Victor Donnay, Professor of Mathematics, Bryn Mawr College
A 2008 SENCER Model
This course for junior and senior math majors uses mathematics, specifically the ordinary differential equations as used in mathematical modeling, to analyze and understand a variety of real-world problems. Among the civic problems explored are specific instances of population growth and over-population, over-use of natural resources leading to extinction of animal populations and the depletion of natural resources, genocide, and the spread of diseases, all taken from current events. While mathematical models are not perfect predictors of what will happen in the real world, they can offer important insights and information about the nature and scope of a problem, and can inform solutions.
The course format is a combination of lecture, seminar and lab. Simulation games, group-work, presentations, and guided inquiry are some of the pedagogies used in this course, which aims to create a community of learners who have the ability to take what they have learned in one situation and apply it to novel situations, and who can pursue information independently. Beyond the capacity to solve mathematical problems, students are expected to be able to communicate their findings clearly, both verbally and in writing, and to explain the mathematical reasoning behind their conclusions. Learning is assessed through pre- and post-tests and a variety of assignments, including short response papers, quizzes, and a final group project involving an oral report and a 10-15 page paper.
Course Learning Goals for Instructors and Students
The instructor will teach students to:
- translate (simple) real world situations into ordinary differential equations (the modeling procedure) and
- extract predictive information about the real world situation from the differential equations.
- solve differential equations in a variety of ways: via traditional analytic methods (formulas) as well as by more modern approaches such as numerical solutions generated by computer programs and by graphical methods that provide qualitative information.
- apply these methods to linear and non-linear equations and systems and see how feedback effects in non-linear systems can lead to unexpected behaviors.
At the end of the course, a student will:
- Comprehend contemporary applications of computer modeling (e.g. what is meant when a newspaper article reports of new developments in the study of climate change that are predicated by computer modeling).
- Be able to communicate, both in writing an verbally, to explain the mathematical reasoning behind their answer, because solving a mathematical equation is only part of the process of using mathematics.
- Develop their ability to work as independent and self-sufficient learners, with the capacity to learn material on their own, and practice and proficiency in “What to do when they do not know what to do?”
- Be able to apply what they have learned in one situation to new and different situations (transfer of knowledge).
- Understand mathematical models are not perfect predictors of what will happen in the real world, but they can offer important insights into key elements of a problem
- Be comfortable with not knowing the answer immediately and learning from peers. Students will become part of a community of learners who support, encourage and learn from one another.
Linking Mathematics and Social Issues
How Ordinary Differential Equations Links Mathematics and Social Issues
|Social Issues||Mathematics Concepts|
|Population growth||Exponential growth model|
|Over-population and carrying capacity||Logistic differential equation, slope field and phase line analysis|
|Population crash caused by over-harvesting of natural resource: ex. Collapse of fish stocks||Logistic equation with harvesting term,
bifurcation analysis, parameter space diagram
|Spread of diseases: ex. AIDS||Modeling diseases via system of differential equations (SIR – Susceptible, Infected, Recoveredmodel), vector fields, linear analysis of stability of fixed points|
|Multi-species interactions||Predator-prey models|
Differential equations and mathematical modeling can be used to study a wide range of social issues. Among the topics that have a natural fit with the mathematics in a course on ordinary differential equations are all aspects of population problems: growth of population, over-population, carrying capacity of an ecosystem, the effect of harvesting, such as hunting or fishing, on a population and how over-harvesting can lead to species extinction, interactions between multiple species populations, such as predator-prey, cooperative and competitive species.
To see how these topics play out in real life, the students read chapters from the book Collapse: How Societies Choose to Fail or Succeed by Jared Diamond. The book examines human societies throughout history that have died out, the factors that led to their collapses, and the lessons we might learn to prevent a collapse of our present day global society. For each chapter that they read, the students are asked to find linkages between what they have read and the mathematics we have been learning in the course.
The first model of population growth that we study involves the exponential function. Students are asked to read the chapter “Malthus in Africa: Rwanda’s Genocide,” which discusses the potential link between genocide and overpopulation. I then give them an assignment that was developed with the assistance of Wen Gao, a Bryn Mawr math major, and was inspired by our participation at the 2006 Mathematics of Social Justice conference at Lafayette College. Using data from the chapter and from international population Web sites, students are asked to calculate for Rwanda the growth rate of population in the decades before the genocide and the population doubling time and then predict what the population will be in later years. For the years after the genocide, they find that their predications significantly overestimate the actual population and are asked to account for the discrepancy. They realize that their overestimates are due to the deaths of hundreds of thousands of people during the genocide period and face the sobering fact that numbers arising from mathematical calculations can have a very human dimension.
