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.