Calculus Reform: What is different this time?
By David Bressoud @dbressoud
This is an excerpt from a longer article that has been submitted to the Notices of the AMS. A draft of the full submission can be found here.
On May 30–31, 2019, representatives of the departments of mathematics at 24 public and private universities met in Lincoln, Nebraska to share department-wide efforts to improve teaching and learning in their precalculus through single variable calculus sequence. These departments are working either with Progress through Calculus (PtC, NSF # 1430540), run by the Mathematical Association of America (MAA), or Student Engagement in Mathematics through an Institutional Network for Active Learning (SEMINAL, NSF #’s 1624643, 1624610, 1624628, and 1624639), a project of the Association of Public and Land-Grant Universities (APLU). Representatives reported on course coordination, collection and use of data, variations in course structure, training for graduate teaching assistants, culturally responsive teaching, use of active learning, and efforts directed at changing departmental culture.
The conference generated a great deal of energy and excitement, a sense that real change is happening. And yet, to many this activity seems reminiscent of the Calculus Reform efforts of the late 1980s and early ‘90s. I am often asked what is different now. Is this simply another iteration of a doomed effort to change these pivotal courses?
It is important to recognize that the Calculus Reform effort was not a failure. It made a real difference as can be seen by comparing textbooks of the 1980s and today. It emphasized the importance of graphical representations, verbal descriptions and communication skills, as well as projects and deep explorations of selected topics. It also served to initiate or accelerate efforts that are bearing fruit today such as Project NExT, the Scholarship of Teaching and Learning, and the explosive expansion of scholarly research into undergraduate mathematics education.
Nevertheless, those who worked at the forefront of the Calculus Reform movement had a vision that has not been realized, a vision that lives on in our current efforts. The goal is of calculus classes that engage all students in the joy of mathematical exploration and the satisfaction of deep learning, not just the memorization of procedures but the ownership of them so that their principles can be applied flexibly in unfamiliar situations.
In 1997, The Chronicle of Higher Education published its post-mortem of the Calculus Reform movement. The article concluded with a discouraging comment from Ed Dubinsky, one of the fathers of this effort, “Except for a small number of isolated pockets, it will be hard to tell that there was a calculus reform. [In a few years] we'll become upset that very few people are really learning calculus and we'll have another round of reforms. I hope that round survives.”
I see three reasons why this is not just a repeat of what happened thirty years ago.
First, the reform agenda that drove Calculus Reform did not disappear. Rather, it developed a lower profile that saw an acceleration of undergraduate mathematics research and a continuation of experimentation leading to a better understanding of the critical elements for improved undergraduate instruction. Today we are building on thirty years of experience. We have a more sophisticated sense of what works and what does not, of where technology can be a support and where it is a hindrance. We have accumulated data to back up assertions of best practice. Part of this building process has involved making connections to educational researchers in other STEM fields, especially the physics education research community, but also in biology, chemistry, geology, and engineering education.
Second, in 1990 the argument could be made that undergraduate mathematics education was working. The work of CUPM in the 1960s that shaped or current curriculum was directed toward students who would be ready to start calculus when they got to college, predominantly middle class white males. In the 1960s, they constituted over half of college graduates. While the percentage of Bachelor’s degrees going to white males had dropped to 38% by 1990, we were still producing adequate numbers of scientists and engineers who were able to meet the challenges they would face. Moreover, the experiments of the early Calculus Reform could and sometimes did go wrong. Departments were often reluctant to run the risk of changing the approaches then in place. Adding to this reluctance was the widespread belief that whatever was wrong was the fault of K-12 education, not the colleges and universities.
Today, the flaws of traditional methods for teaching calculus are far more apparent. The reaction to Calculus Reform that asserted high failure rates to be the price of improving K-12 education is now totally unacceptable. Presidents, provosts, and deans have come to recognize the cost to their institutions of high failure rates. There is pressure both to improve passing rates and to ensure those students go on to succeed in their subsequent courses. In too many cases, the status quo accomplishes neither. This is the economic argument for embracing changes that we know work. Such economic imperatives carry a lot of weight.
Combined with these pressures is the recognition that white males now barely exceed a quarter of college graduates, and demands for the employ of the mathematical sciences have been expanding and changing in fundamental ways. ASA’s GAISE report for undergraduate statistics (ASA, 2016), COMAP and SIAM’s GAIMME report (COMAP and SIAM, 2019), and especially The Mathematical Sciences in 2025 (NRC, 2013) make it clear that today’s undergraduate preparation in mathematics must be more than a proving ground where students demonstrate that they can survive the curriculum of the 1960s. It must actually begin the process of equipping them for the challenges they will face in the changing landscape of 21st century workforce demands.
Third, the goals this time around are very different. There was a naiveté to the Calculus Reform movement, believing that if we built the ideal calculus curriculum then mathematicians would embrace it and adapt to the demands it made on how they taught calculus. Today the focus is on training and support for new generations of educators. We are learning that we must demonstrate how to promote student engagement in the kind of learning that will lead to ownership of the concepts and methods. We are now learning what it takes to prepare and support graduate students and new faculty as well as experienced faculty to teach in this way.
While a bit simplistic, the third thing that is different this time can be summarized as a shift from an emphasis on what is taught to how it is taught. This is combined with the recognition that teaching for meaningful learning is not easy. Building the structures that support it requires buy-in from the dean of science, the department chair, a core of senior faculty, and one’s colleagues both in the department and beyond.
Calculus Reform was not a movement that came and went. It was the opening of a multi-decade effort that only now is truly beginning to blossom.
American Statistical Association (ASA). (2016). Guidelines for Assessment and Instruction in Statistics Education College Report 2016. Alexandria, VA: ASA. Available at http://www.amstat.org/education/gaise.
Consortium for Mathematics and Its Applications (COMAP) and Society for Industrial and Applied Mathematics (SIAM). (2019). Guidelines for Assessment and Instruction in Mathematical Modeling Education, 2nd edition. Philadelphia, PA: SIAM. Available at https://www.siam.org/Portals/0/Publications/Reports/GAIMME_2ED/GAIMME-2nd-ed-final-online-viewing-color.pdf
National Research Council (NRC). (2013). The Mathematical Sciences in 2025. Washington, DC: The National Academies Press. https://doi.org/10.17226/15269.