by Shane L. Larson
Let’s talk about “numeracy” — to be clear, I define this to be the ability to use and understand numbers and data. It is similar to “literacy” but related to quantifiable things, things that can be measured, and things that obey well defined and incontrovertible rules. Specifically, I want to talk about your numeracy.
I have spoken with many thousands of you at public lectures, and many more thousands of you have sat through my introductory physics and astronomy classes. I certainly hope there are many more thousands of you reading this. At this moment, my former students have a certain advantage over all of you — they know better than to utter the words “I’m not good at math,” lest I get out an actual soapbox (I’m quite short), stand up on it, and wax on and on about how that simple phrase, “I’m not good at math” is a lie you have been taught to repeat. You might call such a soapbox speech a “rant,” or so my teenage daughter tells me (she also knows to never utter the words “I’m not good at math”).
I can already hear your objections. I know you will insist that you are bad at math based on a poor past experience in a calculus class, or the fact that you struggled with solving cubic equations, or because you don’t have a foggy clue what tensors are. Fair enough. But I don’t remember any of the soliloquies Shakespeare wrote in Hamlet; I can’t diagram a sentence, nor do I precisely remember the definition of a transitive verb, and I suspect the only reason I know about conjunctions is from singing Schoolhouse Rock songs. But I would not deem myself “bad at English” or “illiterate” as a result (my former English teachers might disagree; sorry Billie Wright!).
But this is precisely my point about math. You are not “innumerate” and “bad at math” — there are sophisticated and complex advanced topics you may have been taught at one point in your life and perhaps didn’t fully grasp and may not remember today. They were taught to you in order to develop the neural and cognitive framework of your brain, just like you were taught about diagramming sentences and iambic pentameter. Today, decades after you were last in a classroom, you may not remember all the details but you still have that cognitive framework. You are perfectly capable of using it. You are “numerate” and I know you are because I see you be numerate every day.
For example, suppose your book club is going to meet next Saturday and you need to order pizza. What do you do? You count up how many people are coming, you estimate how many pieces of pizza each person will eat (based on prior observations), you add a few extra pieces in that your partner and kids will take, and based on the number of slices in a pizza, you place an order. You pretty much get it right every time. Seldom have you ordered 78 pizzas for 7 people because you are “bad at math.”
What if you are trekking across the country on your National Parks road trip. Your dependable late 90’s economy car gets maybe 28 miles per gallon, you’ve got 3/4 of a tank and are heading out onto the long stretch of I-90 between Sioux Falls and Rapid City and quickly calculate when to stop for gas so you don’t run out. You pretty much get it right every time and have seldom been stranded out in the middle of South Dakota.
You are perfectly capable of assimilating data (data are things like previously knowing how many pizzas are eaten or how far you drive on a tank of gas) and using current conditions (things like how many people are coming to your club meeting, or how much gas is left in your tank) to make a numerical calculation (how many pizzas to get or how much gas to buy). You are fine at math. More to the point, you are numerate. You don’t think about it, of course, because the risk associated with over-ordering pizza is low; you seldom have to make a 400 mile run in your car without hope of seeing another gas station soon. It doesn’t change my point that you are, in fact, numerate.
The value of being numerate cannot be overstated in the face of the crisis the world faces today. Understanding what COVID-19 numbers are telling you, and perhaps more importantly what they imply about your personal risk, is critical to safely weathering the Pandemic so you can emerge on the other side. In the sea of numbers we hear each day, how do we absorb those numbers and use them?
Numbers have a certain implacable relentlessness to them, a modicum of unassailable truth that is regularly at odds with the distinctly human need to rationalize. That being said, a number’s implication for how it impacts your life requires context, otherwise it’s just a number devoid of how it relates to the world. We can use your numeracy to illustrate how context is important, and then apply it to understanding the current crisis.
Let’s begin with a learning experiment — a simple example that illustrates how information combines together to inform you about the world. Imagine I have a carpeted living room with a nice square grid pattern in the carpet, 20 squares by 20 squares (400 squares total). For some extravagant reason beyond the scope of this blog post, imagine I have dropped some nickels on the carpet; a lot of nickels. You are enjoying your lunch, and trying to decide if it is worth interrupting your delicious sandwich to go pick up all the nickels before someone else does. So you send me in to check the situation out.
CASE A: I come back with two nickels. Do you go pick up nickels or not? Without any context you really can’t decide. Unhappy with me, you send me back into the room and I come back with five more nickels! Now you have a total of seven nickels, or 35 cents! Do you go pick up the nickels or not? This is all about context of the data — what do the nickels I brought back to you represent? Did I bring you all the nickels, or just a fraction of the nickels? How many more nickels might there be?
CASE B: This time when I come back, I bring you 7 nickels, but I tell you they were all the nickels in one square of carpet. This is context. Context allows you to start figuring things out, because you are numerate. In particular, if every square has 7 nickels on it, and the room is 20 by 20 squares (400 total squares), then the room would have 400 x 7 = 2800 nickels, or $140! This is good context, but we could still do better.
