Why People hate science
Herman Rubin
>In article <CFrxC...@mentor.cc.purdue.edu> hru...@snap.stat.purdue.edu (Herman Rubin) writes:
......................
>>You have now given a positive testimonial to his teaching. The goal of
>>a course should not be to be able to do certain things at the end of
>>the course, but that the material be usable in the long term. The
>By this criterion, I can give "positive testimonials" to the teaching of many
>high school teachers who couldn't teach their way out of a paper bag.
>Not to be unduly snide or anything, but please reread the first sentence of
>the last paragraph. *During the course*, I actually had a rather poor idea
>of what was going on. I ended up doing well overall, but as far as I can
>tell this was short term memory at work here.
>It was only *after* I had some time to actually think, as opposed to simply
>try to keep up with an overenthusiastic lecturer, that the ideas began to
>make sense (as a whole, that is, rather than hastily remembered and easily
>forgotten bits and pieces). This step came after I had taken this class.
>>>Moral: Small, seemingly insignificant differences in presenting style
>>>make a *big* difference. Good lecturers pay attention to both signal
>>>strength AND impedance matching.
>>Good teaching is what gets the students to learn the material in such
>>a way that it will be retained and usable in new situations. Now can
>Exactly. At the end of the class, I had neither retained as much
>as I would have liked, nor was I as flexible with it as I would have liked.
>Certainly not as much as I am now. Any subsequent improvement was by my
>doing, not the professor's.
If you retained as much as you liked, or could use it as well as you would
like, the course would have been below your level. They usually are below
the level of a good student.
>>you say, after the above, that you could clearly identify it? Remember
>>that you learned a lot from that "bad" teacher.
>
>No, I *learned* a lot from studying and reviewing, which are independent of
>the lecturer's ability as a teacher. Good teachers make this approach
>both easier and more productive, of course.
When I was a student at the University of Chicago, there was a professor
who most of the students thought was going way over their heads and that
they were not getting it while they were taking the course. But when they
took the following course, they found that they could use the material
well, and even had gotten the ideas of the subject, so that the following
course was much easier.
--
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)
Herman Rubin
>In article <CFqoC...@mentor.cc.purdue.edu>
>hru...@snap.stat.purdue.edu (Herman Rubin) writes:
>
>> In article <CFqGI...@dartvax.dartmouth.edu> Benjamin...@dartmouth.edu (Benjamin J. Tilly) writes:
>> >In article <1993Oct29....@Virginia.EDU>
>> >jp...@Virginia.EDU (James Prescott Ruffner) writes:
>>
>> I will put my comments at the beginning, as good editing of the material
>> quoted is quite difficult. It is all here.
>>
>> There are several problems involved. I believe that the major one is
>> attempting to get understanding of the concepts and structure. Now we
>> really do not know how to teach it, and even before the curriculum
>> decline about all that could be said is that it was somewhat presented
>> and reasonably well tested.
>>
>> Now when I teach a course in mathematics or statistics at ANY level,
>> I attempt to emphasize the structure and concepts. Do not think that
>> these efforts are well received by the students; they clamor for the
>> instructor to tell them what will be on the exams. Far too many of them
>> are so thoroughly brainwashed well before they get to college that they
>> consider it "unfair" to expect them to do problems of a type different
>> from those explicitly presented in the course, or to have to put several
>> aspects together. These people, even if they become scientists, and
>> many do, seem to think that everything is in essentially non-communicating
>> compartments. Compartmentalization is quite useful, but only if what is
>> in one compartment can be used elsewhere.
>
>What are the structure and the concepts? From my experience a
>mathematician is likely to give a different answer from a physicist. In
>truth I think that the answer should lie in between. For the student
>who is learning the material the important concepts IMO should *not* be
>the formalisms that are used to justify the procedures, but the ideas
>that the procedures attempt to let them deal with.
The formalism is rarely the concepts, and this confusion is far too
common. Also, concepts are only learned by being used, not by being
memorized. The student who can look at a concept and understand
"immediately" how to use it is already a potential mathematician.
Therefore I would be
>more happy with a group of students who remember that the integral that
>they run across in first year is a way to get at their notions of area,
>and who could give a reasonable explanation from that point of view why
>integration "undoes" differentiation, than I would be with ones who
>remember the entire formulation of the Riemann integral well enough to
>do that question on the test, but who have no idea *why* the integral
>would be defined that way.
A slightly redone Riemann integral captures the concept, but only it
more general measures are used. The concept is the sum of values
multiplied by measures, and limits obtained from that. Those who
learn that integral is antiderivative will have to have it beat
into them that area can be computed by integration, and never to
take that as the idea. But the special case of the length measure
obscures the basic concept, and as for choosing a partition for
approximation by a discrete measure, only the existence is needed.
When it comes to probability, there is no new basic concept needed;
expectation is automatically integration. But formalism should
never be the goal, and those not intending to be mathematicians
need not know the proofs.
The idea of approximating an area by counting squares, and getting
bounds by the inclusion principle, used to be presented in high school
geometry. This immediately generalizes to integral, and computing the
amount of a bill is much closer to the concept than merely the area
under a curve.
Nasser Abbasi
I want to throw this comment here.
from my experience in studying in US universities, is that the emphasis
seems to be on the quantities on the materials covered, weather students
really understand or not, do not seem to matter to the teacher, especially
the new, young teachers who go their PhD and are too eager to go out
show off to the students how much they know in that particular area,
so, in one quarter or one semester a huge large text is covered, one
chapter each lecture, the goal of the teacher is to get to the end
of the book before the end of the quarter because this is what the syllables
says to cover.
it is madness, students learn to memorizes how to do certain methods
without really knowing why because there is no time to learn why, just
do it, it works this way, and just to pass the exam and get that piece of paper
and get out.
I am not saying every course I took was like this, but many were.
I think these new PhD's who go to teach need to go first and learn
how to teach, and what teaching really means, and to learn what is
important is quality not quantity of material covered
It is better to learn 5 chapters and understand what is going on that
to cover 20 chapters and leave having little clue what you learned just so
the new teacher can show off to the head of the department how good they are.
Teaching in schools is becoming like a drive through burger places,
the important thing is how fast and how many burgers they can cook and throw
out of the window.
I have decided that the best way to learn is NOT to go to a university or
to a school, but to go to the public library instead. you learn more,
and it is much less expensive, actually it is free.
\nasser
John Novak
[...]
>Now when I teach a course in mathematics or statistics at ANY level,
>I attempt to emphasize the structure and concepts. Do not think that
>these efforts are well received by the students; they clamor for the
>instructor to tell them what will be on the exams. Far too many of them
>are so thoroughly brainwashed well before they get to college that they
>consider it "unfair" to expect them to do problems of a type different
>from those explicitly presented in the course, or to have to put several
>aspects together.
If I may, I think the 'unfairness' is less a matter of being
forced to think independantly than a matter of being forced to be
creative or inspired at a certain time and place.
One of my general complaints with the engineering curriculum at
Bradley Univeristy (as regards 500 and 600 level courses) is the
lack of emphasis on project, and the overemphasis of short
homework problems and one hour test grades.
If you want your students to display an ability to connect
concepts, give them enough time. Give a take home test, or a two
week project, if the class size and composition warrant.
MHO, of course.
>The content of calculus courses is decried below. Many of the professors
>have given up trying to fight what the students demand. What is criticized
>here will bring complaints by the engineering students, and even many of the
>physics and chemistry students. And it is difficult to teach concepts in
>courses to students who only know manipulation, and to whom the notions of
>function and variable are ancient Sumerian, and of no importance.
I'm an engineering student, and my _only_ complaint is that a one
hour, high pressure exam is not the right arena for me to display
any particular creativity. It is an adequate arena for me to
display my familiarity and competance at the material through
which I have already worked.
--
John S. Novak, III
dark...@camelot.bradley.edu
dark...@cegt201.bradley.edu
m...@waikato.ac.nz
[...]
>
> One of my general complaints with the engineering curriculum at
> Bradley Univeristy (as regards 500 and 600 level courses) is the
> lack of emphasis on project, and the overemphasis of short
> homework problems and one hour test grades.
>
> If you want your students to display an ability to connect
> concepts, give them enough time. Give a take home test, or a two
> week project, if the class size and composition warrant.
>
> MHO, of course.
>
[...]
>
> I'm an engineering student, and my _only_ complaint is that a one
> hour, high pressure exam is not the right arena for me to display
> any particular creativity. It is an adequate arena for me to
> display my familiarity and competance at the material through
> which I have already worked.
>
Presumably you will have no objection to paying the extra fees to
support the much more labor-intensive evaluation teaching and
evaluation methods that you favour?
--
Murray Jorgensen, Maths & Stats, U of Waikato, Hamilton, NZ [m...@waikato.ac.nz]
-------------------------------------------------------------------------------
Doubt everything or believe everything: these are two equally convenient strat-
egies. With either we dispense with the need for reflection./ Henri Poincare'
Cameron Randale Bass
Nasser Abbasi <abb...@star.enet.dec.com> wrote:
>
>I have decided that the best way to learn is NOT to go to a university or
>to a school, but to go to the public library instead. you learn more,
>and it is much less expensive, actually it is free.
What university, and what public library? I might have sympathized
if you had said 'University library'.
However, it is easy to become badly mislead when reading books
without the benefit of experience. Consider professors an
adjunct to your reading, and force them to teach you if you
are not satisfied. Ask questions...
dale bass
stephen voss
> In <CFqoC...@mentor.cc.purdue.edu> hru...@snap.stat.purdue.edu (Herman Rubi
>
> [...]
>
> >Now when I teach a course in mathematics or statistics at ANY level,
> >I attempt to emphasize the structure and concepts. Do not think that
> >these efforts are well received by the students; they clamor for the
> >instructor to tell them what will be on the exams. Far too many of them
Have you ever considered that many of your students(especially with
the lower level courses) are not math and science majors. The only
reason they are in the class is because they have to be. These are
students whose real goal is graduate school(law,business,etc.) and
cant get in without extremely HIGH grades and when you throw in unplanned
stuff it lowers their grades and they resent it.
Herman Rubin
The comments made here are completely relevant, but the poster has the
reasons all wrong. The pressure to follow a specific syllabus and
concentrate on memorization does not come from the faculty. New PhD's
have mostly been teaching assistants and told that this is the way it
is done.
But the biggest problems in remedying the madness you mention in your
third paragraph, with which I agree fully, comes from the students and,
in service courses, from the other departments. The students ask, demand,
clamor for "relevance" and "how do I do the problems on the tests."
Attempts to teach WHY are met with active hostility. The engineering
departments want the mathematics and physics and chemistry departments
to prepare the students to do the "standard" problems from those fields;
the physics department wants the calculus course to cover the manipulations
before they will occur in the physics course: everybody wants the first
course in statistics to teach the students how to solve an arbitrary
statistics problem, without having the slightest idea of what anything
means, etc.
I do not believe in following a syllabus, in that I will not do such-and-such
a topic on a given day, but if the material is not covered, what are they
going to do in the next course?
Herman Rubin
>In <CFqoC...@mentor.cc.purdue.edu> hru...@snap.stat.purdue.edu (Herman Rubin) writes:
>[...]
>>Now when I teach a course in mathematics or statistics at ANY level,
>>I attempt to emphasize the structure and concepts. Do not think that
>>these efforts are well received by the students; they clamor for the
>>instructor to tell them what will be on the exams. Far too many of them
>>are so thoroughly brainwashed well before they get to college that they
>>consider it "unfair" to expect them to do problems of a type different
>>from those explicitly presented in the course, or to have to put several
>>aspects together.
>If I may, I think the 'unfairness' is less a matter of being
>forced to think independantly than a matter of being forced to be
>creative or inspired at a certain time and place.
>One of my general complaints with the engineering curriculum at
>Bradley Univeristy (as regards 500 and 600 level courses) is the
>lack of emphasis on project, and the overemphasis of short
>homework problems and one hour test grades.
>If you want your students to display an ability to connect
>concepts, give them enough time. Give a take home test, or a two
>week project, if the class size and composition warrant.
You will get no arguments from me on this. Now how do we implement this?
In a low-level course, the cheating problem is bad enough on in-class
tests; take-home is essentially impossible. This is not something
which an individual teacher can do much about.
Herman Rubin
>In article <2b28fb$l...@cegt201.bradley.edu>, dark...@cegt201.bradley.edu (John Novak) writes:
>[...]
>> One of my general complaints with the engineering curriculum at
>> Bradley Univeristy (as regards 500 and 600 level courses) is the
>> lack of emphasis on project, and the overemphasis of short
>> homework problems and one hour test grades.
>> If you want your students to display an ability to connect
>> concepts, give them enough time. Give a take home test, or a two
>> week project, if the class size and composition warrant.
>> MHO, of course.
>[...]
>> I'm an engineering student, and my _only_ complaint is that a one
>> hour, high pressure exam is not the right arena for me to display
>> any particular creativity. It is an adequate arena for me to
>> display my familiarity and competance at the material through
>> which I have already worked.
>Presumably you will have no objection to paying the extra fees to
>support the much more labor-intensive evaluation teaching and
>evaluation methods that you favour?
If we want the teaching and the grades to mean anything, we must be
willing to spend the time grading the exams. If this means that the
students will not get their grades the next day, or even in the next
three days, so what? If something is worth doing, it is worth doing
well, and the students and society should get as meaningful evaluations
of what the students know and can do as we can manage.
Herman Rubin
>> [...]
I very definitely consider it. I am also quite aware that the only way
that they are likely to use the material, and many of them are, is to
formulate problems and interpret the answers that are obtained with the
aid of computers. Do the "real world" problems they will encounter follow
the routine in the textbook? Not at all! But I do not teach the course
to them as I would to science, and especially math, majors.
As for lowering their grade, how does this happen if all are graded the
same way? I grade according to standards for that particular course, as
I interpret them. Those who are taking a course, if they are honest
students, will try to acquire an understanding of that course.
Craig Graham
In article <1993Oct29.1...@kpc.com>, a...@kpc.com (Alberto Moreira) writes:
> Have you tried reading the book and spending some 2-3 hours
> a day on it ? Science - Math - they're just like learning
> to play a difficult classical instrument such as violin or
> cello. You can't possibly do it unless you put a lot of
> commitment and weight on it. And that's I suppose why a
> lot of people don't like science, and they don't like
> classical music either: it's very hard to conquer.
>
> -Alberto-
And like an instrument, maths & such are Talents. If you are great at them,
you will love them and find them easy. Few do. But people can be shown how to
appreciate them all the same, without having to slog it out. I love music,
but I don't pretend I could ever do it myself - this doesn't lessen my
appreciation of it. My talent for Maths is on a par with this - but I despise
maths as it has been forced on me at a level I could not comprehend.
Teaching people to appreciate, does not mean forcing them to hate a subject
^^^^^^^^^^
that they have no talent for, which is something which many lecturers & teachers
should bare in mind if they want more sympathy from Joe Public.
Craig Graham (Masters Degree, Microelectronics)
Craig Graham. /\ E-mail:Craig....@newcastle.ac.uk
BAe Space Systems || Phone : (UK) 091 4883098
Stevenage,England.|| Snail-mail: 2 Sun Street, Suniside, Newcastle, England.
-----------------/__\-----------------------------------------------------------
SPACE the final^^^^frontier. Seen it,done it,read the book,got the T-shirt....
Mike Dowling
you all think that people hate science? In fact, since most of you seem to be
actively involved in science, the whole theme seems to be rather paranoic!
My students, who are mainly maths and computer science students, seem quite
well motivated. Perhaps I am fortunate, as, if they did not like what I teach,
they are under no compulsion to attend.
I have taught engineering maths repeatedly over the years. They tend to be
quite a mixed bunch. My experience again is that they are mostly well
motivated, despite the fact that they are a captive audience. There are quite
a number present who don't want to persue a mathematical career, and many of
these have difficulties. In my experience, though, even these don't hate maths
and/or science. They generally recognise that maths and science have their
merits even if they don't share my enthusiasm for it.
-----------------------------------------------------------------------------
Dr. Michael L. Dowling (__) moocow.math.nat.tu-bs.de
Abteilung f|r Mathematische Optimierung (oo)
Institut f|r Angewandte Mathematik \/-------\
TU Braunschweig || | \
Pockelsstr. 14 ||---W|| *
D-3300 Braunschweig ^^ ^^ Ph.: +49 (531) 391-7553
Germany
on.do...@zib-berlin.de
----------------------------------------------------------------------------
James Prescott Ruffner
> Now when I teach a course in mathematics or statistics at ANY level,
> I attempt to emphasize the structure and concepts. Do not think that
> these efforts are well received by the students; they clamor for the
> instructor to tell them what will be on the exams. Far too many of them
I know exactly what you are talking about, and you are quite
right. Most students (I know a lot of my cohorts were this
way) were interested in learning by rote only what they were
going to be required to regurgitate on an exam, this hardly
seems a scholarly attitude to say the least.
> The content of calculus courses is decried below. Many of the professors
> have given up trying to fight what the students demand. What is criticized
I wouldn't say I was decrying the content of Calc. courses. In
fact, I'm very sorry I didn't work harder and learn more of the
things which were covered. No doing so only meant going back
and pulling out the textbook and learning it later when I
needed to use it. I am saying that I wish more of the
professors and grad. students who taught me in my first calc.
courses had made as much of an effort to explain/emphasize
structure and concepts as you seem to.
> 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)
Scott Ruffner
Ron Maimon
|>
|> Teaching in schools is becoming like a drive through burger places,
|> the important thing is how fast and how many burgers they can cook and throw
|> out of the window.
|>
|> I have decided that the best way to learn is NOT to go to a university or
|> to a school, but to go to the public library instead. you learn more,
|> and it is much less expensive, actually it is free.
|>
I heartily agree.
I would hate physics if I would have learned it in school.
Ron Maimon
Ron Maimon
|>
|> And like an instrument, maths & such are Talents. If you are great at them,
|> you will love them and find them easy. Few do. But people can be shown how to
|> appreciate them all the same, without having to slog it out. I love music,
|> but I don't pretend I could ever do it myself - this doesn't lessen my
|> appreciation of it. My talent for Maths is on a par with this - but I despise
|> maths as it has been forced on me at a level I could not comprehend.
|>
|> Teaching people to appreciate, does not mean forcing them to hate a subject
|> ^^^^^^^^^^
|> that they have no talent for, which is something which many lecturers & teachers
|> should bare in mind if they want more sympathy from Joe Public.
|>
Oh come on!
There's no such thing as talent.
Believe me, I know. I met a lot of supposedly talented people and they're dumb
as posts for the most part.
I was also called talented at math/physics at one point in my short life, but
I could remember the hours I spent at the physics library trying to understand
a concept I could just rattle off in class, and I got a lot humbler. I remember
when I was a kid, I didn't know _anything_, I had to learn it all, damn it, and
I didn't do it any faster then anyone else (I just seemed to 'cause I spent my
entire waking life learning physics)
There is no math talent, there is no physics talent. There is interest in math
or physics, or art for that matter, and there is luck.
But you will never find the "math genes"
Ron Maimon
MADIGAN KEVIN M
I took Abstract Algebra (in grad school) from a man that made us work damn
hard and demanded we see things his way. Most students didn't like that
approach, and dropped. Two years later, when I took my written Algebra prelim,
I found I didn't have to study very hard, because this man had taught me
all I really needed to know. Figuring out the rest was a piece of cake.
Josh Vander Berg
%Oh come on!
%
%There's no such thing as talent.
Oh, really?
%Believe me, I know. I met a lot of supposedly talented people and they're dumb
%as posts for the most part.
