Note: Those of you who
would rather read this in French may
transfer to <http://www.sauv.net/raimi.php>, where a couple
of French math
warriors have translated it very nicely.
WHY LEARN TRIGONOMETRY?
The following question came into a public email list called
"math-teach", mainly
read and written by school teachers of math and
professors of mathematics or of
mathematics teaching, some time in 1996:
> You are teaching a group
of skeptical high school students trigonometry and
> they need to know
"Why do we learn Trigonometry?"
>
> Unacceptable answers:
> 1. It's the next unit in the book.
> 2. The curriculum committee says you have to.
> 3. It's on the SAT.
> 4. Mathematicians find it "elegant."
> 5. In case you ever need to know the height of
a flag pole.
When this question first came up I regarded it as a joke, the
humor
being contained in the list of
four popular and one fatuous answers to
the question. The true answer to the question, "Why
trig?" seemed obvious
to me, and I assumed it was
obvious to the person who posed the question.
I answered light-heartedly,
making reference to another fatuous question
of about 1942, when I was a
physics student:
That question, in imitation of the sort of thing that too
often
appears on exams intended to
sound practical and true-to-life, was, "How
would you use a barometer to
measure the height of a building?"
Talk
about open-ended
questions! The expected answer, as any
experienced
high-school test-taker knew,
was that you measured atmospheric pressure
with the barometer at the
bottom and the top of the building and applied a
scale (found in a book
somewhere) that tells you how pressure falls off
with altitude. Airplane altimeters used this principle,
which was thought
to be of utmost practicality in
the days before radar and World War II.
But "How would we use the barometer to
measure the height of a
building" gave rise to a
lot of merriment among us teachers of math or
physics (as I later joined
those ranks), and some of the following
proposals remain in my memory:
1. Measure the length of the barometer and apply it repeatedly and
vertically up the side of the
building. Multiply its length by the
number
of times it is needed to reach
the top.
2. Carry the barometer to the roof and drop it to the ground. The time
it takes to crash is related to
the height by the well- known Galileo
formula, s=(1/2)gt2.
3. Sell the barometer and hire a lawyer with the proceeds. Send the
lawyer to the County Real
Estate Records Office for the desired
information.
I must have seen a dozen equally valid answers,
"practical" as
anyone committed to current
educational theory might ask. It is not
hard
to make fun of current educational
fashions, in every era. And of idiotic
examination questions, in
particular, there is no end.
So I mentioned this to the math-teach people, though lightly,
thinking my point had been
made: Ask a silly question, you get a
silly
answer. But more recent communications to
"math-teach" on the thread of
"Why trig?" have
persuaded me that the question was serious, and intended
to elicit real explanations
such as might appeal to high school students
condemned to study that
apparently obsolete subject, and to their parents.
Despite all the efforts of the
scientific professions in the years since
C.P. Snow first deplored the
Two Cultures phenomenon as he saw it in 1960,
much of the
"educated" world still regards science and mathematics as
trades like lace-making or
piloting an airplane, activities of value for
which we pay when we have need,
but certainly not something the rest of us
should learn anything about in
detail. So I have a few serious replies
of
my own.
First, I agree that it is a pity such questions come up at
all.
Those (math teachers) who liken
the question to "Why read Shakespeare?"
are quite right. Trigonometry
is part of the equipment of an educated person,
as much as history, literature,
biology and the rest. Students usually
don't ask
what good Shakespeare is, or
the story of the American Civil War, or
Hitler. References to such things pervade everything
we read, or see in
the movies, or hear our
relatives talking about, from early childhood; and
so we -- as children -- simply
understand that knowing about them is like
knowing how to speak and read
and write at all. Similarly, children
tend to know the necessity of
counting change when making a purchase with
a five dollar bill, and
counting in general, since they see it around them
every day, just as they see
people speaking English.
When I say "it is a pity" that the trig question comes
up at all,
what I really mean is that
children usually get the mistaken idea,
illustrated by this question,
that there are two different justifications
for learning things, by which
the things themselves are classified:
The first class consists of things that are obviously to be
learned as part of general
culture they see about them as soon as they
learn to read. They need to
know what is a big bad wolf, a princess, and
change from a five-dollar bill.
These may or may not be practical things
(they will never be frightened
by a wolf, or see a princess), but they are
so ordinary they bring up no
questions of the form, "Why must we learn
this?"
Then there are things that are not seen every day. Trigonometry
and Shakespeare seem to be
examples of the second sort. These are
just as
much part of our culture (I
will say more about this below) as the big bad
wolf and the five dollar bill,
but since they are not often seen in early
years they tend (in the
schools) to be regarded either as impositions on
childhood and its ineffable
spirituality, or as needing a different sort
of justification from what is
so obviously relevant in earlier childhood
experience.