CASE C: In the last example, you made an assumption. Assumptions are neither good nor bad — assumptions are limited. The important thing about assumptions is that when you make them, you are clear about what the assumption is, so if your understanding of the situation improves (you get more data), you know how to update what you think is going on. Above, you assumed every square had 7 nickels. Is that true? You send me back into to find out. I come back and tell you I looked at three more squares, and they had 23, 18, and 20 nickels in them respectively. This is greatly improved context, because you have many pieces of data. There are simple and complex ways of looking at data, even when you have only a few bits of information. One of the easiest is the average. What is the average number of nickels on a square? Based on our observed data:
Average = (7 + 18 + 20 + 23)/4 = 17 nickels per square (on average)
So now you can estimate that in the room there would be 400 x 17 = 6800 nickels, or $340 dollars! It is definitely looking like you should be collecting those nickels.This image above shows the data this example was drawn from — a 20 x 20 carpet grid, with 8000 nickels ($400) dropped on it. The first experiment where I brought you only 7 nickels told you something, but by collecting more data you developed a clearer picture of what was going on in the living room.
Now let’s use this example to help us understand something about the Pandemic.
As the coronavirus Pandemic has surged in the United States, considerable noise has arisen around testing and what the number of tests and results mean. Fortunately, you can use your numeracy to understand what the data is telling you. Two common testing numbers are reported for most states:
- The number of tests administered
- The number of new daily cases (number of positive tests)
In and of themselves, these numbers have no context, except that most of us have some rudimentary knowledge of our state to provide context — the critical knowledge here is the population. Population provides a simple way to understand how widespread the disease is: 500 cases in a county with 20,000 residents has different implications than 500 cases in a state with 1 million residents.
One of the most common points of discussion in COVID-19 testing is whether or not the number of cases is rising just because we are testing more. At the heart of this talking point is the more fundamental question, the question we really want to know the answer to: how do we know if the coronavirus is spreading and growing in our state or not?
Testing is just like our nickel example above, and you can use the nickel example to help guide you in your thinking.
NICKELS: Each square has some random number of nickels in it. If I look at one square, I get some sense of how many nickels there might be. If I randomly look at many different squares, I get a better, more reliable picture of how many nickels there are in the entire area of the carpet. If I get 7, then 23, then 18, then 20 nickels, there are on average (7 + 18 + 20 + 23)/4 = 17 nickels per square.
COVID-19: Take a fixed number of people, say 100. If I test those 100 people, I get some sense of how many COVID-19 infections there might be. If I randomly pick many different groups of people, I get a better, more reliable picture of how many COVID-19 infections there might be. If I get 7, then 23, then 18, then 20 infections, there are on average (7 + 18 + 20 + 23)/4 = 17 infections per 100 people.
Reporting the number of infections together with the number of tests given is called the positivity (or, more correctly, the positivity rate), and is a way of giving context to the data. Another way to give context is to report the total number of cases divided by the population (typically reported per 100,000 people, rather than the full population; this is more similar in size to a typical community and helps personal visualization about how widespread COVID-19 might be in a small city. Cases per 100,000 also is easier to talk about without making arithmetic errors!). Most state health departments and most major COVID-19 tracking sites that report daily data report both of these important numbers, giving you a better way to understand the risk.
So how do you tell if things are improving, holding steady, or getting worse? You watch how a number like the positivity changes over time. The number of known cases does increase with time. The number of known cases does increase with the amount of testing deployed. But the positivity rate accounts for that fact by always thinking about the data in fixed, similar sized chunks. In our examples above, deploying more tests means more groups of 100 tests to include in the average. Just like counting more squares on the carpet gives a better idea of the number of nickels, increasing the number of tests improves how well we know the positivity rate, which more accurately captures how COVID-19 is spreading in our communities. So the rule of thumb is:
- If the positivity rate is increasing, then for any random group of people you pick, more of them are sick with COVID-19
- If the positivity rate is holding steady, then for any random group of people you pick, the disease is not increasing rapidly
- If the positivity rate is decreasing, then for any random group of people you pick, the disease is slowly being eradicated
You could also replace “positivity rate” in these rules of thumb with “cases per 100,000” if that is an easier number for you to relate to. The story the data is telling you will be the same either way.
Now keeping all of this in your head can be hard, even for those of us who “do numbers” every day. Use your mental examples, like the nickels on the carpet, to keep you grounded. Tactile, hands on examples that you could actually recreate on your living room floor are often easier for your brain to work with, since they are easily visualized or even created, making it easier to stick in your mind.
We will come back to using simple mental models to keep our reasoning grounded in some more of our upcoming discussions. Until then, be safe, be well.
This is the second in a series of posts about scientific reasoning, instigated by the Global Pandemic of 2020. The first post and links to the rest of the posts in this series are:
- Pandemic 01: Learning in a Time of Crisis
- Pandemic 02: Numeracy and Data (this post)