WHY JOHNNIES (AND MAYBE YOU) HATE MATH AND STATISTICS
Julian L. Simon
Many folks who are not good at math believe it means that
they are not "smart". And math-clever people often regard the
math-allergic as stupid. "Among mathematicians and statisticians
who teach introductory statistics, there is a tendency to view
students who are not skillful in mathematics as unintelligent",
says the instructor's guide to the best-selling introductory
statistics text package (McCabe and McCabe, 1989, p. 2). Indeed,
a book on The Psychology of Learning Mathematics says:
Surely the main ability required for mathematics [is]
the ability to form and manipulate abstract ideas; and
surely this ability coincided closely with what we mean
by intelligence[.](Skemp, p. 15)
This is an unfair charge against many capable human beings,
and a nasty, destructive conceit of the math-clever. Many of the
math-allergic hate math for the best of reasons: Their teachers
demand that they perform tasks by rote that they do not
understand. And they do not understand because the tasks simply
cannot be understood intuitively. People are forced to
manipulate x's and y's whose meanings are difficult to remember
from one moment to the next, for mathematical reasons that even
their high-school and university teachers have forgotten - if the
teachers themselves ever understood the reasons.
But is all this conventional abstract complexity necessary?
Surprisingly, no; much of the work that needs doing can be done
in less-complex ways that are just as good theoretically and more
effective in practice.
Some would respond that it is better to handle mathematical
tasks with the usual mumbo-jumbo in order to learn to think
better. No one has ever shown this claim to be true, however.
And in fact this sort of abstract training may damage one's
mental capacities in some crucial ways.
In brief, we have here what may be the greatest intellectual
fraud of all time: Folks are being sold a false bill of goods
when they are told that it is good for them to learn mathematics
the way it is customarily taught in courses in statistics and
calculus. Of course this is a radical assertion, but read on to
see if this claim will stand up to your scrutiny.
Even as I am promising to skin off the emperor's clothes,
the question may be forming in your mind: If this assertion is
true, why do we hear and believe that the difficult higher
mathematics taught in school and college is necessary to learn?
Here is a temporary short answer: Instructors of
mathematics tell you that math must be done the conventional
formulaic way because they themselves love the mysterious
machinery of equations and proofs, and because they enjoy the
puzzle-solving nature of deductive proofs - but also because it
is to their personal advantage to have you believe that they are
the high priests of the intellect. They convince you to collabo-
rate in your own confusion and humiliation because you share with
the mathematicians the belief that being clever and quick with
symbols is the sign of a capable and respectworthy person. You
are a member of the cult of being smart. You admire your captors
and torturers as prisoners too often admire their keepers. You
accede to your hoodwinking because you, too, share the belief
that doing something the clever way is better than doing it in a
way that is more effective but seems less clever.
Now let's get down to specifics. I will address the two
subjects that are studied by the most students at colleges and
universities - statistics, and calculus. I'll use statistics as
my main case study because this is the field I've been trying to
plow for more than a quarter of a century, and it may well be the
most egregious fraud of all.
Let's keep in mind the most basic difficulty with
mathematics: Remembering what the abstract symbols stand for, so
that when you manipulate the symbols you can understand what you
are doing. There are other difficulties with equational
mathematics, too, but it should be pretty clear that if you do
not understand the meaning of the symbols you are working with,
you can't possibly surmount any other difficulties.
GEOMETRY, AND THE INTEGRAL CALCULUS
Do you remember from high school geometry the formula for
the hypotenuse (the slant line) of a right triangle? Bertrand
Russell tells us that "No formula in the whole of mathematics has
had such a distinguished history" (1925/1985,p. 96). For those
of you who do not remember the formula, how about just measuring
the hypotenuse?
Interestingly, Russell says that "It is true that the
'proof' proved nothing, and that the only way to prove it is by
experiment. It is also the case that the proposition is not
quite true - it is only approximately true" (p. 96) because of
the curvature of the earth. So what do we gain by learning to
"prove" the formula?
Is the formulaic answer more "exact" than a measurement? If
the two legs truly are perpendicular, and if they are measured
perfectly, than the formula will give a better answer. But these
two conditions are never met in real life because there always is
some error, and hence the formulaic answer will never be exactly
exact. Whether it or a measurement will produce a better answer
will differ from situation to situation. The formulaic answer is
exact only after we move from real life into the mathmatician's
never-never land of abstracted idealized magnitudes.
Well, you say, what about when you cannot easily measure it,
as is sometimes the case when measuring across water, or as was
the case in ancient Egyptian farming, Bertrand Russell tells us?
(p. 96). You could then make a model of the actual situation by
drawing a picture with the two known legs to scale on paper, and
measure the hypotenuse there - which is exactly what you do in
the Navy to calculate distances at sea.
Now how about the circumference of a circle? You probably
remember that formula, but if you don't but you do you have the
habit of making a simulation model - no problem; draw the figure
on a piece of cardboard, cut it out, and measure it around with a
tape measure. For a less regular shape you could use a sticky
tape measure that will follow the in's and out's, or one of those
neat pen-like devices where you trace the line with a little
wheel-counter in the tip. That does the job even where no
formula is available.
In high school one learns to calculate the areas of
straight-line forms with formulae whose proofs the student learns
in geometry. But if the form includes curved lines, matters
become more difficult. How about the area of a circle? If the
formula does not come to you, the practical mathematician may
turn to an ancient method that has been available to us since at
least Archimedes the Greek: Draw the figure on a piece of graph
paper, and count the number of squares within the figure.
When you squeeze and elongate the circle into an ellipse,
the formulas for the circumference and area no longer are part of
that stock of knowledge that you (perhaps) brought with you from
high school. Does this mean you are stuck? Not at all; the same
cut-the-cardboard and count-the-squares methods that worked for
the circle will work nicely here, too.
Again, these formulaic methods seem attractive because they
are thought to be "exact". But consider this: The formulaic
calculation for the area of a circle is not exact; it is only an
approximation, because any value of pi that is used - even a
value with 100 digits - is only an approximation. And the
methods that mathematicians have used to approximate pi for more
than two thousand years since Archimedes have been variations on
the method of count-the squares. (The Archimedean method that
ruled for 2000 years puts one polygon - say, a hexagon - inside a
given circle and circumscribes another hexagon around it,
calculates the areas of the two hexagons, splits the difference
between the two areas, and makes the result the basis of pi.
Subsequent methods that counted the areas of thin rectangles
within a hemisphere are even closer to square-counting in spirit.
For more on the history of pi, see Beckmann, 1982.)
When you squeeze and elongate the circle into an ellipse,
the formula for the area no longer is part of that stock of
knowledge that you (perhaps) brought with you from high school.
Does this mean you cannot find its circumference? Not at all;
the same count-the-squares method that worked for the circle will
work nicely here, too. And if the curved figure is complex, even
the skilled mathematician may not be able to find an appropriate
formula with the integral calculus. In that case, the practical
mathematician may turn to the count-the-squares method. That
method would work fine if you own a saw mill and you must compute
the area of the cross-section of a very irregular large tree.
