The Historical Approach
By Bobby Neal WintersIntroduction: The Historical Approach
As with so many things, it came into my hands through the
recommendation by a colleague of a book she had not actually read. The
book was A Radical Approach to Real Analysis
by David Bressoud. The thing contained therein was the historical
approach to teaching mathematics.
Mathematicians
are by their natures optimizers. Perhaps the greatest mathematician of
all time, Karl Friedrich Gauss (1777-1855), said, “A cathedral isn’t a
cathedral until the last piece of scaffolding is removed.” His personal
motto was “Pauca et Matura,” few but ripe. This is his testimony to
the fact that after mathematicians have scaled to the top of the
mountain with ropes and spikes and what not, they try to build trails,
roads, or even rails roads up for the rest of us.
This
is a great service for the rest of us, but in looking at the road,
there is a tendency to forget that this is simply the endpoint of a
process which as included numerous researchers, teachers, and students.
Claude Shannon (1916-2001) is known as the father of
information theory. In his theory, he imagined information going from
one location to another via symbols. The teacher/learner relationship
is a system of communication, but not necessarily in the way one might
naively imagine it following on the heels of that sentence. A teacher is
a facilitator of the communication process. It is here that I want to
thread carefully, because I dwell among those who kill upon hearing the
phrase “Not a sage on the stage but a guide on the side.”
In communicating from a transmitter to a receiver, each has
his (or her) on set of symbols. Practical communication theory tells us
there will probably be information lost even if both sets of symbols
are the same. In the teaching/learning process, the symbol sets are
probably not the same. The transmitter may be a German scholar who was
doing his best work when Napoleon was marching across Europe and the
learner might be a wannabe scholar growing up in the Oklahoma oil fields
when Jimmy Carter was in the White House.
My
point is there is a living system in place that transmits between those
two points and more. The Gaussian attitude of removing the scaffolding
is an important part of presenting that beautiful cathedral, but we as
teachers must keep in mind that the scaffolding is still out back in the
shed. Knowing that the beautification of mathematical results (or
indeed the corresponding acts in any discipline) is an important part of
pedagogy, should empower us as teacher-scholars.
Origins in Applications
One piece of scaffolding that often remains hidden is the Primum
Movens in mathematics is physics. I use the word
physics here to be broad enough to include engineering. This Prime
Mover has interfered with mathematics multiple times in history.
The Greeks did pursue geometry for its own sake as an
intellectual game, but Euclid (circa 300 BC) was known as Euclid of
Alexandria. Alexandria was the city of Alexander the Great who, while
not a Greek’s Greek, did spread Greek culture to lands the old fashioned
way: by conquering them. It is not difficult to imagine Euclid and his
like codifying the discoveries of those who’d been involved the many
construction activities of the Egyptians. In effect, they were creating
an orderly way for students to learn the geometry without to actually
build the pyramid first.
Fourier Series
were developed by Joseph Fourier (1768-1830) for problems connected
with the theory of heat. Fourier, while getting great results, didn’t
have a theoretical foundation for his mathematics. Providing those
foundations gave birth to new fields and new directions to old.
We like to teach the pursuit of knowledge for its own sake,
but knowledge has an end, a purpose. Mathematics need not teach mental
masturbation. It should, in fact, teach the opposite: the delay of
gratification. The delay of gratification, I am convinced, is one of
the cornerstones of civilization.
While
in the process of learning to delay gratification, it is helpful to the
student to know there is some gratification to be had at the end of the
process. The engineering student will endure Heat and Thermodynamics
because he knows he must have it to attain his engineering credential.
He will endure Engineering Mathematics because he knows he must have it
to pass Heat and Thermo. He will endure Fourier series because he has
to understand them to pass Engineering math.
As
a mathematics teacher, I can help him to build that narrative. That
process will be aided if I know the story myself in broad terms.
Whereas the details of the story are not a necessary part of my
intellectual equipment as a mathematician, per
se, they are an important part of my tool box
of teacher as scholar.
Bridges
Another value to the use of the historical approach is in
helping the student cross the bridge from where he is now to where he
needs to be. One commonality shared by a modern student of mathematics
before taking up Fourier series and a historical figure like Fourier he
began his work is an ignorance of Fourier series. We can begin at the
same place Fourier did and go from there.
Fourier
didn’t spend much time on the theory. The results he got worked very
well for him when they worked. When they didn’t work, well, that could
be taken care of later.
And it was.