A second kind of justification therefore gets invented, the
so-called
"practicality" of mathematics in particular, as if mathematics
were different from other forms
of literature in furnishing the alert mind
with ways to understand and
maybe control the universe. It is a
curious
and unfortunate feature of our
present civilization that nobody thinks it
necessary to question or defend
the value of the corresponding knowledge
of literature or music on
similar grounds.
If we defended the reading of Shakespeare on the grounds that
novelists and poets need it to
help them write their own books, we would
be as bad off as we now are in
defending mathematics. Why study
Shakespeare when you just know
you're not going to be a professional
writer?
We don't defend Shakespeare on such crass grounds of
practicality,
because the question is not
even posed. If it were, if someone
really
were to ask, "Why study
literature? What do I need it
for?", we would say
in all seriousness, "There
is simply no telling when or whether you will
need it. You need it every day. It is part of your intellectual
equipment."
There might be those who are still not satisfied. Certainly the
analogous answer in the case of
trigonometry is not widely believed. We
might go on (in the case of
Shakespeare, or other literature, say) with
something like this:
"Literature is like some obscure muscle in your hand: if it is
weak, your hand won't work as
well, but it would be very hard to specify
just which task would be
rendered just how much more difficult by its
weakness. You want your hand to be strong in all its
parts, and to be
able to call on reserves of
strength whenever you need them, even while
not at the time recognizing
which reserves they are.
"Thus an athlete exercises his whole body, and not just
those
muscles he fancies are the ones
he will be using in next week's game.
With proper exercise his whole
body is more capable; that is enough
explanation, and it works. Why
isn't the same argument also given for
the mental exercise of charting
the geometric relationships by which we
all view the world?"
This argument is not, by the way, the "transfer of
training" thesis
popular a hundred years ago,
saying that strengthening the mind with
mathematics also strengthens
its other abilities ipso facto.
The
analogous physical argument
would say that exercising the fingers somehow
carries over into stronger
abdominal muscles. Mental exercise was
once
thought to carry over in this
sense, but later psychologists made
experiments to show this was
not so, and that discipline in algebra did
not, for example, strengthen
the power to memorize phone numbers.
Actually, that question is still not entirely settled, but it is
not
needed, either. We don't have to justify teaching
trigonometry on the
grounds that it helps you argue
cases in court if you become a lawyer.
(Actually, I myself believe it
does. I have a daughter who is a lawyer,
and who studied calculus in
college, and she is not sorry she did.)
So "transfer of training" is questionable, and
measuring flagpoles
is laughable. And to say that engineers and scientists
need trigonometry
is not enough either, since
those with no intention of becoming scientists
and engineers remain
unanswered. If there is an answer, it
cannot be
career training.
The true answer is that trigonometry is part of our culture, and
should be as visible in daily
life as anything in Dickens or
Shakespeare. Why can we not
make it so? Why is our daily vocabulary
so
poor that while we can speak of
Communism, or the national debt, no
newspaper ever prints an
algebraic formula and no columnist has ever
been known to mention the
secant of the angle of inclination?
These
things are within our power,
and would increase our ability to describe
the world and make reference to
its properties quite as much as does our
ability to quote Mr. Bumble or
Hamlet.
We would even see the world a little differently. One who has
learned trigonometry walks a
little taller as a result. Like any
other
part of mathematics, or
literature, or history, trigonometry furnishes the
mind with frameworks that
render the experienced universe more
understandable, every day.
Part of it is
practical, to be sure, but it is a disastrous mistake
to present this part as if the
only, or most important value. How can
we
present it otherwise,
then? Merely telling the kids they will
walk a
little taller won't do the
job. (It might even be called HEIGHTISM.)
The subject itself has somehow
to be presented in such a way as to
convey the lesson of its own
interest. Just as an athlete told to
exercise only needs to know his
body feels better and stronger as a
result, without necessarily
testing the results of each day's exercise
by counting his score that day
on the golf course, the intellectual
exerciser should get his
exercise in a way that makes him feel
mentally stronger and
happier. If the exercise is given
without context
it will usually fail. In the past, with most people, it has
failed.
It sometimes happens that age and experience alone will provide
the context, and that a person
who has been out in the world a while will
see the beauty, and indeed
utility, of something like trigonometry simply
because he has more to hang it
from, as it were. After World War II,
for
example, the returning veterans
came to college three years older, on
average, than the usual college
students, and their work showed it.
There
was never a generation of
students like that one. Alas, to delay
education
until it is that late entails a
great loss of time, and we must find
another way.