If the graph-paper method works for complex figures, why not
use it for simple figures as well? It surely works, and there is
no danger of using the wrong formula. Furthermore, if a young
person is taught to calculate areas the graph-paper way, and then
applies it to (say) a rectangle, the young person may well learn
the formula by him/herself. The young person may then gain not
only the skill to calculate areas, but also the habit of
discovering for him/herself workable solutions to problems - and
maybe a deeper knowledge of the deductive formulaic rule, too,
because its nature was self-discovered.
But even if the proof of the formula is not discovered -
what then? To answer that, ask yourself if you have carried
through life from school days and into your present knowledge the
nature of the proof of the formula for the area of a
parallelogram or even that of a rectangle. Is that knowledge
necessary for you - other than to help your kids learn the same
thing you learned, so that they can teach their kids, so that
their kids can teach their kids, so ... ?
Circumference, area - now the volume of a solid. Yes, the
volume of a cube is probably easy to calculate. But if you don't
know how, or if the solid is irregular or curved, turn your
memory to (supposedly) Archimedes, who went flying out of his
bathtub and into the street stark naked shouting "Eureka", "I've
found it", when the king wanted him to determine if a gold
ornament had been debased with a cheap metal. (If you know the
weight of an object, you know how much volume an amount of gold
of that weight displaces.) To get the volume of solid in which
you are interested, fill the bathtub to the level at which it
overflows a drip hole, then insert the solid and catch the water
that overflows. The overflow is a measure of the volume. (Of
course the solid must not float, an easy thing to arrange in most
cases.)
Does this mean that youths should not study the rules of
geometry and their deductive proofs, and instead devote the time
to such subjects as history, computer science, and languages? I
won't answer that question other than to say that anyone who
contemplates a future of being a professional mathematician, or
even an engineer, almost surely can benefit from the conventional
training. But I would ask you to ask yourself: Who would you
prefer working with you in most practical enterprises - a person
who has learned and forgotten the rules of geometry, or the
person who has the habit of finding a practical method of
modeling and measuring the problem at hand?
Here we may find evidence of that curious phenomenon known
as "trained incapacity". It would be interesting to actually
collect data from a randomly-drawn sample of artisans who have
never studied geometry and a randomly-drawn sample of scientific
academics who have studied that subject, and pose to both groups
this problem: "Consider this curved shape (say an ellipse).
Would you please give me your best estimate of its
circumference?" I'll bet that the artisans perform better.
Martin Gardner, for decades the mathematical puzzle editor
of Scientific American, suggests that all mathematics is
inherently difficult to teach.
A teacher of mathematics, no matter how much he loves
his subject and how strong his desire to communicate,
is perpetually faced with one overwhelming difficulty:
How can he keep his students awake? (Gardner, 1977, p.
x)
This difficulty vanishes when the teaching does not include
formulas.
Mathematical physicist John Barrow invented a revealing
scenario about proof-based mathematics. He imagined what might
happen if we were to receive a response from Martians to an
Earth-transmitted extra-terrestrial messages. Those messages
depend heavily upon mathematics, on the assumption that that will
be the easiest for the Martians to decode. Barrow writes first
about the excitement:
There is great excitement at NASA today. Years of
patient listening have finally borne fruit. Contact
has been found. Soon the initial euphoria turns to
ecstasy as computer scientists discover that they are
eavesdropping not upon random chit-chat but a systemat-
ic broadcast of some advanced civilisation's mathemati-
cal information bank. The first files to be decoded
list all the contents of the detailed archives to come.
Terrestrial mathematicians are staggered: at first they
see listings of results that they know, then hundreds
of new ones including all the great unsolved problems
of human mathematics....
Soon, the computer files of the extraterrestrials'
mathematical textbooks begin to arrive on earth for
decoding and are translated and compiled into English
to await study by the most distinguished
representatives of the International Mathematical
Congress. Mathematicians and journalists all over the
world wait expectantly for the first reactions to this
treasure chest of ideas.
Then he writes about the next peculiar reaction:
But odd things happened: the mathematicians' first
press conference was postponed, then it was cancelled
without explanation. Disappointed participants were
seen leaving, expressionless, making no comment; the
whole atmosphere of euphoria seemed to have evaporated.
After some days still no official statement had been
made but rumours had begun to circulate around the
mathematical world. The extraterrestrials' mathematics
was not like ours at all. In fact, it was horrible.
They saw mathematics as another branch of science in
which all the facts were established by observation or
experiment (Barrow, 1992, pp. 178-179).
The key element in Barrow's story is the disappointment.
Terrestrial mathematicians are not excited by a method that
simply offers answers or solutions. To be acceptable, the method
must also meet the aesthetic test of deductive proof.
CALCULUS MADE EASY
Now the differential calculus. To put you in the proper
mood, take note that the World Book Encyclopedia publised at the
time of my youth had just a single paragraph on calculus, saying
that it is "the highest branch of mathematics, studied only in
colleges and universities after a thorough preparation in
algebra, geometry and trignometry". In other words, it is too
difficult a subject for the encyclopedia to teach you. Does that
impress and scare you?
If (when) the following discussion of the calculus - which
inevitably uses the algebraic notation that is such a barrier to
understanding - loses your interest, please skip the rest of it
and go on to the following subject, statistics; the discussion
there is entirely free of algebraic notation.
Consider a famous book called Calculus Made Easy: Being a
Very-Simplest Introduction to Those Beautiful Methods of
Reckoning Which Are Generally Called by the Terrifying Names of
the Differential Calculus and the Integral Calculus, written at
the turn of the twentieth century by Silvanus P. Thompson
(1910/1946, third edition) F.R.S. The "F.R.S" means "Fellow of
the Royal Society", one of the highest honors a scientist can
achieve in Great Britain, so we can be sure that Thompson was no
slouch at his profession. In the book's prologue we read:
Being myself a remarkably stupid fellow, I have had to
unteach myself the difficulties, and now beg to present
to my fellow fools the parts that are not hard. Master
these thoroughly, and the rest will follow. What one
fool can do, another can. (italics added)
Calculus Made Easy has been damned by every professional
mathematician I have asked about it. So far as I know, it is not
used in any calculus courses anywhere. Nevertheless, almost a
century after its first publication, it still sells briskly in
paperback even in college bookstores. It teaches a system of
approximation that makes quite clear the central idea of calculus
- the idea that is extraordinarily difficult to comprehend using
the mathematician's elegant method of limits. And by "difficult
to comprehend" I mean that the invention of the calculus required
the powers of Isaac Newton and Gottfried Leibniz - the greatest
mathematical minds in Europe for centuries. And even they did
not fully understand their own invention, as we shall see.
Thompson starts his book as follows:
The preliminary terror, which chokes off most fifth-
form [eleventh or twelfth grade in the U. S.] boys from
even attempting to learn how to calculate [use
calculus], can be abolished once for all by simply
stating what is the meaning - in common-sense terms -
of the two principal symbols that are used in
calculating.
These dreadful symbols are:
1) d which merely means "a little bit of" (p.3).
Thus dx mdeans a litle bit of x".
(Thompson's second "dreadful symbol" is "merely a long S", the
integral sign, which we can leave aside here.)
Then Thompson goes on: "All through the calculus we are
dealing with quantities that are growing, and with rates of
growth (p. 8)... right through the differential calculus we are
hunting, hunting, hunting for a curious thing: a mere ratio,
namely the proportion which dy bears to dx when both of them are
growing" (p. 10). The quantity dy is a "little bit" of some
"variable" y whose change depends in some way upon the extent of
change in the variable x.