A topic that I have made a particular focus of my differential equations course is modeling population growth where the population being studied also undergoes harvesting. As an illustrative example, imagine fishermen in the Grand Banks region near Newfoundland who each year harvest (catch) some amount of the fish population. To start with, there are a certain number of fisherman involved who each year catch roughly a constant amount of fish. Should we allow more fishermen, perhaps equipped with sophisticated fishing technology, to join the hunt? A reasonable response might be that, to avoid the danger of over-fishing, we could allow a small number of additional fishermen to join in. We expect that such a change would increase the catch by a relatively small amount and hence decrease, by a similarly moderate amount, the level of fish remaining in the Grand Banks. However it turns out that such a seemingly reasonable strategy can be dangerously misguided.
Mathematically, one can model population growth with harvesting via a differential equation of the form: where P(t) is the population, k is the growth rate, N is the carrying capacity and is the harvesting level. A study of the solutions of this equation for various harvesting levels shows the existence of a critical fishing level; technically, it is called the bifurcation value. If the fishing level is increased beyond this critical value, even very slightly, then the model predicts that there will be a drastic crash in the fish population, potentially leading to extinction or near extinction.
The moral of the story is that, if one happens to be unlucky enough to be close to the critical harvesting value, then even a small additional increase in the harvesting level can have cataclysmic implications for the population. Thus great care needs to be taken when increasing harvesting levels even by small amounts, lest we inadvertently cause a population crash. Here is an example where mathematics provides us with a key insight that runs counter to our natural intuition.
Sadly, the phenomenon of over-harvesting is not limited to fishing situations. In its general form, it is often referred to as the “tragedy of the commons.” Consider a community whose citizens let their sheep graze on a shared tract of land, the commons. In this situation, no one individual has any incentive to limit the amount of grazing done by his sheep. Over time, the commons will become depleted of grass and cease to be usable for grazing. In the language of our previous example, over-harvesting has caused the population of grass to crash. To prepare my students to better appreciate the amazing ability of mathematics to explain and predict population crashes, I want them to first experience for themselves how seemingly reasonable human behavior can lead to over-harvesting.
The students read the chapter “Twilight at Easter” that examines the collapse of the society on Easter Island, home to the famous stone statues. They learn that a major factor in the collapse was the complete deforestation of the island, and they are left to wonder how a society could be so shortsighted as to cut down all of its trees. Did no one notice that the tree population was drastically diminishing? Why did no one take steps to address the issue? They feel, a bit smugly, that they would be smarter than the Easter Islanders.
We then have a special three-hour evening meeting of the class in which we play the simulation game Fishing Banks, Ltd., created by Dennis Meadows. In this game, teams of students manage their own fishing fleets with the goal of maximizing profit. Over time, what invariably happens is that the teams build up large fishing fleets to maximize their short-term profit, over-harvest the fish population and cause the fish stock to crash to extinction. At this point, with no more fish to catch, the fish companies go bankrupt and hence fail to meet their goal of maximizing profit. The population crash happens even though the teams get feedback after each round on the amount of fish they have caught. By the time they notice that the stocks are decreasing, the corrections they make are too little and too late to stop the extinction. As we debrief this experience, the students realize that they have fallen into the same trap as the Easter Islanders: by over-harvesting a valuable resource, they have driven it to extinction.
Now that the students have a visceral understanding of the over-harvesting phenomenon, I introduce the differential equation mentioned earlier, that models the situation, and we undertake its mathematical analysis. Students learn that mathematical modeling can be used to predict and explain the population crash phenomenon and can thereby serve as a counterweight to the many pressures encouraging over-harvesting of resources.
We finish the unit with a discussion of the interplay between mathematical modeling and government and business policy making. Why is it that even though modeling can predict negative consequences, as with over-fishing or climate change, it is so hard to get society to take preventive action? Society might be better served by leaders with a firm understanding of mathematics in the context of policymaking. By including in our math courses components that link mathematics to issues of social relevance, we can prepare and inspire our students to become these future leaders.
A major priority in the design of this course is the engagement of students as scientists and citizens. This is accomplished through the variety of techniques described below.
The course meets twice a week for 80 minutes. I prefer this format, as compared to meeting three times a week for 50 minutes, as I regularly have the students engage in interactive group activities during the class and the longer time block facilities such activities.
The text for the course is Differential Equations by Blanchard, Devaney, and Hall, 3rd edition, published by Brooks/Cole. The authors are all researchers in the field of dynamical systems and they apply a dynamical systems perspective to their presentation of differential equations. There is a strong emphasis on quantitative analysis of equations using graphical and numerical methods and a corresponding decrease in emphasis on analytical techniques. The text includes a strong focus on mathematical modeling.
Formats and Pedagogies
A computer disk comes with the text. This disk, that can be used on both PC and MacIntosh computers, contains a variety of easy to use simulations and demonstrations that illustrate many of the ideas in the course. Most of the programs are menu driven, with the user selecting from a set of pre-programmed examples, so there is no learning curve required to use them. The output is displayed in a beautiful visual form. In a few important cases, such as to graph slope fields or vector fields and draw their associated solutions curves, the user can enter her own formulas into the programs.
