Magical Thinking, Math Intuition and the Crime Decline
Public discourse has gone off the rails and its time for new approaches
Understanding the Crime Decline
Last month I published an article that argues that most of the evidence-based explanations for the crime decline in the 1990s had nothing to do with the criminal justice system. And, that these explanations can be grouped into categories of crime decline explanations—some categories are important but largely academic, and some categories can be directly translated into policy action. And further, it argues that the types of crime decline mechanisms that reduced crime in the 1990s and which can be translated into policy and which could reduce the present spike in violence again, by and large, have little to do with the justice system. It’s pretty interesting, you should check it out (or email me if you would like a copy). Here’s a nugget.
Now on to the show…
Mathematical Intuition as the Antidote to Magical Thinking
Recently the world has seen a rapid, and perhaps unprecedented, increase in magical thinking. Much of it is about conjuring defenses for morally untenable political positions. This was likely true in earlier eras of magical thinking. But, back then magic was used to explain physical realities we did not understand—today magic is driven by emotional reactions to metaphysical realities we fully understand but don’t like. Conjuring a reality you can actually get behind…
Among the cognoscenti, the solution to magical thinking is more science. After all, it is a disbelief in science that the audience at the carnival of illusion must suspend. But our thinking about science in general and STEM, in particular, has become its own form of magical thinking. I want to make an argument here for thinking about problems scientifically as the overarching goal of scientific pedagogy rather than learning the mechanics of solving problems with science. This may seem like a subtle distinction, but it is actually quite seismic.
What follows is absolutely a defense of science and its role in society. But for a groovy kind of science that people get. For most people, science in general, and math in particular, is a foreign language. We should acknowledge that most people who study a language do not become fluent, that at best they become marginally accomplished at conversation. But the difference between a visitor who can get around a foreign place in their halting version of the local language and the visitor who storms about in their own is profound. So this is an argument for nudging people toward a groovy intuition rather than hammering on verb conjugation. Because intuition is where you start before you think about something—it is where your bias lives. More simply, intuition is where the magic happens. So, let’s focus on changing the intuition.
If you change your intuition, you change your mind.
The Good and Bad of STEM
STEM is hot, there is no debating that. STEM literally means science, technology, education, and mathematics, but the acronym itself suggests a deeper purpose. Like stem cells, proponents see STEM as the raw material from which knowledge is generated, etiology and ontology united, under the banner of science. Science is good for the country, yielding innovation and productivity gains, and good for individuals, who use it to become healthy, wealthy, and wise. A little hyperbolic perhaps, but all for the good so far.
STEM gets very little criticism but perhaps it deserves a morsel. Let’s start at the top. At the top, we see STEM, but what we don’t see is the forced dialectic cleavage of all knowledge into STEM and not-STEM. In order to privilege STEM, we must deemphasize something else (not-STEM). That is, we seek to privilege science and to deemphasize not-science. That bit does get a fair bit of criticism. But it presumes a duality—that not-STEM can also be important, which misses the point that both STEM and not-STEM exist along a continuum, and what is STEM and what is not-STEM is really murky everywhere but the extremes.
For example, I like to think that I do science for a living. But since I do social science, proponents of ‘hard’ science and ‘natural’ science scoff and eye roll at my claim. They might say, calculus is a hard science, and so are physics and chemistry. Running some regressions on people, which is what I do, is not hard science. But that hard science distinction is already problematic for STEM proponents because this hard science thinking definitely takes education out of STEM and probably most technology as well.
Then it gets even trickier. Because all of this is just saying what real science is not. To be a good scientist, you have to define your terms. And when you define your terms, then you have to say precisely what science is—what science is in terms of its own attributes and whatnot, rather than defining it by what it isn’t.
This turns out to be really hard. Standing in the way of defining science is the Demarcation Problem. The Demarcation Problem asks, essentially, what are the immutable, invariant attributes of knowledge? What are the unchanging bits of knowledge that are permanent and true that we can absolutely say real science has? And once we know these attributes we can then ask, what are the boundaries of science?
What are the boundaries of science? Now, that’s a hell of a good question, right? If you were so inclined, you could pull out your well-thumbed copy of Aristotle’s Posterior Analytics or Popper’s Conjectures and Refutations to find some steady wisdom on this daunting question. But fortunately, you don’t have to, because Larry Laudan wrote the Demise of the Demarcation Problem in 1983 and solved the problem, or at least enough for a Substack newsletter. Laudan’s take is that we are confounding two questions: when is a belief well-founded and reliable? And, when is a belief scientific?