For those of you who have had an encounter (perhaps painful)
with the calculus, as well as for those of you who are entirely
innocent of the subject, it may be helpful to provide a bit of
history (which students are ordinarily deprived of, to their
great detriment in understanding what the calculus is all about).
But if the next few pages and their geometry and algebra get too
tedious for you, skip them and move on to the rest of the story
which is entirely free of all such paraphernalia.
Isaac Newton's professor and predecessor at Cambridge
University, Isaac Barrow, drew a graph like Figure 1, which
portrays the flight of a cannonball. Say that we want to know
the speed of the ball at point P (a typical simple problem that
calculus answers for us). One can approximate the speed by
drawing a small triangle with horizontal and vertical legs
centered below P, and counting the (perhaps fractional) number of
units of increase in the vertical axis portraying variable y, per
each unit of increase in the horizontal axis portraying the
variable x. The result is a perfectly acceptable approximation
of the typical result that we seek from the calculus. When we
have done this WE HAVE ACHIEVED WITH A CRUDE EARLY APPROACH TO
THE PROBLEM WHAT THE CALCULUS ACHIEVES. THE REST IS ELABORATION.
Figure 1
This method has some defects, however. Unless the triangle
is very small, it may yield too crude an approximation,
especially if the curve is changing direction at that point. And
carrying out this method means that we must be able to draw a
neat, sharp graph.
Barrow refined this method and got within an ace of the
modern calculus. He, like most others, chose as a first example
a parabola (the cannon-ball path, usually drawn as a U-shaped
figure that plots the equation y = x2). He drew a line tangent
to the curve at P, and extended it to the baseline. He could
then use geometry (trignometry, if you like) to calculate the
ratio of the increase in y to a unit increase in x, which may
then be written as a ratio of the "dreadful symbols", dx/dy.
But perhaps we are not good at drafting, and can't
accurately sketch in the curve between the points that you plot.
And if there are more than two variables involved or if things
otherwise get complicated - for example, if you want to know the
rate at which water is rising in a cone-shaped vessel - a graph
may not be workable. And you may desire a general rule for
making the computation, and not just the specific number at a
given point. For those reasons, as well as for esthetic reasons,
mathematicians wanted a more powerful method. Thompson explains
to us the nature of that more powerful method, but manages to
avoid entirely the major mystery of the usual way of approaching
the modern method.
You could compute the tangent at different points along a
parabola, and you might notice that at every point the rate is
2x. Good try. But that inductive method will not satisfy
mathematicians, and it will not work where you cannot construct
the graph.
Now let's see how Thompson proceeds. In his initial chapter
entitled "On Different Degrees of Smallness", Thompson writes:
[W]e have to deal with small quantities [that are] of
various degrees of smallness.
We shall have also to learn under what circumstances we
may consider small quantities to be so minute that we
may omit them from consideration. Everything depends
upon relative minuteness (p.3).
Thompson begins work with the standard example of a
parabola, and the equation
y = x2.
Like others before him, he considers what happens when you
pick a point on the parabola and make a small addition to it on
the x axis. He tells us to add a "little bit" of x and the
corresponding little bit of y to the equation, to get
y + dy = (x + dx)2.
Then he actually does the algebra of squaring to get
y + dy = (x + dx)2 = x2 + 2x*dx + (dx)2, where the
asterisk means to multiply 2x by (dx); "dx" can be an ambiguous
symbol.
Thompson then subtracts the equal quantities y and x2 from
the left and right sides of the equation to get
dy = 2x*dx + (dx)2.
Now comes the crucial part of Thompson's method: He points
out to us that (dx)2 is only a little bit of a little bit, which
means that it must be insignificantly small (he has early taught
us how to decide when things are insignificantly small), and
therefore we can throw it away - just ignore it from here on.
After throwing awat the little bit of a little bit, we can re-
write the last previous equation as
dy = 2x*dx.
The act of throwing away the little bit of a little bit
leaves us with one more tiny step to get to the "derivative" we
seek. We tidy up algebraically by rewriting as
dy/dx = 2
which is what we are looking for, and exactly what you obtain
with the conventional method. Thompson uses a method of
approximation, but he arrives at the same exact formula as Newton
and Leibniz.
If you have followed what has just been done, you now
understand the heart of the differential calculus. All the rest
is application of this very same method to more complicated
algebraic constructions.
If you read Thompson's book you will grasp what calculus is
all about - what it is useful for, and the nature of its power.
After you understand Thompson's explanation you are better
prepared to understand the conventional approach - if you still
need to or want to.
Now contrast the way the subject is taught - the way it was
in the text I tried to learn calculus from by correspondence when
I was in the Navy, and later again in graduate school. (I
flunked most of the lessons I sent from my ship to the University
of Wisconsin, and you might say that my attack on the
conventional method is just because I myself failed. Well, if
so, why not? What's wrong with learning to sympathise with the
difficulties of others by experiencing the same difficulty
yourself? One of the drawbacks of having this subject taught by
professional mathematicians is that they have not suffered the
same difficulties as the rest of us.)
Instead of throwing away the term (dx)2 as Thompson did in
the equation we had above,
y + dy = x2 + 2x*dx + (dx)2.
the standard method first cancels the y and the x2, and then
divides by dx to get
dy/dx = 2x*dx + dx.
Here is where all the fancy reasoning about limits enters -
in discussing what happens as dx becomes smaller and smaller.
The endpoint (more or less) is that dx can be made to disappear
logically, either by saying "It is plain" that this happens, or
with exceedingly tight mathematical logic about "neighborhoods"
around the points x and y2.
So we get to
dy/dx = 2dx
which is exactly where Thompson got to.
You may object that throwing away the little bit of a little
bit, as Thompson does, is "merely" an approximation. But in fact
(and despite the assertion of many texts that the standard method
is "exact") the standard method of "limits" is "merely" a super-
rigorous approximation, not in essential quality different from
Thompson's method of approximation. The difference is that
Thompson deals straightforwardly with relatively small quantities
whereas the standard method twists itself into a pretzel dealing
with nearly infinitely small quantities. (When I say "nearly" I
mean small to any standard of comparison, but never so small as
to be non-existent. How's that for neat mathematical logic?)
Consider this analogy: The houses in a new development are
arranged along a curve as in Figure 2. The curve has the shape y
= x2. Consider x to be the house number, and the price will
equal y. Very unusually for houses (but demanded by the calcu-
lus), all houses are infinitely thin. There are houses at all
places along the curve from 100 to 200 except at number 150,
where no house exists. The question is: What is the rate of
change of price at house number 150?
Figure 2
Peculiar as it may seem, the calculus will answer this
question about the rate at house 150 even though house 150 does
not exist. That is because the rate of change of price at house
150 is the limit of the process, and it is calculated on the
basis of the (almost) infinitely thin houses next to it. This
shows you that even the standard method of deriving the calculus
is only an approximation rather than an "exact" method. If it
were exact, it could not calculate the rate at house 150 because
that house does not exist.
