**Steve Hurst, University of Illinois**

The two major problems in teaching complex systems as I define them are 1.) student expectations of simple "cause and effect" relationships are eliminated for the most part, and 2.) the systems are not amenable to analysis by looking at small parts and then reassembling the system. The first is a problem because complex systems have feedbacks which typically obscure or remove the cause and effect ideology that students have built up through years of science education. Feedback in many systems removes the ability to say what caused what - the chicken and egg problem. So getting students to understand feedback is a primary task at the beginning of the semester. I work at this through analogies with everyday examples and use the STELLA modeling program in which feedback is visible to the student within the model.

Working with students to understand feedback is an ongoing process that typically takes the whole semester. Starting with simple feedback systems such as bank accounts, we work up through environmental models of CO2 cycles adding more feedbacks as the models become more realistic. Eventually, we study a socio-economic model of the world, the World 3 model, that has 2 major driving positive feedbacks and numerous negative feedbacks. In many complex systems, delays are intrinsically related to feedback and so it is important to look at models that contain both explicit and implicit delays. Implicit delays are not obvious from the how the models look or the underlying equations but are found in the characteristic time that is takes for a process to run. Such delays are often indicated in terms such as half-life, residence time, equilibrium time, mixing time, compounding and others.

The second major challenge to teaching about complex systems is that they are non-linear, they are more than the sum of the parts. So the typical Cartesian method of breaking down a system into small parts, solving each separately and reassembling into a whole does not generally work. This method works so well that even the modeling program STELLA that solves the non-linear differential equations uses it by breaking the equations into very small linear steps that approximate the non-linear solution. In fact, we have no general way of solving non-linear problems. For many, or even most non-linear systems, approximations are acceptable and we can often come very close to the "real" solution. Using the visual STELLA modeling program allows me to put off discussion of the problems of non-linearity until later in the course. The modeling program successfully hides the problems in the simpler models that the students work with at first.

The non-linear aspects of complex systems often first come up for discussion when students start validating and calibrating their models. They note that changes in a variable do not result in consistent or equivalent changes in the results. This leads naturally into working with error analysis and randomness. Randomness is a pervasive part of natural systems and must be built in to most environmental and economic models. Again, the World 3 model is a good complicated model that demonstrates the synergistic effects found in non-linear systems. Students work on analyzing sensitivity of the model to changes in various parameters and typically find that changing any one parameter results in practically no changes in the result. Only by changing multiple parameters and enjoying the multiplying effects of their synergy does the result show changes.

Using a visual systems modeling program such as STELLA and Vensim seems to aid in teaching complex systems analysis to students for many reasons. It reduces the learning curve in discussion of feedbacks because the feedbacks are explicit and visually displayed in the models. The programs allow me to skip discussion of many aspects of non-linearity until later in the course when students have absorbed earlier lessons. Discussions of the "holy trinity" of complex systems; feedback, delay and randomness is what ultimately brings most students to the realizations of non-Cartesian behavior and the fuzziness of the idea of cause and effect. Finally, by building models all through the course, students are able to build their own models of complex systems that they run across during their education and work.