Yesterday afternoon, a friend mentioned that in his experience people are usually embarrassed to be exposed as illiterate, but they don’t seem to care whether they are exposed as mathematically incompetent. That observation resonated with me. In fact, not only aren’t people embarrassed about being mathematically illiterate, but many people seem proud of being mathematically incompetent. They use their mathematical incompetence to socially bond with other people who are mathematically incompetent. More than a few times, someone in the room has mentioned that they’re not very good with numbers and several other people in the room immediately come to their rescue indicating that it’s okay to be mathematically incompetent because they too struggled with mathematics.
I don’t think it’s any coincidence that American students are so deficient at mathematics compared to the students in many other countries while, at the same time, Americans have such bizarre public policy priorities (e.g., a zero tolerance policy toward terrorism at the same time that thousands of Americans are dying needlessly of treatable medical conditions and while millions of American children are subjected to terribly underfunded schools that will ruin their lives).
After yesterday’s conversation, I pulled out an 1988 book by John Allen Paulos, Innumeracy: Mathematical Illiteracy and Its Consequences. here’s what Paulos has to say right in his introduction:
Innumeracy, an inability to deal comfortably with the fundamental notions of number and chance, plagues far too many otherwise knowledgeable citizens. 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. I remember once listening to someone at a party drone on about the difference between “continually” and “continuously.” Later that evening we were watching the news, and the TV weather forecaster announced that there was a 50% chance of rain for Saturday and a 50% chance for Sunday, and concluded that there was therefore a 100% chance of rain that weekend. The remark went right by the self-styled grammarian, and even after I explained the mistake to him, he wasn’t nearly as indignant as he would’ve been had the weathercaster left a dangling participle. 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.”
Paulos suggests that part of the reason for this ignorance of mathematics is that the consequences are often not as obvious as those of other weaknesses. On the other hand, the problems caused by innumeracy are serious, often times matters of life and death. Paulos lists the following examples:
Stock scams, choice of a spouse, newspaper psychics, diet and medical claims, the risk of terrorism, astrology, sports records, elections, sex discrimination, UFOs, insurance and law, psychoanalysis, parapsychology, lotteries, and drug testing…
Why do people struggle so much with mathematics? Paulos points to natural psychological responses to uncertainty, to coincidence, and how problems are framed, as well as anxiety, romantic misconceptions about nature and the importance of mathematics. One of the biggest consequences of innumeracy are “unfounded and crippling anxieties” and “impossible and economically paralyzing demands for risk-free guarantees.” Paulos mentions that politicians are rarely helpful, because they are often “loathe to clarify the likely hazards and trade-offs associated with almost any policy.”
It’s been a while since I read Innumeracy, but I highly recommend it. It is a timeless book filled with examples to remind us of the importance of a precise understanding of mathematics. Paulos indicates, “The book will have been well worth the effort if it can begin to clarify just how much innumeracy pervades both our private and/or public lives.”
By the way, if you know someone who is struggling with mathematics, Paulos book is a good place to start. He is an excellent teacher of math as well as a clear writer. If you know someone who wants to understand basic math, refer them to the many free video lessons at Khan Academy.
Once we master math, I would suggest that we turn to biology. It is my firm belief that all of us would be much better off with an understand of human beings based on the understanding that humans are human animals.
While math incompetence is troubling, I find widespread ignorance of logic far more disturbing. With no more math than it takes to balance a checkbook, most people can get through life without causing too much harm. However, without the ability to comprehend logic, even well-meaning people can do enormous damage. Voters, and candidates, who can’t fathom circular arguments, ad hominem attacks, or any of the other logical fallacies, can wind up causing all sorts of harm. Just consider the breathtaking gaps in logic that led George W. to invade Iraq. Top of the long list: arguing that Saddam had to prove that he didn’t have WMDs — Saddam couldn’t prove the negative because it’s not logically possible to prove a negative. No matter what proof he offered, Bush could simply claim (as he did) that Saddam was hiding the WMDs somewhere else. It’s like Christians insisting that their opponents prove their god does not exist. It’s a logical fallacy. Other examples from the fallacy land of Christianity: all the innocent people the Christian church burned to death for witchcraft, all the innocent people the Christian church tortured (often to death) to drive out evil spirits, and all the prayers that their god has supposedly answered. All are logical fallacies. Another example: virtually every political ad ever made, including those that blame, or credit, a particular candidate for economic performance that happened during that person’s term in office (unless the performance can be proven to result from specific actions by that candidate). (That last topic is very clear to me after seeing all the idiotic claims being made in Scott Walker’s campaign ads for tomorrow’s recall election here in Wisconsin. His claims are utterly without logic, yet I’ve seen no one on the local news point this out.) Kids shouldn’t be allowed to graduate high school without a better understanding of logic.
