B. Burt Gerstman (June 2001)
The inability to deal with numbers and with probabilities has some unfortunate consequences, contributing to misinformed government policies, confused personal decisions, and increased gullibility to frauds of all kinds. If nothing else, the fear of numerical reasoning is almost as stifling as the inability to reason numerically. Far too often we learn to manipulate numbers without understanding what they mean. This is counterproductive, counter-intuitive, and down-right silly. What is the point of manipulating an equation? Why "solve" an equation when you have no idea what it means? Only after the problem is understood, can a solution be found.
One of the first things needed when trying to understand a problem is to understand the "thing" being measured. Sweeping such things under the rug is an enemy to understanding. Are we dealing with counts of people, some measure of time, a physiologic parameter, a psychological index, or a rate of change? How is this "thing" being measured? The point is that you need to know something about the thing being measured or you're likely to come up with a meaningless numerical "answers."
After the measurement is understood for what it really is, you should get a sense of its "order of magnitude." By order of magnitude, I mean its general size. Is the number big or small? If it is big, how big? If it is small, how small? It is helpful to start by thinking in terms of "10s." That is, 1, 10, 100, etc. on the "large side" and one-tenth, one-hundredth, one-thousandth, etc. on the small side. Think of this as a Richter-type scaling. In quantifying the magnitude of an earthquake, a Richter value of 7 is 10 times more powerful than a Richter value of 6. One order of magnitude up is a 10 times increase. Don't sweat small differences, at least to start. Just get in the ballpark. This prevents unnecessary distractions and helps with the proper degree of focus.
Once we have a general notion of the order of magnitude of a measurement, we can consider measurements accuracy. How accurate an answer is needed? In school, we are trained to answer to some arbitrary number of decimal places. This is nonsense. In practice we would consider what is reasonable for the thing being measured. As a rule, common sense suggests that final answers should be reported to only three or four significant digits.
So what is a significant digit? Consider two values: 0.0000566 and 566,000. Both have three significant digits (leading zeros do not count). In scientific notation, these are 0.0000566 = 5.66 x 10-5 and 566,000 = 5.66 x 105. Scientific notation serves several purposes. First, it forces us to focus on the order of magnitude of a number. (The order of magnitude of the first number is 10-5; the order of magnitude of the second number is 105; there is 10 orders of magnitude difference between these numbers.) Second, it allows us to focus on the number of significant digits used for the measurement.
Although numerical reasoning has solved many human problems and added to understanding, emphasis on numerical manipulations without reasoning adds to problems, rather than solving them. Here's Richard Feynmann's take on the silliness of numerical manipulation without understanding (The Pleasure of Finding Things Out, pp. 5 - 6):
My cousin, at that time, who was three years older, was in high school and was having considerable difficulty with his algebra and had a tutor come, and I was allowed to sit in a corner while the tutor would try to teach my cousin algebra, problems like 2x plus something. I said to my cousin then, "What're you trying to do?" You know, I hear him talking about x. He says, "What do you know -- 2x + 7 is equal to 15," he says "and you're trying to find out what x is." I says, "You mean 4." He says, "Yeah, but you did it with arithmetic, and have to do it by algebra," and that's why my cousin was never able to do algebra, because he didn't understand how he was supposed to do it. There as no way. I learnt algebra fortunately by not going to school and knowing the whole idea was to find out what x was and it didn't make any difference how you did it - there's no such thing as, you know, you do it by arithmetic, you do it by algebra - that was a false thing that they had invented in school so that the children who have to study algebra can all pass. They had invented a set of rules which if you followed them without thinking could produce the answer: subtract 7 from both sides, if you have a multiplier divide both sides by the multiplier and so on, and a series of steps by which you could get the answer if you didn't understand what you were trying to do.