Can We Really Understand Exponential Growth?
By Keith Devlin @profkeithdevlin
With a life-threatening pandemic, the only way to get a picture of the (invisible) virus’s past, current, and future spread is by mathematics; in particular, tabulating data and drawing graphs. As a result, newspapers, magazines, and news websites are replete with graphs and tables these days. But how well is the average citizen — or the average lawmaker or civic administrator — able to adequately digest those displays and make good use of them?
A display from the Washington Post, May 31, 2020. See below for details
For understanding the past, the tables and graphs give a fairly accurate picture of what happened. (Only “fairly” accurate, since there are often problems with collecting the data. It can be incomplete, it can be numerically inaccurate due to collection problems, and it can be incorrect due to erroneous classification, such as, with the current pandemic, the decision how to classify a death due to pneumonia brought on by COVID-19, which can vary from one jurisdiction to the next.)
Still, on the whole, the tables and graphs give us a fairly good understanding of what happened. The pictures we get of the current situation are less definitive since the data is still coming in, so we typically simply project to today based on the trend over the most recent days. Since viruses follow known physical and biological laws, our knowledge of today is essentially as good as for the recent past.
When it comes to the future, however, things are less certain. Yet as long as the world is in the middle of this pandemic, reliable predictions of the possible futures are what we need most of all. We can use those predictions to take appropriate actions now, to guide us towards a more favorable future.
The use of math, in particular graphs, to predict the future, is the topic I wrote about in last month’s post. I gave examples of how even smart, well-educated decision makers were having enormous troubling interpreting the graphs correctly, and explained why this is the case; the main problem being that, in the world of prediction models, graphs are used differently than the way they are used to represent data in the familiar math or science classroom.
But even in the case where graphs are used in the familiar way, to represent data from the past, it’s not clear that average citizens, let alone key decision makers, are able to glance at a graph and get a reliable picture of what is going on. Indeed, it’s not clear if they can understand and make good use of a graph even if they devote some time to it, including playing around interactively for a while, if the graph permits interaction. Doing that requires a good dose of number sense, a crucial mathematical skill in the 21st Century—as I have argued in this blog and elsewhere (google “devlin number sense”).
To start with an “easy” example, on May 30, The Washington Post published an excellent article, with an interactive presentation of graphs and tables, that claimed to show that the pandemic’s overall US death toll, which had just been widely reported as having crossed the 100,000 threshold, had likely surpassed that level some weeks earlier. See the image at the top of this post for a sample.
I spent a short while exploring the Post’s data, and not only found it pretty convincing, I came away with a good sense of why the reporting had lagged so far behind the reality. I also learned a lot more as well. It was all there in the data. Everything needed was well presented, and I did not have to work hard to follow and understand.
That establishes that it can be a useful resource for someone, like me, who studied mathematics to the Ph.D. level and spent my entire career doing mathematics! But how well can others make use of this data? It was, after all, published in a national newspaper, hence intended to be used by mathematically-lay readers. Was a typical middle or high schooler—whose mathematics knowledge is still fresh—likely to be able to read and understand it, for instance?
Fortunately, as I was working on this post, an answer dropped right in my lap, in the form of the May 31 post by Mike Lawler on his Mikesmathpage blog.
Lawler is a former university mathematician (he was on the MIT Putnam Exam team as a student) who left academia but found his way back into teaching when, in 2011, he began homeschooling his two children, now aged 16 and 14, regularly posting videos of his math sessions with them. Those videos now provide a good view of what learning math involves for school-age children when their teacher is a mathematician dad—a mathematician with a teacher’s gift, I should add.
The May 31 post I just linked to presents the videos of the Lawler boys, with their father’s help, trying to make sense of the Washington Post resource. Check it out and ask yourself if, for example, you think your Congressional representative could do as well. (If you suspect they cannot, you might want to make sure that they have a good math expert on their advisory team.)
And that was for an article intended to be read by an average reader of a quality newspaper!