Laudan argues that the latter question is pretty worthless because it presupposes that there are some invariant attributes of science out there waiting to be discovered. Only, given two thousand years of searching, we have not found them, suggesting that their very existence should be in question. And, further, he argues that if we focus instead on figuring out what makes a belief well-founded we are very likely to include almost all of the things that make a belief scientific, so why continue to beat our heads against that wall? Then he takes some shots at people from Edinburgh and says some derisive things about “Scottish sociologists of knowledge” which sounds like a personal beef and rather unscientific.
But then he says this, which is quite profound:
However we eventually settle the question of reliable knowledge, the class of statements falling under that rubric will include much that is not commonly regarded as 'scientific' and it will exclude much that is generally considered 'scientific'.
I think ultimately that is quite a swipe at STEM supremacy and also exceedingly reasonable. So, let’s think a little bit more about what reliable knowledge is, and how we can instill reliable knowledge into students and not instill unreliable knowledge.
***
Lately, it seems intellectually fashionable to criticize STEM curriculum, particularly high school math curriculum, as woefully outdated. We are entering a data science world, they say, and we must teach data science things. In a data science world, understanding the nuances of correlation, association, and causation will make us 10% happier and 26% more prosperous. Or something. But the point is the critics of the present math curriculum want more statistics and less trigonometry. Personally, I have a better intuition for single variable algebra than trigonometry, so it’s easy for me to root for the critics and so I’ve been slow to recognize how misdirected these arguments are.
I don’t that the main problem conspiracy theorists and other magic makers are experiencing is a lack of statistical knowledge. I think it is more about a lack of basic math intuition that leads them to fail to reject ideas that should be immediately rejected. Their problems stem from a fundamental lack of intuition about how the world works. That’s a bigger problem than not understanding probability or sampling.
It turns out though that mathematical intuition[1] is a complex scholarly topic! And one that lives at the intersection of mathematics and philosophy. It also turns out that most people who study mathematical intuition think about it in terms of numerical ability and refer to it as numeracy. And say things like, “numeracy is to mathematics as literacy is to language.” And, other things like “making good decisions in the real world requires some numerical ability.”
Now, before diving into numeracy and thinking about the critical issue of framing and what we might teach school kids so they don’t get trapped in bad frames, I want to address the word ‘ability’. I don’t want to just skate by the term and pretend not to see it. It’s hard to think of a more loaded term. So let’s see how it fits.
The first thing to know is that there are a huge number of factors that affect ‘ability’. Xie, Fang, and Shauman (2015) wrote a tremendous article in the American Sociological Review on STEM Education that details the myriad sources of differences in ability with respect to STEM. There are any number of social determinants of education in general and STEM in particular—neighborhood and other contextual factors, family influences, peer effects, and individual attributes like physical and mental health—that all affect ability.
The second thing to know is that “social psychological factors are more important influences on participation and achievement in STEM versus non-STEM education.” More affirmatively, ability in STEM is driven by social psychological factors. And social psychological factors in this case are explicitly defined as non-cognitive.
The third thing to know, as Robin Williams says while he hugs it out with Matt Damon in Good Will Hunting, is that innumeracy is not your fault. If you too are rather innumerate, it’s not your fault. It’s not your fault.
Many otherwise well-educated persons are innumerate…Unless the mathematics studied in schools is understood with confidence…it will not be used in any situation where the results really matter. When apprehension, uncertainty and fear become associated with fractions, percentages, and averages, avoidance is sure to follow. (Steen, 1990:216)
The next thing to know is that numeracy can be taught. A study of effective numeracy teachers reported that the key is that
[h]ighly effective teachers believed that being numerate requires having a rich network of connections between different mathematical ideas.
It turns out that a lot of innumeracy is the result of poor framing. But hold that thought for a minute. First, one last thought on numeracy.
All too often, schools teach mathematics primarily as a set of skills needed to earn a living, not as a general approach to understanding patterns and solving problems. The distinction of mathematical study from other school subjects—from history and sports, from language, and even from science—is one of the major impediments to numeracy in today’s schools (Steen, 1990: 222).
