TITLE: Math and Computing as Art
AUTHOR: Eugene Wallingford
DATE: July 02, 2008 10:23 AM
DESC: 
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BODY:
<P ALIGN="right"><EM>
So no, I'm not complaining about the presence <BR/>
of facts and formulas in our mathematics classes, <BR/>
I'm complaining about the lack of <STRONG>mathematics</STRONG> <BR/>
in our mathematics classes.  </BR>
-- Paul Lockhart</EM></P>

A week or so ago
<A HREF="http://www.cs.uni.edu/~wallingf/blog/archives/monthly/2008-06.html#e2008-06-23T15_56_51.htm">
   I mentioned</A>
reading a paper called
<A HREF="http://www.maa.org/devlin/devlin_03_08.html">
   A Mathematician's Lament</A>
by Paul Lockhart and said I'd write more on it later.
<A HREF="http://www.cs.uni.edu/~wallingf/blog/archives/monthly/2008-07.html#e2008-07-01T16_21_27.htm">
   Yesterday's post</A>,
which touched on the topic of teaching what is useful
reminded me of Lockhart, a mathematician who stakes
out a position that is at once diametrically opposed
to the notion of teaching what is useful about math
and yet grounded in a way that our K-12 math curriculum
is not.  This topic is especially salient for me right
now because our state is these days devoting some effort
to the reform of math and science education, and my
university and college are playing a leading role in
the initiative.

Lockhart's lament is not that we teach mathematics
poorly in our K-12 schools, but rather that we <EM>don't
teach mathematics at all</EM>.  We teach definitions,
rules, and formal systems that have been distilled away
from any interesting context, in the name of teaching
students skills that will be useful later.  What
students do in school is not what mathematicians do,
and that's a shame, because mathematicians is fun,
creative, beautiful -- art.

As Lockhart described his nightmare of music students
not being allowed to create or even play music, having
to copy and transpose sheet music, I cringed, because
I recognized how much of our introductory CS courses
work.  As he talked about how elementary and HS students
never get to "hear the music" in mathematics, I thought
of Brian Greene's
<A HREF="http://www.nytimes.com/2008/06/01/opinion/01greene.html">
   Put a Little Science in Your Life</A>,
which laments the same problem in science education.
How have we managed to kill all that is beautiful in
these wonderful ideas -- these powerful and even
<EM>useful</EM> ideas -- in the name of teaching useful
skills?  So sad.

Lockhart sets out an extreme stance.  Make math optional.
Don't worry about any particular content, or the order
of topics, or any particular skills.

<BLOCKQUOTE><EM>
Mathematics is <STRONG>the music of reason</STRONG>.
To do mathematics is to engage in an act of discovery
and conjecture, intuition and inspiration; to be in
a state of confusion--not because it makes no sense
to you, but because you <STRONG>gave</STRONG> it
sense and you still don't understand what your creation
is up to; to have a breakthrough idea; to be frustrated
as an artist; to be awed and overwhelmed by an almost
painful beauty; to be <STRONG>alive</STRONG>, damn it.
</EM></BLOCKQUOTE>

I teach computer science, and this poetic sense resonates
with me.  I feel these emotions about programs all the
time!

In the end, Lockhart admits that his position is extreme,
that the pendulum has swung so far to the "useful skills"
side of the continuum he feels a need to shout out for
the "math is beautiful" side.  Throughout the paper he
tries to address objections, most of which involve our
students not learning what they need to know to be
citizens or scientists.  (Hint: Does anyone really think
that most students learn that now?  How much worse off
could we be to treat math as art?  Maybe then at least
a few more students would appreciate math and be willing
to learn more.)

This paper is long-ish -- 25 pages -- but it is a fun
read.  His screed on high school geometry is unrestrained.
He calls geometry class "Instrument of the Devil" because
it so thoroughly and ruthlessly kills the beauty of proof:

<BLOCKQUOTE><EM>
Other math courses may hide the beautiful bird, or put
it in a cage, but in geometry class it is openly and
cruelly tortured.  
</EM></BLOCKQUOTE>

His discussion of proof as a natural product of a
student's curiosity and desire to explain an idea is as
well written as any I've read.  It extends another idea
from earlier in the paper that fits quite nicely with
something I have written
<A HREF="http://www.cs.uni.edu/~wallingf/blog/archives/monthly/2007-02.html#e2007-02-08T18_23_50.htm">
   about computer science</A>:
Mathematics is the art of explanation.

