Minimum standards for math "competency"
Shawn Willden
B.S. in math and CS) there is a movement to change (lower) the
standards for math competency. The current policy states that every
student recieving a degree from Weber State must demonstrate his/her
mathematics competency by either 1) scoring a 19 or better on the
mathematics portion of the ACT or 2) passing a course in basic algebra
skills (called math 105) as taught by the mathematics department. The
proposed change (which, incidentally, was *passed* last year by the
faculty senate and only came up for reconsideration through the
efforts of myself, some other students and math dept. faculty) would
allow any department on campus to submit their own courses as
substitutes for math 105 in fulfillment of the math competency
requirement. The proposal specifies a list of topics that must be
covered by the course in order to qualify but makes no mention of any
standards with which the teaching of these topics is to be judged.
The proposal is championed by the departments of various social
sciences (Sociology, communications, etc.) who feel that math 105 is
simply too difficult for their students to pass. (Math 105 is
essentially a high-school algebra course).
I am completely opposed to this proposal on the grounds that it will
weaken degrees from WSU. (Just so there's no question about *my*
opinion of the matter :-)
I have heard that many other schools around the country are facing
similar problems. I would like to hear experiences from anyone who
has faced this situation and arguments both for and against such
proposals.
Also, (and this is really why I'm posting) I would like to find out
what standards are at other schools. I would like to be able to use
this information as ammunition for my arguments, so I would appreciate
it greatly if you would include your name, position and possibly even
a phone number where you can be reached if I need to verify any of the
info. So, please take a moment and answer the following questions
about your school's math competency requirements.
I am especially interested in colleges and universities with a makeup
similar to Weber's. Weber is basically a four-year educational
institution (we only recently became a university and though there are
plans for many master's programs in the works, currently we have only
master's in education and business programs) with a studentbody of
approximately 14,000 students, most of whom commute to school and many
of whom are non-traditional students (I believe the average age of a
WSU student is ~26 years).
Questions:
School information and demographics:
What is the name of your school and where is it
located?
Describe your studentbody (size, age distribution,
etc.)
What is the "purpose" of your school (i.e. education,
research, is your school an "elite" school, etc.)
Math competency requirements:
What math is required for entrance?
What math is required for graduation?
Is there an additional requirement for B.S. degrees
as opposed to B.A. degrees?
Are all math courses that count towards these minimum
requirements taught by the mathematics dept.?
If not, how are standards enforced?
General thoughts about math competency requirements:
What do you think should be a minimum standard for
a graduate of a four-year institution?
Do you think allowing departmentalized mathematics
courses is wise? Why or why not?
Any other comments?
Thanks for your time,
respond either on the net or by e-mail (e-mail preferred unless you have
something you would like to discuss with the net at large).
--
Shawn Willden
swil...@icarus.weber.edu
--
Shawn Willden
swil...@icarus.weber.edu
Robert J Frey
>At Weber State University, Ogden, Utah, U.S.A (where I am working on a
>B.S. in math and CS) there is a movement to change (lower) the
>
>The proposal is championed by the departments of various social
>sciences (Sociology, communications, etc.) who feel that math 105 is
>simply too difficult for their students to pass. (Math 105 is
>
>
>Also ... I would like to find out what standards are at other schools...
>
As someone with a Ph.D. in applied mathematics who has spend time as both an
academic and an applied mathematician in industry, I would like to offer you
an alternate question: What mathematics will students need when they get out
into the world REGARDLESS of what is or is not required in your or any other
university?
The answer in that case is clear. They need a lot more than they're getting!
It is serendipitous that I recently gave an invited paper dealing with this
issue at a recent conference. The title of the talk was "A View from Outside:
The Challenge of Mathematics Education." What it dealt with is the failure
of educational institutions in this country to produce graduates, at both
the secondary and postsecondary levels, who have the mathematical skills
required of workers in a modern industrial society.
We're not talking about math majors here, but of factory workers, maintenance
personnel, etc. who lack the skills to work with flexible manufacturing
systems, computer aided manufacturing, just-in-time inventory policies and
statistical quality control techniques. It is incomprehensible to me that a
professor of sociology, who is well aware of the role statistics plays in
his or her discipline, can not only suggest that someone graduate with a
B.A. in sociology without a solid statistics course but without even a solid
command of "high school" algebra.
If you want I can snail mail you a copy of the presentation. Here however are a
few facts:
o The Jobs Almanac's top 5 jobs are all mathematically based:
actuary, computer programmer, systems analyst, mathematician
and statistician.
o The Workforce 2000 report from the BLS estimates that 41% of
all new jobs will require "high" levels of skill in language,
mathematics and reasoning, compared with only 24% of current
jobs.
o Between '73 and '90 real per capita GNP increased 28% but real
hourly wages for non-supervisory personnel fell 12%. There
were many reasons for this, but the discrepancy was due in
large part to lower rates of productivity growth in the U.S.
compared to Europe and Asia.
o The math scores of the top 1% of American high school students
would place them in the 50th percentile in Japan (that's not
a typo). How can we expect to compete with Japan in high tech
manufacturing?
Thus, even if a sociology or communication major didn't already need a certain
amount of mathematical training, he or she would need competency in math simply
to qualify to work in a modern office or factory. At the conference, which BTW
was "A SUNY Conversation in the Disciplines - Applied Mathematics: Prospects
for the 1990's," a training manager for LILCO, the local power utility, stood
up and said that his company often had to put new workers through several
months of training before they could be productive.
My recommendations to you are as follows:
o Hit 'em with facts, HARD. Good sources: Lester Thurow, Head to
Head, Morrow, 1992; and National Academy Press, A Challenge of
Numbers: People in the Mathematical Sciences, 1990. These
contain extensive references which will point you further.
o Get the support of local industry. A representative from the
Really Big Corp. who is willing to support your position by
saying that innumerate graduates won't be getting jobs with
them is going to be VERY persuasive.
None of this means your original idea of comparing your university with others
is wrong. I'm simply putting these ideas forward as a more or less orthogonal
strategy that will greatly enhance your arguments.
Good luck!!!
--
Dr. Robert J. Frey
Renaissance Technologies Corp
25 East Loop Rd.
Stony Brook, NY 11790
email: rjf...@rentec.com -- voice: (516)246-5550 -- fax: (516)246-5761
Randy Crawford
>[...] an alternate question: What mathematics will students need when they get
>out into the world REGARDLESS of what is or is not required in your or any
>other university?
>
>The answer in that case is clear. They need a lot more than they're getting!
>If you want I can snail mail you a copy of the presentation. Here however are a
>few facts:
>
> o The Jobs Almanac's top 5 jobs are all mathematically based:
> actuary, computer programmer, systems analyst, mathematician
> and statistician.
A greater need is anticipated for systems analysts and mathematicians than for
nurses or accountants? First time I've heard that. Sounds dubious.
>
> o The math scores of the top 1% of American high school students
> would place them in the 50th percentile in Japan (that's not
> a typo). How can we expect to compete with Japan in high tech
> manufacturing?
Hoo boy! Does this statement need qualification!
In effect, this states that EVERY american student would be below average
mathematically in Japan.
I'll step out on a limb here and say: RUBBISH! Such a statement _has_ to be
completely wrong. If the margin between the US and Japan were that great, we'd
see vast differences between us in virtually every form of technology, including
patents and major prizes for scientific research, which we don't. We'd find
ten Japanese students for every American in every non-american and non-japanese
university, which (I'll bet) we don't.
But perhaps it's just my gross inadequacy in mathematics speaking...
>
>My recommendations to you are as follows:
>
> o Hit 'em with facts, HARD. Good sources: Lester Thurow, Head to
> Head, Morrow, 1992; and National Academy Press, A Challenge of
> Numbers: People in the Mathematical Sciences, 1990. These
> contain extensive references which will point you further.
More power to references.
--
| Randy Crawford craw...@mitre.org The MITRE Corporation
| 7525 Colshire Dr., MS Z421
| N=1 -> P=NP 703 883-7940 McLean, VA 22102
Steve Cunningham
the original posting by rjf...@rentec.com (Robert J Frey) speaking of the
critical shortage of mathematically literate people (as well as mathematical
specialists) and doubting the low standing of American students in mathematics.
Without going to my references (I'm in my office late to staunch a roof-leak
flood and can't dig them out from the piles of materials under plastic) let
me assure Mr. Crawford that Mr. Frey's general points are entirely valid. For
references, consult the large literature on the subject from the Mathematical
Association of America and the National Research Council, who have documented
this situation well. It's real, and it's frightening -- if you can make time
to work with your local schools in bringing a sense of the reality and fun of
mathematics (oh, dear -- I said mathematics is fun, didn't I? It is, and it's
one of the most rewarding of human endeavors, IMHO...) to the students, you'll
do yourself, the students, the nation, and perhaps even humanity a favor...
Pierre VonKaenel
>In article <13...@kepler1.rentec.com> rjf...@rentec.com (Robert J Frey) writes:
>>
>> o The Jobs Almanac's top 5 jobs are all mathematically based:
>> actuary, computer programmer, systems analyst, mathematician
>> and statistician.
>
>A greater need is anticipated for systems analysts and mathematicians than for
>nurses or accountants? First time I've heard that. Sounds dubious.
>
Yup, that's what the article states, I've seen it.
>>
>> o The math scores of the top 1% of American high school students
>> would place them in the 50th percentile in Japan (that's not
>> a typo). How can we expect to compete with Japan in high tech
>> manufacturing?
>
>Hoo boy! Does this statement need qualification!
>
>In effect, this states that EVERY american student would be below average
>mathematically in Japan.
>
>I'll step out on a limb here and say: RUBBISH! Such a statement _has_ to be
>completely wrong. If the margin between the US and Japan were that great, we'd
>see vast differences between us in virtually every form of technology, including
Perhaps you haven't visited a technical university lately. I recall
professors complaining that most of their students are oriental, and
where are the American kids? I'm not sure about the statement above,
but it's quite clear here in education land that American students are
pitifully deficient in math. Why a good number of them can't add
fractions together or interpret what a percent means. As to our
brightest.. there are way too few of them!
>
>But perhaps it's just my gross inadequacy in mathematics speaking...
>
--
Pierre von Kaenel | Skidmore College | pv...@skidmore.edu
Math & CS Dept. | Saratoga Springs, NY 12866 | (518)584-5000 Ext 2391
"Experience is the name everyone gives to their mistakes." Oscar Wilde
Steven E. Landsburg
>In article <1992Nov23.1...@scott.skidmore.edu>, pv...@scott.skidmore.edu (Pierre VonKaenel) writes:
>|> Perhaps you haven't visited a technical university lately. I recall
>|> professors complaining that most of their students are oriental, and
>|> where are the American kids? I'm not sure about the statement above,
>
>sure about that"). But I think this kind of racist crud is intolerable-
>a good number of the Asian students ARE American kids.
>
I applaud the denunciation of the racist crud. But it doesn't go far
enough. So what if they AREN'T American kids?
Steven E. Landsburg
land...@troi.cc.rochester.edu
Polygon
>>
>> o The math scores of the top 1% of American high school students
>> would place them in the 50th percentile in Japan (that's not
>> a typo). How can we expect to compete with Japan in high tech
>> manufacturing?
>Hoo boy! Does this statement need qualification!
>In effect, this states that EVERY american student would be below average
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>mathematically in Japan.
^^^^^^^^^^^^^^^^^^^^^^^^
>I'll step out on a limb here and say: RUBBISH! Such a statement _has_ to be
>completely wrong. If the margin between the US and Japan were that great, we'd
>see vast differences between us in virtually every form of technology, including
>patents and major prizes for scientific research, which we don't. We'd find
>ten Japanese students for every American in every non-american and non-japanese
>university, which (I'll bet) we don't.
>But perhaps it's just my gross inadequacy in mathematics speaking...
>>
>>My recommendations to you are as follows:
>>
>> o Hit 'em with facts, HARD. Good sources: Lester Thurow, Head to
>> Head, Morrow, 1992; and National Academy Press, A Challenge of
>> Numbers: People in the Mathematical Sciences, 1990. These
>> contain extensive references which will point you further.
>More power to references.
>--
>| Randy Crawford craw...@mitre.org The MITRE Corporation
>| 7525 Colshire Dr., MS Z421
>| N=1 -> P=NP 703 883-7940 McLean, VA 22102
If you think the Japanese are tough, try to beat the Singaporeans
first. Then you will know how far behind are the average American
high school students. When most of high school students in certain
Asian countries or areas start learning calculus in grade 10, some
of the American college students are still struggling with
trigonometry, if not algebra.
However, it is rubbish to claim that Asian students take advantages
over American students "absolutely". In the real world, it takes
a lot more than calculus to take advantage of others. Let's not
forget college student in Japan don't study hard-- at least not that I
heard of. I am not from Japan but all of my Japanese friends told
me that college students in Japan party most of the time. Doesn't
it explain something about the real world situation?
Peter, UIUC
David Rector
>In article <1992Nov23.1...@scott.skidmore.edu>, pv...@scott.skidmore.edu (Pierre VonKaenel) writes:
>|> Perhaps you haven't visited a technical university lately. I recall
>|> professors complaining that most of their students are oriental, and
>|> where are the American kids? I'm not sure about the statement above,
>I trying to imply you said the above (I even left a bit of your "I'm not
>sure about that"). But I think this kind of racist crud is intolerable-
>a good number of the Asian students ARE American kids.
>--
>--- It is kind of strange being in CS theory, given computers really do exist.
>John Mount: jmo...@cs.cmu.edu (412)268-6247
>School of Computer Science, Carnegie Mellon University,
>5000 Forbes Ave., Pittsburgh PA 15213-3891
Your somewhat incoherent statement above makes even a staunch liberal
bemoan political correctness. Mr. VonKaenel's remarks imply no racist
views; they may simply reflect a current defect in American English
terminology: how to refer to the once dominant Euro-American cultural
group. You might try sticking to the subject.
Here in California the problem of poor math education is particularly
acute since California schools are a year or two behind the more
competent school systems in the nation. One characteristic of
California's much esteemed Asian American subculture is high respect
for education. Many parents, therefore, devote great personal effort
to overcoming the appalling defects in the educational system. In my
experience they succeed no better than the rest of the population in
overcoming the deficiencies in content, and may even exacerbate the
tendency of our schools to teach for the short answer test.
Incoming students to the University of California--all cultural groups--
share several characteristics:
1. They are bone ignorant.
2. They perform very well on short answer tests where they are
asked to regurgitate facts.
3. They will not reason.
4. They are willing to work very hard, but they are not willing
to be diverted by "theory" or the enjoyment of anything beyond
the required course syllabus.
5. They are totally at sea when asked to work independently.
6. They are very bright and can perform well if (big if) you
can dynamite them out of their careerist fortress.
7. They have no sense of humor--or wander--or beauty--or life.
We used to be able to beat some of the deficiencies out of the
students by the junior year (or at least get rid of some of the
students), but that is no longer possible. Since students come to us
without the prerequisite information or attitudes, we have inevitably
lowered our own standards so that much of the junior year is spent
(re)teaching freshman mathematics.
"Reforms" in education seem to have made things worse. California's
minimum standards tests seem to have become maximum standards. The
demise of the New Math, for all its faults, has meant that we can no
longer count on students having the basic vocabulary of mathematics.
Students have had no experience in numerical calculation, geometry, or
applying mathematics to a practical problem. Many have had calculus
in highschool and not understood it--hardly surprising since they have
had none of the experiences, mathematical or practical, that motivate
it. Most important, students are convinced that the point of
education is to collect isolated facts to parrot on a short answer
test so that they can get certification to apply for a high paying
job. Many--perhaps most--do not even like the course they study.
Some needed changes:
1. Competent teaching is an exhausting enterprise--about like
acting--requiring enormous emotional energy and extensive
preparation. No school teacher can perform well teaching more than
three hours per day--about half what is required in California
schools. University teaching is even harder. Teachers must be
professionals and paid--more important, respected--accordingly.
2. Education is not the same as job training and is much more
important. The great expansion of American industry in the nineteenth
century was based on well a educated (comparitively), flexible work
force. Business (and the Republican party) supported--AND PAID FOR--
education then--why not now?
3. Knowledge is not devided up into neat little packages that
can conveniently be translated into departments, bureaucracies,
and grant programs.
4. The most important requirement for good education is having
fun--both students and teachers. (One reason: a human brain does not
remember events unless signaled to do so by a certain control center.
Fun and fear are the most powerful ways to turn on that control center.
Fun motivates a person to repeat the experience, fear to avoid it.
The choice of motivation, therefore, ought to be obvious.)
5. Education is a social enterprise. Teachers need to talk to their
colleagues and students; teachers need to talk to each other. Class
discussion is very important to education. It has almost completely
disappeared from the California schools I am familiar with. Most of
my students would rather die than talk in class.
--
David L. Rector dre...@math.uci.edu
Dept. of Math. U. C. Irvine, Irvine CA 92717
John Mount
|> jmo...@CS.CMU.EDU (John Mount) writes:
|>
|> >In article <1992Nov23.1...@scott.skidmore.edu>, pv...@scott.skidmore.edu (Pierre VonKaenel) writes:
|> >|> Perhaps you haven't visited a technical university lately. I recall
|> >|> professors complaining that most of their students are oriental, and
|> >|> where are the American kids? I'm not sure about the statement above,
|>
|> >I trying to imply you said the above (I even left a bit of your "I'm not
|> >sure about that"). But I think this kind of racist crud is intolerable-
|> >a good number of the Asian students ARE American kids.
|> Your somewhat incoherent statement above makes even a staunch liberal
|> bemoan political correctness. Mr. VonKaenel's remarks imply no racist
|> views; they may simply reflect a current defect in American English
|> terminology: how to refer to the once dominant Euro-American cultural
|> group. You might try sticking to the subject.
Just because you refuse to read carefully- doesn't mean what I said
was inchorent. Also, you are putting words in VonKaenel's mouth- he is
only reporting what he heard. Read the quote- do the professors seem
at all pleased that their classes are full of Orientals?
It is racist to imply that Asians can not be Americans. You are right
in that one of the most offensive aspects of PC is that they try to
redefine words on the fly- but I don't think "American" ever meant the
dominant Euro-American cultural group. It has always meant "citizen"
and the term you are looking for is "WASP".
How do you think Asian Americans feel when somebody says to white kids
"why can't you American kids do as well as these Orientals?" I'll bet
they don't feel proud that they are busting the grade curve, they
probably feel excluded and we are not talking nit picky little
language questions ("should I say Asian or Oriental?"). Many of them
are Americans and have the right to be lumped in with the local
idiots (if that is what they are into).
Real harm has been done by the "Asians are smart" myth. An example:
At UC Berkeley there are a two really good minority tutoring programs:
PDP and MEP. One is a tutoring program for minority high school
students (so they will be competent enough to get into UCB) the other
is a minority peer tutoring for engineers. Both were based on Uri
Triesman's (sp?) work on how successful Asian students formed study
group. Both groups work very hard, and do not do remedial tutoring.
Neither group (to my knowledge) has the ability to admit students.
This was great. Triesman recently got a MacArthur award and you are
correct when you say the group he studied does (as a whole) perform
much better than average.
