Newsgroups: sci.math
Subject: Re: HELP! I got a contradiction!
From: jxh@math.ksu.edu (James C. Hu)
Date: 27 Sep 1992 05:02:12 -0500
Organization: Dept. of Mathematics, Kansas State University
NNTP-Posting-Host: hilbert.math.ksu.edu

bs@fatima.mitre.org (Robert D. Silverman) writes:

>In article <cannon.21.717524597@scs.unt.edu> cannon@scs.unt.edu
>(Dustin Cannon (Lab 3460)) writes: 
>>
>>First off, bear with me.  I'm only a senior math major.
>>OK, heres the deal.
>>
>>	e^ix  = cos(x) + isin(x)
>>	e^-ix = cos(x) - isin(x)
>>
>>	Let x = 
>>
>>	e^i  = -1
>>	e^-i = -1
> 
>rest deleted.......
> 
>Please don't take what I am about to say as a flame or insult. It is
>not intended as such. Rather it is outrage at an educational *system*
>that can allow one to become a senior math major while still holding
>the misconception that is implied by your post.

[...]

>How do people get degrees while still lacking understanding in (some)
>basic fundamentals?

>U.S. high schools and colleges seem to be grossly watering down their
>mathematics courses.

I am not definite, but I think in Dustin's case, as is the case with
many usenet posts, a person will think they have something neat to
say, and just post it without thinking about whether or not it really
makes sense.  An obvious example has been this 0.999... = 1 thread
that kept persisting, despite the fact that it is in the FAQ list.  He
probably could have figured out what was wrong on his own, but just
thought he'd post for fun.

As to the system as a whole, I am really not qualified to say.  I came
out of the public school system in my little town in Manhattan,
Kansas.  I was fortunate, however, in my background.  My parents
taught me math early, and I studied college mathematics while in high
school.  This was made possible with the University being here in the
same town.  I do know that other school systems are not as fortunate,
but that did not mean that the students were less capapble, just that
they did not have the same resources that were available to me.

I can believe that high schools are watering down courses.  There is
tremendous pressure to pass kids and get them on through.  I remember
a decade ago, the big deal was kids coming out of high school not
knowing how to read or write.  It isn't really a surprise that there
is a problem with mathematics.

On the college level, I think that the amount of information learned
is more the resposibility of the student, under the assumption that
the professor is qualified to teach the subject.  In your particular
example, I don't believe rings and fields are discussed explicitly
until one has had an abstract algebra course.  Of course, other
subject areas use properties of rings and fields, but sometimes
teachers will forget to try to tie together some of the subjects.  I
sometimes wish that physics and first year calculus would be taught
simultaneously.  I think it would make certain concepts clearer for
both topics.  I also think linear algebra ought to be taught before
differential equations.  It just makes more sense now that I have
taken them both.  But, anyway, that's water under the bridge.

Anyway, my point is that if a student only had an introduction to
abstract algebra, I can see him/her forgetting exactly what a ring or
a field is.  Yet, when you ask them if the integers have a
multiplicative inverse, I hope that they will say, "No, of course
not!", and I would bet most math majors will.
-- 
James C. Hu (jxh@math.ksu.edu), 1804 Denholm Dr., Manhattan, KS 66502
I speak for me, the whole me, and nothing but for me.  So help me me.