Grumpy-point well-taken. I think all high schoolers should take a course in cognitive fallacies and biases. http://en.wikipedia.org/wiki/Cognitive_biases But I do also believe that Paulos is correct to warn against innumeracy, because math incompetency gives us no way to compare alternatives and set priorities.
In addition to understanding mathematics and logic, I would add comprehension of large numbers. Not too long ago I was trying to explain to someone why it was unlikely that Earth had ever been visited by aliens, arguing that there are so many planets in the universe that the odds against even finding Earth are astronomical. My argument went like this, but I still don’t think that she was convinced:
– Number of planets per star: Current estimate is that there is somewhat greater than 1 planet per star, on average; assume 1 planet per star.
– Number of stars per galaxy: Estimates vary, as well, but 100 billion seems to be a reasonable number.
So the number of planets per galaxy is approximately 100 billion.
– Number of galaxies in the universe: Estimates vary from ~100 billion to ~1 trillion; use 500 billion as a reasonable number.
So the number of planets in the universe is approximately 100 billion planets-per-galaxy times 500 billion galaxies = 50 sextillion planets.
If we wanted to name all of those planets, and we assumed that there were 26 letters and 10 numbers available with which to do so (case-insensitive), then we would need to use 15-character names just to have enough unique names for all of them.
If we assigned a unique 15-character name to each planet, then we would need 750 billion 1-terabyte hard drives just to hold all of the names.
Thanks, Edgar. Terrific illustration of an incredibly large number.
If you want really big numbers, delve into combinatorics. How many combinations of those planets would we have to check to see if any have contacted another? On the order of a sextillion factorial, or a number greater than the number of atoms in the universe.
“You do realize that ‘a billion’ isn’t what it used to be, right? … Even ‘a trillion’ doesn’t take your breath away the way it once did. The good news about all this conceptual inflation is that the average person may soon be able to truly grasp the concept of infinity. The bad news is that we’re gonna be pondering it while living under a bridge.”
— Evan Morris, The Word Detective
One numerical example that impresses me is right in front of you.
Let’s say you have an average computer, with 4GB ram and a 2GHz clock. It only works if every value of every bit of memory (8x4x109) is correct 2×109 times a second.
In an hour, your computer has to be perfect for 3,600 x 2×109 x 8x4x109 states,
or 230,400,000,000,000,000,000,000 perfect states.
We have come to expect it to do this unerringly for several years.
And yet we complain when one of those states goes bad, and something crashes.
That is why there is extensive error detection and correction built into all computer operations. Computers DO exhibit errors, but there are mathematical techniques to detect and correct them. (And they are far more sophisticated than simple redundancy.) The mathematical techniques are even aware of their own limitations, i.e., we can implement algorithms that deliver any desired level of error detection and correction performance. The tradeoffs are speed, complexity, and size.
Further on the above thread, the important point we should focus on is not that people are bad at math, logic or large numbers; the important point is that ignorance of these subjects can have troubling, sometimes even disastrous, consequences. Bush’s ignorance of basic logic (or his willful failure to employ it) was the proximate cause of death for 4400+ American troops and more than 100,000 innocent Iraqi civilians. It also resulted in the devastation of Iraq’s infrastructure, the looting of priceless objects from Iraq’s museums, and a massive financial debt for U.S. taxpayers (in the trillions of dollars, which per Edgar’s comment most Americans probably can’t begin to comprehend).
This all ties together into the main point of this thread. Given that the world in which we live requires sophistication of thought greater than the capabilities of the average person, what do we do about it? There was a time in the past when much of the population was illiterate and uneducated. Literacy and education were necessary for success; not only for individual success, but for cultural success. The solution was that we educated people. It took a long time, but overall the literacy rate now is pretty high, and our culture has definitely benefitted from it.
The world has now become even more complicated, to the point where simply providing more education is no longer up to the task — some (quite a large number, actually) people will never understand “sextillions”, or even logical fallacies or basic statistics, no matter how much education they receive. In short, the magnitude of the problem has outgrown our ability to solve it by brute-force means. So what are we to do?
Perhaps the computer example above shows the way — when the problem becomes so large as to be untenable, such that the brute-force solution is a bigger problem than the problem itself, one must find simpler ways to deal with it. In the computer world, the solution was to address the problem indirectly, by understanding the fundamental underpinnings of the problem, and controlling them. So what is the analogous approach to enabling an unsophisticated populace to make sophisticated decisions without having to understand every aspect of those decisions?
I am sorry to say that I do not know.