Can We Bridge the Gap? Mathematics and the Life Sciences, Part 1 – Calculus-Based Modules, Programs, Curricula

Abstract

This editorial serves as an introduction to Part 1 of the Special Issue Mathematics and the Life Sciences – a collection of articles showcasing ideas, examples, pedagogical frameworks, and curricular materials aiming to bridge the stubbornly persistent gap at the undergraduate level between the mathematical and the life sciences. The special issue features authors from public and private institutions of diverse types, sizes, and geographic locations: community colleges, liberal arts colleges, and research-oriented universities. We hope this special issue will serve as a resource to faculty who seek to make changes to their own course(s) or initiate curriculum reforms at their own schools. Part 1 focuses on educational initiatives that are appropriate for Calculus classes or require calculus as a prerequisite. Part 2 of the special issue features course materials and programs based on discrete mathematics, computational approaches, and statistics. Part 2 also includes articles on internship programs and co-curricular opportunities.

KEYWORDS:

• Mathematical biology
• computational biology
• mathematical biology education
• interdisciplinary education
• curriculum reform
• bio-calculus
• calculus modules

This is the first of two Special Issues on Mathematics and the Life Sciences. Each features a collection of articles showcasing ideas, examples, pedagogical frameworks, and curricular materials aiming to bridge the stubbornly persistent gap at the undergraduate level between the mathematical and the life sciences. The first issue, Mathematics and the Life Sciences: Part 1, features programs, initiatives, and materials for calculus or calculus-based courses (e.g., courses in differential equations). The second one, Mathematics and the Life Sciences: Part 2, focuses on courses in discrete mathematics and statistics, as well as internship programs and co-curricular opportunities.

It may seem surprising that nearly two decades after the publication of the BIO 2010 report [12], the expected transformative integration of mathematics and biology in the undergraduate curriculum is still far from the recommended ambitious reform. Programmatic, administrative, and pedagogical challenges have often been mentioned as possible culprits – problems with establishing teaching loads for interdisciplinary and team-taught courses, narrow disciplinary criteria for tenure and promotion, and general disconnects between mathematics/statistics, biology, and computer science departments provide just a few examples [9]. The diversity of quantitative and methodological approaches used in the variety of biological subdisciplines is another contributing factor. At a time when the NSF urges the community to develop groundbreaking programs that “prepare the next generations of scientists to navigate the breadth of biological sciences, training in multiple disciplines without sacrificing depth of learning or innovation” [14], the highly linear and compartmentalized undergraduate biology curriculum still enforces rigid sub-disciplinary divisions.

In the 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences, the life sciences were clearly identified as a key path through the mathematics major to graduate programs and the workforce. This account echoed many prior high-profile reports (e.g., Vision and Change [1], A New Biology for the 21st Century [11], BIO2010 [12], and The Mathematical Sciences in 2025 [13]) that had previously discussed the changing landscape at the interface of mathematics and biology and had issued urgent calls for broadening students’ exposure to mathematical methods for the life sciences. Recognizing that students entering medical school would need to “apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world” [3, skill E1], the joint report from the HHMI and AAMC [3] specifically underscored the need for engaging students in “using data and mathematical models for making inferences about natural phenomena” [3, skills E1.2 and E1.5]. Further, the lack of quantitative training has been recognized as a significant impediment for those pursuing graduate studies in the life sciences and for practicing biologists [4, 8]. The ongoing data science revolution has only amplified this problem due to the urgent demand in the life sciences for a cadre of professionals that are skilled at working with big data and can successfully operate at the intersection of mathematics, statistics, and computer science.

The lack of quality educational materials does not seem to be the issue here – a wealth of high-quality educational materials for mathematics, biology, and interdisciplinary courses have been developed in response to the above initiatives, and there are many organizations and communities that encourage and support programmatic innovation [2, 7]. Last year, the Bulletin of Mathematical Biology published its first-ever special issue on education [10]. The issue reviews some early efforts for reform, applauds the progress made since then, celebrates the diversity of programs and institutions driving curricular changes at the interface of mathematics and biology, highlights some stubborn roadblocks to success, and outlines new ones that have developed in the era of big data and technology.

Indeed, we feel that so many educational materials have already been developed – textbooks, modules, labs, student research projects, etc. – that further efforts would be better directed toward gathering those materials in easily accessible hubs, publicizing them more effectively, and providing appropriate faculty development and training. There is a significant momentum for advancing such efforts, as well. Many governments and community initiatives, including The National Sciences Digital Library, QUBES, BioQUEST, and the Intercollegiate Biomathematics Alliance, have already amassed (and continue to gather) high-quality materials available to educators in their searchable databases. Through regular faculty workshops, lectures, seminars, conferences, and student research experiences, they also facilitate the growth of a national network of educators with similar interests and goals. The invited and contributed paper sessions that the Chapter in Mathematical and Computational Biological Science of the Mathematical Association of America (BIO SIGMAA) offers annually at the Joint Mathematics Meetings and MathFest provide another venue for dissemination and for exchange of ideas. Many of the articles in this special issue were solicited at these forums in 2018 and 2019.