Now here comes the funny (unless you think all people are individuals
deserving a chance at success) part. Around 1986 or 1987 Filipino
Americans were dropped off some "minority list" by the university
administration. This meant they were no longer eligible for tutoring
from PDP or MEP. The nasty part is the Filipino Americans even though
they look like all the other Asians have one of the worst retention
rates at UCB- so they really needed the help.
Read Amy Tan or Maxine Hong-Kingston about how much it sucks to be an
artisticly inclined Asian American when all your teachers think you
should be taking extra math classes.
|> Here in California the problem of poor math education is particularly
|> acute since California schools are a year or two behind the more
|> competent school systems in the nation. One characteristic of
|> California's much esteemed Asian American subculture is high respect
|> for education. Many parents, therefore, devote great personal effort
|> to overcoming the appalling defects in the educational system. In my
|> experience they succeed no better than the rest of the population in
|> overcoming the deficiencies in content, and may even exacerbate the
|> tendency of our schools to teach for the short answer test.
I am of European ancestry and I went to UC Berkeley undergrad, so you
can consider my experiences as a data point- or you can ignore it so you
can safely draw any conclusion you want.
|> Incoming students to the University of California--all cultural groups--
|> share several characteristics:
|>
|> 1. They are bone ignorant.
Not all- I took night classes in 2nd year college DiffEqs while in
high school so I wouldn't make a fool of myself in college. I *never*
took any course (public or private) on how to take the SAT or any
other test.
|> 2. They perform very well on short answer tests where they are
|> asked to regurgitate facts.
|>
|> 3. They will not reason.
|>
|> 4. They are willing to work very hard, but they are not willing
|> to be diverted by "theory" or the enjoyment of anything beyond
|> the required course syllabus.
I was pure math- so there was nothing but the "theory".
|> 5. They are totally at sea when asked to work independently.
I did a large senior project.
|> 6. They are very bright and can perform well if (big if) you
|> can dynamite them out of their careerist fortress.
You think there is big money in theoretical CS?
|> 7. They have no sense of humor--or wander--or beauty--or life.
|>
|> We used to be able to beat some of the deficiencies out of the
|> students by the junior year (or at least get rid of some of the
|> students), but that is no longer possible. Since students come to us
|> without the prerequisite information or attitudes, we have inevitably
|> lowered our own standards so that much of the junior year is spent
|> (re)teaching freshman mathematics.
In my senior year almost half my course work was graduate courses in
math and CS.
I have no desire to talk any further with you either...
|> --
|> David L. Rector dre...@math.uci.edu
|> Dept. of Math. U. C. Irvine, Irvine CA 92717
--
Randy Crawford
> >In article <13...@kepler1.rentec.com> rjf...@rentec.com (Robert J Frey) writes:
> >>
> >> o The Jobs Almanac's top 5 jobs are all mathematically based:
> >> actuary, computer programmer, systems analyst, mathematician
> >> and statistician.
> >
> >A greater need is anticipated for systems analysts and mathematicians than for
> >nurses or accountants? First time I've heard that. Sounds dubious.
> >
>
> Yup, that's what the article states, I've seen it.
I don't doubt the accuracy of the quote, only the accuracy of the article.
Just because the book of Timothy says the Bible is the word of God doesn't
necessarily make it so.
>
> >> o The math scores of the top 1% of American high school students
> >> would place them in the 50th percentile in Japan (that's not
> >> a typo). How can we expect to compete with Japan in high tech
> >> manufacturing?
> >
[...]
> >I'll step out on a limb here and say: RUBBISH! Such a statement _has_ to be
> >completely wrong. If the margin between the US and Japan were that great, we'd
> >see vast differences between us in virtually every form of technology, [...]
>
> Perhaps you haven't visited a technical university lately. [...]
> Why a good number of them [US students] can't add
> fractions together or interpret what a percent means. As to our
> brightest.. there are way too few of them!
I don't doubt that US students don't perform as well in math as they should, but
when a scholarly article equates 99% here with 50% there, that is one hell of an
assertion in itself. This implies that the entire bell curve of US student
scores belongs in the 0-50% range in Japan. Before we all start self-flagellating,
why don't we ask for clarification?
What test was this, administered in both english and japanese? SRA? SAT?
Do other tests reflect this level of disparity? Was it repeated over several
years using large samples of students? Exactly what did it test? Or is this
just a magic number meant by the author to strike fear into the hearts of stout
men (and women)? Where oh where is Alan Bloom when we need him?
It seems to me that ready acceptance among educated americans of so improbable a
claim may be evidence that the claim itself is true.
To misquote two for the price of one: "If it's in print, then however improbable,
it _must_ be true."
Frank Adams
> actuary, computer programmer, systems analyst, mathematician
> and statistician.
It appears that a number of people are doubting this, based on the
assumption that these are the jobs which [will] have the most jobs opening
up. On that basis, it is very dubious. A more plausible assumption is that
these are the jobs with the greatest *shortfall* of qualified applicants
compared to positions to be filled. (Although I would still wonder about
including "mathematician". Maybe it's the *ratio* of jobs to applicants.)
Then again, the Almanac may be using some other criteria entirely.
Michael Somos
Is everyone else fed up with this thread too? There have always
been and always will be students who don't have any clue when it
comes to mathematics? Aren't you just quibbling over the relative
percentages? And finally, if the situation is really so bad, then
why are there so many people with a lot of mathematical knowledge
who are underemployed or unemployed? For details look at recent
issues of Notices of the AMS. Something is very wrong, but it is
not what people are saying it is.
Also realize that there is no consensus on this. There is a wide
variety of opinions. If everyone agreed on what the problem was,
there would be rapid progress on solving it. This is absolutely
not the case now. I won't bore you with any of my own opinions,
but point out that this has nothing to do with mathematics per se.
It is a general cultural problem and pervades many areas. Let us
drop this thread, and let us enjoy mathematics while we can. At
least that is what I think this news group is for. There are many
other news groups for other kinds of discussion. Shalom, Michael
--
Michael Somos <so...@alpha.ces.cwru.edu> (* No, I don't work for CWRU *)
USENET News System
enough good university students. Whether or not you agree with this,
there is nothing racist about it. The fact that there are still able foreign
students is praise to the education system in other countries and evidence
that it is possible to produce students of a sufficient standard. I have no
idea how many Asian students ARE American, but if the situation is
similar to New Zealand, you will be getting a lot from Asia. The aim
was not to complain about getting too many of these, rather, it was to
complain that the American education system is not producing enough
local students. Again this is in no way racist - it is a simple statement
of the perception of the writer. (This may be fact or fiction).
It seems to me that any mention of race seems to raise a howl from
the politically correct. Perhaps the politically correct readers can
try to read the context and not just pick out isolated sentences.
Shawn Willden
: Is everyone else fed up with this thread too? There have always
: been and always will be students who don't have any clue when it
: comes to mathematics? Aren't you just quibbling over the relative
: percentages? And finally, if the situation is really so bad, then
No, I don't think we're "quibbling over the relative percentages."
The present discussion concerns (among other things) the validity of
an article that claims that 99% of American students would be considered
below average in Japan (I'm ignoring the discussion about Asian American
students vs. Asian students, which should be dropped). Those percentages
are downright frightening if true.
: why are there so many people with a lot of mathematical knowledge
: who are underemployed or unemployed? For details look at recent
: issues of Notices of the AMS. Something is very wrong, but it is
: not what people are saying it is.
This is a different issue entirely. We aren't talking about the
demand for people with a lot of mathematical knowledge, we're
talking about the requirement of an industrialized society that
all individuals have an understanding of basic mathematical
concepts -- something that is just not true in America today.
: Also realize that there is no consensus on this. There is a wide
: variety of opinions. If everyone agreed on what the problem was,
: there would be rapid progress on solving it. This is absolutely
I disagree that posession of a definition of the problem implies
rapid progress toward a solution. Sometimes solutions aren't that
easy to come by. A mathematician above all others should know
that :).
[ Other (somewhat valid) complaints about the crosspost to sci.math
deleted. ]
: --
: Michael Somos <so...@alpha.ces.cwru.edu> (* No, I don't work for CWRU *)
Now, to see if I can turn this thread in a more useful direction, here
are some questions.
1) Do you disagree that innumeracy is a problem?
Now, supposing the answer to that question is no:
2) What math do you think an "average" citizen should
know?
3) What math skills should an employer be able to expect of
a college graduate?
--
Shawn Willden
swil...@icarus.weber.edu
Keith Ramsay
| o The Jobs Almanac's top 5 jobs are all mathematically based:
| actuary, computer programmer, systems analyst, mathematician
| and statistician.
In article <1992Nov23.2...@Cookie.secapl.com>
fr...@Cookie.secapl.com (Frank Adams) writes:
|It appears that a number of people are doubting this, based on the
|assumption that these are the jobs which [will] have the most jobs opening
|up.
...
|Then again, the Almanac may be using some other criteria entirely.
I suspect that this is based on the same job survey as was cited in
the newspapers, which rated these five jobs highest on some
combination of pay scale, working conditions, stress, possibility for
advancement, and job security. I don't think availability of the job
was among the criteria, just security (once one has it). I dimly
remember wondering whether the method of combining the criteria was
reasonable.
I suspect that mathematics rated well as an occupation because of
tenure (for academic mathematicians), and because it is an indoor job.
By contrast migrant farm work and fishing were near the bottom of the
same list.
Keith Ramsay
ram...@unixg.ubc.ca
David Rector
>In article <drector....@math.uci.edu>, dre...@math.uci.edu (David Rector) writes:
>|> jmo...@CS.CMU.EDU (John Mount) writes:
>|>
>|> >In article <1992Nov23.1...@scott.skidmore.edu>, pv...@scott.skidmore.edu (Pierre VonKaenel) writes:
>|> >|> Perhaps you haven't visited a technical university lately. I recall
>|> >|> professors complaining that most of their students are oriental, and
>|> >|> where are the American kids? I'm not sure about the statement above,
>|>
>|> >I trying to imply you said the above (I even left a bit of your "I'm not
>|> >sure about that"). But I think this kind of racist crud is intolerable-
>|> >a good number of the Asian students ARE American kids.
>|> Your somewhat incoherent statement above makes even a staunch liberal
>|> bemoan political correctness. Mr. VonKaenel's remarks imply no racist
>|> views; they may simply reflect a current defect in American English
>|> terminology: how to refer to the once dominant Euro-American cultural
>|> group. You might try sticking to the subject.
>Just because you refuse to read carefully- doesn't mean what I said
>was inchorent. Also, you are putting words in VonKaenel's mouth- he is
>only reporting what he heard. Read the quote- do the professors seem
>at all pleased that their classes are full of Orientals?
Normally I would consign your posting to /device/null, but you have
read into my previous posting all sorts of racist garbage that I did
not intend and cannot leave unanswered. Could you perhaps turn down
termperature of your postings and stop deliberately misunderstanding
people? You have even, in your last line
>I have no desire to talk any further with you either...
stooped to being intentionally rude.
Perhaps there was a transmission error but you said:
>I trying to imply you said the above
which is incoherent. Pierre VonKaenel has already denied the intent
you ascribe to him. His language was perhaps ill chosen, but he
does not rate your worst case assumptions. I appreciate your
desire--which I share--to combat racism, but you went too far.
(I regret to say I have done the same in other circumstances.)
It is impossible to function if one has to examine every casual
phrase for the worst case someone might read into it.
"If you have to watch everything you say, you won't get much
said." --Lucy Van Pelt-- --(Charlie Schultz)--
>Real harm has been done by the "Asians are smart" myth. An example:
>At UC Berkeley there are a two really good minority tutoring programs:
> ...
>Now here comes the funny (unless you think all people are individuals
>deserving a chance at success) part. Around 1986 or 1987 Filipino
>Americans were dropped off some "minority list" by the university
>administration. This meant they were no longer eligible for tutoring
>from PDP or MEP. The nasty part is the Filipino Americans even though
>they look like all the other Asians have one of the worst retention
>rates at UCB- so they really needed the help.
You have an excellent point. The essence of racism is to treat
individual cases according to membership in some group. All Asians,
and all Asian nations are not alike. Indeed, since Asia extends from
East Asia--the current common usage in California--to Istanbul, there
is at least as much difference between Asian groups as between those
groups and Europeans. ALL words are subject to the same problem, and
we simply have to muddle through. AVERAGES DO MATTER in some
contexts. For example, in
>|> Here in California the problem of poor math education is particularly
>|> acute since California schools are a year or two behind the more
>|> competent school systems in the nation. One characteristic of
>|> California's much esteemed Asian American subculture is high respect
>|> for education. Many parents, therefore, devote great personal effort
>|> to overcoming the appalling defects in the educational system. In my
>|> experience they succeed no better than the rest of the population in
>|> overcoming the deficiencies in content, and may even exacerbate the
>|> tendency of our schools to teach for the short answer test.
I implied and intended NO universality of behavior. But enough
people behave as I stated to make a large difference in the
ethnic composition of California's Universities--as a number of
very vocal ethnic lobbying groups will point out.
Cultural differences are real and cannot be ignored, or we cannot
deal with a highly varied world. Besides, cultural differences
are interesting and often enjoyable. They add spice to human
interaction. The cultural diversity of California is one of
its strong points.
>I am of European ancestry and I went to UC Berkeley undergrad, so you
>can consider my experiences as a data point- or you can ignore it so you
>can safely draw any conclusion you want.
One cannot draw general conclusions from individual cases or even
refute statements based on averages. I used "all" nowhere in my
attempt to indicate current conditions in UC.
>|> Incoming students to the University of California--all cultural groups--
>|> share several characteristics:
>|>
>|> 1. They are bone ignorant.
>Not all- I took night classes in 2nd year college DiffEqs while in
> ...
Average conditions strongly influence a classroom. A few years ago
I decided to stop lecturing to my lower division classes since lecturing
is a poor way to teach a routine course. I had tried that a few years
before with great success. To my surprise, the class would have none
of it. They were not interested in working problems and asking questions
about those the did not understand. There were several outstanding
students in the room, including an eleven year old boy, but their
presence was not enough to overcome the general tendency of the class
to be spectators. This tendency has gotten worse since then.
David Petry
>>If you want I can snail mail you a copy of the presentation. Here however are a
>>few facts:
>>
>> o The Jobs Almanac's top 5 jobs are all mathematically based:
>> actuary, computer programmer, systems analyst, mathematician
>> and statistician.
>
>A greater need is anticipated for systems analysts and mathematicians than for
>nurses or accountants? First time I've heard that. Sounds dubious.
Maybe the jobs are "top" jobs in the sense of the satisfaction they bring to
those who have the jobs?
>> o The math scores of the top 1% of American high school students
>> would place them in the 50th percentile in Japan (that's not
>> a typo). How can we expect to compete with Japan in high tech
>> manufacturing?
>
>I'll step out on a limb here and say: RUBBISH! Such a statement _has_ to be
>completely wrong.
It wouldn't surprise me much if the MEAN score of students in the American
high schools which rate among the top 1% of all American high schools is
about equal to the MEAN score of students in the Japanese high schools which
rate in the 50th percentile among Japanese high schools. That's the kind of
statistic which appears in the recent Scientific American article comparing
Asian and American students.
David Petry
Martyn Thomas Quigley
[,,,]
>"Reforms" in education seem to have made things worse. California's
>minimum standards tests seem to have become maximum standards.
This is inevitable. It has been a fact of life for at least 3000 years.
Mart
John Mount
I would never stoop to such an excuse- it was a dumb (by me) typo that
killed off the "am not" that obviously should have been there- but you
should have been able to read through it.
Carl Zmola
>In article <drector....@math.uci.edu>, dre...@math.uci.edu (David Rector) writes:
>It is racist to imply that Asians can not be Americans.
^^^^^^ ^^^^^^^^^
It is not racist, just incorrect. Asians are the
inhabitants of Asia, Americans are the inhabitants of the Americas:-)
Isn't it fun to take things out of context:-)
Mark Purtill
> actuary, computer programmer, systems analyst, mathematician
> and statistician.
currently in no shortage of mathematicians, as the 12% of last years
class of Ph.D.s who are unemployed can attest. (Source: latest issue
of the Notices of the A.M.S., which also mentions that many of the
employed had part time or one year positions). People have been
predicting shortages for years and years, but they haven't happened
(and there's really no sign that they ever will).
^.-.^ Mark Purtill, pur...@ccr-p.ida.org || purtill%ida...@uunet.uu.net
((")) \@_: IDA/CCR-P, Thanet Road, Princeton NJ 08540; (609) 924-4600.
Alternate email: purtill%ida...@princeton.edu UUCP: uunet!idacrd!purtill
Robert J Frey
>
>> >> o The Jobs Almanac's top 5 jobs are all mathematically based:
>> >> actuary, computer programmer, systems analyst, mathematician
>> >> and statistician.
>> >
>> >A greater need is anticipated for systems analysts and mathematicians than for
>> >nurses or accountants? First time I've heard that. Sounds dubious.
>> >
You are misinterpeting what I (and the BLS) meant by "top". They used a scoring
system which took into account such variables as pay, demand, stress-level and
working conditions to measure the overall desirability of each career. It is
under this measure that these jobs came out on top.
Obviously, one can argue about whether this weighting or that factor was
handled correctly, but it is suggestive that these occupations did score so
well given what was probably at least a reasonable scoring scheme.
>>
>> >> o The math scores of the top 1% of American high school students
>> >> would place them in the 50th percentile in Japan (that's not
>> >> a typo). How can we expect to compete with Japan in high tech
>> >> manufacturing?
>> >
>
>> >I'll step out on a limb here and say: RUBBISH! Such a statement _has_ to be
>...This implies that the entire bell curve of US student
>scores belongs in the 0-50% range in Japan. Before we all start self-flagellating,
>why don't we ask for clarification?
>
I've posted a response to this elsewhere on this thread. This claim, however,
is not unreasonable given the nature of secondary and postsecondary
education here versus Japan.
BTW, a little constructive self-flagellation is exactly what we need. We are
doing an absolutely awful job educating and training (different things) our
youth. We had better wake up.
We spend only slightly less of our GNP on education on education than do
Germany and Japan. We screwing up because we are not spending that money
wisely, because we treat teachers like dirt and students like idiots, because
we lack any coherent form of educational standards, because we have too many
dollars spent in administration and too little in teaching, because we have
lost the will, the vision and compassion to do the damn thing right.
--
Regards,
Robert
Allan Adler
People are taught in school that the way capitalism works is that people
make a living by providing goods and services. That is not true. Actually,
people make a living by withholding goods and services. The proof of this
is that if you give away food, you are providing goods and services, but
you are not making a living. It is only when you make it clear that you
are prepared to let everyone starve to death that you start to make a living.
Naturally, this only works if you have something to withhold and that is
the motivation for manufacturing the goods and preparing oneself to provide
the services.
Well, if you withhold food, that works pretty well, because people die without
it. Ditto for shelter. But no one dies if they can't get enough mathematics,
so when you withhold it, you don't make a living (at least I didn't when
I tried it).
However, that doesn't mean that there is no shortage of mathematicians.
