I previously posted on the magic of "Power of Ten," a book that inspired me with images of things both very large and very small. Here's another wonderful set of images to illustrate the relationships among things that are large and s
I'm in the process of reading Stopped Getting Ripped off: Why Consumers Get Screwed, and How You Can Always Get a Fair Deal, by Bob Sullivan (2009). He starts off by asking you to pretend that you are in a restaurant and you are presented with a menu that he illustrates on page 5 of his book. You are asked to assume that you ordered the onion soup (the price is clearly listed on the menu as $.60) and the "Lancaster Special Sandwich" (the price is clearly listed on the menu for $1.95). The question he asks is this: "How much should you leave for a 10% tip? I'll wait for a bit while you do your calculation in your head. No calculators, please. What did you come up with? [more . . . ] The answer is 25.5 cents, so either 25 cents or 26 cents would be an acceptable answer. What Sullivan next states is shocking:
If you answer this question correctly, consider yourself part of an elite group, because when the US Department of education asked US adults to answer it as part of a nationwide study, only 42% answered correctly. Less than half of American adults were able to pick two numbers from the list, add them, then perform the most basic of all percentage calculations--simply moving the decimal point one column to the left to calculate 10%.
Innumeracy is literally killing us. Try to think of a major issue facing our country that does not require a basic proficiency in mathematics that most of us don't seem to have. Think of the environment, energy, national budget, climate, health care, evolution being taught in public schools, space exploration, public health issues (e.g., the importance of vaccinations), the true cost of the "war on drugs," reform of financial institutions or taxation policy. Since most Americans cannot understand how to calculate a 10% tip, there is little chance that they could meaningfully participate regarding most of the big issues facing our country. These are truly painful words to write. Just think of the many math-related claims that got math-ignorant voters excited during the last presidential election, including Sarah Palin's claim that American could live long and prosperously on Alaskan oil (when straight-forward calculations based on known reserves showed that there is only enough Alaskan oil to supply America's current rate of use for six months). Imagine how different things would be if most Americans could actually calculate the minimal chance that they would be affected by an act of terrorism, and if they were able to compare that risk to the immense numbers of lives that could be saved by much more modest expenditures. But it's not even clear whether most Americans can benefit from further training regarding statistics. It's certainly true that many health care professionals don't adequately understand basic problems involving risk. The reasons so many of us are innumerate are not easily addressed. We desperately need proficient math skills to tamp down our fears. I know it has been tried (and abused) before, but a sinister thought enters my mind. The information presented by Sullivan makes me wonder whether we should make voters take and pass a rudimentary math test before allowing them to vote. How indignant could a rejected voter be if he/she can't figure out a ten percent tip? Understanding the many math-based claims asserted by candidates is sometimes the only way to see past their slick acting abilities. I'm not seriously suggesting a poll quiz, though I'm sure that my frustration is showing through. What we really need to do is provide better math education all the way through school. It appears that we are paying dearly for the many grade schools that fail at math education, individually and as a country.I'm not trying to rain on anyone's parade, or should I say, snow on anyone's parade. The Christmas season can be a terrific opportunity to hear extraordinary music and to catch up with the people we care about. But there is something I'd like to discuss that perplexes me, especially at this time of the year. Those who read this blog know that I am a skeptic and that I don't believe that a divine man named Jesus saved the world. Nor do I think most people who say they believe these things actually believe them, based upon the fact that most people who say they believe in the divinity of Jesus spend very little time learning about the origin of the Bible. Almost none of them take the time to learn Hebrew or Greek, the language used by the earliest manuscripts of writings that they claim to be the direct word of God. Almost none of them pride themselves on being highly informed about the content of what they claim to be the most important book in the world. In short, the behavior of most Believers suggests that they don't deeply believe the things they say they believe about the alleged existence and importance of the man they call Jesus. I don't want to sound too harsh, because this is the Christmas season, and I am well aware that numerous people find inspiration in their religious beliefs and they are motivated by those beliefs to do impressive acts of kindness. Nonetheless, I am on the outside looking in with regard to Christian religious beliefs. From my viewpoint, it is difficult to understand how anyone could claim to believe that a man who was actually God was born at all. One reason I have such trouble is that I don't see the Christmas story as a single belief. Rather, I see "it" as a nested hierarchy of highly improbable events. In order to believe the Christmas story, one must actually believe a long series of events that depend upon each other in order for the entire story to be true. Let's start at the beginning. Did the universe always exist (perhaps as a pulsing series of big bangs or as a huge mostly invisible network of multi-dimensional strings that occasionally bud in the form of individual universi)? Or was there a first clause of the universe, a prime mover? I find the first option to be much more likely, but I'll admit that it's possible that there could have been a first cause, some sort of entity that created the universe such that before the creation, there was no universe at all. What are the odds that there was some sort of entity that created the universe? I would think it highly unlikely, about as unlikely as the Norse claim that four dwarves held up Ymir's skull to create the heavens, or any of the creation myths of any of the other religions of the world. Nonetheless, let's assume that it's 60% likely that the universe had a first cause. We're still a long way from locking down the entire Christmas story. The next step is considering the likelihood that the creator of the universe is sentient (conscious), as opposed to the insentient "God" of Einstein. [more . . . ]
