National Center for Education Statistics (NCES), which is collected annually. Two trends are particularly interesting: the number of African-Americans has remained pretty much unchanged since the mid-1990s, and the number of nonresident aliens has exploded since 2007. It should be noted that NCES began allowing the designation “two or more races” in 2011. In 2011, 216 Mathematics or Statistics majors chose this designation, growing to 684 in 2016. These numbers are not reflected in Figure 3.

 Figure 3. Number of Mathematics or Statistics majors by race, ethnicity, or resident status.
Source: NCES.

The following graphs, Figures 4–7, look closer at each of these four groups, comparing their percentage of all Bachelor’s degrees, of Bachelor’s degrees in Mathematics or Statistics, and of Bachelor’s degrees in Engineering. Again, these do not include students who designated as two or more races after 2010. We see that until 2000, African Americans were well represented among Mathematics majors in the sense that their representation was comparable to their representation among all undergraduates, but since then their percentage has noticeably dropped off. Hispanic Americans are underrepresented, but the trend is promising. Not surprisingly, Asian Americans are well represented among Mathematics and Engineering majors. Non-resident aliens are growing as a percentage of all Bachelor’s degrees and all Engineering degrees, but their growth among Mathematics majors is remarkable. This attests to the importance of student visas in maintaining our mathematical workforce, but it also suggests that more could be done to attract U.S. citizens to the pursuit of Mathematics, especially African Americans.

Figure 4. African Americans as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

Figure 5. Hispanic Americans as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

Figure 6. Asian Americans and Pacific Islanders as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

 Figure 7. Non-resident aliens as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

Sunday, April 1, 2018

Gaps in Student Understanding of the Fundamental Theorem of Integral Calculus

By David Bressoud

You can now follow me on Twitter @dbressoud

I have long held the belief (Bressoud, 2011) that we should revert to the original name, the Fundamental Theorem of Integral Calculus (FTIC), for what in the 1960s came to be known as the Fundamental Theorem of Calculus (FTC). The reason is that the real importance of this theorem is not that integration and differentiation are inverse processes—for most students that is the working definition of integration—but that we have two very distinct ways of viewing integration, as limits of Riemann sums and in terms of anti-differentiation, and that for all practical purposes they are equivalent.

Figure 1. Students working on integral as accumulator, reproduced from the homepage of  Coherent Labs to Enhance Accessible and Rigorous Calculus Instruction (CLEAR Calculus

A recent paper by Joseph Wagner (2017) is an insightful study of the confusion experienced by most students about the nature of integration. As he points out, this is not about student deficits, but about common misconceptions that can be traced to the way we teach integration.

Previous work by Sealey (2006, 2014) and Jones (2013, 2015a, 2015b) has shown that
there are three ways in which students describe the meaning of the definite integral,

  •     as an area,
  •     in terms of an antiderivative, or
  •     in terms of a summation.
Overwhelmingly, students employ the first, the second is common, the third is rare.
Nevertheless, when confronted with a problem in physics that requires integration, the interpretation in terms of a summation is more common. Jones (2015b), after reminding second term calculus students that force is pressure times area, asked whycalculates the total force. Of 150 students, 61 (41%) produced an argument that involved summation, although only 25 of them (17%) indicated that any product was involved.

Following up on this insight, Wagner explored the understanding of definite integrals by physics students. He interviewed eight students in an introductory calculus-based physics course focused on classical mechanics and seven third-year physics majors. Of the students in the introductory course, five had completed both single and multi-variable calculus, two were currently enrolled in multi-variable calculus, and one was still in single variable calculus. All were in majors that required this physics course.

When students in the introductory course were asked what Riemann sums have to do with definite integrals, they split evenly between two types of answers: either as something that accomplishes the same task as an integral (usually finding areas) or as a means of approximating definite integrals. As we shall see, the connection between integration as a limit of Riemann sums and in terms of antiderivatives was hazy at best and not recognized as significant. As Wagner reports, several were mystified why they had to learn about Riemann sums, “Because like when they were teaching this, they were kind of like oh, like you’ll do this for the first test, and then you get rid of it and never have to do it again.”

On the other hand, the third-year physics students were much more inclined to explain the meaning of the definite integral in terms of a summation. They were conversant with how to convert an accumulation problem into a definite integral. As Wagner suggested privately, this appears to be the result of repeated exposure to problems from physics in which definite integrals arise from “slice and add” procedures.