The house-price example is related to the famous paradox of
the Greek philosopher Zeno: If a turtle covers half the distance
to its goal in each successive period, does this mean that it
will never reach its goal? The standard method of the calculus
explains the paradox because the calculus is founded on the same
concept as the paradox. As the Grolier Encyclopedia (1973)
asserts:
The fundamental concept of calculus, which distingishes
it from other branches of mathematics and is the source
from which all its theory and appplications are
developed, is the theory of limits of functions of
variables (vol. 4, p. 452).
Thompson's method never has to employ the concept of the limit or
grapple with paradoxes like that of Zeno.
The text I was assigned in graduate school typically
typically teaches you none of the history above. First the
author, Donald Richmond, goes into great detail about the nature
of the concept of a "limit", upon which the refined modern method
depends, and then he strips down the explanation of finding the
derivative to the shortest possible space. To make the matter
even more obscure, his first (and central) graph shows decreases
in x and y instead of increases, in order to make his algebra a
bit briefer later on (Richmond, 1950, p. 70). That undoubtedly
seems like a trivial change to Richmond. But it is no accident
that the famous mathematician-philosopher Alfred Whitehead de-
fined calculus as the "systematic consideration of the rates of
increase of functions" (1911/1948?, p. 164, italics added), a
definition quoted widely (e. g., Allen, 1956). In practice we are
more often interested in an increase than in a decrease. And
increase is easier to understand algebraically. So shifting to
the examination of decrease is costly to the student in ease of
understanding.
But author Richmond makes clear that understanding by the
average non-mathematician is not his central goal. He uses a
different notation (in D's) instead of dx/dy, and after mention-
ing the dx/dy notation he says:
It should be emphasized... that convenient as the
differential notation is for remembering results... the
D notation lends itself better to the construction of
satisfactory proofs and was therefore adopted at the
outset (p. 100).
And we remember from Barrow's example of the Martians that
logical proof all-important to mathematicians.
Here is another illustration of the unnecessary but brain-
breaking complexity in the standard method: The Encyclopedia
Brittanica (1946 edition) begins work with the notation [trian-
gle] x and [triangle] y and writes them as a ratio. When the
author finishes the derivation he shifts to the dy/dx notation
but abjures you never to think of that notation as a ratio -
because the meaning has shifted as the limit is approached. In
other words, [triangle] y /[triangle] x approaches dy/dx as a
limit, but somewhere along the line it becomes something differ-
ent than what it was - no longer being a ratio. How? When?
Why? Get someone else to answer those questions for you. Those
questions don't arise for Thompson at all, because he has no need
whatsoever for the concept of a limit.
Question: why don't high school and college kids get to
learn calculus the Thompson way? Answer: Thompson's system has
an unremediable fatal flaw: It is ugly in the eyes of the world-
class mathematicians who set the standards for the way
mathematics is taught all down the line; the run-of-the-mill
college and high school teachers, and ultimately their students,
are subject to this hegemony of the aesthetic tastes of the
great. Thompson simply avoids the deductive devices that
enthrall mathematicians with their beauty and elegance. It does
not enhance the "construction of satisfactory proofs" that is the
goal of professional mathematicians such as Richmond, quoted
above.
One more item before we leave the calculus: According to
Whitehead (who should know), Newton and Leibniz did not fully
understand the method they created. The entire business of
limits, the heart of the modern explanation, is very deep
philosophically, almost mysterious.
[W]e are tempted to say that for ideal accuracy an
infinitely small period is required. The older
mathematicians, in particular Leibniz, were not only
tempted, but yielded to the temptation (p. 167).
Leibniz held that, mysterious as it may sound, there
were actually existing such things as infinitely small
quantities, and of course infinitely small numbers
corresponding to them. Newton's language and ideas
were more on the modern lines; but he did not succeed
in explaining the matter with such explicitness so as
to be evidently doing more than explain Leibniz's ideas
in rather indirect language. pp. 168-169).
Whitehead tells us that
"The real explanation of the subject was first given by
Weierstrass and the Berlin School of mathematicians
aobut the middle of the nineteenth century. But
between Leibniz and Weierstrass a copious literature,
both mathematical and philosophical, had grown up round
these myserious infinitely small quantities which
mathematics had discovered and philosophy proceeded to
explain...the general effect...was to generate a large
amount of bad philosophy... The relics of this verbiage
may still be found in the explanations of many
elementary mathematical text-books on the Differential
Calculus" (pp. 169-170).
Of course when Whitehead goes on to explain the calculus
himself, he gives us what he considers to be a clearer
explanation of the Weierstrass approach - rather than something
like Thompson's approach. (Interestingly, Thompson's book was
published almost contemporareously with Whitehead's book.)
Given that the standard route to the "modern method" is so
difficult that even its inventors Newton and Leibniz did not
understand it thoroughly, does it make sense to press it upon
non-mathematically-minded young people when a better method of
stating it is available?
DO YOU NEED BEAUTY, OR DO YOU NEED PRACTICAL TOOLS?
If you ask them about Thompson, mathematicians talk about
gaining "insight" with the conventional method. And they will
mention the holy mathematical trinity of "elegance", "rigor", and
"sophistication". But Albert Einstein said that in mathematical
physics "matters of elegance ought to be left to the tailor and
to the cobbler" (1916, p. i).
What the mathematicians will not admit is that what (maybe)
gives insight to them, and pleases them with its elegance and
sophistication, only gives the non-mathematician a fast headache
and a long-continuing case of brain fog.
Here is the mind set of the mathematician, as expressed by
one of the great mathematicians of the century:
A mathematician, like a painter or a poet, is a maker
of patterns...The mathematician's patterns, like the
painter's or the poet's, must be beautiful (C. H.
Hardy, A Mathematician's Apology, 1967, pp. 84-85,
italics in original)
Here is a more detailed statement of the same mind set:
[T]he mathematician... may be driven to creative
activity, as is the poet or painter, by pride in his
reasoning faculty, the spirit of exploration, and the
desire to express himself ...
It offers intellectual delight and the exaltation of
resolving the mysteries of the universe...
Mathematicians enjoy the excitement of the quest,
the thrill of discovery, the sense of adventure, the
satisfaction of mastering difficulties, the pride of
achievement or, if one wishes, the exaltation of the
ego, and the intoxication of success...
As man's greatest and most successful intellectual
experiment, mathematics demonstrates manifestly how
powerful our rational faculty is. It is the finest
expression of man's intellectual strength. (Kline,
1985, pp. 551-3)
Consider this analogy: A group of people is given the
problem of escaping from a formal box-hedge maze on an English
country estate. A maze is a classic mathematical problem. The
mathematician would have you use your deductive problem-solving
powers to find that path through the maze that leads to the exit
with the fewest steps, or in the shortest time. And when set the
problem, most students will dutifully set out to do what the
mathematical teachers expect, and accept their teachers'
judgments about their success or failure.
But another person might do differently: S/he whips out a
machete and slashes ports through the hedges directly from the
starting point to the finish. Or perhaps s/he gets a ladder and
climbs over. Or swings over on a rope from a high tree. Or
simply runs back out the entry point and then around the maze to
the end point.