The first part of the mathematical program consists of
calculus and other courses that are of use to engineers, physicists,
chemists, biologists, and economists who are interested in mathematics
as a tool rather than as an end in itself. Mathematicians go through
these ourselves because we do value the utility of the subject and want
to equip ourselves in its more practical aspects; this is true, but
there is more. In addition, the number mathematics majors is typically
too small to pay for a separate track for our majors.
This having been said, our majors get a somewhat skewed view
of the subject. Many absorb the view of a subject that consists of
methods that have already been worked-out, methods that they need only
memorize and master. Mathematics is a living subject and our students
need a bridge from the civilized, cultural center of the subject across
the river of uncertainty to the frontier.
The
historical approach supplies such a bridge.
As
someone who has lived most of his life in the center of the country,
first Oklahoma and then Kansas, I will claim some familiarity in the
frontier. Just like in the movies, there are times on the frontier
where the rules are temporarily...um...ignored. We can see Newton,
Euler, and Cauchy doing things we would rap our student’s knuckles for.
Isaac Newton (1642-1727) developed series presentations for
sine and cosine. His work is absolutely brilliant, but it’s also
stupid. This is an exercise today for a freshman using Taylor’s Series.
Seeing the insight behind Taylor’s Series takes half a second, but
Newton, as brilliant as he was, didn’t have it. This didn’t stop
Newton, however, as he experimented, found patterns, and verified the
patterns to his satisfaction.
Leonhard Euler
(1707-1783) played--and I will stick by that verb--with infinite series
and obtained tremendous results. Toward the end of his career, some of
his contemporaries thought he was going off the deep end because some
of his series didn’t converge. It was more than one hundred years after
his death before the foundations were laid that justified these wild
calculations.
Augustin Cauchy
(1789-1857) began to put analysis such as was done by Newton and Euler
on firmer foundations, but in doing so he was himself faced with the
dangers of virgin territory. In Calculus I, we teach the Mean Value
Theorem whose formulation is due to Cauchy. When we prove it in the
introductory analysis course, its prove can be contained in a single
paragraph because the modern theory is so well developed. Yet Cauchy’s
proof is much longer and contains some jumps within it that would cause
us to paint a student’s paper red: but it’s brilliant!
It is good for students to see that our subject is an
adventure. While as undergraduates, they might not have received the
preparation to go into the subjects of current research interests, they
have had enough to appreciate the research of the eighteenth century.
Seeing the spirit of that frontier might inspire them to explore
current frontiers.
The Lasting Effect of History
As scholars who teach, we should be aware of
the effects of history upon us. We are part of a larger world that has
been growing and changing for many centuries. In many cases we bemoan
the publish or perish culture that has grown up around us in modern
academe, but, in doing so, we forget that publishing results is our gift
to civilization. Great minds like Archimedes (c.
287 BC – c. 212 BC) sent letters
to others whom they thought might understand them. Many times they
played tricks upon their rivals to confuse them.
Others
like Niccolò Fontana Tartaglia (1499/1500–1557) solved important
problems like the general cubic equation, but kept the results to
themselves so that they could pose challenges to others and build up
their own reputations.
When Isaac Newton
wrote Principia Mathematica
he invented the language to describe his new system of the world, and
while there might be a jot or a tittle moved here or there, we still use
basically the same language today. I am not sure he was thinking about
this three hundred years ago, but the effect of his work, his words
still stand. Without learning history, however, we might not understand
that he had to be pushed, bullied, and cajoled into writing his book
and he made Edmund Halley pay to have it published for him.
When groups such as the Royal Society and the various
journals arose as means of disseminating research results and
establishing priority, progress hastened. It is difficult to
over-emphasize the value of these steps toward openness and publication.
Yet, history should also show us that we are continually changing.
I believe we need to broaden our ideas of scholarship to
include activies which are, in fact, necessary to teaching, learning,
and research continuing in our universities. Ernest Boyer explored this
generally in his book Scholarship Reconsidered: Priorities of
the Professoriate. I believe that an exploration of the
history of my subject, mathematics, vindicates this idea.
Research and teaching are inextricably linked. Research
discovers the mysteries of creation, but teaching, in a continuum of
forms, transmits those discoveries to the ages. I can have
conversations with Archimedes, Newton, Euler, and Gauss because teachers
who were themselves scholars have worked to make that happen. They
have learned the language of the countries of the past and have shaped
the language of a boy from the Oklahoma oil field so that he may
understand, if only just a little.
That
is what teachers can do and that is little short of magic.
