EnvironmentalThemed Modeling Offers Students RealWorld Perspective
By Lina Sorg
Realistic modeling problems are powerful tools that advance students’ understanding of mathematics’ applicability in everyday life. For this reason, the incorporation of current news articles and practical exercises in classrooms encourages appreciation of the relevance of mathematics in a wide variety of ongoing, practical scenarios.
Environmental deterioration—due to human activity—is a particularly timely scenario. Mathematical modeling can address many environmental issues, including agricultural runoff, groundwater flow, release of carbon into the atmosphere, and changing climate. At the SIAM Conference on Applied Mathematics Education (ED16), held last week in Philadelphia, PA, four participants of a minisymposium entitled “Environmental Modeling in the Classroom, Across Curriculum” discussed their application of mathematical modeling to various situations.
Emek Kose (St. Mary's College of Maryland), for example, leads various climate modeling exercises with her students, including the creation of a piecewise function for calculating albedo (a unitless quantity that measures how well a surface reflects light). At the end of each exercise, her students describe and critique their findings as if they were writing for the science section of the Washington Post. This helps them remember that not every reader has studied differential equations, Kose said, thus increasing communication skills with broader audiences.
Emek Kose (St. Mary's College of Maryland) details a modeling problem about agricultural runoff that connects climate science and mathematics for her students. Staff photo.
Kose then described a class project about agricultural runoff. A significant quantity of raw sewage (wastewater that has not yet been treated) spills into and pollutes waterways, much of it from agriculture. The runoff contains large amounts of animal waste as well as nitrate and phosphate from fertilizer, causing the formation of oxygendepleting algal blooms that lead to aquatic dead zones along the coast.
To begin, Kose’s students read “Boss Hog: The Dark Side of America’s Top Pork Producer” in Rolling Stone to glean realworld context. She then provides a structured worksheet, and asks students to create two models and write a system of differential equations that describes the relationship between algal blooms and dissolved oxygen in waterways. The project concludes with class discussion about which student models are closest to reality, and why.
“It’s using math to predict the future,” Kose said. Because the project pertains to the larger topic of social and environmental justice, students are simultaneously honing their modeling skills while applying math in a meaningful context.
Ellen Swanson (Centre College) does similar modeling activities with her students, except she offers an entire course on just “Environment Modeling.” The class only runs during the college’s winter session, and thus meets three hours a day for three weeks. The only mathematic requirement is differential calculus, an intentionallybroad prerequisite that allows Swanson to bring a substantial environmental perspective to the conversation; about half of the students are from the Mathematics Program, while the others have environmental backgrounds. The course’s five goals are as follows:
 Recognize the role of mathematics
 Create a math model describing an [environmental] process
 Perform a modeling revision process
 Communicate ideas in oral and written form
 Appreciate the role of math in making future decisions
All students must produce a lengthy paper at the end of the course. Students complete the paper after receiving feedback from their classmates; this feedback is often valuable, Swanson said, given the students’ diversified backgrounds.
Classwork usually yields compartmental models, population models, and experimentation with column chromatography, bifurcation, phase lines, and equilibrium in an attempt to better understand the carbon cycle; all students use MATLAB to visualize their models. Swanson also leads a specific lab exercise that applies Darcy’s Law to groundwater flow and the configuration of contour maps. She finishes the course by tying their modeling efforts to a civil action movie about clustered cases of leukemia possibly caused by pollution from a nearby tannery or chemical company. “Ultimately we’re just making connections to the real world,” she said.
Robert McGhee (University of Minnesota) focused his talk on his students’ study of permafrost, a thick layer of soil, sediment, and/or rock that remains frozen yearround and is usually found in the higher latitudes of polar regions.
McGhee explained that he initially provided real environmental context to his problem by discussing the Paris Climate Conference (COP21) with his students. They paid special attention to the Paris Agreement, in which the participating 195 countries decided to keep the global temperature from rising more than two degrees Celsius. Thus, McGhee posed the following question: How much carbon would be released from the permafrost if the global mean temperature rose by two degrees Celsius?
He then presented a specific model—created by one of his students—that measures permafrost’s response to climate change. The model uses Budyko’s equation, and assumes that the permafrost ice line is at a certain temperature, with different albedos for where ice is and is not present. “All parameters are numbers that have to do with what’s happening on the Earth,” McGhee said, “and they have been measured and estimated.” The student used a linear approximation to study how the ice line would vary with a given parameter, and how that parameter would change with a twodegree global temperature increase. In the end, McGhee’s student calculated the updated latitude/location change of permafrost, eventually answering McGhee’s initial question and gaining a better understanding of climate change’s impact in the process.
Jessica Libertini (Virginia Military Institute) closed the minisymposium with a description of her detailed modeling project. Unlike the other presenters, Libertini had no math majors in her noncalculusbased modeling class, which she specifically developed based on the recurring question: How do we feed 9 billion people by 2050? National Public Radio,
The Washington Post, and
The New York Times have all addressed this question, with a wide range of answers including eating local and organic, avoiding GMOs, going vegan, and eating bugs. “There’s enough material for a whole course,” Libertini said.
Her project asked students a slightly different, though equallyimportant question: When is ‘Doomsday?’ When will our population’s food demands exceed the global food production capacity? Libertini broke down the project into three manageable subquestions:
 How many people are there?
 How much agricultural/arable land is there?
 How many people can be fed on one acre?
The class split into three groups, with each team addressing a specific question. Libertini encouraged her students to conduct research, and emphasized the importance of reliability and accurate citations in powerful communication. The team answering question 1 developed a math model based on growth rates and used Excel to propagate the model forward in time to predict future rates. The students also accounted for the actions of some countries to influence population growth, i.e., China’s onechild policy, birth control education, etc. The second team created a model of discrete dynamical systems based on land use trends; these students also used Excel to propagate their model forward. And the third team investigated how many people one acre of land can feed by studying oatbased diets and the number of calories necessary for human growth and sustenance.
Eventually the teams came together to combine their individual results. They overlaid their work on the same graph, with population as the yaxis, and created a model that included a plotted “bestcase” forecast of the number of people who could be fed each year, taking into account both optimistic and pessimistic predictions. Afterwards the students discussed how different policies could cause growth or decay and change the projected outcome.
In addition to increasing their knowledge of discrete dynamical systems and curve fitting, the students developed a plethora of transferrable skills: finding, understanding, and referencing reliable resources; thinking critically; developing sound arguments; and sharpening oral and written communication techniques.
Libertini also presented the audience with possible ways to modify the project’s difficulty level. For example, students in a less rigorous setting could use only recent data and linear models instead of discrete dynamical systems, while more advanced students could use piecewise models to account for past policy changes or consider specific nutrient requirements rather than just caloric intake. She emphasized that many related projects about climate change could stem from this one, including brainstorms about realistic, implementable policies to manage population and global food production.
Ultimately, the project is meant to increase the awareness of the applicability of mathematics to complicated, real problems, Libertini said.

Lina Sorg is the associate editor of SIAM News. 