Really? I have met people like that also. I have also met people whose
ability to grasp complex concepts FAR exceeded my own, and not through any
hard work on their part.
%I was also called talented at math/physics at one point in my short life, but
%I could remember the hours I spent at the physics library trying to understand
%a concept I could just rattle off in class, and I got a lot humbler. I remember
%when I was a kid, I didn't know _anything_, I had to learn it all, damn it, and
%I didn't do it any faster then anyone else (I just seemed to 'cause I spent my
%entire waking life learning physics)
So everyone does it at exactly the same speed? There is no one on this
earth who is any faster at comprehending complex subject material than
anyone else? Hmmm... I KNOW that such is not the case.
%There is no math talent, there is no physics talent. There is interest in math
%or physics, or art for that matter, and there is luck.
And then there are people who just seem to "understand" math, who grasp
concepts intuitively. I have met people like this. They were not arrogant
asses trying to pass off their secret hard work as natural talent, they were
TRULY gifted.
%But you will never find the "math genes"
And I suppose genetics have absolutely nothing to do with intellectual
ability...
%Ron Maimon
-josh
//========================================================================\\
|| "Genius: the intelligence that knows ||Josh Vander Berg||Never stop ||
|| its frontiers" ||k08...@kzoo.edu||asking "why?"||
|| -Albert Camus ||KalamazooCollege|| Why? ||
\\========================================================================//
Daniel E. Platt
|> In article <2arpuf$7...@anaxagoras.ils.nwu.edu> cle...@ils.nwu.edu (John
|> Cleave) writes:
|> >$ In article <wa2iseCF...@netcom.com> wa2...@netcom.com (Robert Casey)
|> >$ writes:
|>
|> >$ Math is a wonderful subject. It has the most intrinsic beauty of
|> >$ of anything around. I teach quantum mechanics to chemists. I find that
|> >$ they don't seem to appreciate much of it ... probably because
|> >$ of a poor math background. It's really sad to try to teach
|> >$ such a beautiful subject, year after year, to a bunch of logs.
|>
|> >This is definitely _your_ problem. Math may be "beautiful" to you,
|> >but it is your job as a teacher to make your students enjoy it, which
|> >means showing them how it is relevant to and useful for them. The fact
|> >that you refer to them as "logs" makes me think you aren't taking this
|> >aspect of your job seriously enough.
|>
|> Let me agree with Cleave here and emphasize that the job of the teacher
|> is to figure out why his or her current approach seems to be failing and
|> do something about it. Insulting the students is a sure sign that the
|> teacher is doing his or job wrong. Certainly I can wish that my
|> students were better taught in high school, or were just more
|> intelligent and curious people (like me, the bad teacher mentally
|> adds), but if they are not, TOUGH. One has the students one has and
|> ones job is to teach THEM, not some fictional ideal students.
|>
|>
Well, as someone who has had several years in a broad range of courses,
I can say that there is a problem getting kids who are there via
required (compulsory) credits. I've done it successfully on some
occasions, but failed on others. The worst I've done was just to
help them get by doing something they really didn't want to do.
For example, I taught a modern physics course to chemistry, math,
and engineering majors, and that went over very well. None of them
expected it to be much fun; they rated it as the best course they'd
ever taken to date (I was *very* proud of that). I also taught
college algebra during the summer. That was *painful*. They just
weren't interested, and hoped to get by it without dieing.
The difference between the two groups was astounding. I tried the
same kind of things in both groups, but one went for it, and the
other didn't... Oh well...
I've come to a place where I can be sympathetic to having a bunch
of 'logs.' It literally happens that regardless of what you try,
you just won't get a response. It's worse if they don't bother showing
up (I had a few who did that... and then they tried to pass a final).
If they're not there, there's nothing you can do.
Dan
--
-------------------------------------------------------------------------------
Daniel E. Platt pl...@watson.ibm.com
The views expressed here do not necessarily reflect those of my employer.
-------------------------------------------------------------------------------
Alberto Moreira
>>> In <CFqoC...@mentor.cc.purdue.edu> hru...@snap.stat.purdue.edu (Herman Rubin) writes:
>
>>> [...]
>
>the routine in the textbook? Not at all!
"Real world problems" depend on the eye of the beholder. A partial
differential equation may look outlandish in the math class, but when
you meet it in Electromagnetic Theory or Fluid Mechanics
sometimes later you'd better not have treated it too antagonistically.
Math is a set of techniques and by definition must be general enough
to be usable by people from different paths of life. I'm into
computer science - I could easily make the point that calculus is
is completely irrelevant and that undergrad math courses should consist
of Discrete Math, Lattices and Number Theory. But like a math
teacher once told me, you don't learn this and learn that in a pointwise
way, you must have breadth of scope and master a variety of techniques.
The important thing is not to know what application you're gonna have
for math, but when you meet something that hasn't been met before,
you know what kind of math you're going to try so that your problem
becomes tractable; and when you need to read a paper or a book on
some aspect of your profession, you know math enough to at least be
able to read it.
If you concentrate in just teaching those things that your students
can see an application right in front of their noses, you're not
giving them what they need. Students don't have enough experience
to tell you what is or isn't useful, it's up to you to get a good
balance. "Real life" is as much inventing the next programming
language or developing the next wing shape as it is knowing your
odds in the lottery or balancing your checkbook.
When I was an electrical engineering undergrad, for four years
in a row my math curriculum was dominated by a very demanding
head of department. He got us to learn complex variables, group theory,
topology, logic and advanced geometry. A lot of people squirmed
like mad - "there's no real world application for this stuff!". Funny
enough, what he taught me gave me the basis for my professional
career. It allows me to open any book, read any paper, digest any
information, without being afraid of the math involved. It gave me
focus and became my scientific security blanket for life. I have no
application today for the calculus and the statistics and the trig
and all the other "real life" stuff I was taught, as I have for the
things that that guy taught us.
Bless him...
-Alberto-
Ron Maimon
|> rma...@husc9.Harvard.EDU (Ron Maimon) wrote:
|> %Oh come on!
|> %
|> %There's no such thing as talent.
|>
|> Oh, really?
really.
|>
|> %Believe me, I know. I met a lot of supposedly talented people and they're dumb
|> %as posts for the most part.
|>
|> Really? I have met people like that also. I have also met people whose
|> ability to grasp complex concepts FAR exceeded my own, and not through any
|> hard work on their part.
well, I guess youve been hanging around all these exceptional people, because
I have never met anyone whose ability to grasp complex concepts exceeded my
own by even one iota.
and, believe me, this has nothing to do with me.
|>
|> %I was also called talented at math/physics at one point in my short life, but
|> %I could remember the hours I spent at the physics library trying to understand
|> %a concept I could just rattle off in class, and I got a lot humbler. I remember
|> %when I was a kid, I didn't know _anything_, I had to learn it all, damn it, and
|> %I didn't do it any faster then anyone else (I just seemed to 'cause I spent my
|> %entire waking life learning physics)
|>
|> So everyone does it at exactly the same speed? There is no one on this
|> earth who is any faster at comprehending complex subject material than
|> anyone else? Hmmm... I KNOW that such is not the case.
Well, I know that it is the case that anybody who is born with a normal brain
can know physics as well as Glashow, Witten, Hawking, or any other physicist.
They just have better things to do with their time, and I can respect that.
|>
|> %There is no math talent, there is no physics talent. There is interest in math
|> %or physics, or art for that matter, and there is luck.
|>
|> And then there are people who just seem to "understand" math, who grasp
|> concepts intuitively. I have met people like this. They were not arrogant
|> asses trying to pass off their secret hard work as natural talent, they were
|> TRULY gifted.
I have _never_ met anyone like that. I have never seen anyone "intuit" math
through some oracle. At times, of course, I see someone solve a problem through
an ingenious trick, but this is not so impressive, since its usually a trick they
have seen someplace before, and the same result can almost always be gotten by
a more brute force approach, which is the way I solve things.
|>
|> %But you will never find the "math genes"
|>
|> And I suppose genetics have absolutely nothing to do with intellectual
|> ability...
I certainly don't think so.
Or at least, if they do, count me as one of the dumb ones.
because I know how hard I had to struggle to understand physics.
Ron Maimon
Jon Bell
Mike Dowling <mike@MooCow> wrote:
>I have taught engineering maths repeatedly over the years. They tend to be
>quite a mixed bunch. My experience again is that they are mostly well
>motivated, despite the fact that they are a captive audience. There are quite
>a number present who don't want to persue a mathematical career, and many of
>these have difficulties. In my experience, though, even these don't hate maths
>and/or science. They generally recognise that maths and science have their
>merits even if they don't share my enthusiasm for it.
Aren't university students in Germany a more "select" group than in the
USA, because of the limited number of Studienplaetze? Here, just about
any high school graduate can go to college _somewhere_. Perhaps not MIT
or Michigan, but if they have the money, they can get into someplace like,
well, here... :-) ... and if they don't have a lot of money, there are
many middle- to lower-rung state-supported colleges.
That might account for some of the difference in motivation between
"typical" American and German students.
--
Jon Bell <jtb...@presby.edu> Presbyterian College
Dept. of Physics and Computer Science Clinton, South Carolina USA
Mark Yeck
wa2...@netcom.com (Robert Casey) writes:
>I remember having to fight my way thru calculus classes. I used to like
>Math in high school, but I learned to absolutely hate it in college.
>The calculus profs seemed to go out of their way to make it hard to figure
>out. Especially those who thought showing a proof on the blackboard
>tells you anything useful to do test problems. Proofs don't mean sh*t
>to me. "I enjoyed math so much, I took it twice". I did make it to
>graduate with a BSEE, but I never felt too happy with calculus, in the
>sense that I would ever *want* to try to solve a problem by making up
>calculus equasions and solving them and feel that I didn't screw it up.
>I imagine that those who didn't survive to graduate in engineering
>and did graduate in something else would feel worse than I did with
>calculus.
I had the same experience in my math classes, enjoying them in high school,
and hating them in college. I actually learned very little in any of the
classes that I took in the math department. I did end up learning most of the
math from my engineering classes, though. I took differential equations twice,
but didnt learn them til we reviewed them in some electrical engineering class.
EE classes like Signals and Systems are very math intensive, but I didnt hate
them and actually learned stuff, but if they were taught by math professors,
i'm sure that i would have flunked them or something. Not necessarily a cut
on math profs. Maybe math majors understand them. who knows.
-mark
(my...@andrew.cmu.edu)
John Novak
>>Have you ever considered that many of your students(especially with
>>the lower level courses) are not math and science majors. The only
>>reason they are in the class is because they have to be. These are
>>students whose real goal is graduate school(law,business,etc.) and
>>cant get in without extremely HIGH grades and when you throw in unplanned
>>stuff it lowers their grades and they resent it.
>I very definitely consider it. I am also quite aware that the only way
>that they are likely to use the material, and many of them are, is to
>formulate problems and interpret the answers that are obtained with the
>aid of computers. Do the "real world" problems they will encounter follow
>the routine in the textbook? Not at all! But I do not teach the course
>to them as I would to science, and especially math, majors.
And here, I have to agree with Mr. Rubin.
Whether or not a student resents having to learn some material is
not an excuse for waiving the level of their understanding, nor
an excuse to dumb down the course.
Presumably, some 'wise' administrators somewhere have decided
that the courses are necessary for either general education or
specific major requirements. Presumeably, there are reasons for
this.
>As for lowering their grade, how does this happen if all are graded the
>same way? I grade according to standards for that particular course, as
>I interpret them. Those who are taking a course, if they are honest
>students, will try to acquire an understanding of that course.
Again, I agree.
Had the University not stipulated a non-Western Civilization
requirement for me (for example) I would not have taken a survey
course of eastern religions. However, once in the course, I did
not expect to be graded on a different caliber, or a different
expectation, than the serious RLS students.
Now, this analogy is slightly off, because the requirement I
filled with that course was general education, not
degree-specific. For a degree-specific requirement (like stats
for business and/or psych majors) I would agree that some changes
in the course motivation might be in order, but not the
introduction of a new standard for non-majors.
John Novak
>After reading so many contributions to this thread, I feel compelled to ask why
>you all think that people hate science? In fact, since most of you seem to be
>actively involved in science, the whole theme seems to be rather paranoic!
<Chuckle>
Look at the distribution line... :-)
I for myself give these reasons:
o I know, from experience, that a lot of people _don't_
like math, or science, and I believe that in large part, this is
learned behavior. Granted, its anecdotal evidence, but many of
the people I know are frighteningly innumerate, yet consider
themselves well educated. And the attitude toward math extends
even into some of the technical majors. A CS senior I know is
_constantly_ complaining about the math courses he is required to
take. (Which boggles my mind-- that's all computers _do_ is
math...!)
o Since I found, and started participating in, this
discussion, I've gotten mail from people telling me that any more
math is useless.
>My students, who are mainly maths and computer science students, seem quite
>well motivated. Perhaps I am fortunate, as, if they did not like what I teach,
>they are under no compulsion to attend.
[...]
You may have hit the nail on the head-- you teach technically
oriented people, who in general know the value of mathematics and
science.
However, the disdain for learning cuts into the engineering
departments as well. Most of my colleagues are openly annoyed at
the fact that they have to take the occasional history course, or
(God forbid) humanities course.
And I've heard a number of fellow students laugh off their
English and speech courses with the statement, "I'm an engineer--
I don't know how to write/speak publicly." I would silently
chuckle, do the work to the best of my ability, and I've landed
myself a part time assisstantship as a grad student. Because my
lab and project reports and memos are in readable, uncluttered,
clearly written English, and because I stand up in front of my
peers and professors, speak for a half an hour and answer
technical questions without _looking_ self-conscious, nervous, or
amateurish. (My skill at avoiding run-on sentences is less than
perfect, however... :-)
(My only complaints with most of my non-major requirements was
that they didn't go into enough depth.)
I wonder if your students are as well-motivated in non-major
classes. If they are, then you are a particularly lucky
instructor. I look forward twenty-five to thirty-five years, and
see myself teaching-- I hope my students are as enthusiastic as
yours.
John Novak
>Oh come on!
>There's no such thing as talent.
>Believe me, I know. I met a lot of supposedly talented people and they're dumb
>as posts for the most part.
I disagree with you, very strongly.
There _are_ people who have a talent or a knack for certain
subjects. This can certainly be augmented by, and sometimes
mistaken for, diligence.
I, personally, have a small amount of mathematical talent. I'm
no god, especially not by professional levels, but of my
classmates as an undergrad, I picked up the pure math concepts
more quickly than most other people. I also spent far less time
on the homework, and I got better grades.
But I have no ability for languages, other than English. I can
_learn_ other languages. At one point, I spoke German, and could
puzzle through written (and slowly spoken) Latin. I even got A's
in the classes (this was high school.) But I only learned them
by spending twice as much time as anyone else, and I can't
remember much of them today. An old acquaintance of mine,
however, used to pick up languages like nothing. A true
polyglot.
Anecdotal evidence, but she had a talent for languages.
I have a talent for math.
[...]
>But you will never find the "math genes"
I remember hearing a report, about a year and a half ago, which
tracked math ability in young students. It seems that
mathematical ability (and particularly three-dimensional
visualization and conceptualization) was strongly correlated with
the child's exposure to particular hormones (testosterone?) in
the womb.
If someone else remembers hearing the report, or better yet can
point me to it, that would be non-anecdotal evidence.
Alan Morgan
>In article <1993Nov1.2...@hobbes.kzoo.edu>,
> k08...@hobbes.kzoo.edu (Josh Vander Berg) writes:
>|> rma...@husc9.Harvard.EDU (Ron Maimon) wrote:
>|> %There's no such thing as talent.
>|>
>|> Oh, really?
>
>really.
Do you consider that to be true about physical ability also? If
so, do you really believe that you could have been a good a high
jumper as Javier Sotomayor? If not, why is mental ability so
different?
I have no trouble believing that Carl Lewis is just *faster* than
me and that Anthony Gatto is a better juggler than I will ever be.
Similarly I have no trouble believing that Donald Knuth is smarter
than I am.
>|> Really? I have met people like that also. I have also met people whose
>|> ability to grasp complex concepts FAR exceeded my own, and not through any
>|> hard work on their part.
>
>well, I guess youve been hanging around all these exceptional people, because
>I have never met anyone whose ability to grasp complex concepts exceeded my
>own by even one iota.
>
>and, believe me, this has nothing to do with me.
Maybe it has to do with the people you hang out with. I count myself as
a pretty bright guy all around and I have certainly met people who can
run little tiny mental circles around me. Not many, but they certainly exist.
>Well, I know that it is the case that anybody who is born with a normal brain
>can know physics as well as Glashow, Witten, Hawking, or any other physicist.
How do you *know* this? I haven't heard any final word from the
psychologist over the nature vs. nurture battle. Maybe you should
pop over to sci.psychology and inform them you have the answer.
>They just have better things to do with their time, and I can respect that.
I tutored someone in high school. She was really earnest and wanted
to do well because she was very close to not graduating because she
couldn't pass math. She (and I) spent several hours a day for the
last few weeks of the year going over algebra. You will no doubt
find this hard to believe but SHE JUST DIDN'T GET IT. It wasn't that she
wasn't interested or didn't want to spend the time on it, she just
didn't understand it. She finally passed with a C- and was overjoyed
to get that. I'm sure she would be happy to hear she could have been
the next Godel if she had just had the inclination.
>|> And then there are people who just seem to "understand" math, who grasp
>|> concepts intuitively. I have met people like this. They were not arrogant
>|> asses trying to pass off their secret hard work as natural talent, they were
>|> TRULY gifted.
>
>I have _never_ met anyone like that. I have never seen anyone "intuit" math
>through some oracle.
I presume you have heard of Ramanujan? Most mathematicians that I have
met consider him a "natural" mathematician. The people who he worked
with thought that of him. He certainly could "intuit" math and I don't
believe that just anyone can demonstrate that ability. Even if (by
some miracle) that was possible, I still think that Ramanujan would
become a Ramanujan^2 under those circumstances.
>|> %But you will never find the "math genes"
>|>
>|> And I suppose genetics have absolutely nothing to do with intellectual
>|> ability...
>
>I certainly don't think so.
You don't think genes have *anything* to do with intellectual ability?
I am prepared to believe that nurture plays a major role, but that nature
plays *none*?
>Ron Maimon
Alan
----
EFI agrees with me 100% on matters of fact. The above isn't and they don't.
-----> Mail abuse to: al...@efi.com <-----
Alias 'C Frog'. Keeper of the alt.tasteless song and part-time evil genius.
George M. Kierstein
>a...@kpc.com (Alberto Moreira) writes:
>
>that there are a lot of crummy books for science and math out there where
>the author assumes you have allready failed this class before and therefore
>have a general knowledge of it. It is up to the prof to make the class
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>enjoying and, failing that, at least innovative.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^. !!!!!!!!!!!!!!!!!
Unfortunatly this tells us as much about the reason that people
hate science as his comment. The responsibility lies with the student,
not the prof. If your E+M book sucks, find another (I suggest, found.
of Electromag thoery, by ____, ____, and christy (sorry forgot the first
two names) In any case your responsibilities as a student include
finding any way possible to learn, including going to class, and possibly
even *gasp* doing so many problems that they dribble out your nose. (by the
way, what would the force exerted by the drops be ?) As a wonderful prof.
of mine told me, Teachers are a public resource. A RESOURCE meaning that
you are to use them as an aid to teaching yourself the material. Mr Sutherland's point was that science must be practiced and practiced and practiced.
through problems and problems and then some reading to do more problems,
and perhaps a little thought... OK a LOT of thought, not just re-reading
things. If a teacher is boring to you well that is unfortunate, and if he
makes his class inovative then great, but those things are mainly asides to
where the learning is really done.