I had a very different experience with the pandemic-data resource designed for a more quantitatively-proficient reader, published by the Oxford Martin School, the University of Oxford, and the Global Change Data Lab, analyzing the per capita coronavirus deaths around the world as the pandemic has progressed.
A graph from the Coronavirus Pandemic Data Explorer, produced by the Oxford Martin School, the University of Oxford, and Global Change Data Lab
It’s a phenomenal resource. When people talk about Big Data, this is the kind of thing they have in mind. But coming to grips with this resource requires some expertise. It’s provided for the experts—the people who advise leaders, lawmakers, administrators, and the like.
But to what degree can the experts digest the information that the site provides? On a technical level, an expert can surely make good, safe, productive use of such a resource. But how well do we feel (on an instinctive or emotional level) the data the graphs are presenting? Or does it remain forever theoretical?
Let me home in on just one aspect of the above graph: the vertical axis. It’s a logarithmic scale. That’s not surprising (to math types), since the spread of the virus is (during the initial pandemic phase) exponential. The logarithmic scale of the vertical axis compresses all the mortality figures downwards, in a progressive manner. With normal axes, the plotted points would all be leaping upwards with steeper and steeper slopes.
Exponential growth is something that the evolutionary development of our brains did not prepare us for. We do not have the kind of everyday-world, or kinesthetic, sense we can acquire for linear or polynomial growth. I certainly don’t have that. Sure, I can do the math, and understand it in a technical way, and I can use that math appropriately when I need to. But unlike with addition or multiplication, the math of exponential growth does not formalize my brain’s natural thought processes, it extends them (to a domain I have no everyday experience of, apart from living through exponential growth of viruses, climate change, and the like).
To illustrate the degree to which exponential growth can trick our minds, math educators often present students with a simple little puzzle called the Lily Pond.
After how many days is the pond half full of lilies?
Imagine you have built a swimming pond in the garden, and for decoration you drop a newly acquired lily in the middle. The vendor says you can expect that kind of lily to grow by doubling every day. So after one day there are two lilies, on day three there are four, then eight on the fourth day, and so on. On day thirty, you find that the entire pond is covered in lilies, and you can no longer swim. After how many days was the pond exactly half full of lilies (which is probably when the alarm bells would have rung in your mind that you need to pull some out so you can continue to swim in the pond)?
Many people get this wrong when presented with it the first time. If you are such a person, take solace that people get it right only because they have, either explicitly or implicitly, had the correct answer pointed out to them at some time in the past. I discussed all of this in my recent post in the BrainQuake blog, written with middle-school math education in mind. So you can check it out there. For now, I’ll just say that the reason our brains lead us to the wrong answer is that we naturally think in terms of linear or polynomial growth, whereas the lilies grow exponentially.
Other than using puzzles like the lily pond to help people come to grips with the cognitive challenges posed by exponential growth, the best (actually only) other resource I know comes in the form of two short videos, one a recent remake of the other, a classic produced over forty years ago. Both show the growth of the exponential function 10^x (“orders of magnitude”). Unfortunately, they only connect to our everyday cognitive experience for the first few values, so they don’t really provide us a real sense of the growth. They just show exponential growth gets you very quickly to the beyond astronomical and the ultra microscopic, neither of which we can really grasp.
Powers of ten, created by Charles and Ray Eames for IBM in 1977;
Cosmic Voyage (original in IMAX format), directed by Bayley Silleck, produced by Jeffrey Marvin, and narrated by Morgan Freeman in 1996.
In other words, exponential growth is beyond our natural intuition and comprehension. That’s why any phenomenon involving exponential growth is potentially dangerous, why mathematics provides the only tool we have to handle it, and why leaders and societies can do themselves and us possibly irreparable harm (including racial extinction) if they ignore what the math tells them.
FOOTOTE: To find out a bit more Mike Lawler, see the March 26, 2018 post on the AMS blog.