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.