(Whew' now that that is out.. 8-)) -George
>
>
>
>--
> How many docs does it take to get to the center of AmigaDOS 3.0?
> The world may never know.
> The Fly Boy <| E-MAIL: tay...@hubcap.clemson.edu |>
>+--<| Activating peril-sensitive sunglasses! |>--+
Kouhia Juhana
>cle...@ils.nwu.edu (John Cleave) writes:
>>but it is your job as a teacher to make your students enjoy it
>
>is to figure out why his or her current approach seems to be failing and
>do something about it. Insulting the students is a sure sign that the
>teacher is doing his or job wrong. Certainly I can wish that my
>students were better taught in high school, or were just more
>intelligent and curious people (like me, the bad teacher mentally
>adds), but if they are not, TOUGH. One has the students one has and
>ones job is to teach THEM, not some fictional ideal students.
A few days ago it were posted following by <U27...@uicvm.uic.edu>
(Thaddeus L. Olczyk) which annoyed a bit me -- the above quotes does
save me in posting a reply. :-)
==clip==
5) ( I hope the numbers scrolled passed) With A students a teacher has little
effort. With B and C students you can push them into learning more and
make them into A and B students.
But with D and F students ( not all some fall into the B C category) you have
a problem. Basically they are idiots. I don't mean that they can't handle the m
aterial. I mean that they get up to go to the bathroom every night and each nig
ht they walk into the door. They rarely ask for help. They don't withdraw
from a course when they get 10 and the passing grade is 60 on a midterm, becaus
e they know they can do better on the final. etc...
these people take the fun out of teaching, and unfortunately they are growing i
n number. this is probably a result of our poor primary and secondary
educational system. Unfortunately the higher ups push us to go easy on these
people making them comfortable with their stupidity ( for example
pushing to lower the passing grade on exams ).
==clip==
So, we have good students and poor student, say. I have heard about
this problem since elementary school. The problem was tried to solve
by helding courses of different level.
But, helding courses of different level in material content is no no,
since everyone (at least math major) would like to learn the same stuff.
Since, those good students learns fast, I could try to teach them
separately and fast -- basically, good students hate slowness of the
lectures. And use more time for not good students; for example, more
example exercises.
Natural solution is to held reasonable fast lectures for all, 'explaned
example exercises'-hour for people not getting the stuff, and normal
home-exercise-hour for all.
Good students could get the extra exercises with solutions on paper
for selfstudy.
In above article, it is written: "they rarely ask help".
Well, I will not ask any help in front of tens of people, if there's
possible that the lecturer do not take the simple questions of "idiots"
seriously.
The point is that the above kind of teachers often overlook the
"idiots" and starts blaming or ignoring them.
Good way to solve the asking-problem is to create a local newsgroup
where students could ask any questions they want -- they probably get
the reasonable answers from the other students and staff people
instead of overlooking teachers.
In such newsgroup I have presented few exercises in higher or
different detail than they were presented in the home-exercise-hour
by individual students (corrected after by exercise-keeper).
Those, who read the newsgroup get the extra material offered.
Juhana Kouhia
Jim Carr
Then I feel sorry for you and your limited experience. I have known
several people with this talent, and it is a striking thing to observe.
In one case, it created quite a problem for a graduate student who was
set a proof as a thesis-related problem. The prof *knew* it was true
(and it proved to be true) but could not provide even a hint of guidance
as to how the problem should be attacked. He had not reasoned out any
part of it from some angle, he had seen the answer.
Too little is understood about thinking to consider whether this is
a result of an unconscious process of reasoning or some independent
pattern formation process. I might also add that one person I have
in mind was a truly awful teacher. You see, he could not explain
physics to anyone because his way of knowing it had little to do
with the sort of analogies or images that can be conveyed verbally.
You had to understand the material quite well before you could
talk about it with him.
--
J. A. Carr <j...@scri.fsu.edu> | "The New Frontier of which I
Florida State University B-186 | speak is not a set of promises
Supercomputer Computations Research Institute | -- it is a set of challenges."
Tallahassee, FL 32306-4052 | John F. Kennedy (15 July 60)
Craig Graham
In article <2b3j2n$d...@scunix2.harvard.edu>, rma...@husc9.Harvard.EDU (Ron Maimon) writes:
>Oh come on!
>
>There's no such thing as talent.
>
>Believe me, I know. I met a lot of supposedly talented people and they're dumb
>as posts for the most part.
>
I think you confuse talent & intelligence - the two do not necessarily go hand
in hand.
Craig.
MADIGAN KEVIN M
> of mine told me, Teachers are a public resource. A RESOURCE meaning that
> you are to use them as an aid to teaching yourself the material. Mr Sutherland's point was that science must be practiced and practiced and practiced.
> through problems and problems and then some reading to do more problems,
> and perhaps a little thought... OK a LOT of thought, not just re-reading
> things. If a teacher is boring to you well that is unfortunate, and if he
> makes his class inovative then great, but those things are mainly asides to
> where the learning is really done.
>
In grad school I always viewed the professors as tour guides, giving me
a very guided tour of the material, showing me where it is, what it does,
how to use it. It was up to me to take it in.
-
MADIGAN KEVIN M
>In article <2b3j2n$d...@scunix2.harvard.edu>, rma...@husc9.Harvard.EDU (Ron Maimon) writes:
>>Oh come on!
>>
>>There's no such thing as talent.
>>
>>Believe me, I know. I met a lot of supposedly talented people and they're dumb
>>as posts for the most part.
>>
>>Ron Maimon
You sir, speak from ignorance. Ever met Charles Fefferman?
Ron, you don't have all the answers. I have been to conferences and met
people so smart and so talented at Mathematics it almost makes one feel
inadaquate.
kevin
emil rojas
>In article <2b1v3s$s...@jac.zko.dec.com>,
>Nasser Abbasi <abb...@star.enet.dec.com> wrote:
>>
>>I have decided that the best way to learn is NOT to go to a university or
>>to a school, but to go to the public library instead. you learn more,
>>and it is much less expensive, actually it is free.
> What university, and what public library? I might have sympathized
> if you had said 'University library'.
> However, it is easy to become badly mislead when reading books
> without the benefit of experience. Consider professors an
> adjunct to your reading, and force them to teach you if you
> are not satisfied. Ask questions...
Along this line, while in college, I would refer to my professors
as "coach." I did this mostly with peers, but also with a few
professors that did not take themselves to seriously. For me, a
coach is someone that is helping me develop my own talents,
rather than someone imparting truth.
--
Emil Rojas em...@shell.portal.com There is truth,
(408) 973-0603 but no one knows what it is.
Cognisys -- Software Systems Development My opinions are my own,
San Jose, CA 95129-2205 nobody else works here.
Dave Batchelor, Space Phys. Data Facil. 301/286-2988
In article <2b403i$a...@scunix2.harvard.edu>, rma...@husc9.Harvard.EDU
(Ron Maimon) writes...
>
>Well, I know that it is the case that anybody who is born with a normal brain
>can know physics as well as Glashow, Witten, Hawking, or any other physicist.
>They just have better things to do with their time, and I can respect that.
>
>Ron Maimon
Excuse me, but you may believe such a thing, nevertheless you have no way to
know it. As a physicist myself, I have always wished it were true, but I
don't have anything solid to prove it. I feel certain that you know of no
controlled experiments establishing your assertion. While I think a good
explanation of a physics concept works wonders for someone struggling with
it, you are over-reaching to imagine that no inherent talent distinguishes
a Nobel prize winner from any other normal person. Motivation may be
absent among most normal people to pursue our career, but the existence
of gifted prodigies is well verified in the arts and sciences. I would
not dare to deny that Mozart possessed inherent talents for musical
composition that I lack, neither would I deny Hawking the recognition
that his genius deserves.
------------------------------------------------------------------------------
Dr. David Batchelor Space Science Data Operations Office Mail Code 632
NASA Goddard Space Flight Center Greenbelt MD 20771 USA
batc...@nssdca.gsfc.nasa.gov * personal opinions only, not NASA policy *
Theorem: Consider the set of all sets that have never been considered.
Hey! They're all gone!! Oh, well, never mind...
John Novak
>>>I sympathize quite a bit with the original poster. I've been in
>>>the same kind of math courses described, and remember feeling
>>>much the same way. One, in particular, was our third calculus
>>>course, which concentrated on vector calculus and multivariate
>>>calculus. The prfessor was a technically brilliant man, whom I
>>>have no doubt understood the content of the course (and far,
>>>far beyond!) But he could not teach it in a way which mattered.
>>>The course was filled with physicists, math majors, and
>>>electrical and mechanical engineers. He insisted on giving us
>>>lectures of proof after proof in class, yet providing us with
>>>numerical problems on the tests. This is _non helpful_ to people
>>>who don't already have a grasp of the material.
I wrote the above, so I will respond to the response.
>Are you sure that this is what happened? Now I agree that the
>presentation might have seemed formal, and it is quite possible
>that the proofs may have been overemphasized, but if anyone is
>to understand the use of mathematics, s/he must be able to take
>an idea and apply it to problems which are not like those worked
>out in class or in homework.
Had this professor displayed any tendency to demonstrate the
formalisms through numerical (or better yet, practical numerical)
examples, I would not be complaining. I agree whole heartedly
that every student of mathematics, even engineers, need exposure
to, and understanding of, the formalisms.
I do not think it is overwhelmingly unfair to give problems
slightly different than the material covered in the lectures.
I _do_ think it is unfair to give problems vastly different on an
hour exam. Mainly because, I don't _think_ that fast, when
running over unfamiliar mental terrain. I will try to associate
it with something familiar. I will make wrong turns and
mistakes, and likely have to go back to previous points and try
new things.
None of which I have time for when I am given five problems
of an unfamiliar nature to answer in fifty minutes.
> If he graded on the correctness of
>the numerical answer, which many do, that is bad, but if he graded
>on the application of the ideas, that is what should have been doen
>in elementary school and high school, and it is not his fault, but
>the fault of your previous teachers, that you found this to be
>even slightly unexpected. I suspect also that other kinds of
>manipulations were involved on the test; that much arithmetic
>would be difficult for anyone not a rapid calculator.
He did grade on correctness of the answer, but he also gave
partial credit, as well. Incidenatlly, I got a 'B' out of the
course. I would have gotten an 'A' had I not taken one of the
hour exams under very adverse personal conditions.
The following is no longer a response to my text. My own
following text is merely general commetary.
>You have now given a positive testimonial to his teaching. The goal of
>a course should not be to be able to do certain things at the end of
>the course, but that the material be usable in the long term. The
>problem of prerequisites is also a major one; at least in mathematics,
>I cannot trust anything on the student's transcript. Possibly your
>teachers did too much, and in trying to make it more palatable to you,
>left out too much.
Hmm.
I think I agree, in general.
However, examinations are typically geared to measure what can be
performed and what is understood _right now_. The grades, of
course, reflect that. And every student in the world is
concerned with his grades, out of sheer practicality.
>I remember reading about a small college which one year turned out a
>large number of students who did quite well in graduate work in physics.
>The only explanation found was that the instructor of the senior honors
>physics course was ill most of the year.
<Chuckle>
>Good teaching is what gets the students to learn the material in such
>a way that it will be retained and usable in new situations. Now can
>you say, after the above, that you could clearly identify it? Remember
>that you learned a lot from that "bad" teacher.
I realize that this was not aimed at me, but I will apply it to
my situation in calculus III anyway. I would claim that the
professor who over-emphasized proofs failed in his job by your
criteria above. It was not he who taught me multiple
integration, or spherical coordinate systems, or vector calculus.
Some of it I knew when I walked in the class (I had an
_outstanding_ math teacher in high school) and the rest I learned
on my own. I collect math books, so I had enough sources to
teach myself from a variety of books.
Keith Robison
>In article <2b403i$a...@scunix2.harvard.edu> rma...@husc9.Harvard.EDU (Ron Maimon) writes:
>>
>>I have _never_ met anyone like that. I have never seen anyone "intuit" math
>>through some oracle. ...
>Then I feel sorry for you and your limited experience. I have known
>several people with this talent, and it is a striking thing to observe.
>In one case, it created quite a problem for a graduate student who was
>set a proof as a thesis-related problem. The prof *knew* it was true
>(and it proved to be true) but could not provide even a hint of guidance
>as to how the problem should be attacked.
Let me guess -- his words were "I have discovered a truly remarkable
proof of this, but this margin is too small to contain it" :-)
Keith Robison
Harvard University
Department of Cellular and Developmental Biology
Department of Genetics / HHMI
John Novak
>>If you want your students to display an ability to connect
>>concepts, give them enough time. Give a take home test, or a two
>>week project, if the class size and composition warrant.
>You will get no arguments from me on this. Now how do we implement this?
>In a low-level course, the cheating problem is bad enough on in-class
>tests; take-home is essentially impossible. This is not something
>which an individual teacher can do much about.
Ugh. I just took a look at your address, and I can see why we
might be looking at this from different angles. Purdue, to the
best of my understanding, is a _large_ school. Bradley is a
small school. In our general calculus I course, the average
section size was some thirty students. (Of course, there were
multiple sections of the class.)
I imagine the attitudes and approaches for both students _and_
professors are quite different at Purdue. We're small enough
that at the upper levels (3 and 400 level courses) we can afford
to give take home tests or small group projects, on occasion, and
have them graded in reasonable amounts of time. And small enough
that people working to gether on the tests will be almost
blatantly obvious. I grade papers of a 500 level engineering
course, and I can spot people who work on their homework together
very easily. (They're lucky we don't consider that a crime...)
In some cases, I just take out the three students' homeworks and
grade them together. Usually with the same comments on each
paper.
The only reasonable question left for me to ask at this point is
the following. Is it possible to construct a test with one or
two questions which test the ability to synthesize the
information, one or two difficult, but familiar problems, and the
rest routine problems?
Of course, this brings the question of grading criteria into the
picture. I'd say that being able to perform familiar questions
quickly, under pressure, is worth a 'C' grade, while performing
complex familiar problems is worth a 'B'. Being able to
synthesize knowledge on the spot should merit an 'A'. Again,
IMHO. (My interest in this thread is double edged-- I'm trying
to land a teaching assistantship, and if I actually get it, I'd
like to be the best I can possibly be.)
John Novak
>|> %Believe me, I know. I met a lot of supposedly talented people and they're dumb
>|> %as posts for the most part.
>|> Really? I have met people like that also. I have also met people whose
>|> ability to grasp complex concepts FAR exceeded my own, and not through any
>|> hard work on their part.
>well, I guess youve been hanging around all these exceptional people, because
>I have never met anyone whose ability to grasp complex concepts exceeded my
>own by even one iota.
Personally, I think you're crocked.
While IQ tests are by no means a perfectly accurate or
precise measurement tool, they do have some weight.
Saying that no one has any greater or lesser ability to learn is
tantamount to saying that everyone has an IQ of 100.
>|> So everyone does it at exactly the same speed? There is no one on this
>|> earth who is any faster at comprehending complex subject material than
>|> anyone else? Hmmm... I KNOW that such is not the case.
>Well, I know that it is the case that anybody who is born with a normal brain
>can know physics as well as Glashow, Witten, Hawking, or any other physicist.
>They just have better things to do with their time, and I can respect that.
And how, exactly, do you know this?
And, do you also 'know' that people can learn physics as quickly
Hawking?
I'd also like to see an operating definition of a 'normal' brain.
Whose is the standard?
>I have _never_ met anyone like that. I have never seen anyone "intuit" math
>through some oracle. At times, of course, I see someone solve a problem through
>an ingenious trick, but this is not so impressive, since its usually a trick they
>have seen someplace before, and the same result can almost always be gotten by
>a more brute force approach, which is the way I solve things.
I suspect that you're simply unobservant in this respect.
John Novak
>Presumably you will have no objection to paying the extra fees to
>support the much more labor-intensive evaluation teaching and
>evaluation methods that you favour?
I'm part of it.
I grade homework papers at the graduate level.
And I do it well, because I take my time, trace through the
students' work, find their mistakes, point them out, and grade on
that basis. Not simply on the basis of 'correct' or 'incorrect.'
I may not get their papers back to them by the next class
session, but generally within 10 days, and always in time for an
exam. (People like to study from their homework for this class.)
Bruce Helenb
>Oh come on!
Except in those that invent transistors.
>Ron Maimon
Stan Friesen
|> In article <mcdonald.1...@aries.scs.uiuc.edu>
|> mcdo...@aries.scs.uiuc.edu (J. D. McDonald) writes:
|> >
|> > I enjoyed math VERY much in college ..... especially the proofs.
|> > Many of them are very beautiful.
Ah, a mathemetician.
|>
|> In my opinion, the biggest service that one can do for someone who is
|> encountering proofs for the first time is to explain that what the books and
|> (most) teachers show in the way of proofs has very little to do with the way
|> that you actually do proofs, rather what is shown is just a quasi-standard
|> final form used for presentation only (a fact that I suspect most of the
|> primary and secondary school teachers I had were unaware of).
Well, that would help in *doing* the proofs, for those few courses
where doing proofs is required.
However, for those of us who are taking math to learn a set of *tools*
to use in doing other things, the proofs seem to be *irrelevent*.
I mean, in *applying* a mathematical technique, I generally do not
do proofs, or even use published proofs. At the most, I directly
apply the stated theorems, but more likely I just use a method
that produces the proper result.
Furthermore, in most of the math courses I took, even the tests did
*not* require doing proofs, just the working out of problems [for
instance: compute the integral of some hairy formula]. In such
courses spending hours listening to the prof. do proofs on the board
seems sort of silly.
So the queastion is: what is the significance of the proofs presented
in such classes? Why are they there at all?
|> The second biggest service you can perform is to introduce the person to
|> formal logic, and thereby ...
|> and math professors just assume that
|> everyone else has seen it somewhere as well. However, most of the
|> non-math majors (i.e. most of the students) haven't even heard of formal logic,
|> and therefore are (quite properly) totally confused.
Well, that wasn't my problem, I got the formal logic quite well.
I just didn't see any *point* to the proofs, at least as a non-math
major. [I can see that it might be different for math majors].
--
sar...@teradata.com (formerly tdatirv!sarima)
or
Stanley...@ElSegundoCA.ncr.com
The peace of God be with you.
Stan Friesen
|>
|> I doubt many thought that. The question is whether the prof _cares_ about
|> "anything useful to do test problems." The failure of the system is that
|> the test problems don't tell us anything useful about whether or not you
|> understand proofs, the essence of math (and science).
I must disagree here. Proofs (especially mathematical ones) have
*nothing* to do with the essence of science, only the essence of
math.
Of course, if what is desired is that the student understand the
meaning of proofs, then *that* is what should be taught. *Doing*
proofs does *not* teach what they are *for* - the *why* of proofs.
[I can, or at least could once, do proofs - but I have never managed
to really understand why I should *want* to do them, unless I were
a mathematician trying to prove some new theorem].
What is needed is at least a short course in the philosophy of proofs,
and in what they are supposed to accomplish.
|> >They are very beautiful when you understand what is going on. However
|> >IMHO when a student is trying to keep up with the material, do
|> >problems, and keep up with their other courses, then most of the
|> >students will see anything that is not directly helping their grade as
|> >a pain and they will ignore it. I think that we need to recognize this
|>
|> I agree completely. We need to recognize it and rethink the way we
|> assign grades (or the very fact that we do so). And also the kinds (and
|> quantity) of problems we assign.
