Towards a Forward-Thinking College Calculus Program

2.1 Brief history of calculus

The invention of calculus is traditionally given shared credit to Isaac Newton and Gottfried Wilhelm Leibniz, who each independently developed the theories around infinitesimal calculus in the late seventeenth century. Bressoud points out that their contributions to calculus lie in connecting theories related to differentiation and integration, rather than in developing the theories of the individual ideas [2]. Newton, primarily a physicist, was motivated to pursue calculus in order to provide a scientific description of motion and magnitude. When documenting his ideas related to calculus, he did so primarily for himself, using a mixture of notations that made sense to him. Leibniz, described as a polymath or someone with a wide array of knowledge akin to a renaissance man (specifically, his interests included metaphysics, law, economics, politics, logic, and mathematics), was motivated to pursue calculus in order to provide a metaphysical explanation of change. He purposefully developed a clear and consistent notation system to document his work that we still essentially use today. While these two men came to be interested in calculus for very different reasons (one from more applied motivations and one coming from more pure mathematical interests), today they continue to share the honor of being credited with this field.

Although Newton and Leibniz are credited with the invention (or discovery, depending on your scientific philosophy), many mathematicians came before them, to develop the ideas they built on, and after them to refine their ideas [3]. Even as recently as 2014, researchers are challenging the credit given to Newton and Leibniz by finding evidence that other mathematicians developed the formal ideas of calculus long before the 1670’s. For example, a group of mathematicians in Southern India from the Kerala School developed and published work on many fundamental ideas of infinitesimal calculus 300 years prior to Leibniz and Newton [4]. With this note aside, college calculus today is a direct descendent of the work of Newton and Leibniz, using Leibniz’s notation, and so I consider these as the birthplace of the college calculus we still see today.

2.2 Evolution of calculus education

So how did calculus come to hold the place as the integral (pun intended) component of so many students’ college educations? To answer this question, I draw significantly from Alan Tucker’s “History of the undergraduate program in mathematics in the United States” [5]. In the 1700s and 1800s, mathematics was studied as one of the main topics (along with Latin, Greek, and Hebrew) in college following the English college model. The goal of mathematics in such an education was as a “classical training of the mind instead of the language of science and engineering it is today” ([5], p. 689). The students attending these colleges were mostly male and mostly from the upper-class. Although Newton’s and Leibniz’s work developing calculus into a more systematized and valued field occurred in the late 1600s, calculus wasn’t taught widely in college until the late 1800s. It was during this time period that colleges shifted from delivering a classical curriculum to a more practical curriculum. This is largely due to the fact that land-grant public universities were established in 1862 by the Morrill Act, and calculus became more standard for technically-oriented students.

By the early 1900s, most colleges allowed students to choose the courses for their study, which led to an increase in college enrollment and a decrease in mathematics study. It was during this time that mathematics was no longer viewed as part of a classic education, and instead a tool useful for engineers and scientists (as it continues to be seen today). During this time, calculus became an elective in US college preparatory high-schools and a mandatory subject for college preparatory high schools in Europe. Also around this time, the presentation of calculus changed from following the order of topics in which the theory was developed (first integration, then differentiation, then series, and lastly limits), to the order most beneficial for rigorously proving theorems of calculus (first limits, then differentiation, then limits, and lastly series) [3]. Little has changed since this time. The undergraduate mathematics curriculum for mathematics majors has become more and more solidified, aided by guidance from the Mathematical Association of America and their Committee on the Undergraduate Program in Mathematics (CUPM) guide. Tucker notes that in the more recent history, though the curriculum has not changed much, “the greatest area of change and concern in the past 40 years has been the articulation between high school and college mathematics.” By the early 1980s, “mathematics faculty were dealing with large numbers of students in freshman courses who showed limited knowledge of needed algebra skills” ([5], p. 702). He attributes these changes to states increasing their high school mathematics requirements and watering down the material to meet these higher expectations, as well as the rise of high stakes testing. Near this time, AP Calculus became an increasingly expected course in high school. During the late 1990s (during which time I was in high school and also the earliest data recorded online of AP participation by subject), approximately 100,000 high school students took the AP Calculus AB exam [6]. From my experience, AP Calculus AB was viewed as a course that only students extremely interested in pursuing mathematics or physics as their college major would take. By 2018 that number had tripled [7]. As part of a large, national study on college calculus conducted in 2010, my research team identified that two-thirds of all students in a college calculus class had already taken a course in high school called calculus (many of these being AP Calculus AB or BC), and half of the students we surveyed believed they needed to take a calculus course in high school to be successful in college [8].

