I think math is worth learning because mathematical thinking can help humans understand the world. As I’ve been inundated with news about COVID-19, I’ve tried to practice habits of mathematical thinking to better understand my place in the pandemic.
One of the toughest yet most gratifying things about teaching math is helping students to see the deep structure of a problem and not only the surface structure. The human mind has a built-in bias toward surface structure. We first notice what’s most visible and salient about a situation, even if it’s not the most meaningful. Students might look at three word problems, see cheeseburgers, drag racing, and cell phone plans, and assume they are unrelated concepts. These situations look different on the surface, but have the same deep structure of linear functions. My job is to help students recognize deep structure. Structure is everywhere in math. I want students to connect unit circle and function representations of trig functions, choose fractions or decimals strategically depending on the context, or recognize that solving a system of equations is the same as finding the intersection point of graphs of functions. It’s all structure.
I’ve been trying to practice understanding the pandemic with a similar understanding of structure. What are the surface features that can distract from the deeper structure that I should be paying attention to?
One piece of surface structure I see everywhere is the six foot rule. We’re constantly told to stay six feet away from humans not in our household. We’re redesigning schools to keep students six feet apart, lining up customers checking out at six foot intervals, and rethinking public spaces like subways and planes where that level of physical distancing is impossible. Six feet of separation is clearly a good idea! The evidence seems clear that if we had begun physical distancing sooner, thousands of lives would have been saved. At the same time, staying six feet away from people around me does not make me immune from the virus. I can walk right by someone without getting infected, but catch the virus having a sustained conversation from ten feet away.
The six foot rule is surface structure; the deep structure is that infected people exhale virus particles. If a healthy person inhales too many of those particles, they are likely to get sick. Erin Bromage has a great article that helped me to understand the subtleties of virus transmission. Walking past someone outdoors, away from large crowds, presents almost no risk. Yet if a pre-symptomatic infected person were to spend an hour in an enclosed space with a dozen other humans talking about rational functions, even if everyone wore a mask and stayed six feet apart, infection would be pretty likely. If a pre-symptomatic infected person sneezed in a bathroom and a healthy person walked in a few minutes later, they would have a decent risk of infection without any direct human contact. I don’t think we should get rid of the six foot rule. But we should recognize that, while it’s a useful rule of thumb, staying six feet away from others does not reduce risk to zero. If we can use the six foot rule but also be mindful of risk that is not mitigated by six feet of distance, we can make better decisions about reopening spaces while minimizing risk. If we reopen spaces, for instance schools, with a laser focus on keeping everyone six feet apart but a lack of understanding of the deeper structure of virus transmission, we’re likely to end up in trouble.
I find the habits of mathematical thinking that I try to teach helpful in understanding the pandemic. I’m not arguing that math is the only thing we need or that math is the solution to everything. But I believe mathematical thinking is valuable. Mathematical thinking also really hard to teach. What would math class look like if our we put more emphasis on helping students understand the world around them, and less emphasis on the standards and concepts that we’re used to?