Well, then there is also the fact that a truly mathematics based
calculus course (of the sort you seem to want) would be rather
useless for somebody trying to learn how to *apply* the calculus.
I think what is needed is a seperation between the math course
for math majors, and the math course for non-majors. Most physics
departments do quite well in making such a distinction, for instance,
but I find few math departments really provide much in the way of
applied math courses for non-majors.
Stan Friesen
|>
|> This is EXACTLY right! I wouldn't touch the subject as to why high school
|> students (on average) seem dumber these days, but it seems to me that those
|> that decide to go to college should, by definition, understand their
|> responsibility as students. I've seen more college teachers with inadequate
|> teaching skills than college students with learning disabilities. My first
|> calculus professor couldn't teach a dog to bark. HE made math ugly. Luckily I
|> found another teacher worthy of the title.
|>
I was lucky, I took my first two semesters of calculus in High School,
from the chairman of the math department, and one of the best teachers
in the school. (One of two or three that consistantly got the "Best
Teacher" award).
When I took multivariate calculus in college, I found it very hard
to stay with it, despite my intelligence and background - the teacher
was *lousy*, and managed to make the subject boring.
Ron Maimon
|>
|> I, personally, have a small amount of mathematical talent. I'm
|> no god, especially not by professional levels, but of my
|> classmates as an undergrad, I picked up the pure math concepts
|> more quickly than most other people. I also spent far less time
|> on the homework, and I got better grades.
I have found that everytime I do a problem quicker or easier then someone
else, it's because I have seen it, or something similar, before.
I am sure you will find the same thing working in your case too.
In addition, I don't think that the essence of mathematical knowledge is
being able to come up with neat tricks. Neat tricks are just that, neat
tricks. They don't often help you understand fundamental physics any better.
I find them highly distasteful. I don't learn something unless I could have
come up with it myself.
|>
|> But I have no ability for languages, other than English. I can
|> _learn_ other languages. At one point, I spoke German, and could
|> puzzle through written (and slowly spoken) Latin. I even got A's
|> in the classes (this was high school.) But I only learned them
|> by spending twice as much time as anyone else, and I can't
|> remember much of them today. An old acquaintance of mine,
|> however, used to pick up languages like nothing. A true
|> polyglot.
You are aware that there is a "trick" to learning languages.
The reason most adults fail to learn languages quickly is that they think
too much. They have an overwhelming desire to compartmentalize the language
into rules that they can memorise (i after e except after c) and then the
rules multiply and multiply, and exceptions build up, and they find they can't
learn the language well at all.
This is completely wrongheaded. The only way to learn a language is to _feel_
it, like a native speaker. The way to do this is to stop using rules altogether,
and learn to speak ungrammatically like native speakers speak. You then find,
after a while, that you can _feel_ what the right rules and exceptions are, just
by listening to how they sound.
People who know a lot of languages know this, and this makes it easier for them
to learn a third.
I highly doubt talent enters into the picture at all, but even if it did, it
is much less important then the factors above.
|>
|> Anecdotal evidence, but she had a talent for languages.
|> I have a talent for math.
|>
|> [...]
|>
|> >But you will never find the "math genes"
|>
|> I remember hearing a report, about a year and a half ago, which
|> tracked math ability in young students. It seems that
|> mathematical ability (and particularly three-dimensional
|> visualization and conceptualization) was strongly correlated with
|> the child's exposure to particular hormones (testosterone?) in
|> the womb.
|>
|> If someone else remembers hearing the report, or better yet can
|> point me to it, that would be non-anecdotal evidence.
|>
It would be far worse then anecdotal evidence. I would believe anecdotal evidence,
I would never believe a scientific study.
The way these studies are done is that they take a large sample of babies, measure
hormone levels and genes and stuff, and then give them a test a few years later
and look for correlations.
people really want to find something, so this has been a million times.
999,999 have found nothing.
By a statistical fluke, once in a while 1 does find something, and everyone gets
excited. Of course, whatever it is they find, it turns out that men have it more
then women (proving that women are inherently less capable at math) and if it
was done in the 19th century, that whites have more of it then blacks and asians.
This is total nonsense.
I would much prefer that you give anecdotal evidence, because if there was such
a thing as talent, you would expect there to be an _overwhelming_ amount of
evidence of this type.
e.g. I would have met lots of people so much smarter then me, that I couldn't
even begin to understand them.
I havn't met any
Ron Maimon
Andy Boden
> I would much prefer that you give anecdotal evidence, because if there was such
> a thing as talent, you would expect there to be an _overwhelming_ amount of
> evidence of this type.
>
> e.g. I would have met lots of people so much smarter then me, that I couldn't
> even begin to understand them.
>
> I havn't met any
>
> Ron Maimon
This is getting silly.
I suppose it's entirely possible that you're just the smartest person around.
Maybe you should ask yourself if you're ever known anyone who isn't a quick
as you?
Seriously, you're simply the only person I've ever heard espouse the belief
that everyone is given the *same* amount of horsepower in all different areas
(and yes, I've talked to more than one person about this over the years).
The overwhelming evidence that you ask for...
In my own experience, in *any* given nitch there is an entire spectrum of
aptitudes in the general population -- both by objective and subjective
figures of merit. This *is* a counter-example to your argument. Now one
of us could be wrong (take me for sake of discussion), but we can't *all*
be wrong.
Or maybe we *can* all be wrong and you can correct us (or would that
disprove your point...)
- A.B.
Robert Casey
>The only reasonable question left for me to ask at this point is
>the following. Is it possible to construct a test with one or
>two questions which test the ability to synthesize the
>information, one or two difficult, but familiar problems, and the
>rest routine problems?
>
like: do the derritive of: X to the Xth power d/dx
I tried to answer: (X-1)X to the (X-1) power, which turns out to be wrong.
I figured that it would go the same way that something like
3X to the 4th power d/dx = 12X to the 3rd power (did I get that right?
been a long time since I had calculus :-) ). Chain rule, I think it
was called.
Turns out that Xth to the X power d/dx question was a way for the math
prof to see who really knew what was going on, and who just did
memorized procedures.
One difficulty I remember having was getting used to the notation in
calculus class. like
5
3X d/dx
I first thought that I could cancel out the d's in the d/dx, like as
if it was a regular fraction like in algebra class. Duuuuhh..
The calculus prof probably mentioned what d/dx meant ONCE, but eighter
I missed that class, or daydreamed, or something.
And more notation I had to learn was the ' , like f'(x).
Hard to follow the presentation if you don't know what the notation means,
and I felt too shy and didn't want to embarris myself by asking the
stupid question of "What's the d/dx in that equasion for?
But there were some success in these classes, like "You can find the
area under the curve by doing the intergral. Imagine you cut up that area
into narrow rectangular strips, and add up the area of all those strips.
Now make those strips narrower, and do it again. Less error.
You eventually get to infinitestially narrow, etc.
We also spent time on limits, and something called Lopetal's(sp) rule.
Not sure what this did other than generate correct answers on tests.
And something about convergence or divergence.
And we spent a LOT of time doing antiderritives. Like:
(csc(1/(x+1))tan3(x-2))/(x+2) dx
I didn't have a clue....
My college made me take the second semester calc class even though I
bombed the first semester. Guess they figured I might make sense of
it on my own (I suppose some students might just do that).
Took these again in summer school, at the same time. Imagine taking
calc 1 and calc 2 at the same time! calc 1 is a prerequesite of calc 2!
That didn't work, ended taking calc 1 yet again (I must have been a
glutton for punishment!, no, just wanted to do engineering) and got
a prof who did a decent job of teaching it, and then made it thru
calc 2 also finally. I don't think I ever made it thru a calc class
without repeating it at least once :-(
Wonder if anyone in the Math field ever did research and written papers
like "Successfully teaching students who have failed calculus before
and are repeating it, so they actually understand it" Or " Teaching
students (who did well on SAT's and Math Achievement tests and decent
high school math grades) who are getting D's and F's on their calc
tests, so they can get up to speed and understand it".
Maybe doing simple stuff like making the students DO the homework
(collect and grade it) and spending some class time having students
work some problems on the blackboard (like teachers did in high school
algebra) might make a big difference at the lower level calc classes.
Sort of "jump starting" the students, to make them say "Hey, I'm lost
here, what did I miss?" It's too late when you get a bad grade on
the midterm test. At that point, I could never catch up.
Ron Maimon
|> In article <2b403i$a...@scunix2.harvard.edu>
|> rma...@husc9.Harvard.EDU (Ron Maimon) writes:
|> >In article <1993Nov1.2...@hobbes.kzoo.edu>,
|> > k08...@hobbes.kzoo.edu (Josh Vander Berg) writes:
|> >|> rma...@husc9.Harvard.EDU (Ron Maimon) wrote:
|> >|> %There's no such thing as talent.
|> >|>
|> >|> Oh, really?
|> >
|> >really.
|>
|> Do you consider that to be true about physical ability also? If
|> so, do you really believe that you could have been a good a high
|> jumper as Javier Sotomayor? If not, why is mental ability so
|> different?
|>
|> I have no trouble believing that Carl Lewis is just *faster* than
|> me and that Anthony Gatto is a better juggler than I will ever be.
|> Similarly I have no trouble believing that Donald Knuth is smarter
|> than I am.
There is a slight difference. Saying that a person can run a 100m dash in
10 seconds is sort of analogous to saying a person can understand galois
theory in one lecture. This has to do with how fast people learn things,
and I agree that they do learn at different rates.
This doesn't make them more intelligent, however, since if I start walking, I
will get to the end of the track sooner or later. If it takes me 20 lectures,
if I want to, I will understand galois theory. Once I know it, and there is
no way to make me unknow it.
I don't like arguing by analogy. I don't like saying "A is like B, B is X,
therefore A is X". Since if you know _why_ B is X you can sidestep the analogy
altogether, and if you don't know _why_ B is X, maybe it isn't X after all.
So I will argue directly.
People are not born understanding math. They have to learn. In order to learn
it they need motivation, real motivation.
Not " I need an A so I can get a good job"
Not " I want to look smart to impress my friends"
but something along the lines of "gosh, I am really curious as to why it
is that the angles in all these triangles add up to two right angles!"
The problem with math, however, is that people obscure these types of ideas
behind the "canned problem". These are artificial situations set up so that
people can apply formulas. The number of formulas is just small enough to
make a person who wants to get an "A" can more easily memorize them all rather
then figure out their derivation. It's just large enough, that no human being
can memorize all of them for all his or her classes _without_ understanding
their derivation.
Different people have a different level of understanding of where these
relationships come from. This is 100% due to experience. Sometimes acquired
alone by sitting and struggling with a problem, sometimes from a book, sometimes
from a teacher. But no one is born with anything that helps them in this task
of understanding.
People place too much of an emphasis on being able to "rotate objects in your
mind". I am very good at math and physics, and I can't do that. Whenever they
have those "matching" problems which most people do by rotating things in their
heads, I do by using an appropriate "right hand rule".
Some people would say that that makes me more intelligent, because I am solving
the problem quicker then the people who mentally rotate the blocks. I say that
that makes me dumber, just that I compensate for it.
I think there is no limit to the dumbness I can compensate for. No matter how
dumb I get, there is a way of thinking about things that will let me understand
a given peice of knowledge.
|>
|> >|> Really? I have met people like that also. I have also met people whose
|> >|> ability to grasp complex concepts FAR exceeded my own, and not through any
|> >|> hard work on their part.
|> >
|> >well, I guess youve been hanging around all these exceptional people, because
|> >I have never met anyone whose ability to grasp complex concepts exceeded my
|> >own by even one iota.
|> >
|> >and, believe me, this has nothing to do with me.
|>
|> Maybe it has to do with the people you hang out with. I count myself as
|> a pretty bright guy all around and I have certainly met people who can
|> run little tiny mental circles around me. Not many, but they certainly exist.
I havn't met any.
I hang out with a lot of the physics department at harvard too.
I admit that they know a lot more physics then I do, but this doesn't mean that
I can't know what they know. As a matter of fact, I am well on my way to doing
this.
|>
|> >Well, I know that it is the case that anybody who is born with a normal brain
|> >can know physics as well as Glashow, Witten, Hawking, or any other physicist.
|>
|> How do you *know* this? I haven't heard any final word from the
|> psychologist over the nature vs. nurture battle. Maybe you should
|> pop over to sci.psychology and inform them you have the answer.
I am not saying that nature has nothing to do with mathematical ability, its
just that you will find that even the ones least "gifted" count themselves
among the greatest mathematicians.
|>
|> >They just have better things to do with their time, and I can respect that.
|>
|> I tutored someone in high school. She was really earnest and wanted
|> to do well because she was very close to not graduating because she
|> couldn't pass math. She (and I) spent several hours a day for the
|> last few weeks of the year going over algebra. You will no doubt
|> find this hard to believe but SHE JUST DIDN'T GET IT. It wasn't that she
|> wasn't interested or didn't want to spend the time on it, she just
|> didn't understand it. She finally passed with a C- and was overjoyed
|> to get that. I'm sure she would be happy to hear she could have been
|> the next Godel if she had just had the inclination.
I have tutored dozens of people, some with zero math background and zero
interest, and I found that all of them could understand everything that I
taught them as well as I could. I have never met anyone without a serious
defect who was any dumber then me.
Even if it took 10 times as long as I thought it would, they will get it in the
end.
|>
|> >|> And then there are people who just seem to "understand" math, who grasp
|> >|> concepts intuitively. I have met people like this. They were not arrogant
|> >|> asses trying to pass off their secret hard work as natural talent, they were
|> >|> TRULY gifted.
|> >
|> >I have _never_ met anyone like that. I have never seen anyone "intuit" math
|> >through some oracle.
|>
|> I presume you have heard of Ramanujan? Most mathematicians that I have
|> met consider him a "natural" mathematician. The people who he worked
|> with thought that of him. He certainly could "intuit" math and I don't
|> believe that just anyone can demonstrate that ability. Even if (by
|> some miracle) that was possible, I still think that Ramanujan would
|> become a Ramanujan^2 under those circumstances.
Ramanujan is a much too often cited example of mathematical genius.
Ramanujan spent his entire life reading math!
His favorite part of mathematics was continued fractions, a very obscure branch
of mathematics which is nearly all advanced trickery.
Of course he could impress everyone with his knowledge!
I am not detracting from his contribution to mathematics, but remember, he was
just a normal human being, and did mathematics just like everyone else. He wasn't
"gifted", he wasn't "superhuman", he was a good mathematician, who worked
in isolation and reinvented a lot of mathematics for himself.
If you consider mathematics as an art, then IMO, his was pretty bad. It was way
too tricky for my taste, and far to limited in its applicability.
But that doesn't make his results any less true.
|>
|> >|> %But you will never find the "math genes"
|> >|>
|> >|> And I suppose genetics have absolutely nothing to do with intellectual
|> >|> ability...
|> >
|> >I certainly don't think so.
|>
|> You don't think genes have *anything* to do with intellectual ability?
|> I am prepared to believe that nurture plays a major role, but that nature
|> plays *none*?
Yup.
You can convince me when I meet someone whose understanding of math and physics
surpasses my own by such an overwhelming amount that I could never hope to know
what they know.
Alternatively, show me a normal, interested person who I cannot teach the entire
body of modern physics to, and I'll believe you.
until then...
Ron Maimon
Ron Maimon
|>
|> Excuse me, but you may believe such a thing, nevertheless you have no way to
|> know it. As a physicist myself, I have always wished it were true, but I
|> don't have anything solid to prove it. I feel certain that you know of no
|> controlled experiments establishing your assertion. While I think a good
|> explanation of a physics concept works wonders for someone struggling with
|> it, you are over-reaching to imagine that no inherent talent distinguishes
|> a Nobel prize winner from any other normal person. Motivation may be
|> absent among most normal people to pursue our career, but the existence
|> of gifted prodigies is well verified in the arts and sciences. I would
|> not dare to deny that Mozart possessed inherent talents for musical
|> composition that I lack, neither would I deny Hawking the recognition
|> that his genius deserves.
|>
Well, I for one, hate Mozart's music, and, no, I don't think he had any special
talents.
On the other hand, I don't think Bob Mould could make the music he makes if he
was 7 years old.
Ron Maimon
Ron Maimon
|> In <2b403i$a...@scunix2.harvard.edu> rma...@husc9.Harvard.EDU (Ron Maimon) writes:
|>
|> >|> %Believe me, I know. I met a lot of supposedly talented people and they're dumb
|> >|> %as posts for the most part.
|>
|> >|> Really? I have met people like that also. I have also met people whose
|> >|> ability to grasp complex concepts FAR exceeded my own, and not through any
|> >|> hard work on their part.
|>
|> >well, I guess youve been hanging around all these exceptional people, because
|> >I have never met anyone whose ability to grasp complex concepts exceeded my
|> >own by even one iota.
|>
|> Personally, I think you're crocked.
|> While IQ tests are by no means a perfectly accurate or
|> precise measurement tool, they do have some weight.
|>
|> Saying that no one has any greater or lesser ability to learn is
|> tantamount to saying that everyone has an IQ of 100.
no, it is tatamount to saying that with a little practice anyone can learn how
to get a 200 on their IQ test.
|>
|> >|> So everyone does it at exactly the same speed? There is no one on this
|> >|> earth who is any faster at comprehending complex subject material than
|> >|> anyone else? Hmmm... I KNOW that such is not the case.
|>
|> >Well, I know that it is the case that anybody who is born with a normal brain
|> >can know physics as well as Glashow, Witten, Hawking, or any other physicist.
|> >They just have better things to do with their time, and I can respect that.
|>
|> And how, exactly, do you know this?
|> And, do you also 'know' that people can learn physics as quickly
|> Hawking?
|>
|> I'd also like to see an operating definition of a 'normal' brain.
|> Whose is the standard?
Well, a person can't learn physics as _quickly_ as Hawking, but, who cares? I
just said anyone could know all the physics Hawking knows. It might take more
time then they are willing to put in, though.
Some people have congenital brain defects that don't even allow them to talk!
I mean that anyone who can understand high school algebra can understand quantum
field theory, is this better?
|>
|> >I have _never_ met anyone like that. I have never seen anyone "intuit" math
|> >through some oracle. At times, of course, I see someone solve a problem through
|> >an ingenious trick, but this is not so impressive, since its usually a trick they
|> >have seen someplace before, and the same result can almost always be gotten by
|> >a more brute force approach, which is the way I solve things.
|>
|> I suspect that you're simply unobservant in this respect.
|>
oh no.
Ron Maimon
cy...@msc.cornell.edu
>
> e.g. I would have met lots of people so much smarter then me, that I couldn't
> even begin to understand them.
>
> I havn't met any
>
> Ron Maimon
Ron,
You must not have met many people. I have lived in 3 countries on
3 different continents. I have met enough people who are so much smarter
than me in physics that it makes me look like a moron. People (who are not
physicists) would think I am above average, however, in the company of other
physicists, I'm just a normal nerd, i.e. not too smart. One other thing, I
work as hard as any other physics grad student here at Cornell. However
hard work is not all there is. There are two grad students here, both doing
QFT. One of them is 16 and the other is 18. Both got their first degrees
at the tender age of 14. Tell me, do you think you could have gone off
to college at 12, get first class honours and continue on to grad school?
There ARE people who are more talented.
Cheng-Yang Tan
Ron Maimon
|>
|> I think you confuse talent & intelligence - the two do not necessarily go hand
|> in hand.
|>
Excuse me.
There is no such thing as intelligence.