Mathematicians meet a need that people can get away with pretending is
not a vital need, that's all.
Allan Adler
a...@altdorf.ai.mit.edu
Robert J Frey
>craw...@boole.mitre.org (Randy Crawford) writes:
>
>>>
>>> o The math scores of the top 1% of American high school students
>>> would place them in the 50th percentile in Japan (that's not
>
I'm the one who originally posted this comment. I took it from Lester Thurow's
new book, _Head to Head_. I don't think it is rubbish; unfortunately it's true.
> I am not from Japan but all of my Japanese friends told
> me that college students in Japan party most of the time. Doesn't
> it explain something about the real world situation?
At the high end, the U.S. is still first-class, but of the ~ 4,000,000 students
who take math in the 9th grade only ~400 end up with Ph.D.'s in math -- that's
only 0.01%. It is from that group (with similar, though less dramatic, numbers
for other fields) that most of the new inventions, research results and
prizes come from. That's why the 1% - 50% comparison is not as unlikely as it
first sounds. Our top 0.01% *is* better than almost everyone else's, but
that wasn't the point I was making.
In Japan students who have entered university are just about done. This is
in sharp contrast to the U.S. where students are for the first time asked
to do real work.
--
Regards,
Robert
Jim Wissner
Whoa! Wait a minute folks, wait wait wait... You are missing the point
entirely. I believe the intent of the statement was that the students
coming from oriental schools are much farther ahead than are the students
coming out of American schools. I don't see even a trace of racial
overtone from Mr. VonKaenel; I see him describing the frustration of
professors over the lack of high quality students coming from our schools.
Haven't you been following the thread? I don't understand why you are
so quick to scream "racist?" C'mon, I don't want to see this thread
turn into a flame war.
(Back to the real topic)
I came out of high school with a gross deficiency in mathematics. The
sad part of this was that I /didn't care/. Not only was it easy for me
to slide through without really learning anything, but none of my
teachers ever really explained the importance and depth of mathematics.
It was a remarkably bland, one-dimensional topic. I hadn't a clue.
It wasn't until after two or three years of studying computer science
at a University that I began to realize the importance of math (and
that, gee, it really wasn't that bad. In fact...)
So I guess my viewpoint is that we're missing some key motivation. It's
easy, of course, to speculate about the problem. I don't know, however,
what needs to be done to fix it. I guess I'd be rich if I did. 8-)
But at any rate, I think this is an important topic and as I said before,
could we please keep it focused? I'm very interested in hearing what
others think about what I consider to be quite a crisis that we have.
- Jim Wissner
Brian Harvey
>So I guess my viewpoint is that we're missing some key motivation. It's
>easy, of course, to speculate about the problem. I don't know, however,
>what needs to be done to fix it.
Help is on the way. A couple of years ago, the National Council of
Teachers of Mathematics issued a document called _Curriculum and Evaluation
Standards for School Mathematics_ (everyone calls it the "NCTM Standards")
that calls for less memorization, less focus on algorithms (how to multiply
three-digit numbers), more questions requiring thought, more discrete math,
statistics, more participatory learning, less lecture, more experiment.
In parallel with that, people are inventing lots of ways to use computer
technology to turn mathematics into an experimental science. This includes
both the use of programming languages (a lot of work has been done in Logo
but there is also a new spurt using ISETL, a language that's more directly
"math-like" in its notation) and more narrowly focused environments such as
Geometer's Sketchpad, Derive, etc.
Of course it's going to take a lot of time and money to get these ideas
and these tools into the hands of most teachers, but the money situation
may start getting better in light of the recent election -- who knows?
Herman Rubin
This is almost like the fox telling us how to guard the henhouse. There
is little change; there is nothing on the teaching of concepts; there is
more on frills than on substance.
I have seen some of the suggestions on discrete math; they are still nothing
more than manipulations. I am a statistician of long standing; I do not see
how they can teach any meaningful statistics--it requires an understanding
of the concept of probability, not combinatorics. ISETL is not math-like;
formal manipulations with sets are still formal manipulations.
Teach the use of variables, the notions of function and relation, formulation
of word problems, logic, the structure of the integers and the reals, and how
operations fit into that. A first-grader can understand the Peano Postulates.
Show them that intuition, while useful, is also dangerous, and that careful
proof is needed. However, point out that there are times when they will
have to accept the idea that someone else has found a proof.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hru...@snap.stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Robert J Frey
>dre...@math.uci.edu (David Rector) writes:
>>California's minimum standards tests seem to have become maximum standards.
>This is inevitable. It has been a fact of life for at least 3000 years.
Unfortunately, this is true. It is precisely the reason that "minimum"
standards have to be TOUGH standards if they're to be of any use whatsoever.
The point isn't to needlessly torture students but to prepare them for the
world they'll need to live in.
Regards,
Robert
Robert J Frey
> I'm not familiar with the Jobs Almanac, but there certainly is
>currently in no shortage of mathematicians, as the 12% of last years
>class of Ph.D.s who are unemployed can attest.
First of all, given the current recession it is not surprizing that a
high percentage of new grads are unemployed, even among fields which
are considered desirable.
Second, many Ph.D.'s in math display the distressing tendency of viewing
positions outside of academia with distain. I recall a recent conversation
with a colleague who was bemoaning the lack of vacancies for new Ph.D.'s.
When I gave him examples of firms on Wall St. that were hiring "quants"
who would love to get some these people, he gave me a surprized look and
said, "I wasn't talking about THAT."
Another example. We approached a department about recruiting some
recent Ph.D.'s. The response was, "Sure, maybe the ones not good
enough to get a post-doc will be interested." Sorry, we want the best
ones, and we pay accordingly.
There are a lot of jobs out there that those new Ph.D.'s should be looking
at that they're not!
Brian Harvey
While I agree with everyone who's been posting messages to the effect that
people ought to know a lot of stuff, I don't agree with the establishment
of specific standards, minimum or otherwise, to achieve that. The trouble
is that that makes for assembly-line education in which the standards
become the purpose instead of learning to think.
Although it's important to know a lot, there is no one single thing that
everyone has to know. For example, I've managed to accumulate five
college degrees including a PhD without ever taking a statistics course.
(I mention this example because everyone's been using statistics as an
example.) I *have* learned a lot of math -- back in high school I was in
an NSF-sponsored Saturday program where I studied stuff like Hilbert spaces
and point set topology.
My point is that the math I've studied stretches the same "mental muscles"
as statistics. If I needed to learn statistics I could do it easily because
I speak the language. It would be different if I had no math at all; that
would be a serious failing in my education. But there is no particular
mathematical topic that's crucial.
The same is true about every kind of learning. When people make lists of
Great Books they always put in things like Plato's _Republic_, but they
don't usually list Alasdair Macintyre's _After Virtue_, a book I read in
graduate school that completely changed my life. So should we add that
one to the list? No, it's hopeless to list all the great books. A better
approach is the one my high school took: They had us fill out forms on
which we listed all the books we read, and they didn't care which books
they were, as long as there were *enough* of them. (If it seemed that
someone was only reading one kind of book, the teachers would suggest ways
that that person might branch out, of course.)
Charles Geyer
(Brian Harvey) writes:
> Although it's important to know a lot, there is no one single thing that
> everyone has to know.
I couldn't agree more.
> For example, I've managed to accumulate five
> college degrees including a PhD without ever taking a statistics course.
> (I mention this example because everyone's been using statistics as an
> example.) I *have* learned a lot of math -- back in high school I was in
> an NSF-sponsored Saturday program where I studied stuff like Hilbert spaces
> and point set topology.
>
> My point is that the math I've studied stretches the same "mental muscles"
> as statistics. If I needed to learn statistics I could do it easily because
> I speak the language.
No so, but I'll excuse your ignorance. The primary difficulties in statistics
are not mathematical. That's why it was such a struggle to invent. Much of
what we teach in introductory statistics courses is less than 100 years old
and has no earlier antecedents. The bulk of the subject was created since
1920. That's also why it is such a struggle to learn. Most of it is highly
counterintuitive.
> It would be different if I had no math at all; that
> would be a serious failing in my education. But there is no particular
> mathematical topic that's crucial.
I agree with that. I don't know much about abstract algebra or complex
variables and I've never felt that either was a "failing".
It is much more important to learn some mathematics well than to have
a knowledge of many areas that is too shallow to be of any use.
--
Charles Geyer
School of Statistics
University of Minnesota
cha...@umnstat.stat.umn.edu
Gary Martin
Teach the use of variables, the notions of function and relation, formulation
of word problems, logic, the structure of the integers and the reals, and how
operations fit into that. A first-grader can understand the Peano Postulates.
Show them that intuition, while useful, is also dangerous, and that careful
proof is needed. However, point out that there are times when they will
have to accept the idea that someone else has found a proof.
This is what I thought I was doing for the past 10 weeks in my Discrete
Math course (about half freshmen, 20% sophomores, 20% juniors, 10% seniors,
mostly CS majors, but a large minority of math and electrical engineering
majors). 6 weeks of sets, functions and relations and 4 weeks of logic
(well, maybe I'm exaggerating - perhaps it was 5 and 3), emphasizing
intuition and ideas as well as proofs. After all of that, about a quarter
of the 75 students (at this point the 10 or 15 weakest students had
already dropped the course) convinced me that they understood why if
A is a subset of B then the complement of B is a subset of the
complement of A. I had asked for a proof by whatever method they liked.
The successful ones almost exclusively chose to do it by Venn diagrams.
Several people didn't know what the symbols meant. One claimed that
\subset was symmetric. Two thought that the complementation operator,
which we denote by superscript c, meant exponentiation. Some thought
that the assumption that A is a subset of B implied that A = B.
Some thought that A is a subset of B implied that A is an element of B.
Perhaps 8 or 10 people left the problem blank. One senior math major
couldn't do it. (Note that I asked them for a proof, but only expected
them to give me a clue that they understood the reason, whether or not
they could express it clearly.)
I could go on and on (and I guess I did), but my question is how is
one supposed to respond to this situation? Now that I have tenure,
I feel secure enough to stick to my guns and continue to emphasize
ideas and proofs, but I'm not sure I can deal with my own and my
students' frustration if I do. Last year, which was my tenure year,
I retreated a bit (not completely) from this position, making the
exams more computational, and while a large portion of the class
still couldn't handle it, the frustration level and tension was much
less.
I've begun to get wind of dissatisfaction from several of the course's
constituents, wanting me to move the course in opposite directions.
Upper level math majors reveal in their courses that they haven't
learned what a function is, what "one-to-one" means, how to read or
write a proof, etc. Some CS professors want me to teach boolean
algebra - methods of simplifying boolean expressions, which seems
to be a rather "plug & chug" type of topic.
Well, it's Thanksgiving, so I'd better forget about this for awhile
if I'm going to feel thankful about something.
--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Mar...@cis.umassd.edu
David Rector
In reference to problems he has encountered at UMass, Dartmouth, teaching
students in a finite math course to reason about sets,
mar...@lyra.cis.umassd.edu (Gary Martin) writes:
>I could go on and on (and I guess I did), but my question is how is
>one supposed to respond to this situation? Now that I have tenure,
>I feel secure enough to stick to my guns and continue to emphasize
>ideas and proofs, but I'm not sure I can deal with my own and my
>students' frustration if I do. Last year, which was my tenure year,
>I retreated a bit (not completely) from this position, making the
>exams more computational, and while a large portion of the class
>still couldn't handle it, the frustration level and tension was much
>less.
I have had tenure for much longer, 18 years, and am only beginning to
see how to deal with your problem. Unfortunately, any real solution
to the problem requires reorganization of teaching from kindergarden
up, but we can individually make a small dent it. Here, I believe, is
the basis of the problem:
1. Reasoning, like any other mental skill, requires long practice
in order to form the necessary brain circuitry. Practice is
most effective when we are young. Reasoning has almost totally
disappeared from American schools.
2. Acquiring a skill is easiest when one understands what one is
trying to do. The classic way to teach formal reasoning in
schools was synthetic plain geometry. There, everyone has some
intuitive understanding of the subject and can rely on the
figures to suggest intermediate steps in the proofs. Synthetic
geometry has been greatly downgraded in our schools in favor
of "modern" topics--that is, topics easier for the teachers to
understand.
3. "Easy" subjects, like set theory, are generally the hardest
in which to teach reasoning. The results seem too obvious to
require proof and the real issues are too subtle. A subject
like elementary number theory, where the proofs have substance
and ideas can be tested by experiment, offers more prospect of
success.
4. Teaching reasoning takes time and care, which do not fit into
the schedule of university classes.
5. "Hands on" teaching is more effective that a pure intellectual
exercise. Mathematics is generally taught as an isolated
subject. Young students and children no longer get experience
with surveying, drafting, map making, navigation, gun laying,
civil and industrial engineering that motivated much of
mathematics.
I have been experimenting with Numerical Analysis as a venue for
teaching reasoning. The subject matter is really calculus, in which
the students have had practice with the technical manipulations, but
generally have no understanding. The course is entirely project
oriented; I spend very little time with analysis of algorithms or
formal convergence proof. All of the mathematical phenomina can be
associated with significant numerical experiments. In particular,
estimation, the heart of analysis, can be motivated by practical
considerations of function approximation. Graphing using analysis is
also thoroughly exercised in solution of transcendental equations. To
keep the work load (barely) within the realm of the possible, I use a
very high level, modern programming language (Smalltalk. Scheme would
also have been a good choice). The philosophy of the course is well
summarized by a quote from R. W. Hamming:
The purpose of computing is insight, not numbers.
The success of the course is mixed. On the one hand, the majority of
students cannot hack it and drop out; on the other, those who survive
the year, about one sixth, perform like professionals. I am gratified
that the survivers include many with only mediocre mathematical
talent. All have learned to reason about numbers and functions.
Another successful course, at a higher level, was on the mathematical
foundations of computer programming. Formal logic, set theory, and
the surprisingly difficult problems of variable binding get a thorough
workout in understanding programming language issues. It says
something about the sad state of contemporary computer science that
this course was not cross listed in the computer science department,
despite interest on the part of some of the faculty. (I even had some
initial difficulty getting it listed in mathematics).
>I've begun to get wind of dissatisfaction from several of the course's
>constituents, wanting me to move the course in opposite directions.
>Upper level math majors reveal in their courses that they haven't
>learned what a function is, what "one-to-one" means, how to read or
>write a proof, etc. Some CS professors want me to teach boolean
>algebra - methods of simplifying boolean expressions, which seems
>to be a rather "plug & chug" type of topic.
It is, but it also offers marvelous opportunities to attempt
reasoning, and hands on exercises. (Your students will hate
you if you try of course.) The algorithms of formal languages and
state machines are very beautiful, and reasoning opportunities abound.
Prof. Bill Smoke at U. C. Irvine has written a beautiful set of notes
on finite math that you might find helpful. His combinitorics notes
have a wonderful collection of problems, and his boolean algebra and
formal language notes contain a lovely theory (due I believe to
Kleene) on the equivalence of regular expressions and finite state
machines. Bill has retired but we still use his notes. I can have my
department send them to you (budget permitting) if you want.
>Well, it's Thanksgiving, so I'd better forget about this for awhile
>if I'm going to feel thankful about something.
Me too. Good luck with your efforts.
Igor Rivin
The included posting seems to miss the point entirely:
Most recent PhDs are young and able-bodied, so they could join the
navy, too. They also are often licensed to drive, so they could drive
a taxi. So? Presumably that is NOT why they spent some years of their
lives studying abstract mathematics -- rather, MATHEMATICS is what
they are interested in doing. A job on Wall Street may be lucrative,
and it may require quantitative skill, but it has absolutely nothing
to do with mathematics research. On the other hand, that kind of work
is sufficiently intellectually and socially demanding that it really
gives one no opportunity to think about mathematics in one's spare
time. Driving a cab is clearly superior from that viewpoint.
Brian Harvey
> 2. Acquiring a skill is easiest when one understands what one is
> trying to do. The classic way to teach formal reasoning in
> schools was synthetic plain geometry. There, everyone has some
> intuitive understanding of the subject and can rely on the
> figures to suggest intermediate steps in the proofs. Synthetic
> geometry has been greatly downgraded in our schools in favor
> of "modern" topics--that is, topics easier for the teachers to
> understand.
Although I agree with some of the other points in the article, I think
this is an unfair cheap shot. It's not at all that geometry has been
downgraded; what is happening in schools is a move toward making
geometry more of an experiential topic -- that is, emphasizing the use
of tools such as Geometric Supposer or Geometer's Sketchpad to discover
generalizations empirically rather than emphasizing formal proof. The
idea is that the proof was getting in the way of the geometry. You may
argue against this shift -- I can see that there are two sides to the
question -- but it certainly hasn't made things *easier* for teachers!
Quite the contrary; the new way is harder to teach, and calls for at
least as much understanding of the mathematics.
Len Evens
>In article <17...@idacrd.UUCP> pur...@idacrd.UUCP (Mark Purtill) writes:
>> I'm not familiar with the Jobs Almanac, but there certainly is
>>currently in no shortage of mathematicians, as the 12% of last years
>>class of Ph.D.s who are unemployed can attest.
>
>First of all, given the current recession it is not surprizing that a
>high percentage of new grads are unemployed, even among fields which
>are considered desirable.
>
>Second, many Ph.D.'s in math display the distressing tendency of viewing
>positions outside of academia with distain. I recall a recent conversation
>with a colleague who was bemoaning the lack of vacancies for new Ph.D.'s.
>When I gave him examples of firms on Wall St. that were hiring "quants"
>who would love to get some these people, he gave me a surprized look and
>said, "I wasn't talking about THAT."
According to the latest Notices, job prospects outside academia
for mathematics Ph Ds were even grimmer than in academia.
>
>Another example. We approached a department about recruiting some
>recent Ph.D.'s. The response was, "Sure, maybe the ones not good
>enough to get a post-doc will be interested." Sorry, we want the best
>ones, and we pay accordingly.
Apparently, you have some jobs. Unfortunately, you are only interested
in hiring the `best', and it appears that they can get jobs in academia
and they prefer that. Now, who is whining?
>
>There are a lot of jobs out there that those new Ph.D.'s should be looking
>at that they're not!
There is no evidence that there are `a lot of jobs out there' of the
kind you describe. All the evidence suggests the opposite.
It's the people without jobs who are complaining. Why don't you try offering
some jobs to those people?
It is really amazing how often people who are supposedly trained to
think precisely can't do it when they engage in this type of discussion.
Leonard Evens l...@math.nwu.edu 708-491-5537
Dept. of Mathematics, Northwestern Univ., Evanston, IL 60208
Herman Rubin
While I do not agree that synthetic plane geometry is the best way to
teach formal reasoning, since we have better ones which can, and should,
be used at a much earlier age, I must disagree with just about everything
in Mr. Harvey's comments. One can leave out some proofs as too difficult,
but all of mathematics is in the formal structure and reasoning. What
makes mathematics useful is that the "real world" seems to behave like
the abstract mathematical constructs, about which we can reason.
Geometry is NOT an experiential topic; it is the applications which are.
