**I just finished putting together a second draft for the description of the course I hope to teach next year (for GreenFaith readers, this is part of my GF Fellowship project). Would love to have comments both technical and otherwise. Here are links to other posts in this series: part 1, part 2, part 3, part 4**

John winding up for the 2-minute version of the course at the GreenFaith retreat this January |

**MATH 033 – MATHEMATICS FOR SUSTAINABILITY**

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__

*General*

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.

*Measuring*

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.

(

**For mathematicians**: the math skills involved here are numerical calculation, and the solution of equations in one or two variables, either algebraically or graphically. Dimensional analysis is in fact computation in the real group ring of**Q**^{d}, where d is the number of dimensions, but I am unlikely to explain it like that to students in this class.)*Changing*

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.

(

**For mathematicians:**Exponential and logarithmic functions are staple topics in courses at this level. Here, we introduce them through population and resource questions as well as through the traditional “personal finance” route. The rest of the “box models” material is qualitative theory of ordinary differential equations, treated at a more sophisticated mathematical level in our MATH 417. It would be nice to introduce the phase plane in describing the Lotka-Volterra model, but I am not sure whether that will be possible. The notion of “sensitivity” is really a partial derivative, and calculations of “sensitivities” in the presence of feedback involve linear equations coming from the chain rule for partials. Of course, all of this machinery will be operating strictly behind the scenes.)*Risking*

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).

(

**For mathematicians**: Again, the elementary probability is standard in a course at this level. “Cost-benefit analysis” simply means computing and comparing expectations. I would like to include more material than is usually done about the extent to which probability computations*fail*to capture human intuitions about risk and uncertainty, as this is highly relevant to sustainability questions and is often ignored in a mathematical presentation.)*Networking*

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.

(

**For mathematicians:**There is another MATH 03x course which could be given simply about these concepts – see*Networks, Crowds and Markets*by Easley and Kleinberg for a textbook.)
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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