One can guess the grade s/he would receive from a mathemati-
cian for such behavior: "F" for flunk! Why? Because the maze
was not "supposed" to be dealt with that way. But who decided
what was "supposed" to be done? And was that even stated at the
start? More likely the proper behavior was only implicitly
specified, and people understood what was wanted because we are
so well socialized to work by certain rules, usually deductively.
This analogy reflects on the general value of training in
deductive thinking. Ever since the Greek geometers, and probably
before, it has been assumed that the possession of the sorts of
skills that they taught implied that a person is a "better"
thinker. But whom would you prefer to have with you if cast away
on a deserted desert island, or working in a new factory you are
setting up - the person who is good at the deductive mathematics
of box hedges, or the person who finds another and quicker way to
get to the finish point? I do not dismiss the value of the
deductive thinker; s/he may be very helpful in setting up effi-
cient production schemes in a factory. Rather, it is simply to
say that the non-formal thinking is not necessarily inferior to
the former.
Indeed, by a curious irony, those who preach creativity in
business and elsewhere are fond of the phrase "thinking that gets
outside of the box," which refers to a famous deductive puzzle-
problem of connecting a set of dots that trace a box; the
solution is to extend a line beyond the apparent rectangular
borders - "outside of the box", in violation of an implicit rule
for behavior.
THE NEW STATISTICS AND PROBABILITY
So far I've talked mainly in generalities. I have not
proven my main point with detailed evidence that you can assess
for yourself. For that evidence, let's now talk about statistics
and probability - arguably the most important tool of good
thinking that a person can possess.
In business, in government, in investments, in your personal
life, you need the ability to sensibly answer practical questions
concerning the uncertainties in decision-making. And it doesn't
matter how you get those answers - whether you work with
equations or simpler tools - as long as the answers are sound.
Though of course it is rare for them to say so publicly,
many mathematical statisticians are fully aware that the emperor
is naked, and that the conventional introduction to statistics is
a disaster. As two of them (one a former president of the
American Statistical Association) wrote, "The great ideas of
statistics are lost in a sea of algebra" (Wallis and Roberts,
1956, viii, xi). Another well-known statistician, Bradley Efron,
writes (with Tibshirani, page xiv, 1993): "The traditional road
to statistical knowledge is blocked, for most, by a formidable
wall of mathematics".
Robert Hogg argues that the formal equational approach is
unsound not only because it is difficult, but also because it
points the student away from deep understanding of
scientific statistical problems.
Statistics is often presented as a branch of
mathematics, and good statistics is often equated with
mathematical rigor or purity, rather than with careful
thinking (1991).
The authors of the most-respected introductory text say:
[W]hen we started writing, we tried to teach the
conventional notation... But it soon became clear that
the algebra was getting in the way. For students with
limited technical ability, mastering the notation
demands so much effort that nothing is left over for
the ideas. To make the point by analogy, it is as if
most the undergraduates on the campus were required to
take a course in Chinese history--and the history
department insisted on teaching them in Chinese.
(Freedman et al., 1991, from the introduction to the
first edition)
Indeed, it is well-accepted in the statistical profession
that the attempt to teach the subject conventionally at an
introductory level is a crashing failure. There is much written
testimony to this effect by thoughtful critics of statistics
education. Here are some examples:
1. Garfield (1991): "A review of the professional
literature over the past thirty years reveals a consistent
dissatisfaction with the way introductory statistics courses are
taught" (p. 1). Garfield asserts (referring to her dissertation,
1981, and to work by Wise) that "It is a well known fact that
many students have negative attitudes and anxiety about taking
statistics courses" (p. 1). "Students enrolled in an
introductory statistics course have criticized the course as
being boring and unexciting... Instructors have also expressed
concern that after completing the course many students are not
able to solve statistical problems... (1981, quoting Duchastel,
1974).
2. Dallal (1990, p. 266): "[T]he field of statistics is
littered with students who are frustrated by their courses,
finish with no useful skills, and are turned off to the subject
for life".
3. Hey (1983, p. xii):
For more years than I care to recall, I have been teaching
introductory statistics and econometrics to economics
students. As many teachers and students are all too
aware, this can be a painful experience for all
concerned. Many will be familiar with the apparently
never-ending quest for ways of reducing the pain - by
redesigning courses and by using different texts or
writing new ones. But the changes all too often turn
out to be purely cosmetic, with the fundamental problem
left unchanged.
4. Barlow (1989, Preface)
Many science students acquire a distinctly negative
attitude towards the subject of statistics...As a
student I was no different from any other in this
respect.
5. Hogg: "[S]tudents frequently view statistics as the
worst course taken in college." He explains that "many of us are
lousy teachers, and our efforts to improve are feeble" (1991, p.
342).
6. Vaisrub (1990) about her attempt to teach medical
students conventional statistical methods: "I gazed into the sea
of glazed eyes and forlorn faces, shocked by the looks of naked
fear my appearance at the lectern prompted" .
7. Freedman et al. noting that most students of probability
and statistics simply memorize the rules: "Blindly plugging into
statistical formulas has caused a lot of confusion." (1991, p.
xv)
8. Ruberg (1992):
It seems that many people are deeply afraid of
statistics. [They say] `Statistics was my worst
subject' or `All those formulas'...I wish they had a
deeper understanding of the statistical method...rather
than the general confusion about which formulas are
most appropriate for a particular data set.
9. Based on their review of the literature, Garfield and
Ahlgren say that "students appear to have difficulties developing
correct intuitions about fundamental ideas of probability", and
they proceed to offer reasons why this is so (1988, p. 45).
These sorts of negative comments are not commonly heard
about other subjects and other groups of students; both the
nature and the volume of the criticism with respect to statistics
are unusual. I have been teaching economics, business, and
demography for three decades without hearing such complaints in
those fields.
Indeed, in the last decade or so, the statistics
discipline's graybeards have decided that probabilistic
(inferential) statistics is just too tough a nut to crack, and
have concluded that students should be taught mainly descriptive
statistics -- tables and graphs -- rather than how to draw
inferences probabilistically. But probability and inferential
statistics are the heart of the matter. A statistics course
without inferential statistics is like Hamlet without the Prince
appearing.
ENTER THE SOLUTION: RESAMPLING
The new resampling approach to inferential statistics, using
simulation instead of formulae, mitigates the problem. A
physical process necessarily precedes any statistical procedure.
Resampling methods stick close to the underlying physical process
by simulating it, requiring less abstraction than classical
methods. The abstruse structure of classical mathematical
formulas cum tables of values based on restrictive assumptions
concerning data distributions tend to separate the user from the
actual data or physical process under consideration; this is a
major source of statistical error.
Resampling, and especially the specific device known as the
"bootstrap", has most commonly been used by statisticians in
situations where classical methods are inappropriate. But
resampling has a more important role as the tool of first resort
in everyday practice of statistical inference. Its greatest
advantage is that there is a greater chance that a correct
resampling test will be used than a correct classical test.
It is easiest to understand the issue if we digress with
some history. About 1615, Italian gamblers brought to Galileo
Galilei a problem in the game of three dice. The theorists of
the day had figured as equal the chances of getting totals of 9
and 10 (also 11 and 12), because there are the same number of
ways (six) of making those points -- for example, a nine can be
126, 135, 144, 234, 225, and 333. But players had found that in
practice 10 is made more often than 9. How come?