There is nothing wrong with using these simplistic tools to DISCOVER
geometric PLAUSIBILITIES, which are not geometric facts until they can
be proved. Also, the computer screen is not an accurate description of
the geometric universe; the Greeks were quite aware of the inaccuracies
of their diagrams, and realized that they were only guides.
The teachers can, to some extent, understand these plausibilities, and
teach students to memorize the computer manipulations and the "facts."
None of this will help the student to understand the mathematics, and
might even make it far more difficult to throw off the idea that what
was learned is not mathematics, but some physical properties of an
idealized situation. Evidence for this is that the teachers cannot
understand formal system. Someone who has difficulty with deriving
arithmetic from the Peano postulats should not teach arithmetic, except
as pure manipulation. And that is not a good idea if one does not
understand.
Brian Harvey
>The teachers can, to some extent, understand these plausibilities, and
>teach students to memorize the computer manipulations and the "facts."
This, too, is a cheap shot. Nobody is asking kids to memorize anything
about the constructions they make with Sketchpad etc. I know that you
have a bee in your bonnet about kids learning to reason, and *** I AGREE
COMPLETELY *** about that! But that is no excuse for making up lies
about what teachers do in classrooms.
Charles Geyer
mar...@lyra.cis.umassd.edu (Gary Martin) writes:
> ... about a quarter
> of the 75 students (at this point the 10 or 15 weakest students had
> already dropped the course) convinced me that they understood why if
> A is a subset of B then the complement of B is a subset of the
> complement of A. I had asked for a proof by whatever method they liked.
> The successful ones almost exclusively chose to do it by Venn diagrams.
> Several people didn't know what the symbols meant. One claimed that
> \subset was symmetric. Two thought that the complementation operator,
> which we denote by superscript c, meant exponentiation. Some thought
> that the assumption that A is a subset of B implied that A = B.
> Some thought that A is a subset of B implied that A is an element of B.
> Perhaps 8 or 10 people left the problem blank. One senior math major
> couldn't do it. (Note that I asked them for a proof, but only expected
> them to give me a clue that they understood the reason, whether or not
> they could express it clearly.)
You don't say whether they were doing lots of proofs themselves as homework.
I would guess not. If so, it's not surprising when they were asked to produce
a proof (instead of just nodding their heads while you went through one)
that they choked.
It is also part of the problem that you didn't ask for what you wanted.
Perhaps some less formal statement of the problem (avoiding the word "proof")
would have shown that more students had "a clue" than you thought. Perhaps
Explain why the following statement is true. If A is a subset of B
then the complement of B is a subset of the complement of A.
Also this is a very hard problem *because* it is so easy. The proof
goes "Here's a Venn diagram (two concentric circles). Here's a point
(outside either circle) in the complement of B; it's also in the
complement of A. QED" But no student who is shaky on the very notion
of proofs will ever guess that it is that simple.
This proof relies on a theorem "Venn diagrams model set theory" that I don't
recall having seen stated just that way. Is there such a theorem?
If not, it would seem that that a proof must involve proof by contradiction
which is generally agreed to be highly mysterious. Suppose to get a
contradiction that x in in the complement of B but not in the complement
of A. Then x is in A hence in B. But then it cannot be in the complement
of B. QED. QED???, I hear some student grumbling. We were supposed to get
that???
Just some thoughts. Not really criticising. I don't know how to deal
with this either.
John C. Baez
>A job on Wall Street may be lucrative,
>and it may require quantitative skill, but it has absolutely nothing
>to do with mathematics research.
Apparently this isn't *always* true. I think our very own Andrew
Mullhaupt may be a counterexample.
David Rector
I am happy to see more experimentation in the curriculum; indeed,
experimentation is very helpful in teaching reasoning. But synthetic
geometry is the ONLY venue in the current high school curriculum
suitable for introducing proof technique, and formal proof technique
is absolutely essential in all of mathematics and its applications
(e.g. computer science). Proofs in algebra are mostly too
difficult--because the facts are two obvious--for an introduction to
axiomatics. Furthermore, not just any approach to geometric proof
will do. Indirect proof is a very difficult concept, for deep
mathematical as well as psychological reasons. Axiomatics must start
with simple direct proofs. The purist approach to geometry,
introduced at about the same time as the New Math, that axiomatized
geometry by translation into algebra in order to get a rigorous
treatment, requires indirect proof from the very beginning. Textbooks
should stick to Euclid.
Another important topic in the highschool curriculum that students,
and even some teachers, think is useless is derivation of
trigonometric identities. The facts are indeed useless; the technique
is essential. The technique being taught is extended formal algebraic
manipulation, which is essential to all later mathematics. All of the
other algebra is too easy.
Another potential venue for proof in highschool could be elementary
number theory. The problem is to find room for it. I and most of my
colleagues think that calculus in high school is a waste of time. It
could be partly replaced by number theory, both formal and
experimental (aided by computers). Another good subject for informal
reasoning is theory of polynomial equations--another old topic--again
treated both numerically and synthetically.
By all means, increase experimentation. More important increase
application and numerical computation. Suggestion for a class
project: Survey the schoolyard. Divide the class into groups, each of
which is responsible for surveying a portion of the schoolyard
starting from a point of their own choosing and with a measuring
system of their own choosing. When they are finished, create a
composite map. Assuming the school buildings are more or less
rectangular in shape, the only equipment needed is
a. Balls of string.
b. Measuring tapes.
c. Squares made of knotted string--that is, triangles with sides in
ratio 5/4/3 (a five thousand year old technique).
d. Some small stakes.
Ideas taught: basics of rectangular coordinate systems, translation of
coordinates, construction of a global object by pasting together local
coordinate systems (manifold).
I intended no cheap shot at high school teachers, but in my sad
experience, few high school teachers understand mathematics, and
most--along with those who train them--rigidly resist listening to the
people who DO mathematics.
David Wilkins
>I am happy to see more experimentation in the curriculum; indeed,
>experimentation is very helpful in teaching reasoning. But synthetic
>geometry is the ONLY venue in the current high school curriculum
>suitable for introducing proof technique, and formal proof technique
>is absolutely essential in all of mathematics and its applications
>(e.g. computer science). Proofs in algebra are mostly too
>difficult--because the facts are two obvious--for an introduction to
>axiomatics.
What about group theory? (Of course this may not be in the
current American high school curriculum, though it was in mine,
in the Engish system.) It is surely ideal for introducing the
axiomatic method: a concise set of axioms, plenty of examples
lying around, and the pupils are not going to be confused by
the fact that the results are `obvious'.
> I and most of my
>colleagues think that calculus in high school is a waste of time.
So far as I am aware, students entering universities in Europe
to study mathematics, science and engineering are all expected
to have basic familiarity with basic calculus before they enter
university. It would surprise me if any of my teachers had
regarded it as a `waste of time'. Certainly English students
of my generation had two years of calculus behind them before
entering university. From what I can gather, American universities
appear to take up to two years merely to teach the sort of topics,
like basic calculus, that other countries succeed in covering
in (high) school.
David Rector
>In article <drector....@math.uci.edu> dre...@math.uci.edu (David Rector) writes:
>>I am happy to see more experimentation in the curriculum; indeed,
>>experimentation is very helpful in teaching reasoning. But synthetic
>>geometry is the ONLY venue in the current high school curriculum
>>suitable for introducing proof technique, and formal proof technique
>>is absolutely essential in all of mathematics and its applications
>>(e.g. computer science). Proofs in algebra are mostly too
>>difficult--because the facts are two obvious--for an introduction to
>>axiomatics.
>What about group theory? (Of course this may not be in the
>current American high school curriculum, though it was in mine,
>in the Engish system.) It is surely ideal for introducing the
>axiomatic method: a concise set of axioms, plenty of examples
>lying around, and the pupils are not going to be confused by
>the fact that the results are `obvious'.
I think group theory is an excellent idea. There are lots of good
examples for calculational experimentation, and as you point out, it
is well suited for teaching proof.
>> I and most of my
>>colleagues think that calculus in high school is a waste of time.
>So far as I am aware, students entering universities in Europe
>to study mathematics, science and engineering are all expected
>to have basic familiarity with basic calculus before they enter
>university. It would surprise me if any of my teachers had
>regarded it as a `waste of time'. Certainly English students
>of my generation had two years of calculus behind them before
>entering university. From what I can gather, American universities
>appear to take up to two years merely to teach the sort of topics,
>like basic calculus, that other countries succeed in covering
>in (high) school.
Most high school calculus courses in America are simply "turn the
crank" mechanics. Calculus is totally unsuitable for teaching proof
technique at that level since it requires three level deep quantifiers
just to state the basic definitions. On the other hand, a few calculus
ideas used, say, to enhance reasoning about functions and graphs, or
as part of elementary numerical methods for equation solving, would be
very benificial. I would rather see the last year of high school used
for subjects that enhance reasoning ability and inspire students to
study mathematics, rather than mechanical drudgery. I've seen too many
excellent students totally turned off mathematics by calculus courses, not
because they cannot understand it, but because it is a crashing bore.
C.E. Thompson
(Herman Rubin) writes:
|>
|> Geometry is NOT an experiential topic; it is the applications which are.
|> There is nothing wrong with using these simplistic tools to DISCOVER
|> geometric PLAUSIBILITIES, which are not geometric facts until they can
|> be proved.
The other side to this is that they should be used to discover plausibilities
which are not facts, and never will be, because one can DISPROVE them. This is
absolutely mandatory in order to provide students with an understanding of the
NECESSITY of formal proof. One can do this in the context of synthetic plain
geometry, of course, but it is not as easy as in some others. Herein lies the
pedagogic difficulty behind the perennial student's response to s.p.g.: ``Why
do I have to prove what I know is true (have convinced myself of) anyway?''
Chris Thompson
Cambridge University Computing Service
JANET: ce...@uk.ac.cam.phx
Internet: ce...@phx.cam.ac.uk
Andrew Mullhaupt
In fact, of the seven people who do research in our company, only four are
not former math professors, and those four are an MIT M.S. (EE) and two
physicists (one a former professor presently working at a major national lab)
and a guy who just got his Ph. D. in math who is starting in a junior position.
It might be worth my pointing out that my embattled colleague Robert Frey
is one of the former math professors, and is the only one still active in
teaching as an adjunct.
Later,
Andrew Mullhaupt
Andrew Mullhaupt
>In fact, of the seven people who do research in our company, only four are
>not former math professors,
Did I say this? Oh well; due to recent hiring this should read "of the nine"
before the recent hires it would have been five out of seven former math
professors. Now it's five out of nine about the remaining four:
>and those four are an MIT M.S. (EE) and two
>physicists (one a former professor presently working at a major national lab)
>and a guy who just got his Ph. D. in math who is starting in a junior position.
After I learn to count, I'm going to figure out how to cancel an article
so it won't look this embarassing.
Later,
Andrew Mullhaupt
Robert J Frey
>>In article <17...@idacrd.UUCP> pur...@idacrd.UUCP (Mark Purtill) writes:
>>Second, many Ph.D.'s in math display the distressing tendency of viewing
>>positions outside of academia with distain.
>
>Most recent PhDs are young and able-bodied, so they could join the
>navy, too. They also are often licensed to drive, so they could drive
>a taxi. So? Presumably that is NOT why they spent some years of their
>lives studying abstract mathematics -- rather, MATHEMATICS is what
>they are interested in doing. A job on Wall Street may be lucrative,
>and it may require quantitative skill, but it has absolutely nothing
>to do with mathematics research.
This is precisely the sort of attitude I was talking about. How do you know
that working as a quant on Wall St. has nothing to do with mathematics
research? I've been both a university professor and a quant on Wall St. and
I would disagree strongly with this. Before Wall St. I spent several years
in the defense industry, I think what I was doing was called mathematics
there as well.
What you really want is to do pure research in areas that happen to interest
you. Well, if that's the only definition of research mathematics, then it's
a pretty narrow one, and it is not surprising that there aren't a lot of
positions like that around, and it isn't surprising they don't pay very
well!
Regards,
Robert
Igor Rivin
>In article <1992Nov27....@CSD-NewsHost.Stanford.EDU> ri...@SAIL.Stanford.EDU (Igor Rivin) writes:
>>In article <13...@kepler1.rentec.com> rjf...@rentec.com (Robert J Frey) writes:
>This is precisely the sort of attitude I was talking about. How do you know
>that working as a quant on Wall St. has nothing to do with mathematics
>research? I've been both a university professor and a quant on Wall St. and
>I would disagree strongly with this. Before Wall St. I spent several years
>in the defense industry, I think what I was doing was called mathematics
>there as well.
>
>What you really want is to do pure research in areas that happen to interest
>you. Well, if that's the only definition of research mathematics, then it's
>a pretty narrow one, and it is not surprising that there aren't a lot of
>positions like that around, and it isn't surprising they don't pay very
>well!
>
>Regards,
>Robert
>
applied mathematics. As I see it, pure mathematics studies questions
for their intrinsic interest (snootily put, "canonical questions"),
while applied mathematics studies questions for their extrinsic
interest (snootily put, "useful questions"). Both these forms are
"mathematics", so I have no quarrel with your statement that you do
mathematics research, but, to my mind, they are not at all similar
philosophically, and often (though not always) very different
socially. For example, if one works in the industry, one often has
other people defining one's research goals, unlike the situation in
academia. One also has more of a tendency to work in teams, etc.
As a result, people attracted to (or good at) pure mathematics are
often not at all the same as those attracted to (or good at) applied
mathematics, thus your perception that people who pooh-pooh Wall St.
possibilities are snobs is wrong -- they may simply know enough about
it to know what they don't want to do. Further, your desire to get the
"best people" may be somewhat misguided. Those people may be "best" in
their ability to discover canonically interesting questions and
conduct their own research program, which is very likely not what you
want to hire them for.
Igor
PS: I have worked (and presently work) on both pure and applied
mathematics (though not on Wall St.)
Robert J Frey
>
>According to the latest Notices, job prospects outside academia
>for mathematics Ph Ds were even grimmer than in academia.
>
of similarly trained colleagues. I would be interested in just how the
Notices defines "jobs prospects," but I would be willing to bet that the same
sort of narrowness I was talking about earlier applies.
>
>Apparently, you have some jobs. Unfortunately, you are only interested
>in hiring the `best', and it appears that they can get jobs in academia
>and they prefer that. Now, who is whining?
>
among academics. Fortunately, we don't listen to them and over the past few
years have made a lot of good hires. Some from academic positions, both
tenured and untenured; some right of school; some from Wall St... We take
talent where we find it. One thing certainly isn't true: the best don't
always prefer academia.
>
>There is no evidence that there are `a lot of jobs out there' of the
>kind you describe. All the evidence suggests the opposite.
>
Academy of Sciences. Your problem is that you're defining the mathematical
sciences much too narrowly. That was the whole point of my original criticism.
One specific reference to get you started: "A Chellenge of Numbers: People
in the Mathematical Sciences," National Academy Press, 1990, ISBN 0-309-
04190-2.
>
>It's the people without jobs who are complaining. Why don't you try offering
>some jobs to those people?
>
Ph.D.'s. Most of those were hired in the past three years. Some were hired
out tenured faculty positions. Obviously we are out there offering jobs.
On Wall St. every major firm hires quants at the Ph.D. level and to the
best of my knowledge most have expanded not shrunk those staffs in recent
years. BTW, they get to work on what Norbert Weiner characterized as one
of the two most diffcult problems in applied mathematics (the other one
was weather prediction).
The firm I started with, Morgan Stanley, has a large number of quants.
First Boston is another with large staffs.
In the defense industry, where I previously worked, it is the individuals
with solid quantitative backgrounds who seem to be weathering the current
cuts best. For all the shrinkage in defense contracting and despite the
fact that I have been out of it for more than five years, I still get
offers to do outside consulting.
This does not mean that I am unconcerned about people who are having
trouble finding employment, but I submit that going out with broad
career horizons to find a challenging and interesting job is a heck of
a lot more contructive an attitude than the alternative many people in
mathematics present.
>
>It is really amazing how often people who are supposedly trained to
>think precisely can't do it when they engage in this type of discussion.
>
What is amazing is the confrontational way you approached this subject.
As a result, you sought to refute the original arguments mainly through
ad hominem attacks. When I criticized academic mathematicians I did so
on a factual basis. You may not believe those facts or feel they are
relevant, but unless you intend to refute them in a more coherent fashion
you and I have little to talk about, and my sole purpose in responding
here is to give some hope to all those new mathematicians out there.
To all of them: you chose a worthwhile and interesting career; your
training is relevant and allows you to do things most other people
can't; there are jobs, good jobs, out there, but what is required
is some leg work and common sense to bring them in.
I've worked as an applied mathematician in both academia and industry for
over 15 years. I've enjoyed my work and been well paid for it. I have
a large number of colleagues in the same position and with the same
feelings. Success, in almost any endeavor, depends on not arbitrarily
limiting your opprtunities.
Regards,
Robert
Robert J Frey
Hey, I know that guy! I can tell you from personal experience that he is a
perfect example. When are we going to get away from the notion that applied
math is not real math? Frequently, in fact, it turns out to be a good deal
more interesting, a good deal more 'mathematical' and a good deal more
difficult.
Regards,
Robert
Andrew Mullhaupt
>perfect example.
Actually, Robert and I are card carrying applied guys, although as a Courant
product, my party line (and religious conviction) is that there is no difference
between pure and applied math - there's just math. Maybe it makes sense to
point out that the fields of the other mathematicians and physicists in the
firm are Differential Geometry, Differential Topology, Several Complex
Variables and a High Energy Physicist. The point is - we seem to want smart
people who can ask and answer questions, as opposed to people who know this or
that thing.
Later,
Andrew Mullhaupt
Tal Kubo
>When are we going to get away from the notion that applied
>math is not real math?
The usual prejudice, that applied math is boring math, is
probably acquired (or at least reinforced) as a side effect of
the way math is taught. Traditional mathematics ideology places
value on deep structure and interconnected general theories, and
it's hard not to acquire this as an aesthetic value during the
eight or so years of training leading to a PhD. So it's not
surprising that many "purely" trained PhD's would rate, say,
elliptic curves as more interesting than financial modelling.
>Frequently, in fact, it turns out to be a good deal
>more interesting, a good deal more 'mathematical' and a good deal more
>difficult.
Can you point to actual examples of applied problems which are that
interesting? Since you claim that they arise frequently, you should
have no trouble coming up with convincing examples without recourse
to the usual magic words like "wavelets", "chaos/fractals", or
"hearing the shape of a drum". How many of these interesting problems
would a Wall Street quant be likely to work on? Enquiring minds want
to know.
-Tal Kubo ku...@math.harvard.edu or ku...@zariski.harvard.edu
Herman Rubin
There is mathematics which has been applied and mathematics which has not
yet been applied.
More to be deplored than some refusing to consider applied problems is the
tendency, now even encouraged by the government, of trying to keep "applied"
people out of the pure research community.
Herman Rubin
>In article <13...@kepler1.rentec.com> rjf...@rentec.com (Robert J Frey) writes:
>>When are we going to get away from the notion that applied
>>math is not real math?