But Wagner uncovered an intriguing gap in their understanding. All fifteen students were asked to make up a simple area problem and then solve it. All of them did so correctly, using a polynomial function and antidifferentiation. As an example the area under the graph of y=x^3 from 0 to 2 was calculated as follows,

He then pushed each of these students to explain why this sequence of calculations produced the area. Only one of the fifteen, a third-year physics student, indicated that this was a consequence of FTC. Several of the others struggled to make sense of how the symbols in the definite integral led to the functional transformation implied by the first equality. Wagner argues that many students are looking for algebraic sense-making in that first equality. With two of the third-year students, he documented their growing sense of frustration as they realized that they could not explain why it works. Quoting the first student:

"Yeah, I do it. I don’t–. I’m not proud of it, but I hope there is some way to justify it. […] When I think about integration as a sum of differentials, quantities–. When I think about that, I go, OK, that makes intuitive sense, and it works. Great. But then I wonder, you know, what is, in terms of more modernized math that I’m doing. Because I usually feel like what I’m doing is kind of a trick. And it works. I don’t feel great about doing this, like, intuitively I feel fine."

From the second student:

"So math gives us these sort of weird tools, and they behave differently than any, like, the physical tools we know of, and it doesn’t really make sense to ask why they work or how they work, because they work mathematically, not physically. So this mathematical tool called the integral allows us to change functions, to apply this operation that changes functions into other functions."

Wagner concludes this article with a thoughtful discussion of the distinction between the algebraic equivalence of two expressions, a notion of equivalence with which students are familiar, and the transformational equivalence that is enabled by FTC. As he laments, “Nothing, however, in the standard calculus curriculum prepares students for the sudden transition from making sense of the symbolic processes of algebra to making sense of the symbolic processes of calculus.” He points out that a great deal of attention has been devoted to a Riemann-sum based understanding of the definite integral, but virtually none to helping students understand the transformational aspects of calculus that are so central.

I believe that a shift from FTC to FTIC can help. As Thompson with others (2008, 2013, 2016) has shown, and I have discussed in earlier columns (Re-imagining the Calculus Curriculum, I, and Re-imagining the Calculus Curriculum, II), it makes sense to first develop the definite integral as an accumulator, making it very clear that Riemann sums are neither an introduction to a subject that eventually will be about antiderivatives nor just a tool for finding approximations, but the very essence of what a definite integral is and how it is used. Then, we bring in FTIC to show that there is another—entirely distinct because it is transformational—expression for this same integral and that this equivalent expression facilitates calculation.  Wagner’s third-year physics students were struggling because they failed to realize that integration has these two very different manifestations. It is a very big deal that it does.


Bressoud, D. (2011). Historical reflections on teaching the Fundamental Theorem of Integral Calculus. American Mathematical Monthly. 118:99–115. - page_scan_tab_contents

Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141.

Jones, S. R. (2015a). Areas, anti-derivatives, and adding up pieces: Integrals in pure mathematics and applied contexts. The Journal of Mathematical Behavior, 38(1), 9–28.

Jones, S. R. (2015b). The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann sum-based conceptions in students’ explanations of definite integrals. International Journal of Mathematics Education in Science and Technology, 46(5), 721–736.

Sealey, V. (2006). Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient. In S. Alatorre, J. L. Cortina, M. Sáiz & A. Méndez (Eds.) Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 46-53). Mérida: Universidad Pedagógica Nacional.

Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245.

Thompson, P.W., and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (MAA Notes Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America.

Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools. 30:124–147.

Thompson, P.W., and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline (pp. 355–359 ) Hannover, Germany: KHDM.

Wagner, J.F. (2017). Students’ obstacles to using Riemann sum interpretations of the definite integral. International Journal of Research in Undergraduate Mathematics Education.

Thursday, March 1, 2018

A False Dichotomy: Lecture vs. Active Learning

By David Bressoud

You can now follow me on Twitter @dbressoud

On January 31, I published a piece in The Conversation, “Why Colleges Must Change How They Teach Calculus.” The following is one of the statements that I made in this article:

Active learning does not mean ban all lectures. A lecture is still the most effective means for conveying a great deal of information in a short amount of time. But the most useful lectures come in short bursts when students are primed with a need and desire to know the information.

There is no simple binary choice between an active learning classroom and straight lecture. Furthermore, making a class an effective locus for student learning requires more than just active learning. 

An article by Campbell et al. (2017), “From Comprehensive to Singular: A latent class analysis of college teaching practices,” reports on an interesting study of what happens in college classes (not just STEM classes), adding a few layers of complexity that are useful for anyone thinking about how to be a more effective teacher. The authors observed 587 courses in nine colleges and universities, ranging from Research 1 (public and private) to comprehensive state schools to liberal arts colleges at a range of levels of selectivity. They looked for seven types of activities in the classroom.

One of these is lecture, defined as “A presentation or recitation of course content by the faculty member to all students in the class.”

They split active learning into three sub-categories:

  • Class discussion. Back and forth conversation between instructor and students or among students about the course content.
  • Class activities. A structured activity where students engaged with the course content (e.g., case studies, clickers, group work).
  • Student questions. Students asking individual questions of the instructor about the course content.

