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\begin{document}
\title{The exemplar model in mathematics instruction}
\author{John Kerl}
\date{November 2, 2007}
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\begin{center}
\emph{I was born not knowing and have had only a little time to change that
here and there.} \\
--- Richard Feynman\index{Feynman, Richard} (1918-1988).
\end{center}

I am no math historian; in fact, as a second-year graduate student and a
teacher with only two years' classroom experience, I am only a budding
mathematician.  Yet this much seems to me to be a clear pattern in the
development of mathematics:
\begin{itemize}
\item Real-world problems are studied; mathematical models are developed.
\item After some time goes by, wherein mathematicians collaborate and
	communicate, similarities are discovered between problems which had seemed
	distinct.  (For example, permutations on symbols of letters and Galois'
	original ``substitutions''; systems of linear equations and systems of
	differential equations.)
\item A system of axioms is developed, encapsulating and
	abstracting the common features of the once-disparate systems.  (In the
	above examples, we obtained abstract groups and vector spaces,
	respectively.)
\item We then study the abstract systems, proving general results
	and developing general methods which apply to all the original
	specific situations, \emph{and} any others which might apply
	in the future.  Herein lies much of the power of mathematics:  once we are
	familiar with, say, abstract groups, whenever we encounter a brand-new
	object, if we recognize it is a group then we instantly know much about
	it.
\item After enough time has gone by, we begin to teach the abstractions
	first.  In a pure-math environment, one might forget (or forget to
	mention) the origins entirely.
\end{itemize}

So much for the \emphidx{phylogeny} of mathematics --- its evolutionary
development across the lives of many thinkers.  What about its
\emphidx{ontogeny} --- the development of ideas within the mind of a single
thinker?  Like Feynman, none of us were born knowing calculus; every
mathematics student and future mathematician must painstakingly relearn these
ideas from scratch.

My claim is that \textbf{ontogeny recapitulates phylogeny}.  This is a
biological maxim\footnote{Attributed to the zoologist Ernst H\"ackel.  The
subject area is called \emph{recapitulation theory}.}, reflecting the fact that
the embryonic development of an individual retraces (albeit inexactly) the
evolutionary development of that individual's species.  When I was an embryo,
very early on I had gill slits --- and so did you.  One pair of slits became
our ears; the rest closed.  One might think that we, as modern humans, could
reproduce more humans directly, pole-vaulting over the intermediate stages ---
but we do not.

Likewise in the development of ideas.  When we are young, we experience many
specific situations.  We later learn to abstract and categorize: many
once-different objects begin to go by the common name of \emph{fork} or
\emph{dog}.  (See the Boas quote below.)  One might think we could pole-vault
over our early development, but we do not.

Few would argue that we could hand a two-year-old a dictionary and have him or
her skip over those early learning experiences.  But in the mathematics
classroom --- blinded by the power and beauty of modern methods --- we often do
just that.  In my eyes, the key trait distinguishing the mathematics of the
20th and 21st centuries from that of the centuries before is the
\emphidx{axiomatic approach}.  This is a rightful component of our discipline.
Yet we start the first day of an upper-division or graduate algebra course with
a mantra of the form ``A \emph{group} is a set endowed with a binary operation
\ldots''; we begin a geometry course with an abstract discussion of atlases and
manifolds.  This feels good and it feels right, especially to the teacher:  we
are going straight for the beauty and the elegance.  It is especially easy for
the teacher, who has already seen these concepts put to use in dozens of
concrete situtations:  the teacher is ready for abstraction.

%In our purity, we may never feel a desire to even mention why our subject
%exists in the first place.

But when we do this, we leave our students behind.  We attempt to leap over the
\emph{absolutely necessary} instantiation of ideas that we ourselves went
through (whether we did so consciously or not).  If we are to retain students
in mathematics, and if modern pure mathematics is to remain relevant to the
rest of the world, we must explicitly acknowledge that ontogeny recapitulates
phylogeny.  This does not mean that we need to retrace the full development of
every mathematical concept, with all its false starts and wrong turns.  It does
mean, however, that we \emph{first} need to tell our students \emph{why} a
discipline was invented in the first place.  We need to present the applied
problems which gave birth to the subject we are teaching.  We must give our
students the opportunity to generalize from concrete situations.

One hears the objection:  \emph{I don't have time for such trivialities.
Students should be able to do this on their own.}  I reply, as a graduate
student and teacher:  It is hard enough for our students to learn what we teach
them; it is far harder for them to learn what we do not teach them.  The
training of a new mind is something worth doing right:  it is better that it
happen, not in spite of our teaching methods, but \emph{because} of them.

%A course in differential geometry ought to be more than a series of
%unmotivated definitions and theorems about differential symbology.

\begin{center}
\emph{Suppose that you want to teach the ``cat'' concept to a very young child.
Do you explain that a cat is a relatively small, primarily carnivorous mammal
with retractible claws, a distinctive sonic output, etc.? I'll bet not. You
probably show the kid a lot of different cats, saying ``kitty'' each time,
until it gets the idea. To put it more generally, generalizations are best made
by abstraction from experience.} --- R. P. Boas\index{Boas, R.P.}
\end{center}

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