In the October 29, 2009 edition of Nature (available online only to subscribers), writer Michael Bond considered whether members of the general public could benefit from specialized training so that they could better appreciate risks. Believe it or not, there's a controversy in this field. According to Bond, many specialists think that the public "will never really be capable of making the best decision on the basis of the available scientific information." This pessimistic position advocates that risk-related decision-making should be conducted by paternalistic expert agencies, which should nudge people into making better decisions without educating them deeply as to why they should make the choices they are being encouraged to make. A classic example is changing the default on one's driver's license with regard to whether one would like to donate one's organs after death. Making the default that one will donate one's organs unless the box is checked dramatically increases those who participate in the program. The optimistic camp is represented by a variety of experts, including psychologist Gerd Gigerenzer, who advocates that "people can be taught to improve their decision make and skills." As the Nature article points out, poor decision-making is ubiquitous and it seriously undermines the well-being of people. When faced with unfamiliar emotion-fraud situations, "most people suspend their powers of reasoning and go with an instinctive reaction that will often lead them astray." Bond gives the examples of people refusing to get vaccinations for measles-mumps-rubella, and the unjustified fear many people have with regard to genetically modified crops. He also mentions the statistical deficiencies of healthcare providers, an issue I discussed in an earlier post. We still get all exercised about snakes, even though we rarely encounter them, but we ignore such things as peak oil and the danger of getting into automobiles. Why do people have such a hard time evaluating risks?
The problem, as many researchers in cognitive neuroscience and psychology have concluded, is that people use two main brain systems to make decisions. One is instinctive--it operates below the level of conscious control and is often driven by emotions. The other is conscious and rational. The first system is automatic, quick and highly effective in situations such as walking along a crowded pavement, which requires the near-instantaneous integration of complex information and the carrying out of well-practiced action. The second system is more useful and novel situations such as deciding on the savings plan, which calls for deliberate analysis. Unfortunately, the first system has a way of kicking in even when deliberation would serve best.Gigerenzer argues that proper education and training could assist people to put the rational system in charge of the instinctive one. He claims that even one half-hour of training in statistics significantly improves people's ability to quantify risk. Bond lists several promising methods for improving critical thinking. One method is to train people to look at problems from an outsiders’ perspective. Another is training them to weigh multiple options simultaneously rather than looking at options one at a time. Another trick is to use "actively open-minded thinking," which requires people to intentionally consider more than the first conclusion that comes to their mind. How important is it that people learn better how to evaluate risks? In addition to the examples cited at the top of this article, research suggests that people who suffer from innumeracy overestimate the likelihood of terrorist attacks. They "tend to have a high body-mass index and tend to be poor at managing their own health." Those who believe that people can be trained to better appreciate statistics believe that people need to be taught to better "feel the numbers." They need to use real-life situations to illustrate the statistics. Many students don’t receive any training in statistics at all. In fact, your article mentions that only one law school in the United States requires a course in statistical thinking. This means that many judges and lawyers are not properly prepared to assess risks in our modern world. Younger students are neglected too. They are only taught the mathematics of certainty (such as geometry and trigonometry), not the mathematics of uncertainty. Bond’s article concludes with the suggestion that we now have a society of people who don't understand that they don't understand. He argues that society would see long-term benefits if we would only stress the need for a rigorous education in the statistics of risk.
Have you ever considered snugly wrapping a string around the entire Earth? If you did that, and then you added merely one additional meter of string (which would then raise the string uniformly off the surface of the Earth), how much higher off the ground would that new string be (the original long string, plus one additional meter)? Here's a simple statement of the problem, allegedly first used by Ludwig Wittgenstein. Here the math. Wonderful problem and surprising solution.