>The usual prejudice, that applied math is boring math, is
>probably acquired (or at least reinforced) as a side effect of
>the way math is taught. Traditional mathematics ideology places
>value on deep structure and interconnected general theories, and
>it's hard not to acquire this as an aesthetic value during the
>eight or so years of training leading to a PhD.
This is not as much the case as is usually presented. Many of the most
important discoveries in mathematics are merely a case of seeing the
obvious. Even when the proof is quite difficult, as for example in
Cohen's introduction of forcing to prove independence, the basic idea
is quite simple. There are places where deep results are needed, but
these are not likely to be all that important.
`So it's not
>surprising that many "purely" trained PhD's would rate, say,
>elliptic curves as more interesting than financial modelling.
>>Frequently, in fact, it turns out to be a good deal
>>more interesting, a good deal more 'mathematical' and a good deal more
>>difficult.
Financial modelling is much harder, because for the model to be useful
it must be mathematically tractible, and it also must have a sufficient
resemblance to the real world. In fact, the modelling has nothing to do
with mathematics, per se. So the mathematician dealing with it cannot
simplify assumptions, etc., but must deal with at least an approximation
of the "real world."
>Can you point to actual examples of applied problems which are that
>interesting? Since you claim that they arise frequently, you should
>have no trouble coming up with convincing examples without recourse
>to the usual magic words like "wavelets", "chaos/fractals", or
>"hearing the shape of a drum". How many of these interesting problems
>would a Wall Street quant be likely to work on? Enquiring minds want
>to know.
Now I do not know if a Wall Street person is likely to work on these,
but here are some form probability and statistics, which are applied.
How about the very simple result of von Neumann and Morgenstern about
getting a linear utility scale? All that was required was to allow
randomized strategies, and to use the assumption that only the
distribution of results counts? Or, to toot my own horn, the
observation that self-consistency assumptions, and the Hahn-Banach
Theorem, give a simple derivation of Bayesian behavior? Or the rather
odd result that the generation of non-uniform random numbers from uniform
can usually be done by a procedure which requires only a finite number of
random bits, and of course a finite amount of computation, plus copying?
Tal Kubo
>We're a firm of about twenty five people; we have a research staff of eight
>Ph.D.'s. Most of those were hired in the past three years. Some were hired
>out tenured faculty positions. Obviously we are out there offering jobs.
Of the annual math PhD crop, what fraction do you think Wall Street would
be willing to hire for technical positions? By your own account, they
would be competing against people from other fields, so one vacancy does
not equal one PhD. Also, the demographics at the big firms are hardly the
same as at your company.
>On Wall St. every major firm hires quants at the Ph.D. level and to the
>best of my knowledge most have expanded not shrunk those staffs in recent
>years.
Firms on Wall Street can fire employees at will, and pay according to
performance in ways that give an "incentive" to work long hours. For the
moment the quants are indispensible, but how would their job security and
working conditions be affected by a large-scale infusion of PhD's into the
business? The kowtowing corporate culture at the big firms isn't much of
an advantage over academic politics, either.
> BTW, they get to work on what Norbert Weiner characterized as one
>of the two most diffcult problems in applied mathematics (the other one
>was weather prediction).
Do you mean functional integration, or financial forecasting?
>In the defense industry, where I previously worked, it is the individuals
>with solid quantitative backgrounds who seem to be weathering the current
>cuts best.
Yes, but are they hiring anyone new? Last I heard, Bell Labs had a hiring
freeze, for example. At least in this area, the defense jobs for PhD's
advertised in the paper are predominantly for engineers and computer
programmers, and in general not many jobs are available at all. It's not
clear whether the defense industry can really absord many math PhD's.
>This does not mean that I am unconcerned about people who are having
>trouble finding employment, but I submit that going out with broad
>career horizons to find a challenging and interesting job is a heck of
>a lot more contructive an attitude than the alternative many people in
>mathematics present.
This sounds like "when all else fails, lower standards." By the way,
why is it necessary to present constructive alternatives -- would you have
made the same remark if we were talking about the Black Plague?
>What is amazing is the confrontational way you [L. Evens] approached this
>subject. [...]
On a related note, I find it interesting that all of the quants I have
talked to, about ten, including some very bright people at large Wall
Street firms, have taken great pains to make (unsolicited) comparisons
between their work and what goes on in academia. For example, they speak
of academics with derision, or point out the number and talent of the PhD's
in their company, or how much they prefer the financial markets to the
universities, how difficult and challenging the work is, etc. It rubs off
as somewhat aggressive and completely gratuitous.
>I've worked as an applied mathematician in both academia and industry for
>over 15 years. I've enjoyed my work and been well paid for it. I have
>a large number of colleagues in the same position and with the same
>feelings. Success, in almost any endeavor, depends on not arbitrarily
>limiting your opprtunities.
I'm glad you like it.
Victor Miller
Monthly, entitled "Applied Mathematics is Bad Mathematics". The first
sentence of the article was something like "I really didn't mean it,
but got your attention; didn't I?" The difference between Applied and
Pure is a matter of individual taste. However, this matter of taste
may be that most people who style themselves as "Pure" will see the
pursuit of the mathematical consequences of the problem the most
interesting thing, whereas those who style themselves as "Applied"
will keep an eye (at least for the moment) on the non-mathematical
connection. It's interesting (and rather sad) that we have this
split. Would you call Gauss "Pure" or "Applied" (this shows how
ridiculous the whole dichotomy is).
--
Victor S. Miller
Bitnet: VICTOR at WATSON
Internet: vic...@watson.ibm.com
IBM, TJ Watson Research Center
"Great artists steal; lesser artists borrow" Igor Stravinsky
Marc Green
needing years of math is nonsense. Sure there will be people in specific
specialties who need mathematical sophistication. In general, most people
would be fine if they could simply do simple arithmatic and perhaps have some
intuitive understanding of elementary statistics. The way our society dishes
out the rewards, you'd have to be crazy to go into a technical profession,
anyway.
Who cares what the Japanese score on math tests? Their success has not
depended on math. It's due to a bunch of factors, like their willingness
to work hard, receive few social benefits, have the government underwrite
business and engage in predatory trade policies. None of this has anything
to do with math scores on standardized exams.
Computer types think that the world revolves around formal systems and that if
you don't know math, you don't know anything. It's exactly the opposite: if all
you know is math, then you don't know anything.
Marc Green
Trent University
Andrew Mullhaupt
>More to be deplored than some refusing to consider applied problems is the
>tendency, now even encouraged by the government, of trying to keep "applied"
>people out of the pure research community.
Well there's this invisible hand, see, and these benevolent market forces,
and...
Actually, I remember the extravagant Reagan administration claims about
funding mathematics - almost all the money went to computer science.
Then there was the infamous secret CIA front funding dynamical systems
(Helen Wisniewski [spelling?]) - the mathematicians didn't want the money
when it was thrown at them.
I'm pretty sick of what I know about government funding of research, and
the only thing that's worse is that this is the government's approach to
funding higher education - by trickle down.
I think the recent moaning about journal subscriptions and the thinness of the
jobs market in academia points to one thing: like it or not, a lot of Ph. D.'s
are going to get some applied experience. The really smart ones will make the
most of it. I remember reading once about the fact that UCB once produced
a Ph. D. who was without question the most promising historian of medieval
philosophy of his generation. Unable to find employment, he became a real
estate tycoon with no regrets. I don't know about the other mathematicians
on the planet but I would have some big regrets if I couldn't do math for a
living.
Now when I was a grad student, the advisor was supposed to make it clear what
kind of job prospects you might expect when you got the degree at the end
of the tunnel. What does someone say in (insert unemployable field here)?
The need to produce students in order to be promoted is not an excuse.
We are at the point now where it makes sense to produce fewer Ph. D.'s,
or else change the employment pattern. My favorite idea is to cut back on
the number of graduate student slots, hire more professors to _teach_ and
to fund universities much more as educational institutions. In fact, the
ideal solution would also address the journal quantity/quality explosion,
(we're starting to look more like physics every day) _and_ the content
deficit in high schools. A professor I have tremendous respect for taught
Calculus in a local high school once in a while for several reasons -
it raised the quality of the high school education, it kept him close
to the level of understanding of the entering freshman, and it gave him
great moral authority when teaching courses for high school teachers. I
think this would (on a rotating basis) be quite a good thing.
There's lots of things to do with Ph. D.'s aside from warehouse them in
bloodless pyramidal schemes for inflating the cross section of such and
such a field.
Later,
Andrew Mullhaupt
Alex Lopez-Ortiz
writes:
This and Dan Quayle are the best examples I've seen for the necessity
of revamping of science education in norteamerica ;-)
Mathematics _should_ help teach people problem solving skills,
abstraction,
methodicalness, etc.
Here's an example. I live in a house where there are 5 different phone
lines
coming in. Just recently Bell Canada rewired the whole neighbourhood. So
far
3 crews of untrained construction workers (rednecks :-) have come to my
house
and they haven't been able to device a solution of how to get 5 lines into
my place from a different place.
The solution is quite simple. But they don't even know how to attack the
problem. They can only think of adding cable, which in this case is not
posible.
Of course nobody is proposing to teach advanced topology to six-greaders,
but certainly more than arithmetic is needed (algebra is an essential).
Alex
P.S. The solution is to use some of the cable backwards. That is, part
of the wire where the signal traversed from the 1st floor to the basement
should now carry the signal from the basement to the 1st floor.
--
Alex Lopez-Ortiz alop...@maytag.UWaterloo.ca
Department of Computer Science University of Waterloo
Waterloo, Ontario Canada
Herman Rubin
>If people can add and subtract, that's enough. All this gibberish about
>needing years of math is nonsense. Sure there will be people in specific
>specialties who need mathematical sophistication. In general, most people
>would be fine if they could simply do simple arithmatic and perhaps have some
>intuitive understanding of elementary statistics. The way our society dishes
>out the rewards, you'd have to be crazy to go into a technical profession,
>anyway.
What good is it to know how to do arithmetic if the knowledge of WHEN to
do it is not present? Someone who just knows how can only use it in exactly
the situations taught. But someone who knows when can usually find a way
to get the how done, especially in these days of calculators.
As for having an intuitive understanding of elementary statistics, I would
prefer to keep people totally ignorant. The misuse of statistics by those
who know what is called elementary statistics, and even much more statistical
methods, and know this quite well, is ubiquitious and HAS done great harm.
This is a case where what you know that ain't so can very definitely hurt you,
and lots of others.
Randolph Gregory Brown
>
> Teach the use of variables, the notions of function and relation, formulation
> of word problems, logic, the structure of the integers and the reals, and how
> operations fit into that. A first-grader can understand the Peano Postulates
Yes! A first grader probably even has a better chance of
understanding the Peano Postulates than a high-schooler. If they are
taught the fundamentals from the beginning, instead of wasting their
time drilling arithmatic into their brains to the point that they'd
rather ignore math for the rest of their life, they might have some
clue what is going on in high school.
Randy
Robert J Frey
>
>Can you point to actual examples of applied problems which are that
>interesting? Since you claim that they arise frequently, you should
>have no trouble coming up with convincing examples without recourse
You are of course correct: I should be prepared to put my "money where my
mouth is."
I'll draw from personal experience; I claim that _I_ found these problems
interesting! :-)
* * *
I worked on an automated diagnostic system. The only practical way to solve
such problems is to integrate multiple reasoning techniques so that their
complementary strengths can be exploited. What is needed then is a meta-model
that can express the results of diverse models in a common framework. Those
models include: symptom-fault signatures, dependency graphs, production rules
and even free-form input from operators and technicians.
I rejected such approaches as Dempster-Schaffer, fuzzy set theory and others
I felt lacked any real rigor. I was able to integrate them using information
theoretic techniques. This not only successfully merged these models into a
coherent system, but achieved some things like non-monotonic reasoning that
conventional mathematical models weren't supposed to be able to do without
recourse to some ad hoc AI techniques. The result was $25 million in funding
awarded to the company I was working for.
* * *
I've also used queueing network models to model the performance of real
and proposed computing systems and communication systems. These models
are often fairly simple, but nevertheless capture the important elements
of performance for these systems. Many of these models are very diffcult
to solve and require a fairly high of mathematical sophistication.
* * *
Consider problems in multi-period portfolio optimization. You are given a
collection of instruments, a model which describes the covariance among
them, a forecasting function for their returns, a trade off model for risk
and reward, a transaction cost function and a starting portfolio. From
this information determine the set of trades for the next trading period,
remembering that future values of the forcast are subject to revision as
the stochastic system evolves. This is HARD, and it remains a largely un-
solved problem.
* * *
There are lots of other examples that can be drawn from the world at large.
Consider fluid dynamics, weather forecasting, designing programs for
parallel machines, etc. I will point out agian that Norbert Weiner stated
that the two most difficult problems facing applied mathematicians are
weather forecasting and modeling financial markets.
Regards,
Robert
Robert J Frey
>over academic politics, either.
I don't think it's any advantage either! It just isn't any worse.
>Do you mean functional integration, or financial forecasting?
Modeling financial markets.
>Yes, but are they hiring anyone new? Last I heard, Bell Labs had a hiring
>freeze, for example. At least in this area, the defense jobs for PhD's
>advertised in the paper are predominantly for engineers and computer...
I never said we weren't in a recession, and in a recession jobs, particularly
jobs for inexperienced workers, are scarce. That's a a general characteristic
of the economy right now, not a fault specific to the market for mathe-
maticians. As for jobs being predominantly for engineers and computer
scientists, my point is that mathematicians can not only compete success-
fully for those positions but they often out-perform them
>This sounds like "when all else fails, lower standards."
Broadening your horizons and expanding your options is NOT the same as
lowering your standards.
>... [A]ll of the quants I have talked to ... have taken great pains to make
>(unsolicited) comparisons between their work and what goes on in academia.
>For example, they speak of academics with derision ... It rubs off as
>somewhat aggressive and completely gratuitous.
Well, I can't speak for everyone else, but perhaps I can offer some insights.
If Wall St. types come off as agressive and abrasive, it's because Wall St.
types are often agressive and abrasive. Their reaction towards academics is
partly a reaction to the sort of differences in outlook that we've been
discussing in this thread. I do not speak of academics with distain; I do
criticize individuals, whether they work in a university or an investment
bank, who put forth a narrow vision of what it means to be successful.
Regards,
Robert
Keith Weintraub - dpr2
expression for two (;-)). I agree with my colleagues Mr. Frey and Mr.
Mulhaupt from Renaissance. I won't apologize or compare what we do with what
is done in academia but I will say that *we* do find it interesting.
I also have a recruiting experience to relate. (Generalize if you wish---I
won't). I went back to the Stat department at my alma mater (where I got my
Ph.D.) for a seminar. During the post-talk refreshements I asked a number
(>4) of my former professors if anyone was graduating soon and looking for
employment. They didn't know of anyone. One month later I received the resume
of a soon-to-be graduate from that department from a *search firm*
(head-hunter). When I called the department to find out about the candidate
they were quite embarrassed. I found the whole thing somewhat amusing. I'm
sure that the students in the department wouldn't be as amused as I was if
they found that an avenue of employment was shut off from them due to
ignorance.
Enjoy and happy holidays.
KW
--
Keith Weintraub (KW) -- Citicorp | I told you a trillion times:
kei...@Citicorp.COM | "Don't exaggerate."
uunet!ccorp!keithw |
chrisman
>needing years of math is nonsense.
Let me tell you about a few things that I (math major) can do which
my girlfriend (who can add and subtract) cannot:
* Read a map, and know right from left even I'm going south;
* Cook a meal (in half the time it takes her) and have all the
dishes finish cooking around the same time;
* Balance my checkbook (yes, she does know how to add and subtract,
but she can't do this).
* Proofread a paper for grammer AND organization AND logic.
I'm sorry if most of these things sound incredibly domestic, but consider
the skills involved and ask yourself if workers and bosses need those
skills (understanding diagrams; visualizing spacial relationships;
organizing tasks - indeed having the intellectual curiosity to wonder
if efficiency can be improved!; organizing raw data and processing it;
thinking logically).
I can't say all math majors are like me or that all non-math-majors are
like her; but if you pick me a random math major and a random non-math-major
I'll bet a reasonably large sum of money that the math major will perform
better in these tasks and similar ones.
One more remark: my girlfriend wants to enter law school, and has done
research into success rates at law school. She has found that law
students who majored in math as an undergraduate (and these are few)
typically excel.
> if all you know is math, then you don't know anything.
True! All I know is math, and I admit I don't know anything. But
I know how to think.
Danny Breidenbach
>I have seen some of the suggestions on discrete math; they are still nothing
>more than manipulations.
I too have seen some of the suggestions on discrete mathematics (I assume
we are both referring to the suggestions in the NCTM Standards) -- while
I am by no means a mathematician (maybe *because* I am not a mathematician)
my eyes were opened to mathematics that I never saw in the
traditional high school and teacher-prep curriculum. They are in fact
more than manipulations -- while they don't delve at great length into
theory and proof, they demonstrate clearly how much mathematics is used
in life.
>>but there is also a new spurt using ISETL, a language that's more directly
>>"math-like" in its notation)
>how they can teach any meaningful statistics--it requires an understanding
>of the concept of probability, not combinatorics. ISETL is not math-like;
>formal manipulations with sets are still formal manipulations.
Brian's statement was that ISETL is "more directly 'math-like' in its notation"
More than Logo. This statement cannot be argued. Period. The statement
that "ISETL is math-like" can be argued -- but even then, the debate would
not be empty. ISETL is a programming language -- subject to all the math-
ematical inadequacies of programming languages. This does NOT mean
that it is destined to ruin the mathematical thinking of students who
use ISETL as a tool for learning and doing mathematics. Purdue University
is an excellent (but not the only) place to see that this is true!
>Teach the use of variables, the notions of function and relation, formulation
>of word problems, logic, the structure of the integers and the reals, and how
>operations fit into that. A first-grader can understand the Peano Postulates.
In order to successfully use ISETL in a meaningful context, one must learn
the use of variables, one must struggle with the notion of function and
relation, one must deal with the beginnings of logic. In using ISETL, one
can deal with the predicate calculus, one can explore number systems --
whether integers or other systems. I don't claim that one MUST use ISETL
to deal with these things--that would be senseless. But realize that the
use of ISETL or other innovative tools/projects does not preclude these
things!
Furthermore, I disagree that a first-grader can understand the Peano
Postulates -- unless it is a very intelligent first-grader!
>Show them that intuition, while useful, is also dangerous, and that careful
>proof is needed. However, point out that there are times when they will
>have to accept the idea that someone else has found a proof.
Well stated. Why is it that people with the same desired outcomes in mind
can so effectively close their minds to other ways to achieve those outcomes?
----------------------------------------------------------------------------
| Daniel H. Breidenbach 435 S Chapelle |
| Teacher Rotator Pierre, SD 57501 |
| South Dakota NSF Systemic Initiative 605-773-6400 |
| Internet: dbre...@dsuvax.dsu.edu SD Science/Math Net: danny |
Gary Martin
In article <MARTIN.92N...@lyra.cis.umassd.edu>
mar...@lyra.cis.umassd.edu (Gary Martin) writes:
> ... about a quarter
> of the 75 students (at this point the 10 or 15 weakest students had
> already dropped the course) convinced me that they understood why if
> A is a subset of B then the complement of B is a subset of the
> complement of A. I had asked for a proof by whatever method they liked.
