B. Burt Gerstman (June 2001, August 2013)
�to
our natural and human reason I say that these terms �large,� �small,�
�immense,� �minute,� etc. are not absolute but relative; the same thing in
comparison with various others may be called at one time �immense� and at
another �imperceptible. Galileo
The inability to deal with quantities 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 quantitative reasoning is almost as stifling as
the inability to reason quantitatively. In �math class� we far too often learn
to manipulate numbers without understanding what they mean. This is
counterproductive, counter-intuitive, and down-right silly. What is the point
of using an equation? Why find an �answer� 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 trait, a psychological function,
or a rate of change? How is this �thing� being studies being measured? The
point is that you need to know something about the thing being measured or you�re
likely to derive 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 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 to
start. Just get in the ballpark. This prevents unnecessary distractions and begins
to provide context.
Once we have a general notion of the order of
magnitude of a quantity, we can then consider the accuracy of its measurement.
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.
Far
better an approximate answer to the right question, which is often vague, than
an exact answer to the wrong question, which can always be made precise. � John Tukey, The future of data analysis. Annals of
Mathematical Statistics 33 (1), (1962), page 13.
As a benchmark for beginners studying
statistics (which by nature, is an inexact science) is to seek final answers that have something like three
or four significant digits (depending on the precision of the original
measurements). So what is a significant digit? Consider these 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 and 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 precision of the measurement based on its number of significant digits.
Third, it suggests the number of significant digits we must carry during
intermediate calculations. We generally need to carry at least two more
significant digits than is needed for the final answer. For example, when three
significant digit accuracy is needed, you must carry at least 5 significant digits for intermediate calculations.
Although numerical reasoning has solved many
human problems and added to understanding, emphasis on numerical manipulations
without common sense 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 was 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.
So I suppose the point is, you can invent sets of rules which if you followed them without thinking could produce the answer if you don�t understand what you are trying to do. As Feynmann says, this is �a false thing that they had invented in school so that the children who have to study algebra can all pass.� But is this what we really want?