ORDERS OF MAGNITUDE

narrowed the distance to within about an order of magnitude. Mexico City is at the far end of Mexico, approximately twice the distance to the California-Mexico border, so 2,000 miles is a reasonable estimate.

Now, consider the turtle. A man comfortably walks about four miles per hour. (A world-class four-minute-mile athlete manages to run 15 miles per hour for a total of four minutes, so this upper limit keeps us within the right order of magnitude.) Obviously a turtle walks more slowly than a man.

An ordinary land turtle (perhaps your student's pet turtle providing he is willing to risk imprisonment from some, as yet obscure, federal law) can, however, escape from his captor if left alone in the back yard for an hour. If it is 50 feet to the yard's edge, this would be about 0.01 miles per hour. Allowing the man and the turtle to rest, eat, and enjoy the scenery during their trip, the turtle's speed is, therefore between two and three orders of magnitude slower than the man.

To ask the turtle to go the 300 foot length of a football field in one hour is clearly too much. This sets an upper limit much closer to the escape from the yard. Let's allow our turtle to average 20 feet per hour.

So, (2,000)(5,000) / 20 = 500,000 hours. (Notice that we use 5,000 rather than 5,280 for the feet per mile. This is close enough for our purposes and easier to calculate, since the student is expected to solve this problem mentally, without use of pencil and paper.) Dividing 500,000 by (24)(365) or about 10,000 equals approximately 50 years.

Do we now need to know how long a turtle lives? Not really, our answer is probably correct to within about a factor of two and certainly within one order of magnitude, which overlaps with a reasonable distribution of turtle longevities. A turtle can probably walk from Oregon to Mexico City providing he devotes most of his life to the effort.

Since we have not specified a particular turtle, this estimate of about 50 years is certainly a good order-of-magnitude estimate and much better than might have been expected from a first glance at the vagaries of the formally stated problem.

Facility with orders of magnitude and mental arithmetic is more common among scientists and engineers in the older generations because they did not have calculators and personal computers. Each slide rule calculation required simultaneous mental calculation of the order of magnitude of the answer, since a slide rule does not preserve the position of the decimal point. This is, however, not an obsolete skill.

Everyone, regardless of his ultimate occupation, should be taught to perform mental arithmetic, calculate in orders of magnitude, and think quantitatively. Since a calculator cannot be inserted into a student's brain, his ability to think quantitatively depends upon his ability to perform mental arithmetic. Access to calculators during the early years of training tends to diminish the development of mental arithmetic.

Many of the pseudoscientific hoaxes that are perpetrated on the general public depend upon the widespread inability to comprehend problems in quantitative terms. A thing is classed as "bad'' or "good'' or "higher'' or "lower'' without asking "how much?'' As the tax-financed socialist schools have degraded the teaching of mathematics, they have substituted the claim that they are teaching students "concepts'' and "the ability to see the big picture'' rather than drilling on mere problem solving and boring calculations that are easily done by computers. In fact, without a good education in mental arithmetic and quantitative thought, which requires years of individual problem solving, a student is unable to evaluate concepts or pictures. This leaves him at the mercy of the canned opinions in media propaganda - opinions that are often quite different from the truth.