courses/complexity/teaching-complexity.html
status = draft
date = 2004.02.18
There are five basic components for teaching and learning about systems that show complex behavior. These principles shouldn't be confused with steps, but should rather be seen as approaches that will contribute to student's being able to understand complex behaviors. These five principle components are:
Successful traditional approaches to teaching the sciences share a common hurdle. Students can be lead effectively through a process of learning concepts and making associations but at some point they have to cross an intellectual/pedagogical gap to see the same concepts using the generalized, powerful, deductive tools of the disciplines. For example, the students learn about toxicity, dose response and genetic variability and have to make the leap to apply the standard toxicity LD50 curve to the results. It seems that the dominant approach to this pedagogical gap is simply to lead students to the edge of this canyon multiple times, in multiple practice exercises until they get used to making this transition. This approach has been criticized as being the predominant pedagogy in introductory classes that doesn't prepare students for actually collecting and analyzing data themselves, without the jump to pedagogically pre-determined deductive models (Colander 2000).
Teaching students to understand complex systems has a different set of strengths and weaknesses. Having students wade right into the quagmire of actual complex systems and to experience the multiple levels of thick, rich and messy information. The students don't have to make any pedagogical leap to generalization, but they may have trouble having any confidence that the models that they construct and analyze have any basis in reality. It may seem to them that if a system has multiple possible paths and outcomes, then any model that provides a suitable range of outcomes would be suitable. The only way I know to describe how to work out of this problem is to address the epistemological issues such as the "sense of aesthetic unity", the relationship between structure and behavior, and the "logical typing" that Bateson (2002) addresses in his book "Mind and Nature: A necessary unity". The crux of the argument is how students learn and communicate about systems and the logical typing "is the difference between talking in a language which a physicist might use to describe how one variable acts upon another and talking in another language about the circuit as a whole which reduces or increases differences" (Bateson 2002, pg 100). Students will have to learn that the reason that it is so hard to claw their way out of the swamp of real complex systems is that they are having to simultaneously talk about the pieces (parameters and details) and the whole system.
Even more than in traditional science (where things seem to be what they are, i.e. objective reality), communication about complex systems is enhanced by using appropriate metaphors. It is easier to start describing a city of millions of people by referring to emergent behavior of ants (Johnson 2001). The metaphor calls up salient features of individual agents, communication, simple behavioral rules at one level and emergent behavior at another level of the colony/city. The metaphor has heuristic value because there are apparent similarities and differences.
Studying complex behavior requires a repertoire of good metaphors. Each metaphor should have the chimerical/hybrid/hermaphroditic? characteristics of what I call "traction" and what Hofstadter (1995) calls "slippabilty". Traction is the quality that there are some rough edges or a handle that allow the student to get a hold of the concept and perceive it to have an useful abstract quality from the very beginning. The apparent rough edges on the description indicate that this intellectual object is not meant to be a finished product but is something to be used. Laurillard (1993) calls an obvious feature that indicates how an object can be used an "affordance", such as a door knob or a hammer handle. But "traction" includes the additional feature that as the student uses the concept other features become apparent. Whereas a hammer might have an "affordance" a swiss army knife has "traction". "Slippability" is the idea that a metaphor is useful even though it is not an exact model of the target problem. In fact Hofstadter (1995) argues that good metaphors that have slippability will be most useful when they are quite a bit off target, that is, the greater distance between a perfect model of a system and a slippable metaphor the more valuable that metaphor is for learning.
The construction, collection and use of a set of metaphors is the provence of a good teacher. A catalog of metaphors should be assembled that span the range of possible complex processes. This catalog should contain enough examples to cover the range and each should have enough detail to be understandable and have some traction. However, the set should not have so many examples that it doesn't rely on the slippability between them. Constructing an appropriate catalog of metaphors will require a teacher's judgment and skill. In addition, the presentation and discussion of these metaphors should rely heavily on an improvisational approach which brings out the active process of understanding and demonstration of the use of the concepts.
