As I end my time as the math teaching fellow at Conserve, I would like to share some reflections about one of my major projects this semester. This semester, with the help of a great book called “Quantitative Reasoning and the Environment,” the lead math teacher (Caitlin Lemley) and the rest of the teaching staff, I developed and implemented a lesson about modeling critical systems in nature.
If you’ve read previous stories I’ve written, you may have seen another example of math modeling in my story about modeling pollution flowing through a chain of lakes. That type is somewhat deterministic. If you know the flow rates, you can reasonably accurately predict the concentration of pollutants each year. Many things in nature are far less predictable. When will a lightning strike lead to a devastating wildfire? When will the next snowfall trigger an avalanche, and how massive will that avalanche be? When will the next big earthquake occur? The job of a mathematician is to lean into that chaos and do their best to tease out its underlying structure and to try to understand uncertainty.
The examples I listed are thought to display a property called “self-organized criticality.” These are systems where simple local interactions spontaneously result in complex behavior. One characteristic of such systems is that they respond differently to the same event depending on the state of the system. For example, whether a lightning strike triggers a wildfire, and how large that fire will depend on the fuel load present.
To understand this concept, we played with a toy model called a “bingo box.” A bingo box is a 5 X 5 grid (like you might play bingo on!). We used a random number generator to decide in which box to put a tally mark. When a square got four tick marks, those tick marks were erased, and a tick mark was placed in the box above, below, to the left and the left of the original. This kind of event was called a “bingo-quake.” We kept track of how many boxes had to be emptied in one go to determine the size of the bingo-quake.
If you are interested in the mechanics of this exercise, check out the activity at https://sites.google.com/view/qre/chapter-projects/chapter6?authuser=0
The important thing about this model is that it has the characteristics of a system with self-organized criticality, which are as follows:
1. The system is composed of parts.
2. The system is fed by energy at a constant rate.
3. The release of energy is accomplished suddenly by avalanches of events.
While going through this activity and keeping track of their data, students began to feel the tension of a critical system. They noticed energy building up and began to feel like they could predict how big the next bingo-quake would be, but also noticed that it was challenging to find a pattern and make accurate predictions.
Our next task was to investigate how well our bingo-box modeled actual earthquakes and fault lines. This entailed using regression to see if our data about bingo-quake size followed a power law distribution, which is the type of distribution that characterizes earthquake size. Roughly, a power law distribution contains many small events, some mid-sized events, and a few significant events.
Several things are pretty cool about power law distributions:
The main one is that since power functions are scale-free if a natural event follows a power law distribution, the frequency data from the small, frequent events can be used to predict the frequency of the large, rare events. We got to discuss that this is the phenomenon people are talking about when they say things like “We’ve had five hundred year floods in the past ten years” or “The Pacific Northwest is due for a big earthquake.”
Another neat thing is that when power functions are log transformed, they become linear, which makes it simple to identify power law distributions. We discussed the fact that the Richter scale is a log scale, which makes good sense given that earthquakes follow a power law distribution!
I hope that this lesson helped our students understand chaos and become a little more comfortable working hard to understand things that seem unpredictable and uncertain. Besides being an excellent math skill, this strikes me as the kind of intellectual courage that is required to try to make progress on solving many environmental issues today.
One thing I appreciated in organizing this lesson was all the people at Conserve School who were there to help me learn and grow. Caitlin was a wonderful mentor who encouraged me to try teaching some pretty heavy hitting math concepts to our students. The teachers were all incredibly helpful in helping us test out the lesson before running it with students and providing thoughtful feedback and affirmation. Our students were engaged and curious and were open to finding joy in math class.
Thank you to all!
Photos contributed by Caitlin Lemley, Math Teacher.
Langkamp, Greg, and Joseph Hull. Quantitative Reasoning and the Environment: Mathematical Modeling in Context. Pearson Prentice Hall, 2007.