I would like to illustrate with a story about my father, who
came here
from Poland as a young man,
never having had any mathematical or
scientific education of any
kind. He liked the idea of my
being a
mathematician, but did not
understand what that was, except dimly.
Something like what Einstein
did, maybe, but he knew nothing about
Einstein's work either.
My father was a rough sort of amateur carpenter; as a small
businessman he often built
things for his store: shelving,
counters,
window displays. One time, when he was already sixty years
old, he came
to me with the following observation:
In building a vertical frame for a set of shelves he would
brace the
vertical plank with a diagonal
piece of wood, one end nailed to the floor
a foot away, say, and the other
to the vertical object. The brace was
not
necessarily at a 45 degree
angle, but depended on certain limitations of
the rest of the structure.
When he attached it at a point 8 inches up from the floor, he
explained to me, the diagonal
piece would have to be longer than the 12
inch baseline, of course (he measured
it as being about 2 and a half
inches longer than one foot).
But if he wanted to attach it twice as far
from the floor, i.e. 16 inches
up, the diagonal piece had to be more
than twice as much longer. He
measured it as about 20 inches from the
floor, that is, 8 inches longer
than the 12 inch base line, and not 5
inches longer, which he would
have expected from having attached it twice
as high as the other one. Why?
His language was not technical, of
course, but what he was saying
was that the increments of diagonal length
were not proportional to the
increments of height.
Now this was not really a practical problem, even though it
began
with carpentry. He had had no difficulty all his life making
scale
drawings and cutting pieces of
wood accordingly. He had never heard of
trigonometry. But here he was, at age 60, suddenly seized
with curiosity
about a phenomenon he had not
previously thought about analytically.
Why
should it be that in making a
brace to attach 16 inches from the floor you
increase the 12 inch baseline more
than twice as much as you have to
increase it when attaching one
at 8 inches?
Of course what he had discovered was (if one wished to put it
so)
the behavior of the cosecant of
the angle his brace made with the floor.
More simply, he was observing
the Pythagorean theorem in action, where
even without invoking angles
one can chart the disproportions as height
increases.
I say "of course" here to emphasize that what is being
described
is no more difficult than the
speech Polonius gives Laertes, and should
be as much part of our
vocabulary, if we are to make easy reference to the
world as we know it, even as we
daily see it with our own eyes. To the
question "why" the
cosecant function behaves that way there was really no
answer, except that Euclidean
space is that way, the Pythagorean
relationship being in effect
the definition of the Euclidean space (also a
philosophical observation of
cultural interest!). But I was able to show
him some other examples of the
relationship that he had observed, with a
few diagrams to illustrate the
ratios as the angle approaches zero and as
the angle approaches 90
degrees.
Well, he found that very interesting. He had thought this was
the sort of thing
mathematicians might know about, but hadn't been sure.
What really astonished him was
my telling him that not only had these
facts been known for several
thousand years, but that the results were
tabulated in as much detail as
the tables of interest payments on
mortgages, and had been used by
astronomers since the time of Ptolemy, two
thousand years ago, where they
were essential in the prediction of the
conjunctions of planets and the
like. And that furthermore, without
drawing any pictures or scale
diagrams I could, by means of tabulated
information, predict
the length of his brace for any desired height of
attachment, whether he used a
baseline of 12 inches or any other!
He had no need to see the tables of sines and cosines -- for his
own
rough carpentry such
information was of no value -- but I showed him a
book of tables anyhow. (Today, with calculators available, I would
have
had to use something else to
prove to him that his observations were both
important and old.)
Think what he might
have learned, and how much more he would have
appreciated about the history
of mankind, if he had had a few weeks in his
youth to calculate a few
triangles, with a teacher and textbook presenting
the problem in the way he had
come by himself, in sixty years, to
appreciate it. He had read, in his romantic youth, of the
ancient Code of
Hammurabi; why could he not
also have learned about Pythagorean triples
and their consequent
trigonometric tables as found on the famous Tablet
#322, also from Old Babylonia,
now residing in the Plimpton collection at
Columbia University? We no longer live by the laws of Hammurabi,
but the
Plimpton tables are as valid as
ever.
If I had told him about flagpoles and shelf bracing for their
sakes only, he would not have
been interested; he has known as much as he
needed about shelves and
flagpoles since childhood. He
understood quite
well, as our discussion went
on, that his shelves were merely the language
in which the relationships that
so fascinated him were made vivid.
Even without the actual
numbers, the behavior of the diagonal in the
vicinity of zero and ninety
degrees was more interesting to him than all
the rest put together. This was perhaps the only time in his life
that he
had an insight into what
mathematics is.
What good did it do him?
I wish he were alive today, to answer that
question for you himself.