Framing
There are several generally obvious impacts of innumeracy on personal and professional development. In addition, it is increasingly obvious that there are critical effects of innumeracy on society and culture. And by effects on society and culture, I mean specifically the increasing appetite for misinformation. Misinformation is an interesting field of inquiry and there are lots of ways to approach it, but one understudied area is the confluence of framing and innumeracy.
Framing is pretty much self-explanatory. A food label ‘frames’ the product with a description, for example, a description of a meat product can be framed as either 75% lean or 25% fat (if you are thinking of entering the field of meat label framing, you should know that research suggests highlighting the lean percentage yields more buyers). Now, framing is a standard feature of life in the modern age, notably in marketing and politics, but at home as well, as your little rascals are probably accomplished framers delivering carefully crafted messages to you.
The idea here is that how data is presented—the way it is framed—has different effects depending on numeracy. Classic studies ask survey respondents to rate their preference for something based on the likelihood of its occurrence. And that likelihood is presented in two ways, for instance, asking how attractive a choice is if it has a 20% of occurring and asking how attractive it is if it has a 2 in 10 chance of occurring. These studies generally find that more numerate people are more accomplished at seeing through the frame. And again, the study finds that it is numeracy, not IQ, that matters as “[a]nalyses showed that the effect of numeracy was not due to general intelligence” (Peters, et al., 2006).
Thus, to the extent that people’s susceptibility to misinformation is a result of innumeracy, improving mathematical literacy could have important implications for preserving democracy. (Numeracy is of course only one element in the fight against fake news.)
Some Ideas for Increasing Numeracy
All of that suggests that there is great value in fighting innumeracy, that there are tools to do so, and that a concentration on STEM over not-STEM may beg the question. So, to conclude, I want to build on the idea that “being numerate requires having a rich network of connections between different mathematical ideas.” So let me present some topics for study that can help with the innumeracy problem, that are generally glossed over as small points in math pedagogy, and, that can connect mathematical ideas without getting into computations.
Orders of magnitude
You were almost certainly taught orders of magnitude, probably in middle school, but it’s unlikely that discussion involved policymaking or pop culture where its value plainly resides. So, the groovy intuition here is that orders of magnitude are where the distance from 1 to 10 and the distance from 10 to 100 are made equivalent. So, 1,000 is three orders of magnitude greater than 10. The mathematical approach is to think about base 10 and exponents and logarithms and I dig it, but I bet loads of you do not.
Why does that matter for policy? It matters because there are 330 million Americans and 330,000,000 is just an absurdly big number. At his peak, Elon Musk’s net worth was about $330 billion, which is also $330,000,000,000, which is even more absurd. The original bill for the American Rescue Plan Act of 2021 came in at around $3,300,000,000,000 or $3.3 trillion.
So, how absurd is Elon Musk’s wealth? How big is the cost of ARPA? Well, Elon Musk’s wealth is 1000 times (three orders of magnitude) bigger than the number of Americans. ARPA was 1 order of magnitude bigger than Elon Musk. In my opinion, using orders of magnitude, the distance from all of us to one Elon Musk is fairly measured as much larger than the distance from all of us to a program designed to help…all of us. So, when you are trying to think about which is more concerning, wealth inequality or the size of government, it’s helpful to have some intuition about the size of each.
Putting aside Elon Musk for the moment, if that‘s even possible, let’s talk about commotio cordis, which is a direct blow to the chest that disrupts the hearts beating and appears to be why Bills safety Damar Hamlin’s heart stopped on a Buffalo football field during a game. In my corner of suburbia, a lot of parents are now worried about their kids playing sports—including youth baseball, hockey, and lacrosse where there have also been commotio cordis incidents. Should they worry?
Two facts and some math intuition help answer the question. About 30 Americans each year experience commotio cordis. But it is an extreme event, because the timing of the traumatic blow has to be just right as “[t]here is just a 20-millisecond interval in the heart’s cycle when a strong blow can cause an arrhythmia.” So, is 30 a big number? And, how long is 20 milliseconds (and, is a millisecond one one-thousandth of a second or one one-millionth?)