Galileo then invented the device of the "sample space" of
possible outcomes. He colored three dice white, gray, and black,
and systematically listed every possible permutation. The
previous theorists - including Gottfried Leibniz - had instead
lumped together into a single category the various possible ways
of getting (say) a 3, 3, and 4 to make 10. That is, they listed
combinations rather than permutations, and various combinations
contain different numbers of permutations.
Galileo's analysis confirmed the gamblers' empirical
results. Ten does come up more frequently than 9, because there
are 27 permutations that add to 10 whereas there are only 25
permutations that add to 9.
The use of repeated trials to learn what the gamblers wanted
to know illustrates the power of the resampling method -- which
we can simply call "simulation" or "experimentation" here. And
with sufficient repetitions, one can arrive at as accurate an
answer as desired. Not only is the resampling method adequate,
but in the case of three dice it was a better method than
deductive logic, because it gave the more correct answer. Though
the only logic needed was enumeration of the possibilities, it
was too difficult for the doctors of the day. The powers of a
Galileo were necessary to produce the correct logic.
Even after Galileo's achievement, the powers of Blaise
Pascal and Pierre Fermat were needed to correctly analyze with
the multiplication rule another such problem - the chance of at
least one ace in four dice throws. (This problem, presented by
the Chevalier de la Mere, is considered the origin of probability
theory.) For lesser mathematical minds, the analysis was too
difficult (though now it is an elementary problem). Yet ordinary
players were able to discern the correct relative probabilities,
even though the differences in probabilities are slight in both
the Galileo and Pascal-Fermat problems. Simulation's
effectiveness is its best argument.
One might rejoin that the situation is different after
Galileo, Pascal, Fermat and their descendants have invented
analytic methods to handle such problems correctly. Why not use
already existing analytic methods rather than resampling?
The existence of a correct algorithm does not imply that it
will be used appropriately, however. And a wrongly-chosen
algorithm is far worse than no algorithm at all -- as the
Chevalier's pocketbook attested. In our own day, decades of
experience have proven that "pluginski" -- the memorization of
formulas that one cannot possibly understand intuitively -- may
enable one to survive examinations, but does not provide usable
scientific tools.
Resampling's advantage over deduction is evident in the now-
famous problem of the three doors as popularized in Parade: The
player faces three containers, one containing a prize and two
empty. After the player chooses, s/he is shown by the "host"
that one of the other two containers is empty. The player is now
given the option of switching from her/his original choice to the
other closed container. Should s/he do so? Answer: Switching
doubles the chances of winning.
The thousands of responses columnist Marilyn vos Savant
received prove that logical mathematical deduction fails badly in
this case. Most people -- laypersons and statisticians alike --
arrive at the wrong answer.
Simulation, however -- and hands-on simulation with physical
symbols, rather than computer simulation -- is a surefire way of
getting and displaying the correct solution. Table 1 shows such
a simple simulation with a random-number table. Column 1 repre-
sents the box you choose, and column 2 where the prize is. Using
the information in columns 1 and 2, the "host" opens a box -
indicated in column 3 - and reveals it to be empty. Lastly,
column 4 scores whether the "switch" or "remain" strategy would
be preferable. A count of the number of winning cases for
"switch" and the "remain" gives the result sought.
Table 1
This process is transparent. Not only is the best choice
obvious, but as you do a few random-number trials you are likely
to understand why switching is better. No other mode of
explanation or solution teaches this intuition so well. With
many other problems in probability and statistics, too,
simulation provides not only sound answers but also insight into
why the process works as it does. In contrast, formulas produce
obfuscation and confusion for most non-mathematicians.
SEE RESAMPLING RUN
Resampling is best understood by seeing it being learned.
The instructor walks into a new class and immediately asks:
"What are the chances that if I have four children that three of
those children will be girls?" Someone says "Put a bunch of kids
into a hat and pick out four at a time". After the laughs, Teach
says, "Nice idea, but it might be a bit tough to do...Other
suggestions?"
Someone else says, "Have four kids and see what you get."
Teach praises this idea because it points toward learning from
experiment, one of the key methods of science. Then s/he adds,
"But let's say you have four children once. Will that be enough
to give you an acceptable answer?" So they discuss how big a
sample is needed, which brings out the important principle of
variability in the samples you draw. Teach then notes that to
have (say) a hundred families, it could take quite some time,
plus some energy and money, so it doesn't seem to be practical at
the moment. "Another suggestion?"
Someone suggests taking a survey of families with four
children. Teach applauds this idea, too, because it focuses on
getting an answer by going out and looking at the world. But
what if a faster answer is needed?
A student wonders if it is possible to "do something that is
like having kids. Put an equal number of red and black balls in
a pot, and pull four of them out. That would be like a family."
This kicks off discussion about how many balls are needed, and
how they should be drawn, which brings out some of the main
concepts in probability -- sampling with or without replacement,
independence, and the like.
Then somebody wonders whether the chance of having a girl
the first time you have a child is the same as the chance of a
red ball from an urn with even numbers of red and black balls, an
important question indeed. This leads to discussion of whether
50-50 is a good approximation. This brings up the question of
the purpose of the estimate, and the instructor suggests that a
clothing manufacturer wants to know how many sets of matched
girls' dresses to make. For that purpose, 50-50 is okay, the
class says.
Coins are easier to use than balls, all concur. Someone
wonders whether four coins flipped once give the same answer as
one coin flipped four times. Eventually all agree that trying it
both ways is the best way to answer the question.
Teach commissions one student to orchestrate the rest of the
class in a coin-flipping exercise. Then the question arises: Is
one sample of (say) thirty coin-flip "families" enough? So the
exercise is repeated several times, and the class is impressed
with the variability from one sample of thirty to the next. Once
again the focus is upon variability, perhaps the most important
idea inherent in prob-stats.
WHY RESAMPLING EASILY GETS CORRECT ANSWERS
Resampling is a much simpler intellectual task than the
formulaic method because simulation obviates the need to
calculate the number of points in the entire sample space. In
all but the most elementary problems where simple permutations
and combinations suffice, the calculations require advanced
training and delicate judgment.
Resampling avoids the complex abstraction of sample-space
calculations by substituting the particular information about how
elements in the sample are generated randomly in a specific
event, as learned from the actual circumstances; the analytic
method does not use this information. In the case of the
gamblers prior to Galileo, resampling used the (assumed) facts
that three fair dice are thrown with an equal chance of any
outcome, and they took advantage of experience with many such
events performed one at a time; in contrast, Galileo made no use
of the actual stochastic element of the situation, and gained no
information from a sample of such trials, but rather replaced all
possible sequences by exhaustive computation.
Contrary to what many mathematicians formerly claimed, the
analytic method for obtaining solutions - using permutation and
combination formulas, for example - is not theoretically superior
to resampling. Resampling is not "just" a stochastic-simulation
approximation to the formulaic method. It is a quite different
route to the same endpoint, using different intellectual
processes and utilizing different sorts of inputs; both
resampling and formulaic calculation are shortcuts to estimation
of the sample space and its partitions. As a partial analogy,
resampling is like fixing a particular fault in an automobile
with the aid of a how-to-do-it manual's checklist, the formulaic
method like writing a book about the engineering principles of
the auto; the engineer-author may be no better at fixing cars
than the hobbyist is at writing engineering books, but the
hobbyist can learn quicker than can the author all that he needs
to know for a particular repair.