They also picked up the three practices laid out in Neumann’s (2014) description of cognitively response teaching. Active teaching should be cognitively responsive. Unfortunately, as their observations showed, it often is not. These three practices are:

  • Core subject matter ideas. The instructor introduced in depth one or more concepts that are central to the subject matter of the course, the instructor created multiple representations of “core ideas,” or the instructor introduced students to how ideas play out in the field.
  • Connections to prior knowledge. The instructor surfaced students’ prior knowledge about the subject “core ideas,” or the instructor worked to understand students’ prior knowledge about the subject matter “core ideas.”
  • Support of changing views. The instructor provided a space for students to encounter dissonance between prior knowledge and new course material, or the instructor helped students to realize the difference similarities and sometimes conflict between prior knowledge and new subject matter ideas.

Developing over the past few decades and now accelerating thanks to the work of the community engaged in research in undergraduate mathematics education, there have been remarkable strides in understanding the misconceptions that are barriers to student learning. To cite just two examples that I have discussed elsewhere, students often have difficulty making the transition from trigonometric functions in terms of triangles to the circle definition, and they tend to interpret functions as static objects, impeding an understanding of them as descriptions of the linkage between variables that vary. I discussed this issue of the disconnection between what we say and what students hear in two columns in 2016: What we say/what they hear and What we say/what they hear II. The instructor who does not try to understand the prior conceptions and knowledge that students bring into the classroom is setting a large proportion of the students up for failure.

For the last practice, support of changing views, the physics education community knows how important this is. With their Force Concept Inventory (FCI), Halloun, Hestenes, and Wells (see Hestenes et al., 1992) demonstrated that prior concepts are powerful. Students are reluctant to release them, even in the face of what instructors consider to be clear exposition of the actual state of affairs. Getting students to recognize cognitive dissonance requires skill.

Campbell et al. observed that traditional lecture—what the Progress through Calculus study (Apkarian and Kirin, 2017) has revealed to be standard practice in 72% of all Calculus I classes in university mathematics departments with PhD programs—did a pretty good job on core subject matter ideas, but almost nothing with connections to prior knowledge or support of changing views. And, of course, traditional lecture involved none of the first two active learning sub-categories. Less obvious but not surprising, student questions were seldom observed in traditional lecture.

Active lecture is the second most common form of calculus instruction, found in about 14% of the PhD-granting mathematics departments we surveyed in progress through Calculus (3% of departments relied mainly on active learning practices in the classroom and the remaining departments reported too much variation by instructor to classify their course as one type). These introduced class activities and did not decrease core subject matter ideas. Campbell et al. found that they noticeably increase student questions, but do nothing in and of themselves to improve connections to prior knowledge or support of changing views.

These last two practices were almost never observed in either traditional or active lecture classes. The only classes that were observed to improve these aspects of cognitively responsive teaching were those that made a point of employing all seven behaviors, including lecture. In other words, connections to prior knowledge and support of changing views do not come for free once one is using active learning. They have to be intentionally incorporated, and they rely heavily on carefully guided class discussion.

The lesson is that lecture has its place, and active learning is only one piece of what is needed for a truly effective class. David Hestenes (1998) summed it up nicely in “Who needs physics education research!?”:

Managing the quality of classroom discourse is the single most important factor in teaching with interactive engagement methods. This factor accounts for wide differences in class FCI score among teachers using the same curriculum materials and purportedly the same teaching methods. Effective discourse management requires careful planning and preparation as well as skill and experience … Effective teaching requires complex skills which take years to develop. Technical knowledge about teaching and learning is as essential as subject content knowledge.

 Apkarian, N. and Kirin, D. 2017. Progress through Calculus: Census Survey Report. Technical Report_Final.pdf

Bressoud, D. 2016. What we say/What they hear. Launchings.

Bressoud, D. 2016. What we say/What they hear. II. Launchings.

Bressoud, D. 2018. Why colleges must change how they teach calculus. TheConversation. January 31, 2018.

Campbell, C.M., Cabrera, A.F., Michel, J.O., and Patel, S. 2017. From Comprehensive to Singular: A Latent Class Analysis of College Teaching Practices. Research in Higher Education. 58: 581–604.

Hestenes D., Wells M., Swackhamer G. 1992. Force concept inventory. The Physics Teacher 30: 141-166.

Hestenes D. 1998. Who needs physics education research!?. Am. J. Phys. 66:46.5.

Mathematical Association of America. 2017. Instructional Practices Guide. resources/instructional-practices-guide

Neumann, A. 2014. Staking a claim on learning: What we should know about learning in higher education and why. The Review of higher Education. 37:249–267. Presidents/37.2.neumann.pdf