Over the past nearly 50 years, there has been tremendous attention paid to reforming college calculus, which has resulted in more attention to problem solving in applied contexts, an increased focus on supporting students’ development of conceptual understanding rather than only procedural fluency, and often more active learning techniques employed in the classroom, including student-centered instruction and more technology use [1, 9, 10]. These changes have certainly resulted in more variation in the college calculus instruction across the country, with some programs very much still rooted in these reforms and others holding on to a pre-reform calculus model. That said, the basic content being taught in all of these programs is still essentially a course on Newton & Leibniz’s ideas, taught in the order best suited for proving calculus, regardless of the presentation, the students being taught, the pedagogy, or the contexts for the word problems.

2.3 Current state of college calculus education

For the past decade, I have been a part of a large research team studying college calculus. This research team has been led by David Bressoud, run under the auspices of the Mathematical Association of America, and funded by the National Science Foundation. Our research has come from two projects, the first begun in 2009 and focused on mainstream college differential calculus programs (typically called Calculus I) in all institution types, called Characteristics of College Calculus (CSPCC);the second begun in 2014 and focused on precalculus, differential and integral calculus programs at Masters and PhD-granting institutions, called Progress through Calculus (PtC).Our work has been generally focused on identifying aspects of college calculus programs that are more successful or innovative than comparative institutions, and supporting more mathematics departments to improve their programs based on these findings. For our purposes, success in college calculus is primarily marked by a large percentage of the students who plan to complete both differential and integral calculus (typically called Calculus I and Calculus II; requirements of most STEM-degrees) reporting that their confidence, interest, and enjoyment of mathematics did not decrease after the first course in calculus, and that these students primarily planned to continue studying calculus (and thus continue studying STEM) after taking the first course¹.

Overall, based on these measures, we did not see great evidence of success in college calculus across the country. Among the students surveyed, we saw significant decreases in confidence, enjoyment, and interest in continuing to study mathematics [8], and we found that nearly 18% switched out of the calculus sequence after taking differential calculus [12]. The main reasons for switching out of the calculus sequence given by students were changing their majors and no longer needing to finish the sequence, not having the time and effort to put into calculus to do well, and having a negative experience in differential calculus. Women students switched out at significantly higher rates than men, and disproportionately credited a lack of confidence in their mathematical abilities as the reason why [12].

From the 213 schools that participated in the CSPCC survey, we identified 18 schools that showed promise, including community colleges, Bachelor’s-granting, Master’s-granting, and PhD-granting schools. We conducted case studies at these sites, and based off of these case-studies have identified a number of components of calculus programs potentially related to student success. A collection of these findings can be found on our project website, www.maa.org/cspcc. For this paper, I focus on the findings that have had the most direct impact on the follow up study, PtC. From the five doctoral-granting departments we visited, we identified seven features that were common and that we believed were related to their success [13]. These features are: a coordinated calculus program, collection and use of local data to inform changes to the calculus program, rich and engaging curriculum, support of active learning, teaching preparation of the graduate students involved in the program, tutoring centers and other supports available for students, and adaptive placement systems into the calculus program. Since publishing those findings, we have seen a number of calculus programs across the country use these findings to guide improvements to their own programs, showing the impact that such studies can have on shifting the national landscape of calculus education.