The success of such efforts at any level, however, is conditioned upon the specific goals that each institution sets for its graduates. Those range from incorporating short quantitative/modeling course modules from the life sciences into conventional mathematics and biology courses to learning communities at the intersection of mathematics, statistics, and the life sciences, to creating new fully-fledged cross-disciplinary programs in mathematical biology. Behind every effort, whether initiated by a single faculty, a team, a department, or an institution, is a story shaped by institutional priorities, appetite for change, and individual opinions and perceptions on the nature of mathematical biology. The drivers for change at each institution, as well as the set of expected outcomes vary significantly and, thus, no “one size fits all” approach is likely to be able to elicit effective change.

With this in mind, when we discussed goals for this special issue in early 2018, we agreed to assemble a collection of articles presenting a cross-section of approaches undertaken by different academic communities to meet the challenges of reforming the mathematics and biology curricula at their own institutions. We were interested in showcasing ideas, examples, and pedagogical frameworks, as well as stories of successes and failures, at a spectrum of different institutions. We did not expect for each article to include outreach efforts or cross-institutional components or comprehensive assessment plans. Instead, we felt that to understand how and why some efforts fail is just as important and valuable as finding the underpinnings of a successful project. We welcomed submissions that feature original sources and materials, but we were also interested in presenting creative ways for successful uses of existing materials.

We view these special issues as a celebration of the diversity of the mathematical biology community – the 17 papers it features come from a geographically diverse group of public and private institutions and highlight initiatives at community colleges, liberal arts colleges, and research-oriented universities. The school sizes vary from enrollments as small as 2,000 to as large as 35,000 undergraduate students. Thus, we hope that faculty who seek to make changes to their own course(s) or initiate curriculum reforms at their schools will find examples of interest in the special issue.

1. MATHEMATICS AND THE LIFE SCIENCES: PART 1 – CALCULUS-BASED MODULES, PROGRAMS, CURRICULA

Using Calculus and Difference/Differential Equations is arguably the most common gateway to mathematical modeling at the college level. The special issue features nine articles describing pedagogy, curriculum development, or original educational resources that utilize this approach.

The article Guiding Students to Understand Functional Responses: Holling's Disc Experiment Revisited by Pulley et al. presents a multi-part activity designed to deepen students’ understanding of the mechanisms behind the Holling Type II functional response. This extended activity has theoretical, experimental, and simulation components, each of which helps students connect theory with results from experimental data collection and NetLogo computer simulation. Flexible in design, the activity allows implementations at levels ranging from high school students to advanced undergraduate and graduate students.

In Emphasizing Model Construction in the Classroom, Oremland takes us through a detailed account of an approach to teaching mathematical model construction and exhibits deep pedagogical reflection. Oremland presents activities and discussions that help students navigate the leap from reality to model. The activities presented are designed around the author’s experience of common student challenges in the modeling process. Exercises are thoughtfully scaffolded for students to support students at every stage of designing a mathematical model.

In A Course on Mathematical Modeling for the Life Sciences, Norton presents the result of a holistic renovation of a three-course math curriculum requested by the biology department, which includes Calculus, Statistics, and Modeling. It is in this modeling course that Norton presents first ODE models to also introduce the ideas and approaches of modeling. The author then uses that as a springboard to consider discrete models and introduce computational tools. Besides a diverse selection of mathematical models and application areas, a nice feature of the course is a semester-long student research project, in which students choose their own topic.

A significant disconnect between calculus as a service to biology is biology’s emphasis on data, not usually featured in introductory calculus courses. While some instructors bridge that gap by using calculus to build models that are later fit to data, in his article Estimating the Sensitivity of Fitted Parameters to Perturbations of Data with Calculus Nievergelt takes this a step further and asks students to think about calculus as integral part to the process of data fitting (for example by finding the minimum of a sum of errors). This paper introduces everything you need to add this sort of investigation in your calculus classes using data sets from biology.

In Aligning Calculus to the Demands of Life Sciences Disciplines: The Argument for Integrating Statistical Reasoning, Luque et al. describe an evolution of a redesigned calculus for the life sciences course at a large urban university. The course integrates data analysis and statistical reasoning along with familiar and compelling biological models to help motivate the need for calculus in biology. The redesign is informed by recommendations on the quantitative needs of biology students [6]. The authors provide a detailed analysis of student success and attitudes for several iterations of the course.

