Epsilon sandwiches
Herbert S. Wilf
Even though I've been teaching for n! years, every class is a fresh
adventure -- An adventure. That word is a euphemism for the stark reality,
which is that every class contains some totally shocking development
that I have never seen before and haven't any idea how to cope with.
Just last year, for the first time in many years, I taught a section
of the junior-level mathematical analysis course. You know -- the one
where students meet proofs in analysis for the first time. The one where
students and epsilons meet, eyeball to eyeball, and it isn't the epsilons
that blink. The one where students decide that they really wanted to be
doctors and lawyers after all.
This course is famous for being our rite of passage. Our hazing
ceremony. If you want to join the club, then here is the hurdle that you
have to jump over. Somehow we spend a lot of our time agonizing over
calculus reform, and very little time thinking about how to improve
this pivotal course, which, perhaps more than any other single course,
determines who our majors will be, and therefore who the mathematicians
of the future will be.
When I was an undergraduate at MIT I had smooth sailing through the
first two years of calculus. In my junior year I took this very same
analysis course, and my teacher was a young C.L.E. Moore Instructor
named -- Walter Rudin.
Then, as now, he wasted few words (the thought of Walter Rudin wasting
words is a possible definition of science fiction). He let his epsilon
be greater than zero, he took his capital M to be one-third of the
reciprocal of that, and on the bottom line it all came out right. It
came out to 1 times epsilon. It wouldn't have had the nerve, it wouldn't
have dared to come out as (M+3) times epsilon. Despite the saying that
"for every epsilon there is a delta," (which is really rather romantic,
when you think about it) I often could not get my epsilons paired off
with suitable deltas. I found that course to be rough going.
I started teaching the course last year by concentrating on the concepts,
the mathematical concepts. There are plenty of them there, including the
likes of convergence of series, uniform and otherwise, continuity, uniform
continuity, differentiability, compactness, Heine-Borel theorems, and
so forth. The students, despite being a bright and hard working bunch,
were falling farther and farther behind.
After a while I started to understand a little better what was going
on. The thing is that that course has not one but two major novel features
in it. One is that the subject is just plain difficult and the concepts
are quite deep.
The other is that, for perhaps the first time in the students'
mathematical careers, manipulative skills and pushing formulas around
just will not suffice. Instead what is at the highest premium is the
ability of the students to wrap complete English sentences around their
mathematical thoughts. Not just to use English, and not just to use
epsilons, but to embed the mathematics in epsilon sandwiches, so that a
continuous mathematical thought flows from one end of a long sentence to
the other end, sometimes skimming across carefully chosen English words,
then running through a brief mathematical display, finishing perhaps
with an English conclusion; all woven together; seamless; proving what
is supposed to be proved; grammatically correct and unambiguous; with
the "for all"s and the "there exists" in just the right places; having a
subject, a predicate, punctuation, and all due appurtenances, accessories,
and optional extras.
An epsilon sandwich, then, is a layer of mathematics between two slices
of English; hold the mayo.
"If, on the other hand, x is not equal to zero, then we can find a M for
which |u-u0| is < delta except for a set of measure less than epsilon/2,
and then we would have ..."
What I am speaking to you about today has a lot in common with some of the
works on mathematical writing, by Don Knuth, Len Gillman, and others, that
have appeared in recent years. But they were striving also for lively,
clear and well motivated exposition. In this junior analysis course I
don't care about "lively" and I don't care about "motivated." "Clear" is
important, but "complete and correct and readable" is really what it's
all about. I am not speaking about writing style here. That is a luxury
that comes after the basics of survival. The construction of a correct
and complete mathematical sentence is what I am talking about today.
But where are our students supposed to have learned this ability to
craft such delicately poised sentences?
In science it is a well known principle that we should never change two
important things at once. Instead we should first change one thing,
then let the waters settle for a bit, and then change the other thing. I
had been violating that rule. What I have learned is that in this course
it is worth pausing for a while in the development of the mathematical
concepts to have an intermission in which the students are doing nothing
but learning how to create their very own epsilon sandwiches.
I wish I had a nickel for every student who has told me "I understand
it. I just can't really say it!" That's a very human feeling. The
problem is that unless you can say this thing you won't be able even
to understand the next thing. In this junior analysis course the line
between understanding and saying becomes very much blurred, maybe for
the first time in the student's career. Without the saying, in this
course, understanding is probably not possible.