> The successful ones almost exclusively chose to do it by Venn diagrams.
> Several people didn't know what the symbols meant. One claimed that
> \subset was symmetric. Two thought that the complementation operator,
> which we denote by superscript c, meant exponentiation. Some thought
> that the assumption that A is a subset of B implied that A = B.
> Some thought that A is a subset of B implied that A is an element of B.
> Perhaps 8 or 10 people left the problem blank. One senior math major
> couldn't do it. (Note that I asked them for a proof, but only expected
> them to give me a clue that they understood the reason, whether or not
> they could express it clearly.)
You don't say whether they were doing lots of proofs themselves as homework.
I would guess not. If so, it's not surprising when they were asked to
produce a proof (instead of just nodding their heads while you went
through one) that they choked.
Well, not lots, but a few. With classes as large as they are, it's not
easy to give the students significant feedback on homework. Besides,
our students react the same way to difficult homework as they do to
difficult test questions - they leave them blank or write some nonsensical
guess and leave it at that. The professor who taught this course before
I took it over three years ago told me that he could ask *exactly* the
same question for homework, on a quiz, on an exam, and on the final,
going over it in class after each, and *still* a large portion of the
class wouldn't get it (and looking at a sample quiz of his, I know that
his questions were more straightforward than mine).
It is also part of the problem that you didn't ask for what you wanted.
Perhaps some less formal statement of the problem (avoiding the word
"proof") would have shown that more students had "a clue" than you thought.
Perhaps
Explain why the following statement is true. If A is a subset of B
then the complement of B is a subset of the complement of A.
I did want a proof; I only settled for an indication that they had a clue
after it became clear that this would be the only way to give them some
credit for the problem.
Also this is a very hard problem *because* it is so easy. The proof
goes "Here's a Venn diagram (two concentric circles). Here's a point
(outside either circle) in the complement of B; it's also in the
complement of A. QED" But no student who is shaky on the very notion
of proofs will ever guess that it is that simple.
Part of the reason I'm so depressed about this is that the statement
didn't appear to be obvious to a large number of students. They didn't
even know what the theorem was saying!
This proof relies on a theorem "Venn diagrams model set theory" that I don't
recall having seen stated just that way. Is there such a theorem?
I suppose one could cook up such a theorem if one either generalizes the
notion of Venn diagram, restricts the kinds of sets one deals with, or
defines "model" appropriately. Surely one cannot model large families of
large sets using a finite dimensional Euclidean space if "model" is
interpreted very strictly. However, I'm not sure this is relevant for most
of the students. I think they tend to believe in Venn diagrams. This
could be a problem for the best students, who might realize that there's a
logical difficulty, but those students would surely be clever enough to
do another sort of proof.
If not, it would seem that that a proof must involve proof by contradiction
which is generally agreed to be highly mysterious. Suppose to get a
contradiction that x in in the complement of B but not in the complement
of A. Then x is in A hence in B. But then it cannot be in the complement
of B. QED. QED???, I hear some student grumbling. We were supposed to get
that???
There are other approaches, too. The one I usually wrote on their papers
when they got stuck was: A \subset B means x \in A -> x \in B;
B^c \subset A^c means x \in B^c -> x \in A^c;
the latter says not(x \in B) -> not(x \in A);
the two statements are contrapositives, hence
logically equivalent; if we know one, then we
know the other.
Two students actually surprised me with a clever approach. They couldn't
follow through on it to get a proof, but they got plenty of partial
credit for their ideas. A \subset B implies A \intersect B = A.
Take complements and use DeMorgan to get A^c \union B^c = A^c, which
means B^c \subset A^c. (Actually, one was doing something more
complicated.) But, as I recall, one of them didn't change intersection
to union when using DeMorgan's law.
Just some thoughts. Not really criticising. I don't know how to deal
with this either.
Thanks. It's a very frustrating problem. I spent a whole class going
over the test, even though I'm already two weeks behind my syllabus for
the course (which was planned at half the speed the authors of the text
recommend). One approach, suggested to me two years ago by a senior
member of my department, was to go as slowly as is necessary for the
students to learn. That's somewhat appealing, but it raises questions
about using the course as a requirement for computer science and
engineering majors, whose departments (and accreditation panels) are
expecting certain minimal coverage.
--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Mar...@cis.umassd.edu
Tal Kubo
>>> a good deal more 'mathematical' and a good deal more] difficult.
>>Can you point to actual examples of applied problems which are that
>>interesting?
>I'll draw from personal experience; I claim that _I_ found these problems
>interesting! :-)
> [3 examples: using information theory (and apparent ingenuity) to
synthesize several reasoning systems for automated diagnosis; queueing
network models; a very general and intractable formulation of the
portfolio optimization problem]
The examples sound interesting and difficult, but they certainly don't
seem more 'mathematical' than pure problems. They just reinforce my
impression that pure math and (most of) applied math are as related as
apples and oranges.
>There are lots of other examples that can be drawn from the world at large.
>Consider fluid dynamics, weather forecasting, designing programs for
>parallel machines, etc. I will point out agian that Norbert Weiner stated
>that the two most difficult problems facing applied mathematicians are
>weather forecasting and modeling financial markets.
There are some applied fields where the pure theory fits more or less
directly, because the underlying assumptions are sufficiently realistic
(e.g. physics, computers, and parts of fluid mechanics). There one can
concentrate on developing the mathematics, without regard to the actual
implementation. But that kind of work is done mainly in universities,
anyway.
Jeffrey Wayman
>
> > ... about a quarter
> > of the 75 students (at this point the 10 or 15 weakest students had
> > already dropped the course) convinced me that they understood why if
> > A is a subset of B then the complement of B is a subset of the
> > complement of A. I had asked for a proof by whatever method they liked.
> > The successful ones almost exclusively chose to do it by Venn diagrams.
> > Several people didn't know what the symbols meant. One claimed that
> > \subset was symmetric. Two thought that the complementation operator,
> > which we denote by superscript c, meant exponentiation. Some thought
> > that the assumption that A is a subset of B implied that A = B.
> > Some thought that A is a subset of B implied that A is an element of B.
> > Perhaps 8 or 10 people left the problem blank. One senior math major
> > couldn't do it. (Note that I asked them for a proof, but only expected
> > them to give me a clue that they understood the reason, whether or not
> > they could express it clearly.)
>
suggested this......I've had success teaching these type of
concepts by actually assigning "events" to A, B, C, whatever
(ex: A=you like football, B=you like all sports). I then create
a "population" of students by actually bringing 20 or so to the
front of the room and moving the groups of students or "events"
around to suit the lesson. You could easily illustrate your
problem this way; the trick is to plan out the events so they
are easy to understand. It's a real "3rd-grade" method, but it
works well because it's funny, it's something other than lecture,
so it will get more student's attention and it's easy to re-conceptualize
when they do similar problems.
Tal Kubo
>I never said we weren't in a recession, and in a recession jobs, particularly
>jobs for inexperienced workers, are scarce. That's a a general characteristic
>of the economy right now, not a fault specific to the market for mathe-
>maticians.
Is it the case in the private sector that an average of, say, 500-1000
applications are received for a typical position paying under $40,000 a
year? That is a specific fault of the mathematical job market, and it
seems independent of the recession.
>Broadening your horizons and expanding your options is NOT the same as
>lowering your standards.
It is in the context in which you are proposing it. The question is,
whether the employment opportunities for mathematicians in industry are a
viable alternative to the increasingly scarce positions in academic
research. Everyone knows that other jobs are out there, ranging from
computer programming and engineering to teaching high school or (as Igor
Rivin pointed out) driving a taxi. Lots of these might be interesting and
rewarding, and lots of them provide an opportunity for mathematicians to
gainfully exploit their skills. The question is, how many jobs are
available in which one can work as a mathematician *per se*. The litmus
test for this is, would your employer be happy to pay you if you spent a
couple of months trying to prove a theorem? I suspect the jobs like that
in the private sector are at least as scarce as in academia.
> I do
>criticize individuals, whether they work in a university or an investment
>bank, who put forth a narrow vision of what it means to be successful.
Successful does not necessarily mean successful *as a mathematician*. As
Andrew Mullhaupt pointed out, many people would regret getting having to
settle for a nonmathematical job after getting a PhD in math. How broadly
one defines "mathematical" is more a matter of personal taste rather than
breadth of vision.
kevin.jessup
>As for having an intuitive understanding of elementary statistics, I would
>prefer to keep people totally ignorant. The misuse of statistics by those
>who know what is called elementary statistics, and even much more statistical
>methods, and know this quite well, is ubiquitious and HAS done great harm.
>This is a case where what you know that ain't so can very definitely hurt you,
>and lots of others.
Fourty-three percent of all statistical figures are generated on-the-spot! ;-)
--
Kevin Jessup, kevin....@mixcom.mixcom.com
"Friends don't let friends run DOS."
-- Microware
Jeffrey Wayman
>
>>As for having an intuitive understanding of elementary statistics, I would
>>prefer to keep people totally ignorant. The misuse of statistics by those
>>who know what is called elementary statistics, and even much more statistical
>>methods, and know this quite well, is ubiquitious and HAS done great harm.
>>This is a case where what you know that ain't so can very definitely hurt you,
>>and lots of others.
I'm not sure I follow you here. Could you please give an example
or two of the harm caused? Thanks a lot.
Herman Rubin
The purview of statistics should be decision making under probabilistic
uncertainty. Now this is something which is in principle quite simple,
but in practice extremely difficult. It is necessary to balance all the
consequences of the proposed action.
Now one can easily teach "descriptive statistics." The real question is
not how to use the data to crank out some numbers, but to come up with the
appropriate action to take. There is this great tendency to take the median
of a population as the "typical" value. But saying that the median diameter
of a planet is approximately 7900 miles (Earth is here the median planet)
does not convey much information about the sizes of planets; Earth is by
no means typical. Or the median planet in terms of distance from the sun
is Jupiter. I think that Pluto has the median density. None of these
is a "typical" planet, nor is there one. Nor is there a typical diameter,
a typical distance from the sun, or a typical density.
When it comes to what is taught in statistical methods courses, the
situation is even worse. Scientists, government officials, etc.,
swear by the 5% (or in some cases 1%) probability of rejecting the
null hypothesis if it is true. But can it be true that saccharin
has NO effect on cancer? Remember, NO here means ABSOLUTELY NO.
Hardly; water consumption must have an effect. Well then, you say
that it means an unimportant effect. This can be handled with
difficulty, but the standard test criteria are still wrong.
OK, you say that we should use the P-value. There was a paper which
reported that a particular chemical had a moderate carcinogenic effect,
because of its P-value. This MIGHT be the case, as there might have
been an accidentally large number of rats who became cancerous, but
having it give cancer to 50% of the rats does not seem to be moderate.
With a sample 10 times as large, the same incidence of cancer would
give a far smaller P-value, but the carcinogenicity would not change.
Gary Martin
I'm new to this newsgroup, so I apologize if someone has already
suggested this......I've had success teaching these type of
concepts by actually assigning "events" to A, B, C, whatever
(ex: A=you like football, B=you like all sports). I then create
a "population" of students by actually bringing 20 or so to the
front of the room and moving the groups of students or "events"
around to suit the lesson. You could easily illustrate your
problem this way; the trick is to plan out the events so they
are easy to understand. It's a real "3rd-grade" method, but it
works well because it's funny, it's something other than lecture,
so it will get more student's attention and it's easy to re-conceptualize
when they do similar problems.
This would be a good way to illustrate this particular concept, but I
suspect that then when I ask for a proof that A \subset B implies
B^c \subset A^c, the vast majority of the class would parrot back,
"let A = you like football, ..." Vivid examples may be so vivid that
they are mistaken for reasoning. Still, I do try to use such techniques
when they occur to me. For example, when introducing the notion of
directed graph, I began with an application - a situation that can
be represented as a directed graph. The classroom has two light switches,
one for each of the two banks of lights. I turned the lights off,
drew a vertex on the board to represent darkness (the curtains were
closed to avoid glare), and flipped one switch. Nothing happened
(the switches had been rewired after one bank of lights had been
non-functional for nearly a year), so I drew a loop at the vertex.
I continued flipping switches and drawing edges and the other vertex.
Then I drew the graph that I expected from the switches and explained
how this is a very simple model of a computer, with the vertices
representing internal states of memory and edges representing
transitions caused by executing commands.
If anyone has more suggestions for this type of illustration, I'd
be interested in hearing them. But also, I'm curious how you'd
convince students that it's also important to be able to formalize
these concepts and deal with them logically, regardless of how
clear the demonstrations might make them seem.
Robert D. Silverman
>
> I'm new to this newsgroup, so I apologize if someone has already
> suggested this......I've had success teaching these type of
> concepts by actually assigning "events" to A, B, C, whatever
> (ex: A=you like football, B=you like all sports). I then create
> a "population" of students by actually bringing 20 or so to the
> front of the room and moving the groups of students or "events"
> around to suit the lesson. You could easily illustrate your
> problem this way; the trick is to plan out the events so they
> are easy to understand. It's a real "3rd-grade" method, but it
of 3rd graders? IMHO, if the students are not up to learning the material,
or lack the maturity to deal with any kind of abstraction, then they
SHOULD NOT BE THERE.
This sounds elitist and probably is, but there are far too many students
attending college without any idea of why they are there (except maybe to
party), and without any interest in LEARNING. In my opinion, these students
should be encouraged to leave school and come back when they are ready
to learn. I believe that someone who has passed a college level course in
(say) elementary discete math should have acquired a basic set of skills.
How else is an employer to judge whether a student who has passed a given
course really learned the material? In other words, I believe in certain
minimal standards, and that any student who does not have them should be
flunked.
I believe that when a course listing in a catalogue gives a certain set
of prerequisites, they should be adhered to. Course material should not
be watered down, exams should not be watered down, because students
taking the course do not have the prerequisites. If they need to fill holes
in their background, it is their responsibility, not the professor's to
fill them. This is college, not kindergarten.
Let us stop teaching to the lowest common denominator and start demanding of
students that they take some responsibility for their own education, rather
than expecting to be spoon fed. In other words, let us start demanding a
better "study ethic" and less of a party attitude from students.
--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Doug Moore
Bob> Why on Earth should a college professor teach a class as if it
Bob> consisted of 3rd graders?
<text omitted>
Bob> I believe that someone who has passed a college level course
Bob> in (say) elementary discete math should have acquired a basic
Bob> set of skills. How else is an employer to judge whether a
Bob> student who has passed a given course really learned the
Bob> material?
I am inclined to agree. But a lecturer has every incentive to please
students and no incentive to please potential employers. Students can
complain to department chairmen and deans and write bad course
evaluations and significantly damage the tenure prospects of a young
professor, or the promotion and raise prospects of an older one.
Students can also make the job of lecturer very unpleasant. On the
other hand, potential employers can, over a period of years, come to
believe that a university is producing less able students than it once
did, and thus become disinclined to hire its graduates. No individual
is harmed by this erosion of faith, but only the university as a
whole. So who do you expect a young professor to please?
Doug Moore
(do...@cs.rice.edu)
Robert D. Silverman
>>>>>> On Wed, 2 Dec 1992 00:53:22 GMT, b...@gauss.mitre.org (Robert D. Silverman) said:
>Bob> Why on Earth should a college professor teach a class as if it
>Bob> consisted of 3rd graders?
><text omitted>
>Bob> I believe that someone who has passed a college level course
>Bob> set of skills. How else is an employer to judge whether a
>Bob> student who has passed a given course really learned the
>Bob> material?
>
>I am inclined to agree. But a lecturer has every incentive to please
>students and no incentive to please potential employers. Students can
>complain to department chairmen and deans and write bad course
>evaluations and significantly damage the tenure prospects of a young
>professor, or the promotion and raise prospects of an older one.
>Students can also make the job of lecturer very unpleasant. On the
>other hand, potential employers can, over a period of years, come to
>believe that a university is producing less able students than it once
>did, and thus become disinclined to hire its graduates. No individual
>is harmed by this erosion of faith, but only the university as a
>whole. So who do you expect a young professor to please?
However, there are times when a professor is a truly bad teacher
and student gripes are legitimate. The problem is to separate the
poorly prepared whiners from the real students. Perhaps, in addition to
grades, a professor could turn in an evaluation on each student that
discussed the student's attitude and preparedness. Then the administration
could pair-correlate the students and professors evaluations. The trouble
is that most profs (especially in a large lecture course) don't get to know
their students well enough.
I can recall a class in Advanced Calculus at Harvard, covering vector/tensor
analysis, curvilinear coordinate systems, Fourier analysis etc., that was
taught by a professor who was TERRIBLE. [I won't name him]. His lectures
were poor and mostly incomprehensible. I recall one class on Cartesian
tensors when he spent 20 minutes covering 2 blackboards, then said: No, this
isn't right. He wiped it all out and started over, then damn if he didn't
do the same thing AGAIN, with different errors. Can you say poorly prepared?
The final was worth 350 points. The median grade was about 60 on that exam.
One of the grad student TF's said he could not do the exam in less than 7 hours.
The professor was overheard to say that one of his motives in teaching the
course was to chase half the students out of the Applied Math department,
because he felt they were unmotivated. Out of a class of about 80,
over 15 flunked and there was only 1 A. Now, you don't take a course like
that at Harvard without already having a pretty decent math background.
For 20% to flunk is extraordinary. Furthermore, the 2nd half of the course
was advanced topics in diffeq's. It was taught by a different prof, and
most of the students got A's.
Student evaluations are too potent a weapon in the hands of students who
don't give a shit to begin with. I suspect in such cases that their
evaluations would correlate strongly with their grades in the course.
Under such circumstances, evaluations are meaningless.
I would suggest doing away with them entirely, but as with that class at
Harvard, there are times when student complaints are legit. The problem
is for the administration to know when they are legit and when they simply
reflect a poor attitude or lack of preparedness on the part of the student.
Perhaps schools need to go back to the days when admission was based on
comprehensive entrance exams given by the school, rather than dubious
high school grades and SAT scores. This would winnow out many of the students
who lack the background, study ethic, and motivation to be in college.
It has gotten to the point where I won't even look at a resume of a student
unless that student is from an elite university. I have interviewed too many
computer science B.A.'s [often with high GPA's!] who can't think and have not
learned what they should.
Brian Harvey
>>>> b...@gauss.mitre.org (Robert D. Silverman) said:
>Bob> I believe that someone who has passed a college level course
>Bob> in (say) elementary discete math should have acquired a basic
>Bob> set of skills. How else is an employer to judge whether a
>Bob> student who has passed a given course really learned the
>Bob> material?
>
>I am inclined to agree. But a lecturer has every incentive to please
>students and no incentive to please potential employers.
I think that three different issues are being mixed together here.
Issue #1: YES, we should have high expectations for our students and
not water down what we're teaching.