The much lesser degree of intellectual difficulty is the
source of the central advantage of resampling. It improves the
probability that the user will arrive at a sound solution to a
problem - the ultimate criterion for all except for pure
mathematicians.
A common objection is that resampling is not "exact" because
the results are "only" a sample. Ironically, the basis of all
statistics is sample data drawn from actual populations.
Statisticians have only recently managed to win most of their
battles against those bureaucrats and social scientists who, out
of ignorance of statistics, believed that only a complete census
of a country's population, or examination of every volume in a
library, could give satisfactory information about unemployment
rates or book sizes. Indeed, samples are sometimes even more
accurate than censuses. Yet many of those same statisticians
have been skittish about simulated samples of data points taken
from the sample space - drawn far more randomly than the data
themselves, even at best. They tend to want a complete "census"
of the sample space, even when sampling is more likely to arrive
at a correct answer because it is intellectually simpler (as with
the gamblers and Galileo.)
Resampling offends mathematical statisticians because it
does not meet the main criteria of what has always been
considered truth in mathematics. It butts up against the
fundamental attitude of the mathematics profession toward non-
proof-based "Martian" methods.
[SECTION OUT?] SOME ORIGINS OF RESAMPLING
[It was a case of too much book-learning, too little
understanding. The students had swallowed but not digested a
bundle of statistical ideas which now misled them, taught by
professors who valued fancy mathematics even if useless or wrong.
It was the spring of 1967 at the University of Illinois, in my
course in research methods with four PhD students. I required
each student to start and finish an empirical research project.
Now the students were presenting their work in class. Each used
wildly wrong statistical tests to analyze their data.
All had had several courses in statistics. But when the
time came to apply even the simplest statistical ideas and tests
in their research projects, they were lost. Their courses had
plainly failed to equip them with basic usable statistical tools.
So I wondered: How could we teach the students to distill
the meaning from their data? Given that the students' data had a
random element, could not the data and the events that underlie
the data be "modeled" with coins or cards or random numbers,
doing away with any need for complicated formulas?
Next class I asked the students the probability of getting
three girls in a four-child family. After they recognized that
they did not know the correct formula, I demanded an answer
anyway. After suggesting some interesting other ideas -
empirical experiments and surveys - one student suggested
flipping coins. With that the class was off to the races. Soon
the students were inventing ingenious ways to get answers -- and
sound answers -- to even subtle questions in statistics and
probability by flipping coins and using random numbers.
Meyer Dwass in 1957, and J. H. Chung and D. A. S. Fraser in
1958, had suggested a resampling version of Fisher's permutation
test for deciding whether two sample means differ from each other
(which, in ignorance of their work, I reinvented.) Now I
suggested handling all (or at least most) problems with
resampling. I offered a wide illustrative variety of methods to
cover the field, and taught a systematic procedure for the wide
range of standard problems.
Could even children learn this powerful way of dealing with
the world's uncertainty? Max Beberman, the guru of the "new
math" who then headed the mathematics department in the
University of Illinois High School, quickly agreed that the
method had promise, and suggested teaching the method to a class
of volunteer juniors and seniors. The kids had a ball. In six
class hours they were able to discover solutions and generate
correct numerical answers for all the major problems ordinarily
found in a semester-long university introductory statistics
class. And they loved it.
Students in an introductory university class complained that
coins and random number tables get tiresome. So in 1973 a
computer language was developed to do with the machine what one's
hands do with cards or dice - RESAMPLING STATS.
Kenneth Travers recruited several PhD students at the U. of
I's College of Education for theses on experiments teaching and
comparing resampling and conventional methods, and in 1976 the
successful results were published in American Mathematical
Monthly.
A turning point came in the late 1970s when a major sub-part
of the general resampling method -- called the "bootstrap" -- was
developed extensively by Bradley Efron and Stanford colleagues,
and now has swept the field of mathematical statistics and
provided legitimacy. By 1991 a National Research Council
publication referred to "statisticians' infatuation with resam-
pling methodology", saying that "Resampling techniques and semi-
parametric models freed investigators from the shackles of normal
theory and linear models". And in Science News, the then-
Executive Director of the American Statistical Association,
Barbara Bailar, said:
[Resampling] has had a tremendous influence on the
route statistics is taking," Bailar says. "It's a very
powerful technique -- a major contribution to the
theory and practice of statistics. We're going to see
more and more applications of it."
PROOF THAT RESAMPLING WORKS
Every day of the week people argue that their approach to
education is better than the other fellow's. Few provide solid
evidence for their claims, however. To show that this is not one
of those unproven assertions, here is some evidence:
As far back as the 1970s, controlled experiments (Simon,
Atkinson, and Shevokas, 1976) found better classroom results for
resampling methods than for conventional methods, even without
the use of computers. Students handle more problems correctly,
and like statistics much better, with resampling than with
conventional methods.
The experiments comparing the resampling method against
conventional methods also showed that students enjoy learning
statistics and probability this way.
Recent surveys of student judgments of courses using the
resampling method - including both introductory classes and
graduate students in statistics, and taught at places ranging
from Frederick Junior College to Stanford University Graduate
School - show that students approve the method. They say they
learn from it, would recommend a course taught with the
resampling method, find it interesting, and use what they learned
in years after they finish to course.
The current students, like the students in the 1970s
experiments, do not show the panic about this subject often shown
by other students. This contrasts sharply with the less positive
reactions of students learning by conventional methods, even when
the same teachers teach both methods in the experiment.
These results should constitute a prima facie case for at
least trying out resampling in a wide variety of educational
settings. But more empirical study would be welcome. The
statistical utility of resampling and other methods is an
empirical issue, and the test population should be non-
statisticians.
WHAT WILL HAPPEN WITH RESAMPLING STATISTICS?
As would be expected, even those statisticians who are
critical of the present substance and methods of teaching
statistics, and who recognize how the algebraic devices lead to
confusion in statistical practice, are ambivalent about it. For
example, a recent president of the American Statistical
Association called for research that would provide better methods
- but would not even acknowledge its existence when shown
evidence that the resampling method is indeed provably better.
(Zellner, correspondence of November 12, 1990).
The trouble in statistics teaching is in the product, and
not the packaging and advertising.
The theory that's usually taught to elementary
[statistics] students is a watered-down version of a very
complicated theory that was developed in order to avoid a
great deal of numerical calculation... It's really
quite a hard theory, and should be taught second, not
first. (Efron quoted by Peterson, 1991, p. 56)
Sooner or later the conventional enterprise smashes up against an
impenetrable wall -- the body of complex algebra and tables that
only a rare expert understands right down to the foundations.
For example, almost no one teacher can write the formula for the
"Normal" distribution that is at the heart of most statistical
tests, let alone the students. Even fewer understand its
meaning. Yet without such understanding, there can be only rote
learning.
One must note a certain schizophrenia. The very
statisticians who assert that the problem is the "wall of
algebra" proceed to use closed-form formulas heavily - even in
their discussions of resampling which boasts that it renders much
(if not all) of the formulaic approach nugatory (see for example
Efron and Tibshirani, 1993; Hall, 1992; Westfall and Young,
1993).
