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Antreas P. Hatzipolakis

Posts: 940
Registered: 12/3/04
Posted: Mar 19, 2001 11:22 AM
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Los Angeles Times Friday, February 6, 1998


Math education: Fewer classes require proofs--more whittling away of
exposure to logic and critical thinking.


While I grew up in snow country, I can't tell my kids that I trudged miles
through snow to get to school. But I can tell them I learned proofs in
high school geometry, which could become as much a part of a vanished
virtuous past.

One of the pleasures of being on the faculty at Caltech is interacting with
our bright undergraduates. For the past two years, I've asked the incoming
freshmen in my calculus/probability class whether they had proofs in their
high school geometry course. About 40% have not, and more than half of the
remainder had at best a cursory few weeks. So less than one-third have had
the kind of rigorous theorem/proof course I had back in James Madison High
School in Brooklyn more than 30 years ago. Why do I mourn this loss of
what was a core part of education for centuries? After all, we no longer
require Greek and Latin in high school and Euclid was just one of those
Greeks, wasn't he? While the geometric intuition that comes from the
classical high school geometry course is significant, what is really
important is the exposure to clear and rigorous arguments.

Modern mathematicians don't use the two-column proofs so beloved by my
high school geometry teachers, and real life rarely needs the precise rigor
of mathematicians, but those who have survived those darned dual columns
understand something about argumentation and logic. They can more readily
see through the faulty reasoning so often presented in the media and by

It is not merely a question of good citizenship. In the global economy,
our young people will be in competition with young people the world
over. If I talk about American high school education with scientific
visitors from abroad, they are either aghast or amused. Immigrants I know
from the former Soviet Union tell of fifth-grade Russian mathematics texts
at a higher level than what we teach juniors in high school. For a large
number of jobs in our technologically based world, a solid scientific and
mathematical training is essential and our foreign competitors are beating
us there.

The trend away from theorem/proof geometry seems to be based on the
following reasoning by the educational establishment: Some high school
students are just unable to get this theorem/proof stuff. If we place them
in a separate track, they'll wind up with a low self-image. So to prevent
some students from having a low self-image, we won't ask anyone to
understand it. This summary is admittedly a caricature but it is the core
of the reason that Euclidean geometry is disappearing.

I'm not particularly worried about the lack of Euclidean geometry among my
Caltech freshmen. While my colleagues also have noted a marked decrease in
the quality of preparation of our students over the past 15 years, we see
that these students are just as smart and motivated as Caltech students
ever were. They are forced by us to absorb notions of careful proof at a
less leisurely pace than they would experience in high school and the
process can be painful, for them and their teachers.

But I am concerned about the country as a whole. The dumbing down of high
school education in the United States, especially in mathematics and
science, is a crime that must be laid at the doorstep of the educational
establishment. We must demand that the level of high school science and
mathematics being taught be improved, starting, of course, with Euclidean
geometry. - - -

Barry Simon Is the Chairman of the Mathematics Department at Caltech.

Copyright Los Angeles Times

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