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Re: Blame Teachers, Blame Parents
Posted:
Sep 6, 2007 7:20 PM
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Haim wrote (in part):
http://mathforum.org/kb/message.jspa?messageID=5893586
> I suppose that if a small child asks his parents
> why he needs arithmetic, most parents could give
> some assurance on this point. But what do most
> parents say when their older children ask why
> they have to factor polynomials or solve
> simultaneous equations? That incomprehending
> shrug speaks with an eloquence that no amount
> of edu-speak will ever overcome.
The paper below by Ralph P. Boas, which I've
included in its entirety, should be of interest
in these discussions. My main reason for posting
it is because it dove-tails with issues that
I brought up in the following thread about a
year ago (see especially the last sentence of
the Boas article in this regard), but perhaps
others will find other things of interest in
Boas' paper.
What has technology subtracted? [math-teach, June 2006]
http://mathforum.org/kb/message.jspa?messageID=4796075
http://mathforum.org/kb/message.jspa?messageID=4796145
http://mathforum.org/kb/message.jspa?messageID=4804253
I've read any number of papers/essays/book-reviews
like this (although few as well written) that
date from the mid 1800's (e.g. Augustus De Morgan)
all the way down to the present, and I've
mentioned it off-and-on-again in this group
during the past few years. The papers I've
encountered are not just on the topic that
Boas deals with. You can find them on pretty
much any topic that people are still debating
about. The people are different, the details
vary somewhat, but the essence of just about
every "reform movement" argument, pro or con,
in mathematics education over the past few
decades can be found repeated over and over
again, often in periodic cycles that whose
period length tends to be roughly the average
productive lifetime of scholars and educators,
30-40 years.
Dave L. Renfro
- ----------------------------------------------------
Ralph P. Boas, "If this be treason ...", American
Mathematical Monthly 64 (1957), 247-249.
If I had to name one trait that more than any
other is characteristic of professional mathematicians,
I should say that it is their willingness, even
eagerness, to admit that they are wrong. A sure
way to make an impression on the mathematical
community is to come forward and declare, "You
are doing such-and-such all wrong and you should
do it _this_ way." Then everybody says, "Yes,
how clever you are", and adopts your method.
This of course is the way progress is made, but
it leads to some curious results. Once upon a
time square roots of numbers were found by
successive approximations because nobody knew
of a better way. Then somebody invented a
systematic process and everybody learned it
in school. More recently it was realized that
very few people ever want to extract square
roots of numbers, and besides the traditional
process is not really very convenient. So now
we are told to teach root extraction, if we
teach it at all, by successive approximations.
Once upon a time people solved systems of linear
equations by elimination. Then somebody invented
determinants and Cramer's rule and everybody
learned that. Now determinants are regarded as
old-fashioned and cumbersome, and it is considered
better to solve systems of linear equations by
elimination.
We are constantly being told that large parts
of the conventional curriculum are both useless
and out of date and so might better not be taught.
Why teach computation by logarithms when everybody
who has to compute uses at least a desk calculator?
Why teach the law of tangents when almost nobody
ever wants to solve an oblique triangle, and if
he does there are more efficient ways? Why teach
the conventional theory of equations, and especially
why illustrate it with ill-chosen examples that
can be handled more efficiently by other methods?
As a professional mathematician, I am a sucker for
arguments like these. Yet, sometimes I wonder.
There are a few indications that there is a reason
for the survival of the traditional curriculum
besides the fact that it is traditional. When
I was teaching mathematics to future naval
officers during the war, I was told that the
Navy had found that men who had studied calculus
made better line officers than men who had not
studied calculus. Nothing is clearer (it was
clear even to the Navy) than that a line officer
never has the slightest use for calculus. At the
most, his duties may require him to look up some
numbers in tables and do a little arithmetic with
them, or possibly substitute them into formulas.
What is the explanation of the paradox?
I think that the answer is supplied by a phenomenon
that everybody who teaches mathematics has observed:
the students always have to be taught what they
should have learned in the preceding course.
(We, the teachers, were of course exceptions;
it is consequently hard for us to understand
the deficiencies of our students.) The average
student does not really learn to add fractions
in arithmetic class; but by the time he has
survived a course in algebra he can add numerical
fractions. He does not learn algebra in the
algebra course; he learns it in calculus, when
he is forced to use it. He does not learn calculus
in the calculus course, either; but if he goes
on to differential equations he may have a
pretty good grasp of elementary calculus when
he gets through. And so on through the hierarchy
of courses; the most advanced course, naturally,
is learned only by teaching it.
This is not just because each previous teacher
did such a rotten job. It is because there is
not time for enough practice on each new topic;
and even if there were, it would be insufferably
dull. Anybody who has really learned to interpolate
in trigonometric tables can also interpolate
in air navigation tables, or in tables of Bessel
functions. He should learn, because interpolation
is useful. But one cannot drill students on mere
interpolation; not enough, anyway. So the students
solve oblique triangles in order (among other
things) to practice interpolation. One must not
admit this to the students, but one may as well
realize the facts.
Consequently, I claim that there is a place,
and a use, even for nonsense like the solution
of quartics by radicals, or Horner's method,
or involutes and evolutes, or whatever your
particular candidates for oblivion may be.
Here are problems that might conceivably have
to be solved; perhaps the methods are not the
most practical ones; but that is not the point.
The point is that in solving the problems the
student gets practice in using the necessary
mathematical tools, and gets it by doing
something that has more motivation than mere
drill. This is not the way to train mathematicians,
but it is an excellent way to train mathematical
technicians. Now we can understand why calculus
improves the line officer. He needs to practice
very simple kinds of mathematics; he gets this
practice in less distasteful form by studying
more advanced mathematics.
It is the fashion to depreciate puzzle problems
and artificial story problems. I think that there
is a place for them too. Problems about mixing
chemicals or sharing work, however unrealistic,
give good practice and even have a good deal
of popular appeal: witness the frequency with
which puzzle problems appear in newspapers,
magazines, and the flyers that come with the
telephone bill. There was once a story in
_The_Saturday_Evening_Post whose plot turned
on the interest aroused by a perfectly preposterous
diophantine problem about sailors, coconuts,
and a monkey. It is absurd to claim that only
"real" applications should be used to illustrate
mathematical principles. Most of the real
applications are too difficult and/or involve
too many side issues. One begins the study of
French with simple artificial sentences, not
with the philosophical writings of M. Sartre.
Similarly one has to begin the study of a branch
of mathematics with simple artificial problems.
We may dislike this state of affairs, but as long
as it exists we must face it. It would be pleasant
to teach only the new and exciting kinds of
mathematics; it would be comforting to teach
only the really useful kinds. The traditional
topics are some of the topics that once were
either new and exciting, or useful. They have
persisted partly by mere inertia -- and that
is bad -- but partly because they still serve
a real purpose, even if it is not their ostensible
purpose. Let us keep this in mind when we are
revising the curriculum.
- ----------------------------------------------------
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