Tuesday, April 01, 2008

A Mathematician's Lament by Paul Lockhart

In this essay [ http://www.maa.org/devlin/LockhartsLament.pdf ] Paul argues that mathematics is arts, a highly creative process. A mathematician is a "poetic dreamer".
Few lines copied-pasted from the essay:

"I simply wouldn't have the imagination to come up with the kind of senseless,
soul-crushing ideas that constitute contemporary mathematics education."

"the mathematician's art: asking simple and elegant questions about our imaginary creations,
and crafting satisfying and beautiful explanations. There is really nothing else quite like this
realm of pure idea; it's fascinating, it's fun, and it's free!"

" 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."

Are you really trying to claim that mathematics offers no useful or
practical applications to society?
Of course not. I'm merely suggesting that just because something
happens to have practical consequences, doesn't mean that's what it is
about. Music can lead armies into battle, but that's not why people
write symphonies. Michelangelo decorated a ceiling, but I'm sure he
had loftier things on his mind.

But people need to be able to balance their checkbooks, don't they?
I'm sure most people use a calculator for everyday arithmetic. And
why not? It's certainly easier and more reliable. But my point is not
just that the current system is so terribly bad, it's that what it's missing
is so wonderfully good! Mathematics should be taught as art for art's
sake. These mundane "useful" aspects would follow naturally as a
trivial by-product. Beethoven could easily write an advertising jingle,
but his motivation for learning music was to create something

"You see kids, if you know algebra then you can figure out how old Maria is if we
know that she is two years older than twice her age seven years ago!" (As if anyone would ever
have access to that ridiculous kind of information, and not her age.)

"It is far easier to be a passive conduit of some publisher's "materials" and to follow the
shampoo-bottle instruction "lecture, test, repeat" than to think deeply and thoughtfully about the
meaning of one's subject and how best to convey that meaning directly and honestly to one's
students. "

"Why is it that we accept math teachers who have never
produced an original piece of mathematics, know nothing of the history and philosophy of the
subject, nothing about recent developments, nothing in fact beyond what they are expected to
present to their unfortunate students? "

"We have millions of adults wandering
around with "negative b plus or minus the square root of b squared
minus 4ac all over 2a" in their heads, and absolutely no idea whatsoever
what it means. "

As Gauss once remarked, "What we need are
notions, not notations."

But isn't one of the purposes of mathematics education to help
students think in a more precise and logical way, and to develop their
"quantitative reasoning skills?" Don't all of these definitions and
formulas sharpen the minds of our students?
No they don't. If anything, the current system has the opposite effect
of dulling the mind. Mental acuity of any kind comes from solving
problems yourself, not from being told how to solve them.

Compare your own experience of learning algebra with Bertrand Russell's
"I was made to learn by heart: 'The square of the sum of two
numbers is equal to the sum of their squares increased by twice
their product.' I had not the vaguest idea what this meant and
when I could not remember the words, my tutor threw the book at
my head, which did not stimulate my intellect in any way."

On High School Geometry Proof
"Could anything be more unattractive and inelegant? Could any argument be more
obfuscatory and unreadable? This isn't mathematics! A proof should be an epiphany from the
Gods, not a coded message from the Pentagon.
No mathematician works this way. No mathematician has ever worked this way. This is a
complete and utter misunderstanding of the mathematical enterprise. Mathematics is not about
erecting barriers between ourselves and our intuition, and making simple things complicated.
Mathematics is about removing obstacles to our intuition, and keeping simple things simple."

"Children are expected to master a complex set of
algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part,
and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables
are stressed, as are parents, teachers, and the kids themselves."

On The Standard School Mathematics Curriculum
"Why Geometry occurs in between Algebra I and its sequel remains a mystery."

About the Author:

Paul became interested in mathematics when he was about 14 (outside of the school math class, he points out) and read voraciously, becoming especially interested in analytic number theory. He dropped out of college after one semester to devote himself to math, supporting himself by working as a computer programmer and as an elementary school teacher. Eventually he started working with Ernst Strauss at UCLA, and the two published a few papers together. Strauss introduced him to Paul Erdos, and they somehow arranged it so that he became a graduate student there. He ended up getting a Ph.D. from Columbia in 1990, and went on to be a fellow at MSRI and an assistant professor at Brown. He also taught at UC Santa Cruz. His main research interests were, and are, automorphic forms and Diophantine geometry.
clipped from www.maa.org

After several years teaching university mathematics, Paul eventually tired of it and decided he wanted to get back to teaching children. He secured a position at Saint Ann's School, where he says "I have happily been subversively teaching mathematics (the real thing) since 2000."

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