Or, at least, if there is, I'm the smartest guy in the world.
Ron Maimon
SCOTT I CHASE
I have been watching this newsgroup long enough to be sure that, given the
choice, there must be no such thing as intelligence. :-)
-Scott
-------------------- i hate you, you hate me
Scott I. Chase let's all go and kill barney
SIC...@CSA2.LBL.GOV and a shot rang out and barney hit the floor,
no more purple dinosaur.
Herman Rubin
............................
>The only reasonable question left for me to ask at this point is
>the following. Is it possible to construct a test with one or
>two questions which test the ability to synthesize the
>information, one or two difficult, but familiar problems, and the
>rest routine problems?
My typical hour tests for low-level service courses are three or four
multi-part problems, with the routine parts interspersed; they are not
displayed as such, and the student has to find them. Crib sheets are
allowed, so that memorization is clearly not of importance. Much credit
is given if the student can understand what the problem is. Calculators
are allowed and their use encouraged; it is much easier to figure out
if they understand something if I do not have to figure out what
procedure known only to God was used to get that number.
--
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)
Herman Rubin
>In article <2arned$b...@agate.berkeley.edu>, aephraim@physics3 (Aephraim M. Steinberg) writes:
>|> I doubt many thought that. The question is whether the prof _cares_ about
>|> "anything useful to do test problems." The failure of the system is that
>|> the test problems don't tell us anything useful about whether or not you
>|> understand proofs, the essence of math (and science).
The prof should care about how well you can understand the material years down
the line. The typical test problems are of little use here; learning how to
calculate answers can lead to understanding only in those with mathematical
research ability, and need not even then.
>I must disagree here. Proofs (especially mathematical ones) have
>*nothing* to do with the essence of science, only the essence of
>math.
Proofs, as such, also are not always the way to get understanding.
They are more likely to lead to it than computation. There are,
however, cases in which the only understanding available is that of
the proofs. I have proved such myself, and I also know of a very
important theorem, for which I know at least a half-dozen essentially
unrelated proofs, and only partial understanding. The most commonly
used of those proofs conveys no understanding whatever.
>Of course, if what is desired is that the student understand the
>meaning of proofs, then *that* is what should be taught. *Doing*
>proofs does *not* teach what they are *for* - the *why* of proofs.
Agreed. And constructing proofs is pure artistry, not teachable.
It is as much of an art form as composition, and constrained by
the fact that it must be within rigid rules.
>What is needed is at least a short course in the philosophy of proofs,
>and in what they are supposed to accomplish.
Look at any good book on logic. Everything known in mathematics, or in
any theoretical science, are assumptions and conclusions obtained from
proofs. Observational properties decide what assumptions are made.
............................
>Well, then there is also the fact that a truly mathematics based
>calculus course (of the sort you seem to want) would be rather
>useless for somebody trying to learn how to *apply* the calculus.
A purely proof-based course would not be good for anyone. For those
who will apply it, the first thing they need is to know the concepts
so that they can formulate problems. FORMULATE, not solve. Especially
in the case of calculus, present-day computer programs can do all of
the manipulations in calculus, and can usually even decide if there are
such which can be applied. Unless the student gets to be able to do
them quickly, not much is gained for the non-mathematician by learning
how to calculate the results.
>I think what is needed is a seperation between the math course
>for math majors, and the math course for non-majors. Most physics
>departments do quite well in making such a distinction, for instance,
>but I find few math departments really provide much in the way of
>applied math courses for non-majors.
It is rather that they do not provide much in the way of mathematics
courses for mathematicians.
Gregory Hamlin
most people were smarter than they were...
Greg Hamlin
ham...@ral.rpi.edu
MADIGAN KEVIN M
>|> If someone else remembers hearing the report, or better yet can
>|> point me to it, that would be non-anecdotal evidence.
>|>
>
>It would be far worse then anecdotal evidence. I would believe anecdotal evidence,
>I would never believe a scientific study.
>
>The way these studies are done is that they take a large sample of babies, measure
>hormone levels and genes and stuff, and then give them a test a few years later
>and look for correlations.
>
>people really want to find something, so this has been a million times.
>
>999,999 have found nothing.
>
>By a statistical fluke, once in a while 1 does find something, and everyone gets
>excited. Of course, whatever it is they find, it turns out that men have it more
>then women (proving that women are inherently less capable at math) and if it
>was done in the 19th century, that whites have more of it then blacks and asians.
>
>This is total nonsense.
>
>I would much prefer that you give anecdotal evidence, because if there was such
>a thing as talent, you would expect there to be an _overwhelming_ amount of
>evidence of this type.
>
>e.g. I would have met lots of people so much smarter then me, that I couldn't
>even begin to understand them.
>
>I havn't met any
>
>Ron Maimon
You are one of the most arrogant people I have ever encountered on usenet.
Or you are pulling our chain.
I will give you one anectode, then i will put you in my kill file.
One day this summer, I was helping my Linear Algebra students. Outside my
office was an 18 year old acquaintance of mine (who had never seen Linear
Algebra before), listening to what I was saying. After my students left,
this 18 year old walked in and said, "Kevin, is that Linear Algebra? That
stuff is easy." Based on hearing 10 or 20 minutes of my explanation, he
started explaining the subject to me. I suppose you will tell me that he
learned it elsewhere but forgot. The point is, some (few) people out there
"see" Mathematics, they don't have to try to figure it out until they get
to the really hard stuff. These people exist. Many of us have met them.
Some of them occassionally are on usenet.
You are making the naive mistake of assuming that your experiences are the
only ones. This is silly.
I wish I had the strength at this late hour to flame you appropriately. You
really deserve it.
Kevin Madigan
John Novak
>I have found that everytime I do a problem quicker or easier then someone
>else, it's because I have seen it, or something similar, before.
>I am sure you will find the same thing working in your case too.
I am sure that you overgeneralize.
Sometimes, this is the reason. Sometimes, it is not.
>In addition, I don't think that the essence of mathematical knowledge is
>being able to come up with neat tricks. Neat tricks are just that, neat
>tricks. They don't often help you understand fundamental physics any better.
>I find them highly distasteful. I don't learn something unless I could have
>come up with it myself.
I don't remember mentioning anything about neat tricks. I don't
remember saying anything about physics. I do not trust my
intuition on physics.
>You are aware that there is a "trick" to learning languages.
[...]
>People who know a lot of languages know this, and this makes it easier for them
>to learn a third.
>I highly doubt talent enters into the picture at all, but even if it did, it
>is much less important then the factors above.
How do you know these things?
How many examples to the contrary will be needed to disqualify
your sweeping overgeneralization? (The answer is 'one' which I
have more than provided.)
>It would be far worse then anecdotal evidence. I would believe anecdotal evidence,
I have given it. You persist in your arguments.
>I would never believe a scientific study.
Ah. Then I suspect we have little more to discuss.
>The way these studies are done is that they take a large sample of babies, measure
>hormone levels and genes and stuff, and then give them a test a few years later
>and look for correlations.
>people really want to find something, so this has been a million times.
>999,999 have found nothing.
Oh? Justify that.
>By a statistical fluke, once in a while 1 does find something, and everyone gets
>excited. Of course, whatever it is they find, it turns out that men have it more
>then women (proving that women are inherently less capable at math) and if it
>was done in the 19th century, that whites have more of it then blacks and asians.
>This is total nonsense.
So, you're saying that because you believe that everyone has
exactly the same abilities, it must be so. Do you let your
personal beliefs dictate the results of all your investiations?
>I would much prefer that you give anecdotal evidence, because if there was such
>a thing as talent, you would expect there to be an _overwhelming_ amount of
>evidence of this type.
>e.g. I would have met lots of people so much smarter then me, that I couldn't
>even begin to understand them.
Do the stats-- if you have an IQ of 145 (after which conventional
IQ scores become less than helpful indicators) then a reasonably
small percentage of the population will be 'smarter' than you.
John Novak
>Excuse me.
>There is no such thing as intelligence.
>Or, at least, if there is, I'm the smartest guy in the world.
You expect anyone to buy this?
"There is no such thing as intelligence" indeed.
I'm intelligent. My dog is not.
John Novak
>no, it is tatamount to saying that with a little practice anyone can learn how
>to get a 200 on their IQ test.
Out of curiosity, have you done any reading on the measurement of
intelligence quotients?
>|> And how, exactly, do you know this?
>|> And, do you also 'know' that people can learn physics as quickly
>|> Hawking?
>|> I'd also like to see an operating definition of a 'normal' brain.
>|> Whose is the standard?
>Well, a person can't learn physics as _quickly_ as Hawking, but, who cares? I
>just said anyone could know all the physics Hawking knows. It might take more
>time then they are willing to put in, though.
Sounds like an admission of the 'talent' concept.
>Some people have congenital brain defects that don't even allow them to talk!
Is it such a stretch to leap from "normal brain" and
"congenitally defective" to a set which also includes
"congenitally brilliant?"
And from that set of three to a full spectrum of brains, some of
which fall so far outside the norms that they are noticeably more
or less efficient and successful at certain tasks?
I don't think so.
But apparently, a brain is either normal, or broken.
>I mean that anyone who can understand high school algebra can understand quantum
>field theory, is this better?
This may or may not be true. I am not enough of an expert on
physics to judge whether QFT is understandable by a typical high
school graduate. Namely, because I don't know QFT.
John Novak
>My typical hour tests for low-level service courses are three or four
>multi-part problems, with the routine parts interspersed; they are not
>displayed as such, and the student has to find them. Crib sheets are
>allowed, so that memorization is clearly not of importance. Much credit
>is given if the student can understand what the problem is. Calculators
>are allowed and their use encouraged; it is much easier to figure out
>if they understand something if I do not have to figure out what
>procedure known only to God was used to get that number.
In other words, even though they may not be able to do the
manipulation needed, you'll give credit for a proper set-up and
an attempt on the right track? (Ie, "The answer is obviously the
definite integration of this curve (drawn) from this to that
point (marked) expressed like this. It is this integral, because
the quantity relates tp the area under this curve. However, I
can't remember the form...")
Or, from my undergrad days, "I need to do this and this and this,
combine the answers this way, but I', outta time!"
I couldn't ask for more.
Monisha Ghosh
Ron Maimon writes:
> Well, a person can't learn physics as _quickly_ as Hawking, but, who cares? I
>just said anyone could know all the physics Hawking knows. It might take more
>time then they are willing to put in, though.
Ron, I think you are missing the basic point which is learning or understanding
something _after_ someone else has proven it to be true, versus being the first
person to _conceive_ an idea and prove it to be true. Sure I could know all
the physics Hawking or anyone else knows _now_, but could I (or you) propose
and prove to be true what he has done, before he had done it? That, IMHO,
is what sets apart the averagely intelligent from the 'genius' variety -
the creativity to see something without having to spend years slogging at it.
I _know_ that I don't have the creativity (I prefer that word to intelligence)
of Galois, do you?
The sort of knowledge that you are talking about can be
acquired by sheer diligence and time spent on a subject, be it math, physics
or Latin; creativity and insight are something else which need more than just
time and effort to develop. Integral calculus can be mastered given enough
time, but could you have developed the theory all by yourself?
Monisha
Ron Maimon
|> From article <2b6lra$q...@scunix2.harvard.edu>, by rma...@husc9.Harvard.EDU (Ron Maimon):
|>
|> You must not have met many people. I have lived in 3 countries on
|> 3 different continents. I have met enough people who are so much smarter
|> than me in physics that it makes me look like a moron. People (who are not
|> physicists) would think I am above average, however, in the company of other
|> physicists, I'm just a normal nerd, i.e. not too smart. One other thing, I
|> work as hard as any other physics grad student here at Cornell. However
|> hard work is not all there is. There are two grad students here, both doing
|> QFT. One of them is 16 and the other is 18. Both got their first degrees
|> at the tender age of 14. Tell me, do you think you could have gone off
|> to college at 12, get first class honours and continue on to grad school?
Yeah, but why would I want to?
I have better things to do with my time.
|> There ARE people who are more talented.
well, all right.
Just because they know their field theory at 16 and I know it at 20 doesn't
mean that they know more of it, or know it better.
If you want to call "knowing the same thing that people know at age 30, except
at age 16" talent, then fine, they're talented. Big deal.
I just think they should spend more time partying.
Ron Maimon
Ron Maimon
|>
|> >I highly doubt talent enters into the picture at all, but even if it did, it
|> >is much less important then the factors above.
|>
|> How do you know these things?
I don't, but neither does anybody else. And until proven otherwise, I will
continue to assume that I could learn French as well as other adults.
|>
|> So, you're saying that because you believe that everyone has
|> exactly the same abilities, it must be so. Do you let your
|> personal beliefs dictate the results of all your investiations?
No, I am just saying that I believe that the more tests are done, the fewer
the genetic correlations to intelligence that you will find.
For example, it was once believed that intelligence genes correllated with myopia
genes, since so many "smart" people were myopic.
Now, of course, it has been shown that reading leads to both increased "smartness"
and increased incidence of myopia.
|>
|> >e.g. I would have met lots of people so much smarter then me, that I couldn't
|> >even begin to understand them.
|>
|> Do the stats-- if you have an IQ of 145 (after which conventional
|> IQ scores become less than helpful indicators) then a reasonably
|> small percentage of the population will be 'smarter' than you.
|>
I had an IQ of 130 when it was last measured.
and this is misleading, because it was 110 spatial, the verbal brought it up.
I have met people who score way above 145.
they are no smarter then I am, as a matter of fact, they are usually obsessed
with puns and chess (hence their increased verbal and spatial IQ) and not with
anything important.
I never found any bit of physics that they could understand that I can't, because
of their high IQ and my relativly low IQ. Go figure!
Ron Maimon
Ron Maimon
I agree that there is a huge amount of variability in who comes up with something
first,
but most people, including me, don't care if they come up with something first-
they just want to understand it, and a lot of people, when they find that they
can't understand something, blame their genes instead of their textbook.
I wish they wouldn't.
Ron Maimon
Alberto Moreira
By the sound of it you couldn't tell a music is Mozart's unless
somebody told you so. -Alberto-
Jonathan Dixon
>
>Oh come on!
>There's no such thing as talent.
>
There is indeed such a thing as talent, with various people given
differing amounts in different areas. Some people (Mozart, for
example) can play piano virtuositically (?) at age 4 (or whatever)
while others won't reach that level if they spent their whole lives
that way; some people can hear a piece of music in their heads,
complete with harmonies, and write it directly to paper - but most
have to struggle through analyzing chord structures.
In the same way, there are people who can add long columns of numbers
on sight, visualize in 3D accurately that which is drawn in 2D,
immdediately grasp complex concepts behind physical (or non-physical
even) problems. Most people cannot do these; but this doesn't mean
that people can't develop some of these skills. With practice, you
can improve your abilities in most of these areas; the talent is just
the base you start at, which is not always the same.
For a specific example, I have talents in math, science, and music,
while my sister has talents in drawing, painting, and selling.
Throughout school, I have had to put in little time in most of my
technical classes to get the A, whereas it is often a struggle for my
sister to do well in, for example, math classes. It isn't that she
can't learn the concepts, it's a matter of the amount of time she has
to spend to understand them is longer than mine. On the other hand,
were I to take a drawing class, I'm sure I would spend a lot more time
for less results than she would have to.
This is the difference talent makes - the amount of effort it takes to
be able to do a certain task as opposed to the amount of effort
required of most other people. This is what is refered to when people
are called "geniuses" in science, music, art, writing - they can do
things with little effort that others would have to spend all their
time to even approach.
Now just because somebody is talented in an area doesn't mean that
they won't be dumb in others. A brilliant teacher who can really
inspire their students may be completely absentminded and devoid of
most areas of common sense. But that doesn't mean they aren't
talented teachers. So talent shouldn't be confused with intelligence
(intelligence may be a manifestation of a type of talent, but is
certainly not the only measure for talent).
Jon Dixon
di...@mimicad.colorado.edu
Craig Graham
In article <2b6q0b$q...@scunix2.harvard.edu>, rma...@husc9.Harvard.EDU (Ron Maimon) writes:
>Well, a person can't learn physics as _quickly_ as Hawking, but, who cares? I
>just said anyone could know all the physics Hawking knows. It might take more
>time then they are willing to put in, though.
>
I think, that our 'freind' here is confused. Knowledge is not a measure of talent
or intelligence (other than in the context that a good memory may be considored a
talent). Intelligence and talent are ability. Not 'I have' but 'I can do' or 'I understand'. To KNOW that something is true is not really to understand it, all it needs is a spot of beleif - but no intelligence.
>Some people have congenital brain defects that don't even allow them to talk!
So we should regard speech as a talent - all be it a widespread one. Some people
talk early, some later, a small minority sadley will never be able to do it.
Craig Graham. /\ E-mail:Craig....@newcastle.ac.uk
BAe Space Systems || Phone : (UK) 091 4883098
Stevenage,England.|| Snail-mail: 2 Sun Street, Suniside, Newcastle, England.
-----------------/__\-----------------------------------------------------------
SPACE the final^^^^frontier. Seen it,done it,read the book,got the T-shirt....
Stan Friesen
|> rma...@husc9.Harvard.EDU (Ron Maimon) wrote:
|> %
|> %There's no such thing as talent.
|>
|> Oh, really?
|>
|> %Believe me, I know. I met a lot of supposedly talented people and they're dumb
|> %as posts for the most part.
|>
|> Really? I have met people like that also. I have also met people whose
|> ability to grasp complex concepts FAR exceeded my own, and not through any
|> hard work on their part.
What I have seen, and frequently, is what I call different *aptitudes*
for various types of material. I am not sure that I like the word
'talent' for these differences, but I do agree, they are *real*.
|> %when I was a kid, I didn't know _anything_, I had to learn it all, damn it, and
|> %I didn't do it any faster then anyone else (I just seemed to 'cause I spent my
|> %entire waking life learning physics)
|>
|> So everyone does it at exactly the same speed? There is no one on this
|> earth who is any faster at comprehending complex subject material than
|> anyone else? Hmmm... I KNOW that such is not the case.
What I have seen is people who take to different classes of material
with different speeds.
I can pick up computer or biology related concepts almost immediately
on contact. I can do formal mathematics only with difficulty (although
I can solve applied mathematics problems easily).
Back, years ago, when I tried to tutor some people in computer science,
I found that some of them did not seem to understand even the basics
after literally *hours* of one-on-one discussion. Yet, given the way
some of these people tried to program the computer, I suspect that many
of them may have been truly brilliant at, say, English composition (or
similar forms of expression).
A friend of mine, who has a real ability to deal with people, and to
help them through hard times, could barely understand symbolic logic.
An this was not through stupidity, but because it required a style of
thought that was foreign to him (I think he finally made that break-
through into the appropriate thought mode, but only after two semesters
of effort).
Or, take my cousin, who's intensely visual approach to understanding
gives her a real advantage in dealing with the visual arts (as compared
to my very verbal mode of understanding).
|> %There is no math talent, there is no physics talent. There is interest in math
|> %or physics, or art for that matter, and there is luck.
|>
|> And then there are people who just seem to "understand" math, who grasp
|> concepts intuitively. I have met people like this. They were not arrogant
|> asses trying to pass off their secret hard work as natural talent, they were
|> TRULY gifted.
Yep, and then there are people like me who have the same inuititive
type of understanding for an entirely different realm of knowledge.
[For a good example of an inutitive mathematician - try Dan Ashlock,
or even better a mutual friend of ours, Arthur Parker]
Benjamin J. Tilly
dark...@cegt201.bradley.edu (John Novak) writes:
> In <2b403i$a...@scunix2.harvard.edu> rma...@husc9.Harvard.EDU (Ron Maimon) writes:
[..]
