Imagine a test for cancer that is 98% accurate.
Now assume that 0.5% of the population has cancer.
That means in a test group of 10,000 people, 50 of them have cancer. If the test is 98% accurate, then 49 of those 50 will test positive: that means 49 accurate positive tests.
The other 9,950 people in the group do not have cancer. But if the test is 98% accurate, then 2% of them will get incorrect results, i.e. false positives. 2% of 9,950 is 199 false positives.
Are you surprised that the false positives outnumber the true positives 4 to 1? I was.
Now imagine that instead of testing for cancer, we’re testing for drug use. A misunderstanding of these numbers could have major implications in states where laws require low-income people to pass a drug test before they can receive food stamps or other public assistance.
In his 1988 book Innumeracy: Mathematical Illiteracy and Its Consequences, mathematician John Allen Paulos discusses the many ways that innumeracy, “an inability to deal comfortably with the fundamental notions of number and chance, plagues far too many otherwise knowledgeable citizens” and has real-world consequences for health, public policy, criminal justice, and other areas. “The same people who cringe when words such as ‘imply’ and ‘infer’ are confused react without a trace of embarrassment to even the most egregious of numerical solecisms,” he writes. “In fact, unlike other failings which are hidden, mathematical illiteracy is often flaunted: ‘I can’t even balance my checkbook.’ ‘I’m a people person, not a numbers person.’ Or ‘I always hated math.’” (3-4)
You know who you are.
Paulos points out that our lack of understanding of statistics in particular can affect policy in many ways. The media’s tendency to fixate on every terrorist attack and mass shooting, for example, leads citizens to believe that their risk of dying in one of these is far higher than it actually is, and it leads politicians to focus on policies addressing those issues, but ignoring far more common killers such as suicide and heart disease. (Steven Pinker also discusses this phenomenon at length in his book The Better Angels of Our Nature, which I reviewed here.)
Paulos also shows that the human tendency to attribute meaning to coincidences and other rare events can lead us to fall for frauds such as psychics and quack medicine. TV talk shows make hay from every “correct” psychic prediction, ignoring the dozens of incorrect ones that preceded it. Similarly, if an ordinary person has a dream, and the events of the dream then occur in real life, they remember this and find it significant; they forget the thousands of dreams they had before and since that did not come true. But even an event that is statistically rare will happen occasionally, given enough time and opportunity. It would be much more unusual if you went your entire life without having a predictive dream; it would be much more surprising if a psychic’s random or intuitive guesses never aligned with reality.
Paulos presents one example after another, always spelling out the basic math required for understanding. This might lead you to think that the book is tedious, but in fact the opposite is true. Paulos’s explanations are clear and simple, and he uses humor throughout to lighten the mood. Knowing that many in the audience will be intimidated by chunks of numbers, he writes, “Now is probably a good time to reiterate my earlier remark that an occasional difficult passage may be safely ignored by the innumerate reader. … The occasional trivial passage likewise may be quite safely ignored by the numerate reader. (Indeed, the whole book may be safely ignored by all readers, but I’d prefer that, at most, only isolated paragraphs will be.)” (16-7).
Further, though the book is now 30 years old, every point that Paulos makes is still timely; indeed, prescient. In his conclusion he writes, “I’m distressed by a society which depends so completely on mathematics and science and yet seems so indifferent to the innumeracy and scientific illiteracy of so many of its citizens; with a military that spends more than one quarter of a trillion dollars each year on ever smarter weapons for ever more poorly educated soldiers; and with the media, which invariably become obsessed with this hostage on an airliner, or that baby who has fallen into a well, and seem insufficiently passionate when it comes to addressing problems such as urban crime, environmental deterioration, or poverty” (134). Clearly, the problems Paulos discusses are at least as relevant today as they were at the time of his writing.
What are the causes of these issues? Paulos points to “poor education, psychological blocks, and romantic misconceptions about the nature of mathematics” (72-3). Too many people (including many teachers) believe that one is simply born with or without mathematical ability; you either get it or you don’t. (I find a similar belief about writing among my English students.) He discusses math anxiety and its sources: “The same people who can understand the subtlest emotional nuances in conversation, the most convoluted plots in literature, and the most intricate aspects of a legal case can’t seem to grasp the most basic elements of a mathematical demonstration. … They’re afraid. They’ve been intimidated by officious and sometimes sexist teachers… they’re convinced that they’re dumb” (88). Paulos discusses a few simple solutions to this that can be integrated at any grade level, but too many teachers do not have the time, resources, or sometimes the will to implement them.
This section reminded me of an NPR story that aired a few years ago, “Why Eastern And Western Cultures Tackle Learning Differently,” which discussed differences in our views of struggle in education. Americans tend to believe that struggling with a subject is a bad thing, a sign of failure or stupidity, while Japanese educators treat struggle as a natural and expected part of the learning process. Western schools reward those who don’t struggle, who breeze through school with little effort; eastern schools value perseverance and celebrate those who push through and overcome struggle. Perhaps a more eastern view of struggle in math could help us overcome our math anxieties.
Before reading this book, I would have called myself numerate; I did well in math in school and was never intimidated by mathematical concepts. But as I read, I realized that my numeracy was deficient in a key area, the area to which most of Paulos’s book is devoted: statistics and probability. Among all math concepts, these are the most applicable to our everyday lives, yet most people understand them poorly. Further, statistics are easy to abuse, and they can be manipulated both intentionally and unintentionally to deceive people about important issues like the relative danger of terrorism or the prevalence of other types of crime. A strong grasp of statistics would be a major advantage to individuals and to society at large.
Why, then, was I never required to take a statistics class in high school or college? Algebra, geometry, trigonometry, calculus—that was the expected progression of my math education and the path that I followed. I was not required to take statistics at any point; I was never encouraged to take statistics for any reason; so far as I can remember, no one ever suggested that I take statistics. It seems that this one change in requirements could make at least some small positive difference in numeracy among Americans.
Paulos’s book is short, readable, and important no matter what your current degree of numeracy. Students and parents in particular will find it both informative and encouraging. And concerned citizens of all stripes should take its message to heart and examine how their own innumeracy affects their beliefs, behaviors, and, perhaps most importantly, voting habits, and what they can do about it.