Issue #2: NO, that doesn't mean we should disdain the use of analogies
and other pedagogic tools to help introduce new ideas. I don't think
that constitutes watering-down. It's only watering-down if we *stay*
at the level of analogies instead of expecting students to handle the
formal methods. (In principle you could write an advanced geometry
book with no pictures in it, since the pictures aren't part of the formal
proof, but nobody would do that.)
Issue #3: NO, I don't think we should consider ourselves as part of the
hiring process for employers. I know I'd never hire anyone just because
they have a college degree! I'd want to see what the person did that
indicates initiative and creativity. It's harmless if you interpret the
needs of industry as "send me someone competent"; but it's the first step
down the slippery slope to "send me someone who knows how to program in
COBOL" or "who knows how to use DBASE" or whatever the current ephemeral
demand is. Employers should run their own personnel departments.
Gary Martin
SHOULD NOT BE THERE.
I've been fantasizing about doing this more and more frequently of late,
but it's easier said than done. Many universities rely on tuition for
a large portion of their budgets, and tightening admission standards
and increasing failure rates would exacerbate an already dire financial
situation. So there is some institutional pressure for student
"retention" without the support needed to help these retainees
actually learn something.
Charles Lin
In article <1992Dec2.1...@linus.mitre.org>, b...@gauss.mitre.org (Robert D. Silverman) writes:
[part of article deleted]
>
>Student evaluations are too potent a weapon in the hands of students who
>don't give a shit to begin with. I suspect in such cases that their
>evaluations would correlate strongly with their grades in the course.
>Under such circumstances, evaluations are meaningless.
>
>I would suggest doing away with them entirely, but as with that class at
>Harvard, there are times when student complaints are legit. The problem
>is for the administration to know when they are legit and when they simply
>reflect a poor attitude or lack of preparedness on the part of the student.
I don't like student evaluations for a different reason. The profs.
almost never listen to them. Either you get the good prof., who is good
already, and who may actually try to respond to criticism, or the mediocre
one, who may try to listen, but may have a hard time getting better, or
the incompetent teachers (though not researchers), who often don't listen
to the criticism, and blame the students for not paying attention or working
hard enough. The profs. who teach the worst do so year after year, and
often with a disdain for the evaluation procedure. How many profs. who
are considered poor teachers, honestly attempt to improve their teaching,
when in the end, they feel students don't study as a whole, and so they
feel more satisfation with research.
In any case, how often has it been the case that student evaluations
determined the tenure of a faculty, and once tenure has been received,
what difference does the evaluation make. It should be the case that
profs. meet and perhaps the chairman asks how they intend to improve their
teaching for the upcoming year.
>
>Perhaps schools need to go back to the days when admission was based on
>comprehensive entrance exams given by the school, rather than dubious
>high school grades and SAT scores. This would winnow out many of the students
>who lack the background, study ethic, and motivation to be in college.
>
You know what's going to happen. Most likely such tests are going to
be considered putting too much pressure on students to perform well on
one day, are going to be considered biased (esp. if it is comprehensive
in all areas), and perhaps forcing a national curriculum on the US.
Then, there will be arguments as to what should be the content of this
test (though I doubt the math and sciences would be greatly affected).
While the test may in fact provide the means for improving the quality
of undergraduates, and while it may also remove the subjective decision-making
of an admissions committee (which has also been criticized), I'm sure
other criticisms will also arise to take its place.
>
>It has gotten to the point where I won't even look at a resume of a student
>unless that student is from an elite university. I have interviewed too many
>computer science B.A.'s [often with high GPA's!] who can't think and have not
>learned what they should.
>
Perhaps there needs to be a minimum standard of competency. I am inclined
to think that that will not be enough, as it is the ability to be inquisitive,
and question what you know, and then proceed to figure out those answers,
which eventually is the most valuable lesson to be learned from school.
However, these lessons are often not taught, and is something that is
expected to be learned anyway.
--
Charles Lin
cl...@eng.umd.edu
Charles Lin
Even if we were able to disregard tuition, and assume that schools
are somehow funded, this attitude bothers me. It seems to say, the
school system is screwed up, and so, let's just take the best students
that high schools have to offer, and forget the rest. Part of the problem
is a social one. That is, the US has never been terribly intellectual,
preferring the exploits of athletes to that of mathemeticians. While
education is often stressed, not enough people *really* believe in it,
to the extent they believe it in other countries, with Japan being
one of them. Of course, in a society like Japan, one might say that
education leads to social rewards, and so it is not necessarily love
of learning that motivates students, but fear of not having a job, or
not getting to the best universities, which lead to the best jobs.
While I think the videotape "Where There's a Will, There's an 'A' "
(which I've never seen) may be of dubious value, there is one comment
made on these 30 min. shows that I feel is relevant, which is when
John Ritter says that parents shouldn't just tell their kids to study
harder, they should give them the tools (of which I would guess part of
it is memorization, but...). So, if you expect students to be responsible,
start slowly by giving them the responsibility, and tell them what is
expected. If this should be done in high school, then fine, do it there.
I think it's unrealistic, in this day, to tell high school students,
that "Oh, by the way, even though you were never really told not to cruise
through high school, and never told about the minimum expectations of
college, and were admitted anyway, we're going to decide that we are
sick and tired of your unpreparedness, and yell at you for your
incompetentness", which almost strikes me as having a country that
takes in a bunch of refugees, then decides that they weren't well-trained
in their society, and will therefore be a burden, and how dare they think
of trying to emigrate to a country, when they have nothing to contribute
to that society.
If you want the quality students, I suggest either revamping the
high school system (and everything leading up to it), or providing
some outlines on what is expected of students, and make this
easily accessible, and also provide the means by which they might
accomplish this goal.
--
Charles Lin
cl...@eng.umd.edu
Steven E. Landsburg
Herman Rubin wrote:
>As for having an intuitive understanding of elementary statistics, I would
>prefer to keep people totally ignorant. The misuse of statistics by those
>who know what is called elementary statistics, and even much more statistical
>methods, and know this quite well, is ubiquitious and HAS done great harm.
>This is a case where what you know that ain't so can very definitely hurt you,
>and lots of others.
whereupon another poster wrote:
>I'm not sure I follow you here. Could you please give an example
>or two of the harm caused? Thanks a lot.
One dangerous fallacy I've seen repeated a lot lately (frequently in
the New York Times) goes like this: People
near the top of the income distribution have, on average, experienced more
recent gains than people near the bottom of the income
distribution. (Or, in more extreme versions, people near the top have
recently gained on average whereas people near the bottom have recently
fallen behind on average). Therefore (says the writer) income disparity
is widening.
This kind of thing seems to get taken completely seriously, but it is a
total non sequitur. People near the top of the distribution are,
disproportionately, people having unusually good years. We should
*expect* that a lot of them are doing better than last year,
just as we should expect that a lot of them are doing better than they
will *next* year. Likewise people at the bottom are disproportionately
people having unusually bad years; many of them are doing worse than last
year and many of them will be doing better again next year. The
statistical observations being quoted are *necessary* consequences of
the reasonable expectation that income has both a permanent and a
transitory component.
I think that it does a lot of harm when people are so easily fooled by
statistical fallacies in areas where public policies depend in part on
what people believe.
Steven E. Landsburg
land...@troi.cc.rochester.edu
Danny Breidenbach
Bob Silverman told us about sounds something like Bob --- the prof wanted
to chase half the students out of Applied Math because he felt they were
unmotivated. Bob seems to have the same attitude when he tells us that
instructors shouldn't bring the presentation on material down to the level
that the students we've got can understand. Both the Harvard prof in his
anecdote and Bob himself feel that the proper thing to do is bemoan the
state of affairs, keep the level of presentation at a unnecessarily
difficult level, and damn the students if they won't work hard enough to
keep up.
No offense, Bob, but I see that as a crappy attitude.
--Danny
Robert D. Silverman
>I don't want this to look too much like a personal attack, but the professor
>Bob Silverman told us about sounds something like Bob --- the prof wanted
>to chase half the students out of Applied Math because he felt they were
>unmotivated. Bob seems to have the same attitude when he tells us that
>instructors shouldn't bring the presentation on material down to the level
>that the students we've got can understand. Both the Harvard prof in his
the college freshman level and not at either the junior high school level,
or at the 2nd year grad student level. If college freshman can only be taught
at the junior high level, then they don't belong there. Nor should one expect
them to absorb a course taught at a level for second year grad students.
Both are extremes. And I was not complaining that the Harvard Prof was teaching
at too high a level. I complained because he could not TEACH. He did not
make himself understood to a class that had a great deal of mathematical
maturity. My example about the lecture on tensors was well pointed. It showed
that HE was not adequately prepared to teach such a class. I took this
class as a sophomore simultaneously with a first year grad course in complex
variables [taught by Alfors]. The latter course, although at a higher level
was MUCH easier to learn from because Alfors was a much better teacher.
>anecdote and Bob himself feel that the proper thing to do is bemoan the
>state of affairs, keep the level of presentation at a unnecessarily
>difficult level, and damn the students if they won't work hard enough to
complained that the professor was a lousy teacher.
I used it to illustrate the point that there are conditions where student
evaluations DO have meaning and are not sour grapes from underprepared
students.
As for keeping it at an unnecessarily difficult level, I still
maintain that the students *should* be damned if they are not ready
to be taught at the undergraduate level and one has to resort to
3rd grade "gimmiks" to teach.
Andrew Mullhaupt
>There are some applied fields where the pure theory fits more or less
>directly, because the underlying assumptions are sufficiently realistic
>(e.g. physics, computers, and parts of fluid mechanics).
I think the list is very much longer, and lengthening as every day passes.
You seem to have left out all of modern physics (Quantum theory, Relativity)
and there seems to be no end to the important applications of Quantum
theory. Then there are the huge parts of engineering science which are
essentially mathematical disciplines. (For example how else would you
describe signal processing?)
> There one can
>concentrate on developing the mathematics, without regard to the actual
>implementation. But that kind of work is done mainly in universities,
>anyway.
This is pretty naive about the utility of the process of mathematics
and pretty optimistic about how much of the world's problems can be
solved by dropping in existing techniques, and very much too optimistic
about the cost of computing approximations.
Keep in mind that there are many times more scientists and engineers in
the world than mathematicians, so that even if only one in a thousand
of them does any significant mathematics the result is significant with
respect to the output of mathematicians. The attitude you express seems
to me to be like the guy who went and asked his friend the chemist
if anyone ever proves any theorems in chemistry. The chances that he knows
the correct answer (Yes) are pretty small.
(Yes is the correct answer - for example Walter Stockmeyer in combinatorics.)
Think about all the work in self-avoiding random walks. This stuff is
in some large part a problem in polymer chemistry, and there are very
important applications. The idea that chemists will be largely unable
to contribute to the mathematical understanding is silly.
The other major contributing factor to this attitude is that there is
no lack of bad examples of (for example) chemists who publish what they
_think_ are contributions to mathematics in non-mathematical journals.
This does tend to cloud the issue, but I am counting these guys in the
999 out of a thousand who do not make useful contributions to mathematics.
Later,
Andrew Mullhaupt
Richard Bielak
[...]
>
>Teach the use of variables, the notions of function and relation, formulation
>of word problems, logic, the structure of the integers and the reals, and how
>operations fit into that. A first-grader can understand the Peano Postulates.
>
>Show them that intuition, while useful, is also dangerous, and that careful
>proof is needed. However, point out that there are times when they will
>have to accept the idea that someone else has found a proof.
>
To do this you need teachers that know and love mathematics. Instead
you get people who get an education degree, take a calculus course and
then teach high-school mathematics (a friend of mine is such person,
and she teaches in a public school).
Should someone who hasn't heard of Shakespeare teach high school
English?
...richie
--
* Richie Bielak (212)-815-3072 | *
* Internet: ric...@bony.com | Rule #1: Don't sweat the small stuff. *
* Bang {uupsi,uunet}!bony1!richieb | Rule #2: It's all small stuff. *
* - Strictly my opinions - | *
Richard Bielak
>In article <drector....@math.uci.edu> dre...@math.uci.edu (David Rector) writes:
>>I am happy to see more experimentation in the curriculum; indeed,
>>experimentation is very helpful in teaching reasoning. But synthetic
>>geometry is the ONLY venue in the current high school curriculum
>>suitable for introducing proof technique, and formal proof technique
>>is absolutely essential in all of mathematics and its applications
>>(e.g. computer science). Proofs in algebra are mostly too
>>difficult--because the facts are two obvious--for an introduction to
>>axiomatics.
>
>What about group theory? (Of course this may not be in the
>current American high school curriculum, though it was in mine,
>in the Engish system.) It is surely ideal for introducing the
>axiomatic method: a concise set of axioms, plenty of examples
>lying around, and the pupils are not going to be confused by
>the fact that the results are `obvious'.
>
That's sounds like a good idea. My favorite example of a finite group
is the Rubik's Cube (or more precisely the group of side rotations...).
Tal Kubo
(Andrew Mullhaupt) writes:
>In article <1992Dec1.1...@husc3.harvard.edu>
ku...@boucher.harvard.edu (Tal Kubo) writes:
>
>>There are some applied fields where the pure theory fits more or less
>>directly, because the underlying assumptions are sufficiently realistic
>>(e.g. physics, computers, and parts of fluid mechanics).
>I think the list is very much longer, and lengthening as every day passes.
The list was meant to be illustrative, not comprehensive. I intended
to point out what I consider a misleading feature of the "plenty of
other examples" (of mathematically interesting applied fields) alluded to
by R. Frey in his earlier post: they are from very obviously-mathematical
fields, where the techniques have matured, and the fit between theory
and reality is fairly well understood. In such a situation, there arises
a professional distinction between those who invent the mathematical
techniques and those who use them. The former are just ordinary
mathematicians, who happen to work on an applicable specialty. They
might carry a different title like "physicist", or "cryptographer", but
such titles thinly disguise what amounts to the pure/applied dichotomy all
over again. People receiving degrees in mathematics are well aware of, and
pursue, jobs of this type; the question was whether the other sort of
job represents an unmined lode of mathematically interesting work.
>You seem to have left out all of modern physics (Quantum theory, Relativity)
>and there seems to be no end to the important applications of Quantum
>theory.
I'm afraid I don't catch your meaning. See caret ('^') signs above.
>Then there are the huge parts of engineering science which are
>essentially mathematical disciplines. (For example how else would you
>describe signal processing?)
I used to work in (a subfield of) signal processing, but I'd hardly recommend
it to someone who intends to do mathematics. With a few limited exceptions,
researchers in signal processing are not developing mathematics, although
their accumulated experience may well generate intuitions which can be
exploited in harmonic analysis and elsewhere. Using mathematics is not
the same as doing mathematics.
>> There one can
>>concentrate on developing the mathematics, without regard to the actual
>>implementation. But that kind of work is done mainly in universities,
>>anyway.
>
>This is pretty naive about the utility of the process of mathematics
>and pretty optimistic about how much of the world's problems can be
>solved by dropping in existing techniques, and very much too optimistic
>about the cost of computing approximations.
The same comments apply. There may be a considerable distance to go
between the model and reality, but is the "mathematical engineering" that
bridges the gap really interesting mathematics (as Herman Rubin pointed
out, often it's not even mathematics)? I ask again of those who deny the
depth of the pure/applied split, for convincing examples of problems likely
to arise in the sort of applied math work available in industry, which
would support the claims made earlier that this type of work is at least as
mathematically interesting as anything pure math has to offer.
>Keep in mind that there are many times more scientists and engineers in
>the world than mathematicians, so that even if only one in a thousand
>of them does any significant mathematics the result is significant with
>respect to the output of mathematicians. The attitude you express seems
>to me to be like the guy who went and asked his friend the chemist
>if anyone ever proves any theorems in chemistry. The chances that he knows
>the correct answer (Yes) are pretty small.
>
>(Yes is the correct answer - for example Walter Stockmeyer in combinatorics.)
>
Soddy, of hexlet fame, was a chemist, and I've seen references to
mathematical theorems first published in biology journals. So what?
Would you recommend, say, chemistry, as a career for a recent PhD in math
who'd like to do mathematics? How many jobs like Stockmeyer's are really
available, where (for example) a chemistry department or chemical company
would pay someone to do combinatorics? The lack of awareness that you
mention, where people don't even know that math is done in their field,
means that a recent PhD in math would have a hard time finding such a
job, even if he tried and even if he were not looking for a research
position. This sort of narrow-mindedness is a universal problem, which
limits job prospects for many people, mathematicians or not (if anything,
mathematicians are in a better position to switch fields than most others).
>Think about all the work in self-avoiding random walks. This stuff is
>in some large part a problem in polymer chemistry, and there are very
>important applications. The idea that chemists will be largely unable
>to contribute to the mathematical understanding is silly.
Yes, and there are other obvious examples where intuitions and methods
developed in other (not obviously mathematical) fields can lead to
significant mathematical results. Minimal surfaces and solitons come to
mind. To strike closer to home, it's not inconceivable that the experience
of certain people working in finance, whose methods involve computing
averages over huge numbers of random walks, might lead to advances in
computing path integrals (here's your chance to become the first applied
mathematician to win a Fields Medal!). Once again, so what? How many
positions do you think are available, in industry, where one can do
research on such problems, as opposed to just providing the grist for
other people's mathematics?
Herman Rubin
>In article <DOUGM.92D...@titan.cs.rice.edu> do...@titan.cs.rice.edu (Doug Moore) writes:
>>>>>>> On Wed, 2 Dec 1992 00:53:22 GMT, b...@gauss.mitre.org (Robert D. Silverman) said:
>>Bob> Why on Earth should a college professor teach a class as if it
>>Bob> consisted of 3rd graders?
>><text omitted>
One problem here is that, not necessarily their fault, that many of them
are third graders in mathematical understanding. The days when it could
be assumed that a college student had a high school algebra course which
paid reasonable attention to anything but manipulation, and a high school
geometry course which placed reasonable emphasis on proving theorems and
reasoning are gone NOW. But I do not know of any university which is
willing to tackle this problem head on, provide remedial courses on the
basis that the student is ignorant but not stupid, and proceed accordingly.
>>Bob> I believe that someone who has passed a college level course
>>Bob> in (say) elementary discrete math should have acquired a basic
>>Bob> set of skills. How else is an employer to judge whether a
>>Bob> student who has passed a given course really learned the
>>Bob> material?
>>I am inclined to agree. But a lecturer has every incentive to please
>>students and no incentive to please potential employers. Students can
>>complain to department chairmen and deans and write bad course
>>evaluations and significantly damage the tenure prospects of a young
>>professor, or the promotion and raise prospects of an older one.
>>Students can also make the job of lecturer very unpleasant. On the
>>other hand, potential employers can, over a period of years, come to
>>believe that a university is producing less able students than it once
>>did, and thus become disinclined to hire its graduates. No individual
>>is harmed by this erosion of faith, but only the university as a
>>whole. So who do you expect a young professor to please?
All of this happens to be true. The universities have too much succumbed
to the philosophy of the educationists that one must teach a course at the
level of those who happen to be placed in it, and understandable by them.
The grade system exaggerates the problem; I know of honors students who
drop a course because not getting an A or a B might remove them from the
ranks of honors students. And the idea of taking a course with a reasonable
chance of failure because you want to LEARN has probably always been frowned
upon.