Our newfound math intuition around orders of magnitude is helpful. The US population is 6 orders of magnitude bigger than commotio cordis prevalence which is gigantic, and supports the idea that an occurrence is unbelievably rare. A milli-second is one one-thousandth of second, or three orders of magnitude smaller than a second. For perspective, it takes 5 to 20 times longer to blink your eye than to get hit in just the right part of your chest at just the wrong moment. While some intuition about probability calculations would help to confirm, it’s safe for your kids to play lacrosse. At least on this measure…
What else do we need intuition about? Scads of stuff. Type I and Type II errors are really important in all kinds of venues. Some folks like to think of this as false positives and false negatives and that’s groovy too. I have to look it up every time, but essentially one type of error is when you overestimate the likelihood something will happen and then it does not happen. And the other is when you underestimate the thing and it does happen. The device I like here is the metaphor of the train that approaches while you are on the track—in one case you think the train is not coming and it does, and in the other, you fling yourself into the weeds and no train appears. You have to be thoughtful about this, and carefully consider both types of error, because while the answer is obvious in this case, one type of error does not always dominate. That railroad track might be the only path away from an even bigger risk.
We used to talk about counterfactuals but now we talk about potential outcomes but either way, it’s really important. I think huge amounts of magical thinking occur around this point. This idea here is simply about how to make a comparison. We tend to think really hard about what the implications will be if the world changes in some important way (the treatment), and very little about what no change means (the comparison or the counterfactual). And the trick is that the world is always changing, so you have to be equally thoughtful about the nature of both realities. What I love here is the contemporary transition in language—counterfactual seems to imply ‘against fact’ and dissuades us from making a comparison. The ‘potential outcomes’ language creates a symmetry that makes comparisons more engaging. More language shifts please to promote fair comparisons...
I’ll stop here, but there is a whole cannon of scientific intuition that is helpful. How about a course on fallacies?! The ecological fallacy is my favorite, the idea that if you find a general pattern you cannot, YOU CAN NOT, then affix that general pattern onto an individual. Post hoc ergo propter hoc is the idea that simply because something comes before something else, the early thing does not necessarily cause the later thing. Yo! That’s a fallacy. There are loads more.
Or how about some seminars in distributions? It could have the subtitle: You There, My Friend, Should Know That Very Few Important Things Are Explained Well By A Bell Curve (or how about: She’s Not Gaussian, She’s My Sister). Or how some things are bimodal, and thus when you go to a party where the average age is 25, you should not be surprised to find a room full of small children and their parents. Or the Power Law, which helps us understand that exponential distributions are everywhere, and when they appear in the universe, a small number of things (20 percent) explain most of the outcomes (80 percent). Approximately.
Finally, a plea for the simple focus on teaching the intuition of causality. Magical thinking substitutes emotion for reason. But the process of causal analysis—the intuition—is where the true magic of science lies. You must define your terms. You must state why your test is fair, accurate and reasonable. You must acknowledge your errors. As John J Heckman and Rodrigo Pinto say:
Good policy analysis is causal analysis. It analyzes the factors that produce outcomes and the role of policies in doing so. It quantifies policy impacts. It elucidates the mechanisms producing outcomes in order to understand how they operate, how they might be improved and which, if any, alternative mechanisms might be used to generate outcomes. It uses all available information to give good policy advice.
It systematically explores possible counterfactual worlds. It is grounded in thought experiments – what might happen if determinants of outcomes are changed. In this regard, good policy analysis is good science. Credible hypothetical worlds are developed, analyzed, tested in real world data.
Anyway, all of these ideas appear fleetingly along the way in our scientific education. We should pause and reflect on them more intentionally.
Musical Interlude
In the timeless words of David Bowie, I know when to go out, and I know when to stay home… and get things done. And with that in mind, it’s time to go out. So, the already infrequent External Processing will be hitting the road off and on this year to traipse about with Bruce and the guys and gals from the E Street band. I walk this road with a fiery lantern. I’ll see you when I see you. Here’s some magic for the journey:
[1] If you descended into the footnotes, you are probably looking for a formal definition of mathematical intuition. See Parsons, Charles. 1995. “Platonism and Mathematical Intuition in Kurt Gödel’s Thought.” The Bulletin of Symbolic Logic 1 (1): 44–74. On page 45, footnote 3, Parsons cites “Leibniz's "Meditations on knowledge, truth, and ideas" [11]. Knowledge is intuitive if it is clear, i.e., it gives the means for recognizing the object it concerns, distinct, i.e., one is in a position to enumerate the marks or features that distinguish an instance of one's concept, adequate, i.e., one's concept is completely analyzed down to primitives, and finally one has an immediate grasp of all these elements.