(1)  I think that the burden of proof
that learning mathematics *does not* train the mind
belongs with those who make that (negative) claim.
Similarly the claim that learning Latin grammar
*does not* make one a better speaker and writer of English
is so dubious that it hardly deserves our attention.

Nobody in the athletics department questions
the value of football players high-stepping through
automobile tires.  It is self-evident that this exercise
improves strength and coordination.  The fact that
the tires are removed from the field of play
before the opening kickoff does not detract from
the value of the exercise.

The teaching of grammar in American schools has
completely decayed, and furthermore is against
the prevailing ideology in schools of education,
so that even serious writers no longer know
when to use "might" as opposed to "may."
Possibly the only politically acceptable way
to inculcate the grammatical turn of mind
is to sneak it into the students' heads
through study of a foreign language, whether
Latin, German, or whatever, where the students
can't pass the test until they have thought about grammar.
But I don't see how anyone can sincerely argue
that understanding of grammar, and effective language,
doesn't transfer from one language to another.

I would go so far as to say, and I would be happy to
debate it, that learning to diagram sentences
not only improves writing and speaking,
but probably improves logical thinking,
and thus makes students better prepared for mathematics.

(2)  On another point that has been running through this thread,
I remember, when I was on our mathematics department's
committee on the business courses -- i.e. the courses
in finite math and calculus that were designed for
students from the college of business -- We had always
followed the publishers' lead and designed these courses
to be full of applications -- or actually pretend, fake,
applications.  Then our faculty counterparts in the business college
told us that, rather than segregating their students into
"business sections" to take "business calculus,"
they would prefer that their students enroll in the 
mainstream courses.  Evidently they realized, perhaps
better than we in the math department did, that the
real purpose of the math requirement for business students
was (is) learning qualitative literacy, & mathematical thinking,
and not how to do phony exercises in maximizing the
net income from bicycle sales on the assumption
that sales are a cubic function of price 
(it had to be a polynomial function because
transcendental functions were not in the curriculum).

We could not accommodate this, because, whether they
liked it or not, our courses served as a "filter," and their
students, most of whom were not motivated to learn
any mathematics, would not survive in the mainstream courses.

One book actually had exercises where one used calculus
to optimize some function related to "purple people eaters on Mars."
Talk about practical applications!  I encouraged my business students
to bring their crap detectors to math class, and not to imagine
that they were taking calculus because
some day they would be in the manager's office 
overlooking the shop floor, and a mechanic would come up
to them with a polynomial that needed to be integrated.

At 09:30 AM 5/18/2010, Murray Eisenberg wrote:
>I read Underwood Dudley's article and am skeptical about his thesis that one learns mathematics in order to train the mind, to learn to think, etc.
>
>What evidence is there that particular skills taught and learned in mathematics generalize to other areas of thinking and reasoning?  Or is it just that people who are particularly good at learning and doing the kind of reasoning employed in math happen to be good, too, at reasoning in other subjects.
>
>Dudley's thesis reminds me, in a way, of the claim that one learns Latin in order to better understand English grammar.  But surely learning English grammar is the best way to better understand English grammar (without superimposing upon it some artificial and inappropriate Latin structure).
>
>I do agree with Dudley that many of the purported "applications" foisted upon students (and teachers) in math books is so much nonsense. In many such applied problems, you are given what you could not possibly already know and are asked to determine what you already know. (Exercise for the reader: find 10 such examples in the first three chapters of a current calculus text.)  Then there are the ridiculous problems, again foisted off as having real-world import, that ignore critical real-world constraints.  E.g., finding the dimensions of a window consisting of a rectangle surmounted by a semicircle that maximizes the area given a fixed perimeter (when you really need to take into account architectural limitations not to mention aesthetic considerations); or to minimize the material used to make a circular can given the volume it will hold (without taking into account odd shapes that don't fit shelves or shipping containers, or again without considering appeal to the buyer).
>
>On the other hand, Dudley may be underemphasizing genuine real-world applications which are often not taught because they are too messy. Again from calculus, there's the old chestnut about the lifeguard running along the beach and swimming in the water in order to reach the drowning man; or the "smart" dog who knows how to do the same sort of minimization. Such problems are posed, typically, because the numbers work out tractably.  But too often a significant, real-world application is ignored -- the behavior of light rays in different media, e.g., in passing from air to water, where the numbers are not so nice.  (My one-time colleague Frank Wattenberg taught me to use that application.)
>
>Murray Eisenberg                     [log in to unmask]
>Mathematics & Statistics Dept.
>Lederle Graduate Research Tower      phone 413 549-1020 (H)
>University of Massachusetts                413 545-2859 (W)
>710 North Pleasant Street            fax   413 545-1801
>Amherst, MA 01003-9305

Martin C. Tangora
University of Illinois at Chicago
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