Gordon et al. present the article Developing a Mathematics Curriculum for the Biosciences – a curriculum development effort at Farmingdale State College through a multi-year collaborative project linking mathematics and the biological sciences. The authors summarize lab activities they have developed for an entry-level biology course, as well as detailed descriptions of quantitatively-based exercises in a newly created two-credit lab course restricted to bioscience majors. They share thoughtful reflections on the project’s successes and failures, as well as plans for improvement in future offerings.

Transforming a Calculus for Life Science Course: Moving from Procedural Calculus to Studying Dynamical Systems and Bifurcation Theory by Bennoun features the process of redeveloping the calculus course for life science students at a large research university into a course focusing on modeling using dynamical systems and bifurcation theory. The article describes the course’s active learning model that incorporates clicker questions and small group work. In addition to developing a course to best serve the life science and pre-medical student population, the course developers designed the course that it could be taught by faculty without expertise in mathematical biology.

It is well known that increased student engagement is a key factor for educational satisfaction and success [5, 15]. To heighten their students’ engagement, Stoner and Joyner introduce a calculus project to model the volume, flow, and pressure of air inside the lungs of a patient receiving positive pressure ventilation. It is presented in the article Breathing Life Into Calculus: Using Simulation to Enhance Students’ Understanding of the Relevance of Calculus to Physiology. The authors share their experience, outcomes, and results from implementing this authentic project into a course in differential equations for mathematics majors and into a Calculus I course designed primarily for biology majors.

With COVID-19 as a backdrop, in Using a COVID-19 Model in Various Classroom Settings to Assess Effects of Interventions, Ledder and Homp seek to heighten student interest in epidemiology modeling. The authors present an original collection of materials that demonstrate the need for and the value of mathematical models in epidemiology. Students can use the COVID-19 model to examine different scenarios and assess the effect of public policy and community behavior on the spread of the COVID-19 pandemic. The authors comment on the use of these materials in a broad range of courses – an advanced project-based mathematical epidemiology course, an introductory differential equations course, a general education course, and as a project in a Master of Arts for Teachers Program.

ACKNOWLEDGEMENTS

The guest editors would like to thank the PRIMUS Editors-In-Chief Jo Ellis-Monaghan and Matthew Boelkins for inviting them to edit this special issue and for the support they provided throughout. Their gratitude extends to all contributing authors, as well as the reviewers whose in-depth consideration of submitted manuscripts and constructive suggestions to authors led to many improvements and increased clarity. When in 2020, the COVID-19 pandemic put a temporary hold on this project, necessitated work from home, and increased the challenges of juggling family and professional duties, their contributing authors, reviewers, and PRIMUS staff showed great professionalism and flexibility. The guest editors are grateful to all of them for their patience and dedication.

Additional information

Funding

The work of the first editor was supported in part by the Karl E. Peace Fellowship in Mathematics at Randolph-Macon College, VA. The work of the third editor was supported in part by the National Science Foundation under Grant DUE-1446258. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Notes on contributors

Raina Robeva

Raina Robeva is a Professor of Mathematics at Randolph-Macon College in Virginia. She is the lead author/editor of several textbooks and volumes in mathematical biology, including An Invitation to Biomathematics, Algebraic and Discrete Mathematical Methods for Modern Biology, and Algebraic and Combinatorial Computational Biology. She has led numerous educational and professional development initiatives at the interface of mathematics and biology sponsored by NSF, NIH, and MAA among others. Her service to the community includes serving twice as President of the MAA Special Interest Group in Mathematical and Computational Biology (BIO SIGMAA) and Chair of the Advisory Board of the National Institute for Mathematical and Biological Synthesis (NIMBioS).

Timothy D. Comar

Timothy D. Comar is a Professor of Mathematics at Benedictine University in Lisle, IL. He earned his Ph.D. in Mathematics at the University of Michigan specializing in low-dimensional topology. He is currently working mathematical biology with interests in the dynamics of deterministic and stochastic models for pest management, vaccination strategies for epidemics, and gene regulatory networks using impulsive differential equations, difference equations, Boolean models, agent-based models, and computing-intensive techniques. He taught biocalculus courses for many years and mentored many undergraduate research projects in mathematical biology, some of which have been published in professional journals. He has served in several leadership roles for the BIO SIGMAA and is the Group’s Past-President.

Carrie Diaz Eaton

Carrie Diaz Eaton is Associate Professor of Digital and Computational Studies at Bates College and is a Mathematical Biologist turned inclusive and interdisciplinary undergraduate STEM education researcher. Diaz Eaton also spends time raising two kids and enjoying the beautiful Maine summer and fall weather.