That means, it is a good idea to give the students practice in writing out
mathematical sentences and paragraphs, in full, in which the mathematics
that is in those sentences will be very familiar to them. Do that
before getting into the substantive new mathematics. So only one thing
will change. What will change will be the requirement for expressing
mathematical thoughts in writing, clearly, using good English and good
mathematics. Both. Wrapped around each other. But the mathematics should
be familiar at first.
How can this be carried out?
Pick a few attractive, elementary theorems that your students have
probably seen before, and go to work. Prove that the square root of 2
is irrational. That's a good one. Explain it thoroughly. Go over it a
few times until everyone understands what is going on. End of stage 1.
Then call on a student and ask him to describe the beginning of the
proof, without asking him to write anything down. While the student is
talking about the steps, you might jab at the blackboard, writing down
a sketch of the ideas, but without trying to make good sentences out
of them. Then call on another student. Ask her to continue from where
the first one left off. Continue until, with contributions from several
students, you have jointly talked your way through the proof.
Now comes the fun part. Call on a student and ask her to come to the
blackboard and write out the first few lines of the proof. Insist on
complete sentences! Subjects, predicates, verbs, objects, all that
good stuff. Even punctuation; commas after displayed equations. Show
no mercy. Do not accept any spoken sentences. The focus is on what is
being written on the blackboard. To underscore this point you might ask
for silence in the classroom while those sentences are being written on
the board. After they have been written there, you and the class should
criticize them constructively, polish them and get them to be absolutely
perfect. Not a semicolon out of place.
Then erase the board and ask another student to come forward and write
out the same proof. Go over it all again, insisting on perfection in
writing, accepting no oral input. Never mind style, liveliness and that
sort of thing. Just get the vital signs beeping.
Then go on to another example. Give them the proof that the sum of
the interior angles of a triangle is 180 degrees. Then ask someone to
step forward and write it on the board. Constructively criticize, with
the class, every word, phrase, sentence and paragraph, until the proof
positively glitters on the blackboard (do us all a favor and skip the
numbering of the steps, that many high schools insist on). Then erase
the blackboard and do it again with another student.
What are some other examples of good theorems to practice on, aside from
the irrationality of the square root of 2 and the sum of the angles of
a triangle? There is the fact that the sum of the first n integers is
n(n+1)/2; that the sum of the first n odd numbers is n squared; that if
we cut diagonally opposite corners out of a chessboard then it can't be
covered by dominoes; that a set of n elements has exactly 2^n subsets;
that the number of disjoint ordered pairs of subsets of n things is
3^n; and so forth. The attributes of these examples are that they are
elementary to state and easy to understand, that they have short proofs,
and that it is still not quite a trivial job, in each case, to write out
a complete formal correct proof, as opposed to vigorous waving of hands.
Assign homework problems of this same kind. For one thing ask them
to write out once again exactly the proofs that were done in class,
emphasizing that you will want them to be perfect. Then find a few
other examples of proofs that really are proofs but whose mathematical
content is not so difficult in itself, so that the main problem will be
the arrangement of thoughts into sentences.
With some preparation like this, for a few class hours before you really
get into the compact sets, the uniform convergence and so forth, the
bumps can be smoothed out considerably, leaving a transition that,
instead of being traumatic, has risen all the way up to being merely
painful. A little investment of your time and of your students' time in
the construction of finely tuned mathematical sentences at the beginning
of this course, can reap large rewards in sailing the rough seas of the
junior course in real analysis.
I don't see how any textbook could be written that would help
significantly with this job. It is a classic teacher-and-student
human-to-human interaction situation, and what it is about is the
appreciation of the differences between "I understand it but I can't say
it," and "I can say it in my own words but I can't write it out formally,"
and "I can do it all." A textbook can't do that job. A computer can't
either. Neither can a CD or a videotape. Nor can a ten million dollar
grant from a funding agency.
But you can do it. We can do it.
What I have said here today is surely not in the category of a major
revelation, to be funded by a multi-million dollar NSF grant.
If you follow my suggestion, it won't help a lot.
But it will help a little.
And that's all a teacher can ever do.
Good luck. You'll need it; -- and thank you.