I am confident these features provide concrete aspects of calculus programs that departments can focus their improvement efforts on, and that these are likely to lead to some improvements. However, I have recently argued [14] that it is also likely that focusing on these aspects alone can lead to programs making improvements that better serve the populations of students already being supported through calculus programs. In Table 1, I provide demographic data of the students earning Bachelor’s degrees in any major, and specifically in STEM, from each school near the time of our data collection in 2010 from the five universities visited.

Table 1 highlights the low population of students of color at the institutions visited, and the lack of STEM degrees earned by women of all ethno racial backgrounds and students of color of both sexes. The percentage of students who switched out of the calculus sequence at these institutions varied drastically by institution, from as low as 2% at one technical institutions to 30% at one large, public. However, the trends of women students switching at higher percentages than men and low enrollment by students of color are common across each of these sites. A deficit-oriented interpretation of this data would argue that these differences in interest, success, and persistence by different student populations are due to internal deficiencies of some populations of students, playing into common stereotypes of women and students of color not being as good at or as interested in STEM as white, Asian, and male students [15].

An anti-deficit interpretation rejects this assumption and rather assumes (1) that women and students of color can thrive in STEM, (2) that the disproportionate enrollment of white and Asian students, and disproportionate persistence of men indicate a failure of the system and not of the students, and (3) we can learn how to better support women and students of color to thrive in STEM by studying the women and students of color who are already thriving in STEM [16].

The enrollment and persistence data indicates that the five schools we visited, and that we based our “features of successful calculus programs” (which have come to shape improvements to calculus programs across the country) were based on programs serving a predominantly white and Asian, male student population. That is, the demographics of the students in these calculus programs were predominantly white and Asian men and women students, and the students persisting through the sequence were disproportionately men of all races and ethnicities. Knowing this information and coming from an anti-deficit perspective, I argue that the seven features can only offer possible improvements for calculus programs when considered in conjunction with diversity, equity, and inclusion practices. Diversity practices refer to actions done within the calculus program and mathematics department that attract and retain a diverse population of students. Equity practices refer to actions that (1) acknowledge the multiple ways in which some people face barriers (both visible and invisible) to their success, and (2) work to dismantle these barriers [17, 18]. Inclusion practices refer to actions that support the full participation of a diverse student population within the classroom community and within the broader departmental and institutional communities. By focusing on the original seven characteristics alone, departments may foster inequities by further supporting the populations of students who are already successful in calculus. Instead, departments should explicitly implement diversity, equity, and inclusion practices while also improving their programs through focus on the seven characteristics.

As a follow-up project to CSPCC, the PtC project has identified 12 research-oriented mathematics departments implementing a combination of the seven features in the Precalculus and calculus programs. For the PtC project, we used IPEDS data to very purposefully consider the demographics of the students enrolled at the schools, and the demographics of the students graduating with STEM degrees, when selecting the 12 institutions involved in our study. This resulted in a number of institutions serving a more racially and ethnically diverse student population, and with a few of those institutions implementing programs specifically designed to support women and/or students of color and/or first-generation students to be successful in STEM. This work is ongoing, and we are in the process of learning more about these programs so that we can share more about them with others schools. One disappointing finding in our recent work has been the general lack of programs geared to increasing the diversity in STEM among research-oriented math departments across the country; while there were programs at the university and college level developed to foster diversity, equity, and inclusion in STEM, there were very few programs at the department level [19].

3.1 Background on DIRACC

Project DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus)is an NSF-funded college calculus curriculum developed by Pat Thompson and his colleagues based on years of research on student understandings of calculus (see [21] for a description). This curriculum is self-described as “Newton meets Technology”, focusing on developing meaning for infinitesimals (while utilizing animations and interactive apps) rather than emphasizing the notation and formality of Leibniz. This curriculum is shared online for free, and is currently being implemented in at least two large, public, doctoral granting mathematics departments, including one involved in Progress through Calculus.