> >well, I guess youve been hanging around all these exceptional people, because
> >I have never met anyone whose ability to grasp complex concepts exceeded my
> >own by even one iota.
>
> Personally, I think you're crocked.
> While IQ tests are by no means a perfectly accurate or
> precise measurement tool, they do have some weight.
>
Not much in my books. The basically test your ability on tests. I
happen to be very good on timed tests. But there are people who I would
say are more intelligent who do significantly worse than I do on timed
tests. IMO the difference is that they do not have the same sorts of
skills that I have on tests. Something that I find interesting is that
women do significantly worse than men on the SAT's...and significantly
better in college. I would guess that the reason for this is that women
are less likely to take shortcuts. This will slow them down, which
hurts them on the timed test, but it also will make them do a generally
better job, which helps them in college. I find it interesting,
whatever the real reason is, that there is some set of traits than you
tend to find in women that make them *worse* on a test that is supposed
to tell you how good they are at something that they are *better* at.
> Saying that no one has any greater or lesser ability to learn is
> tantamount to saying that everyone has an IQ of 100.
No. Saying that everyone has an IQ of 100 is a statement about how they
do on certain tests. If the tests do what we want them to then the
tests say something about how well they learn, but it is possible that
everyone could be just the same when it comes to learning even though
there is variation on IQ tests. Furthermore I would say that there are
real differences in how people learn, but as far as I can see the
biggest differences are learned, not innate. For example a lot of it is
in the attitude that they have and in the way that they approach
things. For another example, if I was to say what it is that has made
me good at math I would probably include the following.
1) I am obstinate about *understanding* what is going on. Time after
time I find that this makes me do extra work to start with...and it
pays off every time.
2) I used to play role-playing games and I wanted a good table of
probabilities for the sorts of dice rolls that we were doing. This was
in highschool. As a result of this I wound up creating them myself *by
hand*. In the process of doing that I learned that I could really
understand math, and that math was kind of fun. I also learned during
that that simple is not the same as easy. In addition it was the first
of a number of projects that I took on. The math that I learned in
doing these things for myself gave me a headstart that I have been able
to capitalize on.
3) I had an outstanding highschool teacher, Mr. Bradley. He would
encourage me to do things for myself like that last one. Or the time
that he asked me if I could write a program which could come up with
planes that intersected either at a given point, or on a line through
two points. Remember, this is a grade 12 student who has *no* linear
algebra. Figuring out that one was HARD for me w/o matrices. However
doing these sort of exercises builded up my confidence, and made me
bring math down to my level.
4) I understand that simple is not the same as easy. This is
extremely important. It happens all of the time that you hear some math
and think, "That is simple, it will not be a problem." Sorry. It is
extremely easy to *think* that you understand w/o actually
understanding it. Also you will often not get a problem, and then when
someone explains it you will realize that it was really simple all
along. However that does *not* mean that you are stupid to have not
understood the question. We are not wired to do simple things. We are
wired to do certain complex tasks well. For example think about how you
would get a person and a computer to (a) judge someone elses
approximate emotional state, and (b) do 1000 computations...getting the
right answer w/o needing to double-check. Task (b) is a simpler task,
but it certainly is *NOT* easy for the person. But if you think about
getting the computer to do it, the first task (which is easier for the
person) is actually a lot more complex of a task. :-)
Ben Tilly
Benjamin J. Tilly
s...@tools3teradata.com (Stan Friesen) writes:
[...]
> Well, that would help in *doing* the proofs, for those few courses
> where doing proofs is required.
>
> However, for those of us who are taking math to learn a set of *tools*
> to use in doing other things, the proofs seem to be *irrelevent*.
>
I would disagree, although I would agree that running through proofs
and having nobody follow them (as tends to happen) *is* useless.
> I mean, in *applying* a mathematical technique, I generally do not
> do proofs, or even use published proofs. At the most, I directly
> apply the stated theorems, but more likely I just use a method
> that produces the proper result.
>
What I would want is to have the student *understand* the ideas in the
material. That way the theorems and the techniques would make sense.
This would let them develop a sense for what is true and false in the
topic. The point of proofs is to get the students to see the subjects
insides. To give them a sense of *why* things are true, and what can go
wrong. This, ideally, would give them the sense that I was talking
about before. However IMO mathematicians tend to mix up logical rigour
and clarity of expression in teaching. As a result the actual effects
of doing the proofs is exactly the opposite in many cases from the
desired result.
> Furthermore, in most of the math courses I took, even the tests did
> *not* require doing proofs, just the working out of problems [for
> instance: compute the integral of some hairy formula]. In such
> courses spending hours listening to the prof. do proofs on the board
> seems sort of silly.
>
I agree that there should be more connection between what is tested and
what is presented.
>
> So the queastion is: what is the significance of the proofs presented
> in such classes? Why are they there at all?
I think that I explained that above. I think that it is also clear how
this does not work. So what should mathematicians do? How important is
logical rigour in an introductory course. I have a good friend who
thinks that it is not at all important. I hear him say, "This is not
quite right but this is how it works." And I have also seen him get
through to students and get them to see what is going wrong. As I heard
one student complain, "Why can't we learn how things go right before
being shown every way for things to go wrong?"
What do other people think? Any suggestions for how we *should* do it?
Ben Tilly
hill...@msc.cornell.edu
[IQ tests]
> Not much in my books. The basically test your ability on tests. I
> happen to be very good on timed tests. But there are people who I would
> say are more intelligent who do significantly worse than I do on timed
> tests. IMO the difference is that they do not have the same sorts of
> skills that I have on tests. Something that I find interesting is that
> women do significantly worse than men on the SAT's...and significantly
> better in college.
[...]
Defining better by better grades I suppose. The problem being that there's
a significant difference in the distribution of majors, and grade inflation
has hit math and the sciences much less than it has other areas. Hence woman
get better grades in college.
SEH
Jim Carr
>
> Unfortunatly this tells us as much about the reason that people
> hate science as his comment. The responsibility lies with the student,
> not the prof. If your E+M book sucks, find another (I suggest, found.
> of Electromag thoery, by ____, ____, and christy (sorry forgot the first
Reitz and Milford. I also think it is the best book for independent study
of certain key parts of undergrad E+M.
I know that in my entire career I only walked out of one classroom, and
that guy was having a really bad year with his chairman and refused to
tell us his name -- just his number -- and proceeded to lecture about
escaping from submarines instead of linear algebra. When done, he
started *reading* from the textbook. Did teach me a few things about
preparing for class, I must admit, and about how fast one could change
sections if you really wanted to do so.
What I found most interesting about the teacher-student relationship
was learned while in both roles during graduate school. I noticed
that I was telling my students the same thing the professors were
telling my classes! Hmmmm. Lets see if that shoe fits....
> ... Teachers are a public resource. A RESOURCE meaning that
> you are to use them as an aid to teaching yourself the material. ...
I think of the role as that of guide: helping you avoid pitfalls and
crevasses, and pointing you toward some of the better viewpoints along
the trail. You can go to the public slide show by the famous explorer,
but there is no substitute for taking the hike yourself.
I hope that analogy is not too subtle for this widely posted forum ....
--
J. A. Carr <j...@scri.fsu.edu> | "The New Frontier of which I
Florida State University B-186 | speak is not a set of promises
Supercomputer Computations Research Institute | -- it is a set of challenges."
Tallahassee, FL 32306-4052 | John F. Kennedy (15 July 60)
Herman Rubin
>In article <42...@tdbunews.teradata.COM>
>s...@tools3teradata.com (Stan Friesen) writes:
>
>[...]
>> Well, that would help in *doing* the proofs, for those few courses
>> where doing proofs is required.
>> However, for those of us who are taking math to learn a set of *tools*
>> to use in doing other things, the proofs seem to be *irrelevent*.
The first thing in learning to use mathematical tools, is to learn the
language. The next is to learn the concepts.
To apply mathematics to a "real world" problem, it is necessary to first
transform the problem to a mathematical problem; this involves using
mathematics as a language. To use this language, as any other language,
requires knowing what the terms used mean, i.e., the concepts.
Then the formulated problem can be operated on by the grammar rules of
mathematics, namely, the valid computational methods, theorems, correct
transformations, etc. This produces other mathematical statements which
follow from the formulation. Some of these statements translate back
into the real world results available.
Now it is possible in very simple problems to dispense with the first
step, but should someone trying to learn to use the tools do that?
Furthermore, it often helps to understand the theorems if proofs are
presented. In general, I as a professor would not expect those not
going into mathematics to be able to produce original proofs.
>I would disagree, although I would agree that running through proofs
>and having nobody follow them (as tends to happen) *is* useless.
>> I mean, in *applying* a mathematical technique, I generally do not
>> do proofs, or even use published proofs. At the most, I directly
>> apply the stated theorems, but more likely I just use a method
>> that produces the proper result.
But see above. The student who learns a technique without understanding
what it means is likely to misapply it. Of what use is it for an
engineer to apply a technique of solving differential equations to a
setup which was selected because the solution method was known, or there
is a library function to get it? People applying statistics, including
physical scientists, are prone to do this.
>What I would want is to have the student *understand* the ideas in the
>material. That way the theorems and the techniques would make sense.
>This would let them develop a sense for what is true and false in the
>topic. The point of proofs is to get the students to see the subjects
>insides. To give them a sense of *why* things are true, and what can go
>wrong. This, ideally, would give them the sense that I was talking
>about before. However IMO mathematicians tend to mix up logical rigour
>and clarity of expression in teaching. As a result the actual effects
>of doing the proofs is exactly the opposite in many cases from the
>desired result.
I agree that this is done far too often.
>> Furthermore, in most of the math courses I took, even the tests did
>> *not* require doing proofs, just the working out of problems [for
>> instance: compute the integral of some hairy formula]. In such
>> courses spending hours listening to the prof. do proofs on the board
>> seems sort of silly.
Except possibly for mathematicians, potential theoretical scientists who
will be doing such developments too often to rely on the computer, and
those who will program the methods, I cannot see any point in having
students compute integrals just to compute integrals. But engineering
and science students greatly prefer those problems rather than problems
requiring the students to think about how to go about attacking a problem.
The students want plug-and-chug, and there are professors who seem to
have difficulty in realizing that there is more than plug-and-chug
and formal proofs.
>I agree that there should be more connection between what is tested and
>what is presented.
>>
>> So the queastion is: what is the significance of the proofs presented
>> in such classes? Why are they there at all?
>
>I think that I explained that above. I think that it is also clear how
>this does not work. So what should mathematicians do? How important is
>logical rigour in an introductory course. I have a good friend who
>thinks that it is not at all important. I hear him say, "This is not
>quite right but this is how it works." And I have also seen him get
>through to students and get them to see what is going wrong. As I heard
>one student complain, "Why can't we learn how things go right before
>being shown every way for things to go wrong?"
>
>What do other people think? Any suggestions for how we *should* do it?
Rigor is important in the most introductory course, while completeness
is not. Giving the proofs, especially in full generality, is completeness.
In practical situations, errors are committed by assuming that all functions
are integrable, or even continuous, or that all distributions are normal.
Students have been unable to compute the definite integral of e^{-|x|} over
the entire real line, claiming that the antiderivative "cannot be computed."
To them sgn(x)(1 - e^{-|x|}), which is an indefinite integral of that, is
not a function. But even this is not needed.
The proofs, or indications of them, also help in understanding the concepts,
as well as their use. A student who has seen the proof by induction of the
derivative of x^n is far more likely to be able to use it than one who has
memorized the definition.
Andrew Poutiatine
>|> that they have no talent for, which is something which many lecturers & teachers
>|> should bare in mind if they want more sympathy from Joe Public.
>|>
>
>Oh come on!
>
>There's no such thing as talent.
>
>as posts for the most part.
>
>I was also called talented at math/physics at one point in my short life, but
>I could remember the hours I spent at the physics library trying to understand
>a concept I could just rattle off in class, and I got a lot humbler. I remember
>when I was a kid, I didn't know _anything_, I had to learn it all, damn it, and
>
>There is no math talent, there is no physics talent. There is interest in math
>
>
Ron, this is a crock of shit. I am sorry, but different people learn,
comprehend, apply, make mental connections, grasp concepts, and just plain
think in different ways and in different degrees. You cannot be serious
that if you take two newborn babies and expose them to the exact same things
that they will play basketball just as well, and be just as competent in
math or physics. There certainly *is* such a thing as innate talent, you
are a case in point: if you "spent (your) entire waking life learning
physics," then why haven't you achieved the same level of competency or
contributed the same amount to the field of physics as others who have
spent the same or less time and effort as you? How come most of us have
never heard of your work except here?
You cannot seriously argue that Mozart reached the level of competency
and brilliance that he did by the age of 6, or whatever age he was when he
was composing, by just plain hard work. Many have worked just as hard for
a lot longer without achieving a comparable level of proficiency.
A lot of people try extremely hard to comprehend somthing, or learn to
do something, just as hard as the "greats" in their respective fields
without ever themselves attaining true excellence. It is not a mere matter
of effort, although it is extremely important.
A fair amount of what you write on the net is in my opinion valuable, but
I am afraid the above contention is just jibberish.
-Andrew
Benjamin J. Tilly
hru...@snap.stat.purdue.edu (Herman Rubin) writes:
> Rigor is important in the most introductory course, while completeness
> is not. Giving the proofs, especially in full generality, is completeness.
> In practical situations, errors are committed by assuming that all functions
> are integrable, or even continuous, or that all distributions are normal.
> Students have been unable to compute the definite integral of e^{-|x|} over
> the entire real line, claiming that the antiderivative "cannot be computed."
> To them sgn(x)(1 - e^{-|x|}), which is an indefinite integral of that, is
> not a function. But even this is not needed.
>
I admit to repeating to the students that I tutor, "A function is just
a rule. You give me a number, and I give you one out according to the
rule. That is all that there is to a function." Not *quite* precise,
but good enough at their level.
> The proofs, or indications of them, also help in understanding the concepts,
> as well as their use. A student who has seen the proof by induction of the
> derivative of x^n is far more likely to be able to use it than one who has
> memorized the definition.
Something else that the students do not seem to understand is the value
of examples. If you cannot remember, for example, the second derivative
test, then just remember that it works for the functions x^2 and -x^2,
and you will always be able to get it straight. I have a large stock of
questions that I have. Also I think that we should try to focus more on
ideas rather than formulas. Before anyone flames me, let me give a
specific example. Most textbooks introduce matrices as a rectangular
array of numbers. Then they define a multiplication on them and list
algebraic properties. (Usually w/o a proof that it is associative.)
Then they say something about linear functions being matrices. Here is
where I was at this point in the first course that I took that had
them.
1) Where did that multiplication come from?
2) Why is this of interest?
3) What was that multiplication rule?
4) How are they really linear functions?
Here is the way that I *wish* that I had seen it. (Assume that this is
a multivariable calculus course.) Start with a discussion of the
tangent line formula in single variable calculus, f(x) is approx. f(c)
+ f'(c)(x-c). Discuss why we need to have something for f'(c), but we
have the other things. Explain why we are going to take that to be a
linear function when we come to it. Talk about linear functions and
explain how we need a notation for them. Explain how it is that knowing
what a linear function does to the standard basis tells you what it
does to anything. Then define the matrix representing the
transformation to be [T(e_1) T(e_2) ... T(e_n)] where each vector is
written as a column. Get them to understand the usual matrix
multiplication of a vector as just T acting on the vector. Then define
matrix multiplication as composition of functions. (This would ideally
not just take one class, and there would definitely be a number of
homework exercises that would be related.) Of course once you are done
this then there would have to be a number of exercises in which they
got comfortable with the notation. (But that *is* all that a matrix
is.) (The only book that I have seen come close to this approach to
matrices is Halmos' _Finite-Dimensional Vector Spaces_. Which is,
unfortunately, not IMO a good first introduction to the subject.)
So as you see I am not advocating having them ignore the notation.
Instead I would want them to master the notation *but* I do not think
that we should just throw essentially unmotivated notation at them.
Ben Tilly
Ron Maimon
|> In article <2b3j2n$d...@scunix2.harvard.edu> rma...@husc9.Harvard.EDU (Ron Maimon) writes:
I know I promised to shut up, and I hope this won't cause a flood of replies.
But I need to say a few closing words.
|> >
|> >Oh come on!
|> >
|> >There's no such thing as talent.
|> >
|> >Believe me, I know. I met a lot of supposedly talented people and they're dumb
|> >as posts for the most part.
|> >
|> >I was also called talented at math/physics at one point in my short life, but
|> >I could remember the hours I spent at the physics library trying to understand
|> >a concept I could just rattle off in class, and I got a lot humbler. I remember
|> >when I was a kid, I didn't know _anything_, I had to learn it all, damn it, and
|> >I didn't do it any faster then anyone else (I just seemed to 'cause I spent my
|> >entire waking life learning physics)
|> >
|> >There is no math talent, there is no physics talent. There is interest in math
|> >or physics, or art for that matter, and there is luck.
|> >
|> >But you will never find the "math genes"
|> >
|>
|> Ron, this is a crock of shit. I am sorry, but different people learn,
|> comprehend, apply, make mental connections, grasp concepts, and just plain
|> think in different ways and in different degrees. You cannot be serious
|> that if you take two newborn babies and expose them to the exact same things
|> that they will play basketball just as well, and be just as competent in
|> math or physics. There certainly *is* such a thing as innate talent, you
|> are a case in point: if you "spent (your) entire waking life learning
|> physics," then why haven't you achieved the same level of competency or
|> contributed the same amount to the field of physics as others who have
|> spent the same or less time and effort as you? How come most of us have
|> never heard of your work except here?
that's becuase I just turned 20! I was just a teenager last month, for god's
sake.
Also I feel no drive to discover something new. I am perfectly content in
understanding what has already been discovered. If I understand something
new, I will tell somebody, but what do I care if I get credit? I don't
need to prove anything to the world about my intelligence.
|>
|> You cannot seriously argue that Mozart reached the level of competency
|> and brilliance that he did by the age of 6, or whatever age he was when he
|> was composing, by just plain hard work. Many have worked just as hard for
|> a lot longer without achieving a comparable level of proficiency.
Mozart is just not very brilliant. I don't like his music and I never have. I
think it is pretty bad music, and to answer Alberto, yes I have heard
Mozart, yes I can tell it apart from other classical music, usually by it's
extreme cheese factor, and no, I have not been moved to tears by it.
Yes, I used to be a classical music buff when I was a little boy. Yes I grew
up.
J mascis formed Dinosaur in 1985, when he was in his teens. That's the youngest
I have seen a brilliant musician be. I am not going to argue about this, because
this is a matter of taste, and no one will be convinced.
But it doesn't matter. Whenever you sample a large enough pool, you find someone
exceptional. I just think it has zero to do with innate talent. I have never
found something I couldn't do because my "innate talent" came in the way. I think
the reason other people find these obstacles is because they want to see them.
That doesn't mean that I can do anything. I can't play chess at all, for
instance, and I will never be able to.
That's because I _hate_ that stupid game. It's so boring. I couldn't bring myself
to do all that thinking about something that dull, I'd go crazy.
But a ten year old kid who can bring himself to do this is hailed as a genius. Go
figure!
|>
|> A lot of people try extremely hard to comprehend somthing, or learn to
|> do something, just as hard as the "greats" in their respective fields
|> without ever themselves attaining true excellence. It is not a mere matter
|> of effort, although it is extremely important.
