Today, teaching and ambiguity. The University of
Houston's College of Engineering presents this
series about the machines that make our
civilization run, and the people whose ingenuity
created them.

I've done classroom teaching
ever since January of 1952. Yet I view teaching
with vast distrust. I cannot bear to be told what I
might just as well have read out of a book or,
better yet, figured out myself.

One fall, long ago, I took two classes. One teacher
was known for his clarity. You left his lectures
with perfect class notes -- no thought incomplete,
no question unanswered. The other teacher presented
a wild muddle -- stray thoughts spinning about a
central theme, a great sunspray of disorganized
ideas. Back in my room, I rewrote each lecture as
the good teacher might've delivered it. You can
guess the outcome. The good teacher left me with
only a clean set of notes. The would-be poor
teacher enriched my life.

The subject in both classes was what mathematicians
call complex analysis -- so-called imaginary
numbers. One course dealt with it as math. The
other course was about fluid flow. Let me offer a
little background here. First, think about squaring
numbers:

Two squared is four; three squared is nine; 1.4
squared is a little less than two, and so on. The
square root of two has an infinite number of
digits. You can never write it down with complete
accuracy, but you can come close. Now ask about
negative numbers. What's the square root of minus
four? Square either two or minus two and you get
plus four. Since we can make no obvious physical
sense of the square root of minus four, we call it
*imaginary.*

That may sound like angels dancing on the heads of
pins, but like so much of mathematics, it reveals a
great deal about the nature of things. Describing
the motion of moving air or water is a daunting
mathematical task. In the early 19th century, two
mathematicians -- Cauchy and Riemann -- set up
tools that were later used to explain those
motions. The idea, stunning in its elegance, works
like this:

We solve equations that describe both real and
imaginary numbers. Then the real and imaginary
pieces of the solution describe facets of fluid
motion. An abstraction suddenly turns out to be
very useful. That sort of thing happens all the
time. Again and again, seemingly silly branches of
math become mirrors of reality.

So I spent the Fall of 1956 in two classes,
studying two faces of the same problem. Both
classes demanded that I come to terms with alien
ideas. Both classes were about things I would often
put to use during my long life as an engineer. And,
together, those classes taught me about the
greatest abstraction of all -- the idea that one
person can teach another.

That fall, I rewrote the poor teacher's notes until
they looked like the notes of the good teacher.
Today, I can't even remember the good teacher's
name. But I think about that poor teacher every day
of my life. He added as much to my mental growth as
anyone, short of my parents -- against whose
unexplicit and formative teaching I inevitably
weigh all others.

I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.

(Theme music)