In this chapter, I will draw on my experience at the one university involved in PtC (referred to as Large State University; LSU), where DIRACC is the curriculum used for all calculus courses for science, computer science, and mathematics majors. The undergraduate population of LSU is approximately 50% white students, 20% Hispanic and Latinx students, 7% Asian, and 5% Black and African American. In the DIRACC calculus courses I observed, I estimated that approximately 30% of students were Black, Latinx, and/or Native (based on appearance). At LSU, there is a separate (and more procedurally oriented) college calculus course for engineering majors. The DIRACC courses are taught by instructors, mathematics education faculty, and doctoral students pursuing degrees in pure and applied mathematics and mathematics education. This course is coordinated by a full-time instructor, and this coordination includes weekly meetings for all instructors, where the topics of discussion during the meetings include understanding the mathematics and student thinking related to the mathematics for the upcoming section. Preliminary results from PtC indicate that students in DIRACC outperform students at comparable universities on a calculus content assessment and maintain positive beliefs towards mathematics more than students at other institutions.

3.2 Shift in curriculum

To best serve the students in our calculus classes, we need to learn what is motivating them to pursue degrees requiring calculus – whether future career goals or general interest in learning – and rethink our calculus curriculum to be in line with these interests. It is well established that in today’s economy, STEM jobs pay significantly more, on average, than non-STEM jobs [22]. Given this widespread knowledge, we cannot ignore that one contributing motivation for students to pursue STEM is future job and wage prospects. When sitting in Calculus I classes across the country, it often seems that everyone knows the students are there not to learn deep and interesting mathematics, but to get a grade in the course that allows them to continue pursing whatever STEM degree they are hoping for in order to get a good job. I believe that we are missing a big opportunity in our calculus classes to inspire these STEM-intending students about the magic and beauty of calculus. The great majority of calculus courses I have visited have been “mainstream” courses, meaning to serve all STEM students, although in actuality the great majority of the content is driven by the needs of the engineering students, with occasional word problems being set in other contexts.

In a forward-thinking calculus system, there would be a meaningful connection between the content taught in calculus, the needs of the majors whose students are taking calculus, and the interests and motivations of the students enrolled in our courses. It would be these latter two driving the content, rather than historical precedents. The DIRACC curriculum achieves this by forgoing Leibniz’s precise notation in favor of Newton’s more intuitive ideas – skipping the formalities of ideas such as limit to spend more time supporting students to understand the ideas of infinitesimals and how this can support meaningful understanding of rate of change functions and accumulation functions. This curriculum was designed explicitly to support students in developing rich mathematical meanings, and is thus inherently responsive to how students think about calculus and what todays’ students should be learning in a calculus course. As currently taught, I witnessed this curriculum equitably engaging a racially diverse student population in rich mathematics. This curriculum could go further in the future by engaging the diverse learners as whole people, by situating the mathematical content in contexts that are especially interesting and relevant for them (where these contexts could be identified by talking to students and using local data to identify trends in women and students of color’s majors).

3.3 Shift in pedagogy

Through PtC, I observed three DIRACC calculus courses at LSU, and though the three courses looked different, in each I witnessed a racially diverse group of students equitably engaging in rich mathematics, contributing to constructing mathematical meaning as a class. In one class, the instructor stood in front of a 40-person class, while he randomly selected students to answer questions related to a context problem they worked on. The questions he asked were substantive and open ended, allowing every student to contribute thinking related to the question rather than simply answering correctly or incorrectly. The second class was a 120-person class where the instructor presented a slide presentation wearing a microphone, with three Learning Assistants circulating the room, and students discussing problems in small groups. The third class was a 30-person class where students spent the entire class working in groups of three-four on rich tasks while the instructor floated around the room, visiting with individual groups, and then bringing the class together for a whole-class discussion. The common element of these courses, in addition to the content being taught, was that the instructors authentically cared to understand what their students were thinking related to the mathematics, and that the instructors used this understanding of their students’ thinking to connect the mathematics to the students’ understanding of the mathematics – what Hackenberg has called exhibiting mathematical caring relations [23]. The DIRACC curriculum and its enactment at LSU illustrate a forward-thinking calculus program by centering the mathematics, and every individual student’s construction of the mathematics, as the guiding forces.