It's not effort, but it's not genes either. It's more like "randomness". But
it's cumulative randomness, a kid who knows a little more math then other kids
in the class will be pushed to learn more, the kid who knows less will be
put in a slower class, etc.
|>
|> A fair amount of what you write on the net is in my opinion valuable, but
|> I am afraid the above contention is just jibberish.
well, you can pick and choose what to read. I'm afraid I can't pick and choose
what I believe.
Ron Maimon
Ron Maimon
|> In article <CFz4L...@mentor.cc.purdue.edu>
|> hru...@snap.stat.purdue.edu (Herman Rubin) writes:
|>
|> > Rigor is important in the most introductory course, while completeness
|> > is not. Giving the proofs, especially in full generality, is completeness.
|> > In practical situations, errors are committed by assuming that all functions
|> > are integrable, or even continuous, or that all distributions are normal.
|> > Students have been unable to compute the definite integral of e^{-|x|} over
|> > the entire real line, claiming that the antiderivative "cannot be computed."
|> > To them sgn(x)(1 - e^{-|x|}), which is an indefinite integral of that, is
|> > not a function. But even this is not needed.
|> >
|> I admit to repeating to the students that I tutor, "A function is just
|> a rule. You give me a number, and I give you one out according to the
|> rule. That is all that there is to a function." Not *quite* precise,
|> but good enough at their level.
I think that its misleading to suggest that the rigorous definition is somehow
better then the defintion you gave. They are the same thing in different
languages, and the mathematical language is not necessarily better.
The only thing different about the rigorous definition is that the sets the function takes as values and spits out values into don't have to be numbers, but
thats about it.
|>
|> > The proofs, or indications of them, also help in understanding the concepts,
|> > as well as their use. A student who has seen the proof by induction of the
|> > derivative of x^n is far more likely to be able to use it than one who has
|> > memorized the definition.
I think proof by induction is the most awful way to prove something to a student.
The best way to prove the derivative formula for x^n is to say
"the derivative of a function is a good approximation to f(x+h) which looks
like f(x) + ch where you can adjust c to be as good a fit as possible near x.
then note that (x+h)^n = (x+h)(x+h)....(x+h) = x^n + nx^(n-1) h + (something)h^2
this proves by inspection that the derivative is nx^(n-1).
not to mention that this works to prove every result about calculus easily:
e.g.
f(x+h)g(x+h) = ( f(x) + f'(x) h + (order h) ) (g(x) + g'(x) h + (order h) ) =
f(x)g(x) + h(f'g+g'f) + (order h)
etc.
this is the best way to introduce the derivative, not as the limit of some
quotient, or even as the tangent line, although, of course, these must be
mentioned.
[rest of Ben's really great post on teaching linear algebra deleted. Go read it
if you plan to teach linear algebra sometime in your life]
Ron Maimon
William Tyler
>In article <CFznn...@dartvax.dartmouth.edu>, Benjamin...@dartmouth.edu (Benjamin J. Tilly) writes:
>|> In article <CFz4L...@mentor.cc.purdue.edu>
>|> hru...@snap.stat.purdue.edu (Herman Rubin) writes:
>|> I admit to repeating to the students that I tutor, "A function is just
>|> a rule. You give me a number, and I give you one out according to the
>|> rule. That is all that there is to a function." Not *quite* precise,
>|> but good enough at their level.
>
>I think that its misleading to suggest that the rigorous definition is somehow
>better then the defintion you gave. They are the same thing in different
>languages, and the mathematical language is not necessarily better.
>
>The only thing different about the rigorous definition is that the sets the function takes as values and spits out values into don't have to be numbers, but
>thats about it.
You may think that "that's about it," except that it's wrong - close,
good for teaching the idea intuitively, but not quite correct. The
most serious flaw is not the nature of the sets - that's a natural
extension - but the idea that the process of giving back a number can
be done according to a rule. To most listeners, that implies some sort
of algorithm, but most functions cannot be computed with any
algorithm, or at least any algorithm that can be executed by a Turing
machine.
>I think proof by induction is the most awful way to prove something to a student.
Why? I assume that you understand induction. Induction leaves me with
a more comfortable feeling that the proof is correct than the
handwaving you did in calculating derivatives in your posting.
Oh yeah. Does this really need to go to all the newsgroups that are
getting it? I've set followups to soc.culture.scientists, so that I
don't have to use induction on the newsgroups line.
Bill
--
Bill Tyler wty...@adobe.com
Herman Rubin
>|> In article <CFz4L...@mentor.cc.purdue.edu>
>|> hru...@snap.stat.purdue.edu (Herman Rubin) writes:
..................
>|> I admit to repeating to the students that I tutor, "A function is just
>|> a rule. You give me a number, and I give you one out according to the
>|> rule. That is all that there is to a function." Not *quite* precise,
>|> but good enough at their level.
>I think that its misleading to suggest that the rigorous definition is somehow
>better then the defintion you gave. They are the same thing in different
>languages, and the mathematical language is not necessarily better.
>The only thing different about the rigorous definition is that the sets the function takes as values and spits out values into don't have to be numbers, but
>thats about it.
This is an important point; there is no one precise definition. I personally
would even avoid the use of the term "definition" and rather use
"characterization." Stating that function = rule is precise and correct.
It is important that the students not get misled by sloppiness and lack
of rigor; most of the time, precise formulations are no more difficult.
..........................
>I think proof by induction is the most awful way to prove something to a student.
>
>The best way to prove the derivative formula for x^n is to say
>
>"the derivative of a function is a good approximation to f(x+h) which looks
>like f(x) + ch where you can adjust c to be as good a fit as possible near x.
>
>then note that (x+h)^n = (x+h)(x+h)....(x+h) = x^n + nx^(n-1) h + (something)h^2
>this proves by inspection that the derivative is nx^(n-1).
Now how did you get the binomial formula? Every means of deriving it
involves induction somewhere. Of course, one can just have the students
memorize the formula, but how does that contribute to understanding?
>not to mention that this works to prove every result about calculus easily:
>e.g.
>
>f(x+h)g(x+h) = ( f(x) + f'(x) h + (order h) ) (g(x) + g'(x) h + (order h) ) =
>
>f(x)g(x) + h(f'g+g'f) + (order h)
>
>etc.
>this is the best way to introduce the derivative, not as the limit of some
>quotient, or even as the tangent line, although, of course, these must be
>mentioned.
This does occur in what is called differential algebra. It can be done
in situations where there are not even any limits possible. But it is
a purely formal theory. Using this formalism, one can get, without any
use of limit, that sin'(x) = cos(x) sin'(0) and cos'(x) = -sin(x) sin'(0),
etc. This approach is far less likely to be understood. This is far
more formal than the most formal version of Euclidean geometry, and I
even question the approach for prospective mathematics majors.
Daniel Rubin
>|> QFT. One of them is 16 and the other is 18. Both got their first degrees
>|> at the tender age of 14. Tell me, do you think you could have gone off
>|> to college at 12, get first class honours and continue on to grad school?
Hmm I have to agree with the aspect of creativity being the deciding factor.
I know why half this newsgroup is this thread, people really don't like to
be classified as lacking intelligance. In my personal experiences I have met
many people who pick things up quickly, but are about as creative as rock. I
don't think I have met too many people who are creative but very slow which
leads me to believe that creativity is a better indicator of intellegence
than how fast someone picks things up etc...
IMHO a person is a genious if they both pick this up quickly and are creative
enough to use what they know in totally unique and undiscovered ways.
>Just because they know their field theory at 16 and I know it at 20 doesn't
>mean that they know more of it, or know it better.
Who cares how long it take them to learn it, what are they going to do with
what they know?
Hasn't this all be studied before anyways for things like the IQ tests etc...
- Dan
--
Daniel Rubin ru...@cblpf.att.com _/_/_/_/ _/_/_/_/ Go Bucks
Systems Administrator _/ _/ _/_/ _/ _/
AT&T Bell Labs Columbus, Ohio _/_/_/_/ _/_/ _/ _/
(614) 860-6487 Go Bucks _/_/_/_/ _/_/_/_/
Ron Maimon
I didn't use the binomial formula:
I used what is known as "foil" by high school students and "the distributive
law" by college students.
I had
(x+h)(x+h)....(x+h)
you need to multiply all that out. You want the two terms that are lowest order
in h. It's pretty clear that these are x^n and nx^(n-1)h just by inspection. A
formal proof, would, of course, require induction. But this is the proper place
for induction- to prove statements that it's obvious where they came from, not
to show the validity of formulas pulled out of a hat.
It is possible also to _derive_ the binomial formula from the calculus of finite
differences without any tricks. The way to do this is to call the binomial
coefficients Cmn. That is
(a+b)^n = (sum) Cnm A^m B^(n-m)
then notice thar the definition
(a+b)^(n+1)= (a+b)(a+b)^n
gives you a (solvable) difference equation for the coeffients Cmn. This is
the way I proved it to myself in high school, because I didn't appreciate
the combinatorical arguments that were made for it, I just wasn't smart
enough to understand them. Of course, after solving the difference equations
I said "oh, that's what they mean" and it was all clear.
of course, there are shorter ways of deriving this- like doing a taylor series
once you know the derivative of a power, which can be gotten only from the first
two terms of the expansion, which are obvious anyway.
|>
|> >not to mention that this works to prove every result about calculus easily:
|>
|> >e.g.
|> >
|> >f(x+h)g(x+h) = ( f(x) + f'(x) h + (order h) ) (g(x) + g'(x) h + (order h) ) =
|> >
|> >f(x)g(x) + h(f'g+g'f) + (order h)
|> >
|> >etc.
|>
|> >this is the best way to introduce the derivative, not as the limit of some
|> >quotient, or even as the tangent line, although, of course, these must be
|> >mentioned.
|>
|> This does occur in what is called differential algebra. It can be done
|> in situations where there are not even any limits possible. But it is
|> a purely formal theory.
From what I know of differential algebra, it is just that, algebra. I am doing
analysis when I say that f(x+h)=f(x) + f'(x)h + (order h) because the definition
of order h is that it's limit is zero even when you devide by h.
Using this formalism, one can get, without any
|> use of limit, that sin'(x) = cos(x) sin'(0) and cos'(x) = -sin(x) sin'(0),
I don't see how. You need limits to define sin and cos properly in the first
place. Even if they are just defined algebraically by their addition law, you
still need the values of the function at zero and their derivative at zero
to define them uniquely.
|> etc. This approach is far less likely to be understood. This is far
|> more formal than the most formal version of Euclidean geometry, and I
|> even question the approach for prospective mathematics majors.
This is the way I learned multivariable calculus in my college calculus class. It
was very intuitive, as I recall. I think you are confusing the technique I give
with the horrendously formal algebraic differentiation you mentioned before.
Ron Maimon
Herman Rubin
>In article <CG10v...@mentor.cc.purdue.edu>, hru...@snap.stat.purdue.edu (Herman Rubin) writes:
>|> In article <2bck11$l...@scunix2.harvard.edu> rma...@husc9.Harvard.EDU (Ron Maimon) writes:
>|> >In article <CFznn...@dartvax.dartmouth.edu>, Benjamin...@dartmouth.edu (Benjamin J. Tilly) writes:
......................
>|> >then note that (x+h)^n = (x+h)(x+h)....(x+h) = x^n + nx^(n-1) h + (something)h^2
>|>
>|> >this proves by inspection that the derivative is nx^(n-1).
>|>
>|> Now how did you get the binomial formula? Every means of deriving it
>|> involves induction somewhere. Of course, one can just have the students
>|> memorize the formula, but how does that contribute to understanding?
>I didn't use the binomial formula:
>I used what is known as "foil" by high school students and "the distributive
>law" by college students.
>I had
>(x+h)(x+h)....(x+h)
>you need to multiply all that out. You want the two terms that are lowest order
>in h. It's pretty clear that these are x^n and nx^(n-1)h just by inspection. A
>formal proof, would, of course, require induction. But this is the proper place
>for induction- to prove statements that it's obvious where they came from, not
>to show the validity of formulas pulled out of a hat.
There are too many places where misusing intuition can be deadly. And what is
that difficult about induction? It can be understood by first graders; they
have not mislearned enoubh.
>It is possible also to _derive_ the binomial formula from the calculus of finite
>differences without any tricks. The way to do this is to call the binomial
>coefficients Cmn. That is
>(a+b)^n = (sum) Cnm A^m B^(n-m)
>then notice thar the definition
>(a+b)^(n+1)= (a+b)(a+b)^n
>gives you a (solvable) difference equation for the coeffients Cmn. This is
>the way I proved it to myself in high school, because I didn't appreciate
>the combinatorical arguments that were made for it, I just wasn't smart
>enough to understand them. Of course, after solving the difference equations
>I said "oh, that's what they mean" and it was all clear.
This also involves induction. One does not have to bring in the addition
complication of the number of combinations, which involves induction. Just
about everything here involved induction; there is no point in hiding it.
>of course, there are shorter ways of deriving this- like doing a taylor series
>once you know the derivative of a power, which can be gotten only from the first
>two terms of the expansion, which are obvious anyway.
>
But a Taylor series involves already knowing the properties of derivatives.
>|> >not to mention that this works to prove every result about calculus easily:
>|> >e.g.
>|> >
>|> >f(x+h)g(x+h) = ( f(x) + f'(x) h + (order h) ) (g(x) + g'(x) h + (order h) ) =
>|> >
>|> >f(x)g(x) + h(f'g+g'f) + (order h)
>|> >
>|> >etc.
>|>
>|> >this is the best way to introduce the derivative, not as the limit of some
>|> >quotient, or even as the tangent line, although, of course, these must be
>|> >mentioned.
>|> This does occur in what is called differential algebra. It can be done
>|> in situations where there are not even any limits possible. But it is
>|> a purely formal theory.
>From what I know of differential algebra, it is just that, algebra. I am doing
>analysis when I say that f(x+h)=f(x) + f'(x)h + (order h) because the definition
>of order h is that it's limit is zero even when you devide by h.
>
> Using this formalism, one can get, without any
>|> use of limit, that sin'(x) = cos(x) sin'(0) and cos'(x) = -sin(x) sin'(0),
>I don't see how. You need limits to define sin and cos properly in the first
>place. Even if they are just defined algebraically by their addition law, you
>still need the values of the function at zero and their derivative at zero
>to define them uniquely.
This is not the case. One can merely define them as they were done by the
Greeks 2000+ years ago. The key equations are the equations for sin and
cos of sums. Then use the multiplication and function of a function rule.
The value of sin'(0) depends on the unit; radians are chosen to make this
one. If degrees were used, it would be pi/180.
>|> etc. This approach is far less likely to be understood. This is far
>|> more formal than the most formal version of Euclidean geometry, and I
>|> even question the approach for prospective mathematics majors.
>
>
>This is the way I learned multivariable calculus in my college calculus class. It
>was very intuitive, as I recall. I think you are confusing the technique I give
>with the horrendously formal algebraic differentiation you mentioned before.
It takes so little more to use precise rigorous formulations instead of the
sloppy ones. It is not necessary to use the cute definitions.
Charles Manning
[stuff deleted]
>
> Golly, if you refuse to learn unless your prof spoonfeeds you,
> good luck, you're going to need it.
>
The problem with math
> teaching today isn't bad teachers but lack of work. In math
> it isn't enough to understand the concepts in a broad way but
> you must be able to handle the detail with proficiency. That
> is only achievable by solid, good old hard work. If math
> teaching needs any reform, it is in the area of demanding more
> work from students, making exams real knowledge tests and
> creating an atmosphere of self responsibility.
Complete bosh!!
I am fortunate in having had an easy ride with math + physical science
at school. It was just something I understood well and did well
accordingly. I never had to study much for exams etc, but math was *fun*
for me so I would do extra problems and further work for enjoyment.
However for kids who don't have this luck, a good teacher is required.
At high school we had an excellent teacher. He realised that math isn't
something that you can force into kids brains. Different kids needed
things explained differently to help them 'see the light'. Once a math
concept makes sense (as opposed to just rote learning), it is becomes
easy for the kid to understand/use etc. This teacher was good because he
would try one or two ways to explain something. If he couldn't get
through to a particular kid, he would get another kid who understood to
sit down and explain it. He would watch and learn a new way of
explaining. He *never* used the prescribed text books. These are written
to make teachers' lives easier and are a guide. A real live teacher is a
must for most kids.
I don't know of a single kid who didn't do better under this teacher.
One 15 yr old guy (who had just moved from an exclusive school and had
been written off as a math failure) had his marks go from 17% to 65% in
six months. By the time he took his final exam be was hitting high 70's
and was often helping explain concepts to other kids. This pupil is now
31 and is a successful auditor. His life would have been very different
if he had not had a good math teacher.
I know a lot of kids who have gone through the reverse process under bad
teachers.
Does the story of Garfield High ring a bell?
The start-at-page-one-and-keep-going-till-the-end-of-the-book types
crush kids confidence. A math teacher must have the interest and
patience to try to make math and science accessible and meaningful to
kids. Math requires understanding and can't just be rote learned like
history or other book-based subjects.
> If you don't want to do solid work on the details until you
> can manage them even when you're asleep, mathematical sciences is
> the wrong thing for you. And the same applies for any science.
Sure hard work is involved, but the text books and teachers are far too
often very poor.
> -Alberto-
--
--------------------------------------------------------------------------
Charles Manning Internet: mann...@nz.dialogic.com
Dialogic (New Zealand) Phone: (+64) 9 3021831 Fax: (+64) 9 3021793
--------------- When all else fails, find a scapegoat -------------------
Benjamin J. Tilly
> |> I admit to repeating to the students that I tutor, "A function is just
> |> a rule. You give me a number, and I give you one out according to the
> |> rule. That is all that there is to a function." Not *quite* precise,
> |> but good enough at their level.
>
> I think that its misleading to suggest that the rigorous definition is somehow
> better then the defintion you gave. They are the same thing in different
> languages, and the mathematical language is not necessarily better.
>
> The only thing different about the rigorous definition is that the sets the function takes as values and spits out values into don't have to be numbers, but
> thats about it.
>
Not at all. Very often there is no "rule" to it. Also there are all
sorts of details of set theory that make for differences. Not, of
course, that it really matters at their level. (Or even later if you
are not into set theory and logic. :-)
> |>
> |> > The proofs, or indications of them, also help in understanding the concepts,
> |> > as well as their use. A student who has seen the proof by induction of the
> |> > derivative of x^n is far more likely to be able to use it than one who has
> |> > memorized the definition.
>
> I think proof by induction is the most awful way to prove something to a student.
>
I disagree. However students should think a bit about why it works...
I also like to point out that induction is just recursion run
backwards, and vica versa. (I will let people think about why. :-)
> The best way to prove the derivative formula for x^n is to say
>
> "the derivative of a function is a good approximation to f(x+h) which looks
> like f(x) + ch where you can adjust c to be as good a fit as possible near x.
>
> then note that (x+h)^n = (x+h)(x+h)....(x+h) = x^n + nx^(n-1) h + (something)h^2
>
> this proves by inspection that the derivative is nx^(n-1).
>
> not to mention that this works to prove every result about calculus easily:
>
> e.g.
>
> f(x+h)g(x+h) = ( f(x) + f'(x) h + (order h) ) (g(x) + g'(x) h + (order h) ) =
>
> f(x)g(x) + h(f'g+g'f) + (order h)
>
> etc.
>
> this is the best way to introduce the derivative, not as the limit of some
> quotient, or even as the tangent line, although, of course, these must be
> mentioned.