<BLOCKQUOTE><EM>
By concentrating on <STRONG>what</STRONG>, and leaving
out <STRONG>why</STRONG>, mathematics is reduced to an
empty shell.  The art is not in the "truth" but in the
explanation, the argument.  It is the argument itself
which gives the truth its context, and determines what
is really being said and meant.  Mathematics is 
<STRONG>the art of explanation</STRONG>.  If you deny
students the opportunity to engage in this activity--to
pose their own problems, make their own conjectures and
discoveries, to be wrong, to be creatively frustrated,
to have an inspiration, and to cobble together their
own explanations and proofs--you deny them mathematics
itself.
</EM></BLOCKQUOTE>

I am also quite sympathetic to one of the other themes
that runs deeply in this paper:

<BLOCKQUOTE><STRONG>
Mathematics is about problems, and problems must be
made the focus of a student's mathematical life.
</STRONG></BLOCKQUOTE>

(Ditto for
<A HREF="http://www.cs.uni.edu/~wallingf/blog/archives/monthly/2005-03.html#e2005-03-14T14_47_54.htm">
   computer science</A>.)

<BLOCKQUOTE><EM>
... you don't start with definitions, you start with
problems.  Nobody ever had an idea of a number being
"irrational" until Pythagoras attempted to measure
the diagonal of a square and discovered that it could
not be represented as a fraction.
</EM></BLOCKQUOTE>

Problems can motivate students, especially when students
create their own problems.  That is one of the beautiful
things about math: almost anything you see in the world
can become a problem to work on.  It's also true of
computer science.  Students who want to write a program
to do something -- play a game, predict a sports score,
track their workouts -- will go out of their way to
learn what they need to know.  I'm guessing anyone who
has taught computer science for any amount of time has
experienced this first hand.

As I've mentioned here a few times, my colleague
<A HREF="http://www.cs.duke.edu/~ola/">
   Owen Astrachan</A>
is working on a big project to explore the idea of
<A HREF="http://www.compscipbl.com/">
   problem-based learning</A>
in CS.  (I'm wearing the project's official T-shirt
as I type this!)  This idea is also right in line
with Alan Kay's proposal for an "exploratorium"
of problems for students who want to learn to
commmunicate via computation, which I describe in
<A HREF="http://www.cs.uni.edu/~wallingf/blog/archives/monthly/2004-11.html#e2004-11-06T21_03_42.htm">
   this entry</A>.

I love this passage from one of Lockhart's little dialogues:

<BLOCKQUOTE><EM>
SALVIATI: &nbsp;&nbsp;&nbsp; ... people learn better
when  the product comes out of the process.  A real
appreciation for poetry does not come from memorizing
a bunch of poems, it comes from writing your own.

SIMPLICIO: &nbsp;&nbsp;&nbsp; Yes, but before you can
write your own poems you need to learn the alphabet.
The process has to begin somewhere.  You have to walk
before you can run.

SALVIATI: &nbsp;&nbsp;&nbsp; ...  No, you have to have
something you want to run <STRONG>toward</STRONG>.
</EM></BLOCKQUOTE>

You just have to have something you want to run toward.
For teenaged boys, that something is often a girl, and
suddenly the desire to write a poem becomes a powerful
motivator.  We should let students find goals to run
toward in math and science and computer science, and
then teach them how.

It's interesting that I end with a running metaphor,
and not just because
<A HREF="http://www.cs.uni.edu/~wallingf/blog/archives/monthly/2008-06.html#e2008-06-09T16_48_00.htm">
   I run</A>.
My daughter is a sprinter and now hurdler on her
school track team.  She sprints because she likes to
run short distances and hates to run anything long
(where, I think, "long" is defined as anything longer
than her race distance!).  The local runners' club
leads a summer running program for high school students,
and some people thought my daughter would benefit.
One benefit of the program is camaraderie; one drawback
that it involves serious workouts.  Each week the group
does a longer run, a day of interval training, and a
day of hill work.

I suggested that she might be benefit more from simply
running more -- not doing workouts that kill her, just
building up a base of mileage and getting stronger
while enjoying some longer runs.  My experience is
that it's possible to get over the hump and go from
disliking longs runs to enjoying them.  Then you can
move on to workouts that make you faster.  So she and
I are going to run together a couple of times a week
this summer, taking it easy, enjoying the scenery,
chatting and otherwise not stressing about "long runs".

There is an element of beauty versus duty in learning
most things.  When the task is all duty, you may do
it, but you may never like it.  Indeed, you may come
to hate it and stop altogether when the external
forces that keep you on task (your teammates, your
sense of belonging) disappear.  When you enjoy the
beauty of what you are doing, everything else changes.
So it is with math, I think, and computer science, too.
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