>One solution is to stop having the students rate their professors.
>However, there are times when a professor is a truly bad teacher
>and student gripes are legitimate.
....................
>I can recall a class in Advanced Calculus at Harvard, covering vector/tensor
>analysis, curvilinear coordinate systems, Fourier analysis etc., that was
>taught by a professor who was TERRIBLE. [I won't name him]. His lectures
>were poor and mostly incomprehensible. I recall one class on Cartesian
>tensors when he spent 20 minutes covering 2 blackboards, then said: No, this
>isn't right. He wiped it all out and started over, then damn if he didn't
>do the same thing AGAIN, with different errors. Can you say poorly prepared?
>The final was worth 350 points. The median grade was about 60 on that exam.
>One of the grad student TF's said he could not do the exam in less than 7 hours.
The question I must ask is how well the students learned what they should
have learned. The last two statements do not bother me in the least; all
my exams are like that. What do you think happens on our PhD written exams?
I do not expect a student in a strong course to be able to do that well on
a short exam.
....................
>Student evaluations are too potent a weapon in the hands of students who
>don't give a shit to begin with. I suspect in such cases that their
>evaluations would correlate strongly with their grades in the course.
>Under such circumstances, evaluations are meaningless.
........................
>Perhaps schools need to go back to the days when admission was based on
>comprehensive entrance exams given by the school, rather than dubious
>high school grades and SAT scores. This would winnow out many of the students
>who lack the background, study ethic, and motivation to be in college.
I do not believe that this was ever the case. Nor does even an elite
university have the resources to do so. Now there was a time when New
York State required Regents' exams for high school graduation; I believe
that they are available now but not required. But I doubt that anything
other than an SAT-type exam, possibly with an essay for English composition,
was done by the schools themselves. However, at least the better schools
maintained course standards.
>It has gotten to the point where I won't even look at a resume of a student
>unless that student is from an elite university. I have interviewed too many
>computer science B.A.'s [often with high GPA's!] who can't think and have not
>learned what they should.
I have seen this even in students from quite good universities. The problem is
that students A and B can have virtually identical resumes, and A knows nothing,
while B is really good.
Herman Rubin
>
>In article <MARTIN.92...@lyra.cis.umassd.edu>, mar...@lyra.cis.umassd.edu (Gary Martin) writes:
>>In article <1992Dec2.0...@linus.mitre.org> b...@gauss.mitre.org (Robert D. Silverman) writes:
>> Let us stop teaching to the lowest common denominator and start demanding of
>> students that they take some responsibility for their own education, rather
>> than expecting to be spoon fed. In other words, let us start demanding a
>> better "study ethic" and less of a party attitude from students.
>>I've been fantasizing about doing this more and more frequently of late,
>>but it's easier said than done. Many universities rely on tuition for
>>a large portion of their budgets, and tightening admission standards
>>and increasing failure rates would exacerbate an already dire financial
>>situation. So there is some institutional pressure for student
>>"retention" without the support needed to help these retainees
>>actually learn something.
> Even if we were able to disregard tuition, and assume that schools
>are somehow funded, this attitude bothers me. It seems to say, the
>school system is screwed up, and so, let's just take the best students
>that high schools have to offer, and forget the rest. Part of the problem
>is a social one. That is, the US has never been terribly intellectual,
>preferring the exploits of athletes to that of mathemeticians. While
>education is often stressed, not enough people *really* believe in it,
>to the extent they believe it in other countries, with Japan being
>one of them.
It is not clear that even the American anti-intellectual attitude would
have that effect, if the extension of this to the schools was not fostered
by the present school of educational philosophy. Before they got into
power starting about 60 years ago, the country was just as anti-intellectual
as now, if not more so, but there was at least a strong attempt in much of
the country to have an honest curriculum, and to maintain standards. There
was a separate college-preparatory curriculum in high schools; even in
elementary school, those who could not keep up were held back; there was
more of an effort to advance those who could; and while concepts were not
well taught, they were tested to a large extent.
Equality was equality of OPPORTUNITY, not equality of RESULTS. This is all
that the immigrant groups asked for, and while they did not always get it,
they took advantage of it when present.
Herman Rubin
There is a major difference. The prof Bob was criticising was doing a
poor job of teaching the course at the level at which the course was
intended. If a curriculum is to be meaningful, that must be done.
It is improper to use material in a course which someone who understands
the prerequisites does not know. But this is quite different from
teaching at the level of ignorance and stupidity of those who happen
to have been placed in the class. This ignorance is often due to those
who follow Mr. Breidenbach's suggestion to put the course at the level
of the students who happen to be there.
It is not a matter of how hard students work. "You cannot make a silk
purse out of a sow's ear." Hard work cannot overcome much lack of
ability, or the lack of background which we find too often. Prerequisites
should be stated in terms of knowledge, not credits, and adhered to. But
adjusting a course to the level of the students is why the students in
the elementary and secondary schools only learn garbage, and not even
much of that.
Herman Rubin
>In article <drector....@math.uci.edu> dre...@math.uci.edu (David Rector) writes:
.....................
>So far as I am aware, students entering universities in Europe
>to study mathematics, science and engineering are all expected
>to have basic familiarity with basic calculus before they enter
>university. It would surprise me if any of my teachers had
>regarded it as a `waste of time'. Certainly English students
>of my generation had two years of calculus behind them before
>entering university. From what I can gather, American universities
>appear to take up to two years merely to teach the sort of topics,
>like basic calculus, that other countries succeed in covering
>in (high) school.
The usual "basic calculus" course is 99% calculus manipulations,
and has nothing to do with basic calculus. Learning manipulations
may be useful, but it has nothing to do with understanding.
What should be taught as basic calculus is the understanding of the
notions of continuity, limit, and derivative; the general notion of
measure and integration, not restricted to linear measure; and the
ability to use these notions to formulate and indicate lines of attack
for the solution of associated problems. Calculating the derivatives
and integrals by the mechanical use of formulas, which is what is
emphasized, is far from basic anything. Basic calculus, as I have
described it, is not usually taught except to SOME mathematics majors.
Herman Rubin
>In article <ByA50...@mentor.cc.purdue.edu> hru...@mentor.cc.purdue.edu (Herman Rubin) writes:
>[...]
>>Teach the use of variables, the notions of function and relation, formulation
>>of word problems, logic, the structure of the integers and the reals, and how
>>operations fit into that. A first-grader can understand the Peano Postulates.
>>Show them that intuition, while useful, is also dangerous, and that careful
>>proof is needed. However, point out that there are times when they will
>>have to accept the idea that someone else has found a proof.
>To do this you need teachers that know and love mathematics. Instead
>you get people who get an education degree, take a calculus course and
>then teach high-school mathematics (a friend of mine is such person,
>and she teaches in a public school).
At least who know mathematics.
>Should someone who hasn't heard of Shakespeare teach high school
>English?
Teachers who cannot read Shakespeare do. Having heard of Shakespeare is
not the problem. One of my former colleagues, a professor of English,
claims that most of the high school English teachers cannot even read
the King James Bible, let alone the more difficult English of Shakespeare.
Randy Crawford
In article <1992Dec2.1...@galileo.cc.rochester.edu>, land...@troi.cc.rochester.edu (Steven E. Landsburg) writes:
>
> Herman Rubin wrote:
>
> >As for having an intuitive understanding of elementary statistics, I would
> >prefer to keep people totally ignorant. The misuse of statistics by those
> >who know what is called elementary statistics, and even much more statistical
> >methods, and know this quite well, is ubiquitious and HAS done great harm.
>
> whereupon another poster wrote:
>
> >I'm not sure I follow you here. Could you please give an example
> >or two of the harm caused? Thanks a lot.
>
> One dangerous fallacy I've seen repeated a lot lately (frequently in
> the New York Times) goes like this: People
> near the top of the income distribution have, on average, experienced more
> recent gains than people near the bottom of the income
> distribution. (Or, in more extreme versions, people near the top have
> recently gained on average whereas people near the bottom have recently
> fallen behind on average). Therefore (says the writer) income disparity
> is widening.
>
> This kind of thing seems to get taken completely seriously, but it is a
> total non sequitur. People near the top of the distribution are,
> disproportionately, people having unusually good years. We should
> *expect* that a lot of them are doing better than last year,
> just as we should expect that a lot of them are doing better than they
> will *next* year. Likewise people at the bottom are disproportionately
> people having unusually bad years; many of them are doing worse than last
> year and many of them will be doing better again next year. The
> statistical observations being quoted are *necessary* consequences of
> the reasonable expectation that income has both a permanent and a
> transitory component.
If you're trying to say that those who had $1,000,000 in the bank at 3%
earned more last year than someone who had $1000 in the bank, I agree.
However, as you phrase it, your objection doesn't hold water in either
of the cases you mention, that is, if the measure of a good year were
based on on someone's rate-of-change-of-income or on his/her net financial
worth, as I believe it is.
In the first case, the people who have seen their net worth change rapidly
usually have low net financial worth before and after the change, not high.
In coming off welfare or finding a job after living on unemployment, one's
increase in one's rate-of-change-of-income is enormous. However, those
who have high net worth and make millions more (or millions less) in one
year will not see their rate-of-change-of-income change appreciably, and
they will remain "near the top of the income distribution". Clearly, the
press has been describing the rich as doing well, not the poor.
In the second case, if someone at the top of the income distribution sees
their net financial worth grow more rapidly than the national average,
then either their investments are growing at a better rate than the 3%
I'm getting in the bank, or their 3% is more than my 3% because they have
more principal in the bank and they earned more interest on it.
The latter case isn't worth reporting, and I don't believe anyone in the
press is dull enough to try. But the former case is news. If more wealthy
people became wealthier, or more people became wealthy, or wealthy people
became wealthier at a rate greater than I am, then that is worth reporting.
In each case it means the gulf between their wealth and mine is widening.
Therefore, the rich are getting richer and the poor (or relatively poor,
like me) are getting poorer.
I'd say the press did at least as good a job as you did in interpreting
those statistics.
--
| Randy Crawford craw...@mitre.org The MITRE Corporation
| 7525 Colshire Dr., MS Z421
| N=1 -> P=NP 703 883-7940 McLean, VA 22102
Jeffrey Wayman
>>
>> I'm new to this newsgroup, so I apologize if someone has already
>> suggested this......I've had success teaching these type of
>> concepts by actually assigning "events" to A, B, C, whatever
>> (ex: A=you like football, B=you like all sports). I then create
>> a "population" of students by actually bringing 20 or so to the
>> front of the room and moving the groups of students or "events"
>> around to suit the lesson. You could easily illustrate your
>> problem this way; the trick is to plan out the events so they
>> are easy to understand. It's a real "3rd-grade" method, but it
>
>Why on Earth should a college professor teach a class as if it consisted
>of 3rd graders? IMHO, if the students are not up to learning the material,
>or lack the maturity to deal with any kind of abstraction, then they
>
>SHOULD NOT BE THERE.
Please read the post more carefully. 3rd grade is in quotes. In no way
did I even remotely indicate that I taught the class as if it consisted
of 3rd graders. That's disrespectful to the students; they catch on
to that and get real surly real quick. The point was the illustration
I used was not very
complicated but it got the message across and provided another way to
visualize the concept. Another poster made a good point that sometimes
it helps to use a simple example to provide a solid base so the student
can understand more complicated concepts more efficiently.
This has spawned a disturbing thread about the lack of knowledge our
students have. For the most part, this is a convenient excuse.
All you folks that have bemoaned the attitudes, knowledge and ability
of our students: forget about it. Put it behind you. There is
absolutely nothing you can do about the knowledge they have when they
come into your class. You can, however, teach them as much as you can
in your class. Let them know what level of knowledge you expect them
to have, teach to that level and teach them with respect. I promise
you if your expectations are fair that they will rise to that. Also,
you'll enjoy teaching the class and the students will enjoy you.
My $.02
Jeffrey Wayman
Boy, I'm really glad to hear someone argue this way. I totally
agree with what you said in your reply. I'm sure you've noticed
that even a lot of statisticians don't understand the real use
of statistics.
The point you made is so intuitive and practically so easy --
wouldn't we be better off teaching stats in this manner than
keeping people "totally ignorant"?
Brian Harvey
>It is not clear that even the American anti-intellectual attitude would
>have that effect, if the extension of this to the schools was not fostered
>by the present school of educational philosophy. Before they got into
>power starting about 60 years ago, the country was just as anti-intellectual
>as now, if not more so, but there was at least a strong attempt in much of
>the country to have an honest curriculum, and to maintain standards.
Really, now, this thread, which was never very serious to begin with,
seems to be degenerating to the level of "water fluoridation is a
Communist plot!"
In 1900, only 6% of the US population completed high school, let alone
college. The other 94% were not considered failures or unemployable;
our economy was largely based on manufacuring or farming and the
technology involved was unsophisticated.
Under those circumstances it is very easy to talk about "maintaining
standards" -- no effort on the part of the teachers is required to
do the maintaining.
In the early years of the century, we also had an *expanding* economy,
so those old immigrants you're so fond of could reasonably expect to get
a job if they worked at it.
Today we have a technological base that requires sophisticated workers
for almost all jobs, and today we have the sort of economy you'd expect
us to have after 12 years of deliberate encouragement of unemployment
as the strategy against inflation. These changes are neither the fault
of students nor the fault of progressive educators.
It's true that we could deal with the modern world the way Japan does:
We could have fiercely competitive exams for 10-year-olds that determine
the entire future course of the victims' lives. We could then devote
all of our educational resources to the winners, dump the losers into
custodial institutions, and tell ourselves what good teachers we are
because all of our students already know what we're supposed to teach them.
Instead we try to make it possible for everyone to learn as much as they
can. (By "we" I mean teachers -- more specifically, progressive teachers
who don't blame the victim -- not "society" in general, which exacerbates
the problems by condemning poor kids to schools without textbooks or heat.)
Steven E. Landsburg
interpretation of income statistics.
Maybe the point would be clearer in a context that is less
likely to elicit such an emotional response.
Imagine a community of nomads who spend their time randomly
wandering up and down a mountain. Take a snapshot of the
mountain. Now look at those nomads who are near the top in
your snapshot, and ask what percentage were walking uphill at
the moment when the snapshot was taken. It will probably be
a large percentage. Look instead at those nomads who are
near the bottom in your snapshot and ask what percentage were
walking *down*hill at the moment when your snapshot was taken.
Again, it will probably be a large percentage.
The explanation for the phenomenon is simply that nomads near
the top are likely to be there *because* they've been recently
walking uphill and those near the bottom are likely to be there
*because* they've been recently walking downhill. There is
absolutely no basis for an inference that the variance in nomads'
altitudes is increasing over time.
I leave it as an exercise for the reader to formulate this
in a precise manner, making reasonable assumptions about how
nomads choose their paths and using some elementary probability
theory.
I note also that the variance in altitudes might, on some mountains,
be increasing. I claim only that the statistics I quoted from the
snapshot provide no *evidence* for that assertion. Likewise, the
gap between rich and poor in the US might or might not be
increasing. What I claim is that the New York Times's observation
(that those currently near the top have recently gained and those
currently near the bottom have recently lost) is no *evidence* for
that assertion.
I hope not to have to explain on sci.math that it is possible to
demonstrate that reasoning is invalid without thereby being required
to take a stand on whether the conclusion is correct. I hope also
not to have to explain on sci.math why Mr. Crawford's personal
finances, which he invokes to make his point, are quite irrelevant
to the issue.
Steven E. Landsburg
land...@troi.cc.rochester.edu
Robert D. Silverman
>In article <1992Dec3....@linus.mitre.org> b...@gauss.mitre.org
>(Robert D. Silverman) writes:
>
>:As for keeping it at an unnecessarily difficult level, I still
>:to be taught at the undergraduate level and one has to resort to
>:3rd grade "gimmiks" to teach.
>
>whether they *would* be undergraduates. This is in the realm of
>admissions boards. Doesn't sound like a student problem in this
>context, but a problem with the college/university the student is
In a sense you are right, but if universities today only admitted
those ready to do the work and capable of doing the work, a large
fraction of students today would be unable to attend college.
However, you are also wrong in that there are too many students attending
who should not be because of ATTITUDE. They don't want to have to work
hard and want to avoid anything that might be difficult. Sciences and
math are hard subjects. This is why there is a preponderance of
non-US citizens in these fields within American Universities. Foreign
students tend to have much better attitudes and are willing to work
harder.
When you say they wanted to be undergraduates, it does not mean they
wanted to WORK like undergraduates should. Rather, what it means is that
they want to get a degree while doing as little work as possible.
"Ready to do the work" implies an interest in learning and a study ethic
that says that they are willing to work in order to learn. Too many
students today, are not.
>attending, and with the criteria used for admission. If the student
>has performed the necessary hoops and hurdles to be in the institution
>and is nevertheless not ready, where else would you point toward?
The students. There is no way a university can measure "attitude".
Let me also add a comment about students who lack preparation.
I see nothing wrong with a University providing remedial courses for
those students who lack the background to take courses taught at the
college level. Far from it. I encourage such things. However, such
courses should NOT count towards a degree. Furthermore, I believe that
students should NOT be allowed to muddle through such course. They should
be required to pass with a high grade [I would say at least B+] before
being allowed to continue with normal courses. I would say that those
who can't pass *remedial* courses with at least a B+ probably should
not be in college. Failure to get a high pass would indicate to me either
a lack of inate ability, or a lack of sincere interest in learning.
It must also be made clear to students that if they have deficiencies
and duck the remedial courses, that the regular courses are not going
to be watered down for them.
Unfortunately, as others have pointed out, this scenario is idealistic
and unrealistic. Schools are under too much economic pressure and need
the tuition too badly to turn away or flunk out such students.
So what is the solution? The only thing I can see is letting the
marketplace AFTER college weed out those who failed to learn what they
should. The diploma may get you a job, but there is no guarantee you
will KEEP it, should you prove to lack the skills your diploma says
you should have.
Let me off another alternative: Have "tracking" within Universities.
Provide both Honors and non-honors courses. The goof-offs can then
take the non-honors courses. A distinction should also be made on the
diploma. This should be a separate distinction from the "cum laude"
distinction made today. A student could get C's in honors courses and
not graduate "cum laude" and still be better off than one who took
the non-honors track.
Mike O'Connor
:As for keeping it at an unnecessarily difficult level, I still
:maintain that the students *should* be damned if they are not ready
:to be taught at the undergraduate level and one has to resort to
:3rd grade "gimmiks" to teach.
The students wanted to be undergraduates. The students didn't decide
whether they *would* be undergraduates. This is in the realm of
admissions boards. Doesn't sound like a student problem in this
context, but a problem with the college/university the student is
attending, and with the criteria used for admission. If the student
has performed the necessary hoops and hurdles to be in the institution
and is nevertheless not ready, where else would you point toward?
...Mike
--
Michael J. O'Connor | Internet: m...@fmsrl7.srl.ford.com
Ford Motor Company, OPEO | UUCP: ...!{backbone}!fmsrl7!mjo
20000 Rotunda, Bldg. 1-3001 | Phone: +1 (313) 248-1260
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