Course structure

Here is a rather detailed outline of the structure of the proposed course, which was written a while back.  Please understand that many details may change when the course is finally implemented, but nevertheless I hope that this outline is helpful. 


The course will provide students with the mathematical background and quantitative reasoning skills necessary to engage as informed citizens in discussions related to resources, pollution, recycling, economic change, and similar matters of public interest.   These include the four key mathematical ideas of “measuring”, “changing”, “risking” and “networking”, further detailed below.  Throughout the course, the techniques that are developed will be applied to a range of real-world examples at individual, local, and global scales.


This course is intended to be one of several offered by the mathematics department with the goal of helping students from non-technical majors partially satisfy their general education quantification requirement. It is designed to provide an introduction to various mathematical modeling techniques, with an emphasis on examples related to environmental and economic sustainability.  The course may be used to fulfill three credits of the GQ requirement for some majors, but it does not serve as a prerequisite for any mathematics courses and should be treated as a terminal course.


Course Content and Learning Objectives


These objectives relate to the title of the course and its overall purpose.

  • Understand, describe and critically appraise various expressions of the notion of “sustainability”, including those contained in the Penn State Sustainability Plan.
  • Understand the relevance of mathematical and quantitative reasoning in discussions of sustainability.


This course theme includes representing information by numbers, problems of measurement, units, time and spatial scales, estimation skills, and the notion of dynamic equilibrium.  Learning objectives will include

  • Understand and calculate with large and small numbers in scientific notation.
  • Work with customary and metric units, and conversions between them.  Be able to carry out dimensional analyses and check calculations for unit consistency.
  • Understand the effect of changes of scale on lengths, areas, and volumes, and the resulting constraints on the size of organisms or systems.
  • Make realistic estimates, based on limited information, of real-world data, including a realistic evaluation of the potential range of error in such estimates.  Use online and other information tools critically to refine such estimates.
  • Understand and apply the distinction between “stocks” and “flows”.
  • Calculate equilibria in box (stock-flow) models, using both algebraic and graphical methods.  (One-box and simple two-box models)
  • As examples, compute with simple energy balance models (e.g. for home insulation or for planetary climate).
  • Understand energy flows, different forms of energy and their interconvertibility (including thermodynamic constraints on the conversion of heat into other forms of energy)
  • Understand and apply the notion of residence time in box models. Relate to “characteristic timescales” for environmental phenomena and thus to the notion of sustainability.



This course theme includes quantities changing with time, rates of change, simple dynamic models, interest and discount rates, change of equilibrium as a function of parameters, feedbacks, bifurcations and “tipping points”.  This is the most developed theme of the course and will probably take the greatest amount of course time.

  • Understand that box models exhibit dynamic behavior when out of equilibrium.  Model this using appropriate software.
  • Understand the properties of the simplest dynamic box models (one box, one linear feedback): exponential growth, exponential decay, exponential approach to equilibrium.  Assess qualitative behavior of these models from the sign of the feedback.
  • Understand and apply the notion of doubling time. Relate to the equilibrium residence times discussed earlier.
  • Use exponents and logarithms to compute with exponential models.
  • Work with financial applications: loans, simple and compound interest, present value and discount functions.
  • Work with resource applications: static and exponential reserve indices for a resource.
  • Understand the properties of the logistic model for growth under an environmental constraint.
  • Assess the application of the logistic model to a non-renewable resource (oil and the “Hubbert peak”)
  • Assess the application of the logistic model to a renewable resource (fisheries management)
  • Understand the possibility of oscillation in box models.  Assess the Lotka-Volterra predator-prey model in this context.
  • Understand the possibility of overshoot in box models. Assess its application in real-world examples.
  • Make use of the notion of sensitivity of an equilibrium point to change of parameters.
  • Understand how feedback in a box model may enhance or reduce the sensitivity of an equilibrium.  Quantify this in a simple model of the ice-albedo feedback system for the earth’s climate.
  • Describe bifurcations (tipping points) in an equilibrium system as parameters vary.  In the example of the ice-albedo feedback, describe the “Snowball Earth” bifurcation.
  • Develop awareness of long-term variations in global climate, including the possibility of “Snowball Earth” events during the Neoproterozoic period.




This course theme includes probability, expectation, skew distributions and upside vs downside risks, uses and limitations of cost-benefit analysis, risk v. uncertainty

  • Understand and apply the notions random, event, probability, probability distribution, sample space, conditional probability, independent events.
  • Carry out simple probability computations in standard (discrete) examples: fair coins, dice rolls, etc.
  • Understand and apply Bayes’ Theorem to the computation of conditional probabilities.  Apply to evidence and inference.
  • Compare mean and median as measures of central tendency.  Understand (qualitatively) the notions variance and skewness.
  • Understand and compute with the notions payoff and expectation.
  • Carry out simple cost-benefit analyses involving expected payoffs.  Be prepared to engage discussion regarding the real-world limits of such analyses and the “precautionary principle”.
  • Be aware of nonlinearities in human assessment of risk, as expressed for example in the purchase of insurance, or in the experiments of Kahneman and Tversky.  Apply to human judgments of “acceptable” versus “catastrophic” risk, e.g. in the context of nuclear power generation.
  • Critically consider the methodologies of cost-benefit analyses of global ecological risks (e.g. the Stern report on climate change).




This course theme (likely the shortest) includes the basic language of graph theory, connectedness, components, transportation, and examples.

  • Understand the notion of a network (or graph) and the terms vertex, edge, valence.
  • Be aware of real-world examples of networks including social networks, transportation networks, ecological networks such as the hydrological and carbon cycles, food webs.
  • Understand and apply the notions of connectedness and connected component in a network.
  • Explain why a random network may be expected to contain just one “giant component”.
  • Explain why the diameter of a random network may be expected to be small (the “six degrees of separation” phenomenon)
  • Understand simple network models for Bayesian inference.
  • Investigate real-world examples.


Regular (graded) homework will be delivered via online quizzes using ANGEL or its successor.

There will be two midterm exams and a final exam.

A course blog will be maintained.  Students will be expected to post the results of their own application of the course ideas to real-world problems that they have identified.  They will be graded on their own contributions and on their comments on the contributions of other students.


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