IN THOMPSON'S TITLE: "EPILOGUE AND APOLOGUE"
The final words of Calculus Made Easy say it well:
EPILOGUE AND APOLOGUE
It may be confidently assumed that when this
tractate Calculus Made Easy falls into the hands of the
professional mathematicians, they will (if not too
lazy) rise up as one man, and damn it as being a
thoroughly bad book. Of that there can be, from their
point of view, no possible manner of doubt whatever.
It commits several most grievous and deplorable errors.
First, it shows how ridiculously easy most of the
operations of the calculus really are.
Secondly, it gives away so many trade secrets. By
showing you that what one fool can do, other fools can
do also, it lets you see that these mathematical
swells, who pride themselves on having mastered such an
awfully difficult subject as the calculus, have no such
great reason to be puffed up. They like you to think
how terribly difficult it is, and don't want that
superstition to be rudely dissipated.
Thirdly, among the dreadful things they will say
about "So Easy" is this: that there is an utter
failure on the part of the author to demonstrate with
rigid and satisfactory completeness the validity of
sundry methods which he has presented in simple
fashion, and has even dared to use in solving problems!
But why should he not? You don't forbid the use of a
watch to every person who does not know how to make
one? You don't object to the musician playing on a
violin that he has not himself constructed. You don't
teach the rules of syntax to children until they have
already become fluent in the use of speech. It would
be equally absurd to require rigid demonstrations to be
expounded to beginners in the calculus.
One other thing will the professed mathematicians
say about this thoroughly bad and vicious book: that
the reason why it is so easy is because the author has
left out all the things that are really difficult. And
the ghastly fact about this accusation is that--it is
true! That is, indeed, why the book has been
written--written for the legion of innocents who have
hitherto been deterred from acquiring the elements of
the calculus by the stupid way in which its teaching is
almost always presented. Any subject can be made
repulsive by presenting it bristling with difficulties.
The aim of this book is to enable beginners to learn
its language, to acquire familiarity with its endearing
simplicities, and to grasp its powerful methods of
solving problems, without being compelled to toil
through the intricate out-of-the-way (and mostly
irrelevant) mathematical gymnastics so dear to the
unpractical mathematician.
There are amongst young engineers a number on
whose ears the adage that what one fool can do, another
can, may fall with a familiar sound. They are
earnestly requested not to give the author away, nor to
tell the mathematicians what a fool he really is
(1910/1946, pp. 236-237).
What Thompson says about the calculus and his method also is
true about statistics and the resampling method.
The author is well aware of the quixotic nature of the
campaign described in this article. It might even be called the
most foolish crusade ever undertaken. It is up against the
largest constituency in the world: all those who think they are
clever. And it has no constituency of its own.
Why then undertake it? One might as well as why birds fly
and fish swim.
Some readers will chide me for the aggressive attitude I
display here. They may even tell me that I am my own worst
enemy, because the fighting words written here will anger those
whom I should try to persuade with a positive approach. But for
more than a quarter of a century I have been trying to persuade
those persons with the most conciliatory words and thoughts that
I could dream up, and I have been quite unsuccessful. The reason
is that they see beyond the words to the reality - which is a
threat to their situation. And no amount of statesmanship can
get around that. This controversy is not about communication; it
is about intellectual and occupational turf.
REFERENCES
Allen, R. G. D., Mathematical Analysis For Economists
(London: Macmillan & Co. Ltd; New York: St. Martin's Press,
1956).
Barlow, Roger, Statistics (New York: Wiley, 1989).
Barrow, John D., Pi in the Sky: Counting, Thinking and Being
(New York: Oxford UP, 1992)
Beckmann, Petr, A History of Pi (Boulder, Colo: The Golem
Press, 5th edition, 1982).
Dallal, Gerard E., "Statistical Computing Packages: Dare We
Abandon Their Teaching to Others?", in The American Statistician,
November 1990, Vol. 44, No. 4, p. 265-6.
Efron, Bradley and Tibshirani, Robert J., An Introduction to
the Bootstrap (New York: Chapman and Hall, 1993).
Einstein, Albert, Relativity (New York: Crown Press,
1916/1952).
Freedman, David, Robert Pisani, Roger Purves, and Ani
Adhikari, Instructor's Manual for Statistics (second edition)
(New York: Norton, 1991).
Gardner, Martin, Mathematical Carnival (New York: Vintage
Books, 1977).
Garfield, B. Joan, "Reforming the Introductory Statistics
Course," paper presented at the American Educational Research
Association Annual Meeting, Chicago, 1991.
Garfield, Joan, and Andrew Ahlgren, "Difficulties in
Learning Basic Concepts in Probability and Statistics:
Implications for Research," Journal for Research in Mathematics
Education, 1988, vol. 19, no. 1, 44-63.
Hall, Peter, The Bootstrap and Edgeworth Expansion (New
York: Springer-Verlag New York, Inc., 1992).
Hardy, C. H., A Mathematician's Apology (New York:
Cambridge U. P, 1967).
Hey, John Dn D., Data in Doubt: An Introduction to Bayesian
Statistical Inference for Economists (Oxford: Martin Robertson,
1983).
Hogg, Robert V., "Statistical Education: Improvements are
Badly Needed", The American Statistician, vol. 45, Nov. 1991,
342-343.
Hogg, Robert V., "Statisticians Gather to Discuss Statisti-
cal Education," Amstat News, November 1990.
Kline, Morris, Mathematics for the Nonmathematician (New
York: Dover Publications, Inc., 1985).
McCabe, George P., and Linda Doyle McCabe, Instructors Guide
with Solutions for Introduction to the Practice of Statistics
(New York: W. H. Freeman, 1989).
Peterson, Ivars, "Pick a Sample," Science News, July 27,
1991.
Richmond, Donald E., Fundamentals of the Calculus (New York:
McGraw-Hill Book Company, Inc., 1950).
Ruberg, Stephen J., Biopharmaceutical Report, Vol 1, Summer,
1992.
Russell, Bertrand, The ABC of Relaivity (New York: Mentor,
1925/1985).
Simon, Julian L., Atkinson, David T., and Shevokas, Carolyn,
"Probability and Statistics: Experimental Results of a Radically
Different Teaching Method", American Mathematical Monthly, vol.
83, no. 9, Nov. 1976, pp. 733-739.
Skemp, Richard, R., The Psychology of Learning Mathematics
(New York: Penguin, 1971).
Thompson, Silvanus P., Calculus Made Easy: Being a Very-
Simplest Introduction to Those Beautiful Methods of Reckoning
Which Are Generally Called by the Terrifying Names of the
Differential Calculus and the Integral Calculus (New York:
Macmillan, 1910/1946 third edition).
Vaisrub, Naomie, Chance, Winter, 1990, p. 53.
Wallis, W. Allen, and Harry V. Roberts, Statistics: A New
Approach (Chicago: Free Press, 1956).
Westfall, Peter H., and S. Stanley Young, Resampling-Based
Multiple Testing (New York: Wiley, 1993).
Whitehead, Alfred North, An Introduction to Mathematics
(London, Oxford, and New York: Oxford University Press, 1969.
First published in the Home University Library, 1911).
Zellner, Arnold, correspondence of November 12, 1990.