Especially at the introductory level, there should be continuity between the tools that are used for simulations, building data into information and analysis looking for patterns. I don't know of any multipurpose application that meets these criteria, however Excel might come close. It has the features that allow you to construct simulations, create data streams, analyze and visualize data. STELLA and other dynamic simulation programs are very useful in the simulation phase and can be used for testing the sensitivity of systems to ranges of parameters, but there are no inference tools that have the same feel. The absense of a ideal package for student use is a particularly large barrier for introductory courses, and yet this is where they would be the most useful. There are texts that demonstrate how simulations can be incorporated into understanding complex systems. Roughgarden (1998) has created a very interesting text that integrates Matlab for simulating ecological processes. Ruth and Hannon (1997) have a text that provides many examples of dynamic biological systems using STELLA, including simlations of cellular automata, catastrophe and self-organizaiton processes. Gaylord ann Nishidate (1996) focus on the modeling of cellular automata using Mathematica which could link to the body of work by Wolfram (2002) on patterns and complexity. There is no package or text that I know of makes inferntial data analysis accessible to an introductory course.
The analysis of data from a putatively complex system requires visualization and statistical inference tools. For an introductory course, the visualization of patterns and simple statistical treatments to explicitly consider Type I or Type II errors should be used. Eventually however we would like students to be able to use much more powerful mathematical tools such as the list of about ten so suggested by Brock and Colander (2000) that includes general bifurcation theory and genetic algorithms.Each of these tools would require both computer expertise and curricular emphasis. Thus, where traditional science teaching reaches a common hurdle early in the pedagogical cycle, teaching complex systems will face a serious challenge in the analysis phase.
The following three examples are very briefly sketched out to show how complex processes could be incorporated into environmental science and policy courses.
Example 1: Sand box erosion
Based on the metaphor of a sand pile in one dimension, the question is how erosion might look in two dimensions such as a river basin. The students could gain experience by dripping water over a smooth bed of sand that is slightly tipped. This should result in erosion patterns that would be different each time they do the experiment. The patterns could be simulated with a cellular automata and this could be used to explore parameters such as water flow and the angle of the tip. As the students collect data on the process, they should eventually try to characterize the length of each stream or the area of each basin. This data can be analyzed by plotting the stream length to the basin area or computed outflow of each area with the drainage area. When these data are transformed to log-log and visualized, the plots should be nearly linear, indicating a scale-independent process. Thus, this simple exercise has the five components; a metaphor that needs to be expanded a bit, experience with a test bed and by looking at pictures of natural watersheds, simulation with a cellular automata, data collection that is open ended and not driven by an immediately apparent natural laws, and data analysis by visualization that shows a scale-independent pattern.
Example 2: Order through fluctuation
This second suggested exercise doesn't have a readily available system for data collection and relies on the simulation to generate a data set for analysis. The initial metaphors are for turbulence in water and the castrophe surface for a threshold effect. Simulations can be built easily in STELLA that will demonstrate locally stable behavior, but if a threshold is crossed result in a regime change to another region of behavior. A common example of this behavior is in lake eutrophication. As the input data to the simulation is varied, with imposed random noise, the output data can be collected as a pseudo-data set. Different runs of the simulation will show different outcomes. The students will then be given data sets that they know either resulted in switching or not and asked to analyze these using simple statistics and visualization. Finally, the students will test their ability to find a pattern with partial data sets (generated by the instructor) that go near, but not to the point of the switch. The value of detecting a pattern can be discussed in the context of the consequences of thinking there is a pattern when there isn't and not seeing a pattern when there acutally is.
Example 3: Multi-actor game (economic)
There are several examples of multi-actor games that can be used to simulate complex group decision processes. Stodder (2000) gives three examples that can be used in an introductory economics class (the efficiency of a competitive market, the difference between marginal and average products, and money as a efficient solution to the "double coincidence of wants" problem). Another game that is more involved and requires a mediator is based on the value of lake water for recreation or farming (Carpenter 2002). One point of these games is to collect the data of individual actions during play and then analyze that data later. Stodder describes how the economic games can be compared to traditional economic theories. Carpenter describes the individual actions in terms of the resiliency cycle. One obvious points that these games make is that complex behaviors should be expected, not a suprise, in even the simplest muti-actor game.
The principle components for teaching about complex systems are different than used in teaching traditionally about deterministic systems. There is a pedagogical hurdle faced in both activities. Students are expected to jump from associations to powerful generalizations in the traditional approach whereas students may be restricted by the mathematical analysis required to fully appreciate complex systems.