On 8/16/2010 at 11:54 pm, Domenico Rosa wrote:

> > The truly superb article, "What Is Mathematics
> For?,"
> > by Underwood Duddley has been published in the May
> > 2010 issue of the AMS Notices.
> > 
> > 
> >
> http://www.ams.org/notices/201005/rtx100500608p.pdf
> 
> The August 2010 issue of Notices contains four
> letters about Dudley's article:
> 
> http://www.ams.org/notices/201007/rtx100700822p.pdf


Dom,

I thank you for mentioning this issue of the Notices from
the AMS that contain these four replies to Dudley's article.

David A. Edwards from the University of Georgia says that relatively
few positions even in science and engineering rarely use much
mathematics beyond eighth-grade mathematics (whatever eighth
grade mathematics is exactly, I'm not sure, and that seems
to vary some from state to state anyway) and that they require
technical degrees as merely filters.  He also mentions 
Vivek Wadhwa's statement that America is producing more scientists 
and engineers than there are job openings.  I take that he is referring 
to his article that I had found at
http://www.businessweek.com/smallbiz/content/oct2007/sb20071025_827398.htm.

I'm no scientist or engineer, so I cannot say whether such claims are
true or not.  But I can say that our schools and even colleges are
in such a mess these days that high school diplomas and even many
college degrees are not worth more than the paper they are printed on.
How else do you explain the countless floods of college students
I see whose reading, writing, math, study, and even common sense
skills are still stuck in second or third grade?  It disturbs me
to see all the discussions and assignments in class that appear
as if they were written by the students' children rather than
by the students themselves!  I would conjecture that massive grade
inflation troubles employers enough that many cannot trust that
ones with only a high school diploma or college diploma truly
have meaningful diplomas because of many who do manage to graduate
without learning much of anything.  I suppose that they can test
the skills of those they might be interested in hiring, but I imagine
that such testing is time consuming and expensive.  I thought about
that when I have thought about a job as a statisician, but I believe
I would need a stronger background in statistics to qualify or at
least to give myself a strong chance of getting such a job.  But I
then realized this dilemma if I choose to study some statistics
on my own: How can I show that I have learned more statistics than 
what my degrees and transcripts show?  I would not blame employers
in the least bit if they did not believe me because anyone can
make such claims.  And it would take a lot of their and my time to 
show that I indeed did study on my own.  So this thinking has told
me that I am sure that employers want students with credentials
but that there is tangible proof of such credentials and that the
proof of such credentials is actually meaningful proof and not
simply a fancy version of some scribbled note by someone saying
that John Q. Smith really has these credentials and that I witnessed
this myself.

I doubt these claims myself, but I don't work in science or 
engineering to know how to test this claim or to refute it.  
The best I can do for now is to ask some colleagues I know who work 
in science.        

However, let us suppose Edwards' claim is true.  Does this mean
that students who stop with arithmetic are really competent enough
to understand the mathematical side of science?  Perhaps some are,
but few would be.  I myself would doubt this seriously because
of the meaningless way that most elementary mathematics is taught
and, on top of that, with the massive grade inflation these days
so that many students can finish arithmetic with good grades but
understand very little of it.  And I would venture that most who
work in education know that virtually all students finish arithmetic
with good grades but don't understand much of it.  Those who know
what subject knowledge it takes to teach mathematics effectively
generally realize and agree that teachers should know mathematics
at a higher level than the level they will teach because the extra
mathematics courses help them learn (at least they should, but that
is not always the case) the mathematics they will teach much better
than otherwise.  A few other reasons are often given as well, but
this one reason is an important one and pertinent to the discussion
here.  If any form of mathematics is a significant part of the
job--whatever level that math may be, then the students should learn 
the mathematics and should learn it well.  Furthermore, as Sherman Stein 
mentions in one of these replies, it is better to overprepare in mathematics
than to underprepare in mathematics in case of changing career
goals and also because further preparation in mathematics can help
students understand better what mathematics they will use.  

One of the most important goals of learning mathematics that is
sorely missing from elementary math courses is teaching students
how to study and learn mathematics for themselves and how to learn
to mature mathematically.  A major difficulty I see with students
in mathematics and statistics courses is that students have little
mathematical maturity and little idea of what it means to think
mathematically.  

And few of them understand symbolic reasoning, which makes it difficult 
for them to learn the reasoning behind mathematics.  And that also
makes it difficult for them to learn algebra.  In fact, so many students
I have seen have such little understanding of any form of symbolic
reasoning that they have little idea of what it means to use a formula!
Rather than stressing algebra as we do, why not emphasize more about the
teaching of symbolic reasoning?  Until students can learn to make sense
of symbolic reasoning and learn to make sense of mathematical statements
with letters in them, algebra and other symbolic mathematics will make
little sense to them, and their learning will be wasted.

And we should emphasize logic and reasoning and conceptual understanding
in arithmetic.  Far too many students take arithmetic and "not get it."
And far too many students end up thinking that mathematics is mechanical--
nothing but following recipes and plugging and chugging into formulas.



Jonathan Groves