The class format is an integrated mixture of lecture, seminar and lab. Part of the time I lecture, there is also a lot of group work, often using the computer programs, and classroom discussion. In earlier versions of the course, I would use the computer programs to demonstrate ideas, via a computer projection system, to the class. The class would have a separate computer laboratory component in which students would do assignments in our computer lab. Several years ago, the math department purchased a set of ten laptop computers. Now the students, in teams of two or three, use these laptops during class time to explore the concepts themselves and at present we do not have a separate computer lab component. There are still some more extensive computer assignments that students do on their own time.
For the group work, I have both open-ended discovery work and guided work. For the discovery work, I have the students use the computer programs to investigate a new situation and respond to prompts such as “what do you observe?”, “do you see any patterns?”, “what questions do you have?”, “can you make some predictions or conjectures? “. In the guided work, the students practice a technique that I have presented during lecture.
I regularly assign homework problems from the textbook. Students read out of the book Collapse: How Societies Choose to Fail or Succeedby Jared Diamond, and write short response papers in which they describe the ways that they see the material in our math course applying to the social issues being discussed in the chapter. There is a more focused assignment on over-population and the Rwandan genocide (See Appendix for Rwanda Assignment). There is a final project in which student teams learn about a topic of interest that involves differential equations, give a short oral presentation on their project and write a 10 – 15 page report on their findings. (See Appendix for description of final project and list of potential project topics.)
We have a special three hour class meeting one evening in which we learn about over harvesting of resources by playing the simulation game Fishing Banks, Ltd created by Dennis Meadows. (See Appendix for Fishing Simulation Game).
Below is the course “play-by-play” in which I briefly describe the topic for each class and also have links to the handouts for group work and computer work that we used in class that day. Also below is an example of a group modeling project.
At the first meeting of the class, I give the students a pre-assessment which gauges their knowledge of key topics that we will cover during the course that they might have seen in previous math courses. I used this information to decide what level of knowledge I can assume the students already have attained and how much time I need to spend on (re-) introducing these topics. I give the same set of questions at the end of the term as a post-assessment. (See Appendix for Pre-Course Assessment)
During the term, I regularly use methods of Formative Assessment (adapted from Dylan William, Assessment for Learning. See Appendix for description of Formative Assessment). For example, when students are working in groups, I can circulate among the groups, listen in to their discussions, and get information about the students’ level of understanding.
Part way through the term, I had students fill in a course feedback that asks them what is helping their learning, and what is interfering with it, as well as any changes they would recommend.
Work that is graded and contributes to the final grade:
- Weekly homework
- Occasional quizzes
- Two mid-term exams
- Final project
Students fill in the standard college wide course evaluation form, as well as a questionnaire that I developed using the web-based “Survey Monkey.” I have noticed that they write more extensive comments via the computer than when I had them fill in the questionnaire by hand.
Final Course Questionnaire can be found at:
Background and Context
Victor Donnay, Professor of Mathematics, Department of Mathematics
The differential equations course is taught to between fifteen and twenty sophomore, junior and senior math and science majors. I use the text Differential Equationsby Blanchard, Devaney and Hall. Over the past several years I have been focusing the course more on mathematical modeling than on physics and engineering applications. Since Bryn Mawr is a liberal arts college without an engineering program and our physics department teaches its own mathematical methods course, I have the freedom to replace some traditional topics with material on modeling.
The text has a lot of interesting real world applications of modeling which have stimulated my interest in more applied aspects of mathematics. As my interests and expertise has developed more in these directions, I have added more components to the course that have a civic engagement focus.
Factors that have contributed to my development in this direction include:
- teaching a course for senior math majors on Mathematical Modeling and the Environment, using the text of the same name by Charles Hadlock
- supervising on year long senior these on Mathematical Epidemiology in which we studied the book … (ref)
- attending the 2006 Mathematics of Social Justice conference at Lafayette College
Place in the Curriculum
Mathematics 210, Differential Equations with Applications, is an elective that counts towards the mathematics major. It has as pre-requisites Calculus 1 and 2 and as a co-requisite either Multivariable Calculus or Linear Algebra. It can count as an elective for science majors. Mathematically oriented students in the Environmental Studies concentration are encouraged to take the course. The course is not required for physics majors. There is no engineering program at our institution so we do not need to cover engineering oriented topics.
I have received no funding support for course development. The last time I taught the course, I had an undergraduate teaching assistant who attended the class and helped me with the logistics of the course, the worksheets and the laptop computers. I also have an undergraduate student who grades the weekly homework.
Resulting Projects and Research
I have written two versions of an articles describing this course both titled “Differential Equations and Civic Engagement”
SIGMAA-QL Newsletter, October 2007;
Civic Matters–A Catalyst for Community Dialogue, a publication of the Civic Engagement Office at Bryn Mawr College, Issue 2, April 2008.
(See Appendix for full text)
List of References
Pre-Course Assessment Form
Data for Rwanda Assignment
Data for Rwanda Assignment
Final Project Description
Description and Guidelines for the Final Project
Final Topic Suggestions
List of Potential Final Topics
Donnay, “Differential Equations and Civic Engagement”
Donnay’s Essay on Differential Equations