>
I am not so sure that this is the best way to prove the derivative
formula, but the underlying idea is important, namely that if you write
down a linear function that is a really good approximation to the
function in "close" to c, then it is the tangent line at c, and the
slope of that line is the derivative.
If students could come out of their first calculus course having
learned that, then I would be happy. From it they can remember fairly
easy heuristic derivations of all of the important rules in their first
course, and then they will be in a good position to understand
multivariable calculus if it is taght to them the way that I would want
it taught.
> [rest of Ben's really great post on teaching linear algebra deleted. Go read it
> if you plan to teach linear algebra sometime in your life]
*grin*
Ben Tilly
Kathryn Jean Drennan
*why* it works. Knowing the whys and hows of things allows you to apply it
in a manner that wasn't necessarily presented in class. However, instructors
should try to understand that for undergrads, especially those wanting to go
on to grad school or professional school, high grades are very important.
Because of this, undergrads tend to worry a lot about things like what will
be on the next exam. This is not because we consider the material to be
unimportant, but because we need to put the highest emphasis on the things
which will effect our future directly. Just my $0.02 worth.
--
------------------------------------------------------------------------------
| Kathryn Jean Drennan KJD...@tamsun.tamu.edu | | "Multiple bonds are good bonds." |
------------------------------------------------------------------------------
Ron Maimon
|> >
|> > I think proof by induction is the most awful way to prove something to a student.
|> >
|> I disagree. However students should think a bit about why it works...
|>
|> I also like to point out that induction is just recursion run
|> backwards, and vica versa. (I will let people think about why. :-)
|>
Well, I didn't mean that induction itself is wrong, or somehow difficult, but
it is an awful way to present something non-obvious.
for example, say I wanted to prove that the sum of the integers from 1 to n is
n(n+1)/2. I could do it by induction. This requires, of course, that someone
smarter then me derive the formula first by some "inspiration" and then give
it to me to prove by induction. As a matter of fact, some books actually do
urge students to try to come up with "inspirations" and try to prove them by
induction.
So I see it that there are two ways to teach induction- "useful induction" and
"abusive induction".
Useful induction is induction to prove theorems where the result is clear from a
simple manipulation that is iterative. For example, the proof that mn=nm is quite
a good example of useful induction. This type of induction is necessary. It's
also a very good exercise in building appreciation of formal proof, and it is
a really beautiful way to prove elementary facts about arithmetic.
Abusive induction is induction given as a "derivation" of a formula. For example,
if someone ever states the binomial theorem, a legitemate question I can ask is
"how did you know that it's true" and if he says "well, it's clear by induction"
I get really mad, because that is only a proof that he didn't derive it himself
but read it in a book somewhere, memorized it, and is giving a heuristic
last-ditch proof. If he proves other, more obvious things, by induction - like
deriving the calculus of finite differences, and then deriving the binomial
formula, I would be much more satisfied at the end of the proof.
It's like those expressions for grad, div, and laplacian in polar coordinates
that are given in introductory EM textbooks. You can prove them by expanding them
out and comparing them to what the chain rule would give, or you can just use
tensor calculus to see why they take the form that they do.
People mistakenly think that the way these "neat expressions" were derived was
that someone _really_ smart guessed the form and then proved it by induction.
This is completely untrue. They were derived by some way that is all too often
forgotten, and then the proof was made into a rote exercise. People don't just
sit around and get "inspired" about math, even if that is the way the
mathematical world likes to present its results.
Ron Maimon
Godfrey Degamo
: In article <1993Oct29.1...@hubcap.clemson.edu> tay...@hubcap.clemson.edu (C Taylor Sutherland III) writes:
: >a...@kpc.com (Alberto Moreira) writes:
: >
: >that there are a lot of crummy books for science and math out there where
: >the author assumes you have allready failed this class before and therefore
: >have a general knowledge of it. It is up to the prof to make the class
: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
: >enjoying and, failing that, at least innovative.
: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^. !!!!!!!!!!!!!!!!!
: Unfortunatly this tells us as much about the reason that people
: hate science as his comment. The responsibility lies with the student,
: not the prof. If your E+M book sucks, find another (I suggest, found.
: of Electromag thoery, by ____, ____, and christy (sorry forgot the first
: two names) In any case your responsibilities as a student include
: finding any way possible to learn, including going to class, and possibly
: even *gasp* doing so many problems that they dribble out your nose. (by the
: way, what would the force exerted by the drops be ?) As a wonderful prof.
: of mine told me, Teachers are a public resource. A RESOURCE meaning that
: you are to use them as an aid to teaching yourself the material. Mr Sutherland's point was that science must be practiced and practiced and practiced.
: through problems and problems and then some reading to do more problems,
: and perhaps a little thought... OK a LOT of thought, not just re-reading
: things. If a teacher is boring to you well that is unfortunate, and if he
: makes his class inovative then great, but those things are mainly asides to
: where the learning is really done.
: (Whew' now that that is out.. 8-)) -George
: >
: >
: >
: >--
: > How many docs does it take to get to the center of AmigaDOS 3.0?
: > The world may never know.
: > The Fly Boy <| E-MAIL: tay...@hubcap.clemson.edu |>
: >+--<| Activating peril-sensitive sunglasses! |>--+
Well, I once repeatedly asked my professor if he could recommend
some tutor for me in mathematical logic so that I can understand the
course not just to merely get by. HE LAUGHED AT ME. And jokingly
pointed to a person in his office and snickered "why don't you ask him?"
So, when I took set theory with the same professor, I asked if he
could help me better understand the concept of "for all" (I had a hard
time grasping this concept and still do. I think it's because at the
time I was confusing it with the issue of computability: "But how can
you make a statment of all the members in an infinite set if you can't
see them all at once, and if you check each member to see if they have
property X, then you may never finish because the set is infinite.")
Anyways, my professor told me, "I can't help you with that."
Well from my experience with both mathematical logic and set theory, I
can tell you I lost all motivation for learning the subject.
It is only now, that I can look back and actually appreciate, though
not understand the course I went through.
I use to think that it was the students responsibility for learning
the material, but I've had similar incidents like this with several
professors, and not just math. So, it makes me wonder just how much
of it is students responsiblity or just how much it is the professor's
responsibility for killing his student's spirit.
-G. Degamo,
spac...@acs.bu.edu
Benjamin J. Tilly
kjd...@tamsun.tamu.edu (Kathryn Jean Drennan) writes:
> Wouldn't any "tool" such as math be much more useful if the user understood
> *why* it works. Knowing the whys and hows of things allows you to apply it
> in a manner that wasn't necessarily presented in class. However, instructors
> should try to understand that for undergrads, especially those wanting to go
> on to grad school or professional school, high grades are very important.
> Because of this, undergrads tend to worry a lot about things like what will
> be on the next exam. This is not because we consider the material to be
> unimportant, but because we need to put the highest emphasis on the things
> which will effect our future directly. Just my $0.02 worth.
> --
Then why should we not try to test understanding in some way, therefore
pushing the students to learn what we think is most important?
Incidentally this *is* possible. If there is sufficient interest I will
describe the steps that have been taken at Dartmouth to make people's
grades reflect their ability to understand the concepts.
Ben Tilly
Benjamin J. Tilly
rma...@husc9.Harvard.EDU (Ron Maimon) writes:
> In article <CG1Du...@dartvax.dartmouth.edu>, Benjamin...@dartmouth.edu (Benjamin J. Tilly) writes:
> |> >
> |> > I think proof by induction is the most awful way to prove something to a student.
> |> >
> |> I disagree. However students should think a bit about why it works...
> |>
> |> I also like to point out that induction is just recursion run
> |> backwards, and vica versa. (I will let people think about why. :-)
> |>
>
> Well, I didn't mean that induction itself is wrong, or somehow difficult, but
> it is an awful way to present something non-obvious.
>
> for example, say I wanted to prove that the sum of the integers from 1 to n is
> n(n+1)/2. I could do it by induction. This requires, of course, that someone
> smarter then me derive the formula first by some "inspiration" and then give
> it to me to prove by induction. As a matter of fact, some books actually do
> urge students to try to come up with "inspirations" and try to prove them by
> induction.
>
I disagree, and this is a good example of why. I remember back in
highschool working out how to add the sum from 1 to n of 1, n, n^2,
n^3, n^4, and showing that I could continue on and do it for any
exponent. You know how I did it? BY UNDERSTANDING INDUCTION. What I did
is worked out what conditions I would need for an induction proof to
work, and then wound up with several equations in several variables to
solve.
Furthermore back in highschool I managed to prove a version of Fermat's
little theorem, specifically I showed that n^p-n was divisible by p.
The proof that I came up with was by induction. (In fact in trying to
prove this result I derived induction for myself.) Until I got to
university I did not know that I had been anticipated, that there were
other proofs, or that I was not the first to come up with the principle
of induction.
Try to figure out how I did it. :-)
> So I see it that there are two ways to teach induction- "useful induction" and
> "abusive induction".
>
> Useful induction is induction to prove theorems where the result is clear from a
> simple manipulation that is iterative. For example, the proof that mn=nm is quite
> a good example of useful induction. This type of induction is necessary. It's
> also a very good exercise in building appreciation of formal proof, and it is
> a really beautiful way to prove elementary facts about arithmetic.
>
Agreed.
> Abusive induction is induction given as a "derivation" of a formula. For example,
> if someone ever states the binomial theorem, a legitemate question I can ask is
> "how did you know that it's true" and if he says "well, it's clear by induction"
> I get really mad, because that is only a proof that he didn't derive it himself
> but read it in a book somewhere, memorized it, and is giving a heuristic
> last-ditch proof. If he proves other, more obvious things, by induction - like
> deriving the calculus of finite differences, and then deriving the binomial
> formula, I would be much more satisfied at the end of the proof.
>
Agreed. But there are other situations in which induction is a good way
to figure out new results. Specifically it gets your mind focused on
the inductive step. Often trying to figure out what will work in the
inductive step will suggest a new theorem to you.
> It's like those expressions for grad, div, and laplacian in polar coordinates
> that are given in introductory EM textbooks. You can prove them by expanding them
> out and comparing them to what the chain rule would give, or you can just use
> tensor calculus to see why they take the form that they do.
>
Tensor calculus is a lot newer than those expressions.
> People mistakenly think that the way these "neat expressions" were derived was
> that someone _really_ smart guessed the form and then proved it by induction.
> This is completely untrue. They were derived by some way that is all too often
> forgotten, and then the proof was made into a rote exercise. People don't just
> sit around and get "inspired" about math, even if that is the way the
> mathematical world likes to present its results.
This is also often untrue. Induction is not just a way to prove
unmotivated results that make no sense. Induction is also a way of
thinking that can suggest new results. But here is how it can work in
situations where induction was *not* the motivation. Suppose that you
get a vauge understanding of something after thinking about the problem
for a long time. Then you want to try to formalize it and prove it.
Eventually what you are forced to do is invent some formal notation,
and then do things with an induction proof. This is not because you
thought of the induction proof first, but because you have no way to
describe very well the way that you *did* think of it. Therefore the
first proof that you could write down was the induction proof, even
though induction was *not* how you thought of it. I am not just trying
to be vague here, I am describing the process of discovery that I have
followed myself. And if you think that I am being confusing now, then
you do *not* want to try the way that I thought about the problem. (The
example I am thinking of wound up with a *horrible* notation. I had a
subscript of a subscript of a superscript on a variable with a
subscript of a subscript also. Trust me, it makes sense when you get
there, and it is in a lemma that I described as "technical". :-)
Therefore I believe that your dislike of induction is based on a lack
of understanding of the process of using it in practice. That being
said, it does provide a simple way to crank out answers to a variety of
problems. The fact that the insights that lead to the idea for those
problems may or may not involve induction is not an argument against
it. Especially not in situations where the previous reasoning was only
heuristic in nature. (You would probably have to actually run across
one of these situations before you undersand what I mean.)
Ben Tilly
Herman Rubin
>|> > I think proof by induction is the most awful way to prove something to a student.
>|> I disagree. However students should think a bit about why it works...
>|> I also like to point out that induction is just recursion run
>|> backwards, and vica versa. (I will let people think about why. :-)
>Well, I didn't mean that induction itself is wrong, or somehow difficult, but
>it is an awful way to present something non-obvious.
>for example, say I wanted to prove that the sum of the integers from 1 to n is
>n(n+1)/2. I could do it by induction. This requires, of course, that someone
>smarter then me derive the formula first by some "inspiration" and then give
>it to me to prove by induction. As a matter of fact, some books actually do
>urge students to try to come up with "inspirations" and try to prove them by
>induction.
Two things have been confused here. Following a proof requires knowing what
a proof is, and NOT how to do the original thinking to produce one. In some
cases, as the proof by induction of the sum of successive integers, or of a
geometric series, the carrying out of the proof should give the student little
problem. This is quite different from discovering the theorem; students should
be encouraged to find their own proofs, but inspiration cannot be taught.
One could get the sum of the first n integers otherwise, as the legend claims
Gauss did at age 7. But that method will not work on the sum of the first n
squares or cubes. There is a cute method for cubes, but this was only done
after the formula was known. And the simplification by using factorial
moments did not arise until the binomial theorem, and Pascal's Triangle,
become known.
......................
>if someone ever states the binomial theorem, a legitemate question I can ask is
>"how did you know that it's true" and if he says "well, it's clear by induction"
>I get really mad, because that is only a proof that he didn't derive it himself
>but read it in a book somewhere, memorized it, and is giving a heuristic
>last-ditch proof. If he proves other, more obvious things, by induction - like
>deriving the calculus of finite differences, and then deriving the binomial
>formula, I would be much more satisfied at the end of the proof.
Given the statement of the binomial theorem, proving it by induction is not
at all difficult. Given the statement of Pascal's Triangle, likewise. Even
if we knew how, we do not have the time for a student to derive everything
without being told some of them.
.................
>People mistakenly think that the way these "neat expressions" were derived was
>that someone _really_ smart guessed the form and then proved it by induction.
>This is completely untrue. They were derived by some way that is all too often
>forgotten, and then the proof was made into a rote exercise. People don't just
>sit around and get "inspired" about math, even if that is the way the
>mathematical world likes to present its results.
As a mathematician, who does derive things like this and others, I must
disagree. There are ways to go about guessing forms, and fiddling around
is not that difficult. Those who derive these formulas have a combination
of talent and luck, and putting apparently unrelated things together. It
is most often the case that others who think in terms of structure and
rigorous content will at least consider the presentation reasonable, and
use those ideas to develop new results.
Herman Rubin
>In article <2bfhtj$a...@tamsun.tamu.edu>
>kjd...@tamsun.tamu.edu (Kathryn Jean Drennan) writes:
>> Wouldn't any "tool" such as math be much more useful if the user understood
>> *why* it works. Knowing the whys and hows of things allows you to apply it
>> in a manner that wasn't necessarily presented in class. However, instructors
>> should try to understand that for undergrads, especially those wanting to go
>> on to grad school or professional school, high grades are very important.
>> Because of this, undergrads tend to worry a lot about things like what will
>> be on the next exam. This is not because we consider the material to be
>> unimportant, but because we need to put the highest emphasis on the things
>> which will effect our future directly. Just my $0.02 worth.
>Then why should we not try to test understanding in some way, therefore
>pushing the students to learn what we think is most important?
We should! To do this at the college level runs into extreme opposition,
especially in service courses. After 12 years of being told that the
important thing is to regurgitate and plug-and-chug, it is understandably
difficult for students to switch to trying to understand structure, and
to see that something is applicable when this is not immediately apparent.
They are used to using routine solutions, and never to producing formulations
which then have to be worked on, the solution taking several steps.
I do this, and I can assure you that the students are none too happy
about it.
Benjamin J. Tilly
hru...@snap.stat.purdue.edu (Herman Rubin) writes:
> In article <CG3sq...@dartvax.dartmouth.edu> Benjamin...@dartmouth.edu (Benjamin J. Tilly) writes:
[...]
> >Then why should we not try to test understanding in some way, therefore
> >pushing the students to learn what we think is most important?
>
> We should! To do this at the college level runs into extreme opposition,
> especially in service courses. After 12 years of being told that the
> important thing is to regurgitate and plug-and-chug, it is understandably
> difficult for students to switch to trying to understand structure, and
> to see that something is applicable when this is not immediately apparent.
> They are used to using routine solutions, and never to producing formulations
> which then have to be worked on, the solution taking several steps.
>
> I do this, and I can assure you that the students are none too happy
> about it.
I agree. In another post I am going to describe in detail exactly what
is being tried at Dartmouth to deal with this problem. It is not
perfect, but it is a step in the right direction.
Ben Tilly
Benjamin J. Tilly
calculus is taught at Dartmouth, I have decided to post a fairly
complete description of how the course is structured, along with a
description of the resources that it takes to do this. I will finish up
with some comments about what our experience with this is like, and
changes that we are thinking of making. Comments and ideas are welcome.
The students are the ones who have not been placed out of their first
calculus course, but who do not need remedial math. We have
approximately 300 students in 3 sections of 100. Also assigned to the
course are 6 grad students and a number of undergrad graders. (I do not
know how many offhand butI will try to find out.) The classes here ore
on a quarter system so time is tight. In any case the structure is as
follows. The students meet in lecture 3 times a week for 65 minute
classes. In these classes they hare given homework to be turned in and
marked by the graders, but which are not for marks. (However the
comments are valuble.) In addition the graders have tutorials that they
run for 2 hrs a time, 3 days a week. (Not all of the graders are there
each time.) The students can go there to ask questions about the
material, what they did not get on their homework, and so on. In
addition there are 2 midterms and a final, all multiple choice. If they
need more help with this sort of material they can go to the profs
office hours, and they can get a tutor. (There is an organized way to
arrange for tutors on this campus called the tutor clearing house.)
This brings us to the interesting part, and what the grad students do.
There is one last marked section, which is worth about a third of their
grade. At the start of the year we divide them up into groups of 4-5
for group projects. (They have to then work out meeting with each
other, and here the fact that the campus is heavily connected by
computers helps.) After the first project they are allowed to reform,
and we end up with groups of 4-6. Each grad student has around 10
groups to deal with. The projects are all open-ended. The format is
that they are given a set of problems that relates to a topic, which
will be due in one week. (They have more time for the first one.) They
are supposed to do their best with the problems, go to the clinics run
by the grad students if they need to, then sign up to meet with their
grad student for 15 min, and then the groups submits a project.
Whatever mark goes on that project goes on each persons grade. (That is
usually closer ot 20 min...) In that meeting my job is to ask them how
they are coming along, explain some of the points, indicate further
directions that they can go, and be there for them to ask more
questions of. Typical questions on the projects include things like the
following. "Is there are function which is continuous at 0 that is (3^x
- 1)/x for x other than 0?" Note that this question followed a computer
analysis of the limit involved. In the meeting it was pointed out to
them that they probably did not have the knowledge to answer this
question completely, that the computer did not prove anything, but that
they could carry out an analysis of what the relevant cases were, and
make conjectures about the answer. (This problem was on the first
assignment.) It is also up to the grad students to do the marking of
these assignments.
Which brings us to what goes on from the grad students perspective. We
start the week with a meeting with the profs. There we discuss any
problems that have come up, and the profs ask us if there is anything
that we want to discuss about the projects. At the start there were
also some meetings about how to conduct group work, and the profs were
more likely to point out details of importance. Later they accepted
that we would have thought of these, and any important questions would
come up in the discussion. Then the grad students meet to do the
marking from last week. We mark together to standardize the results. In
a typical session we will not only mark our own papers, but we will
check over a number of questionable ones that others have. (Should I
give this person this mark...) The marking system is based on our
overall impression of the papers, not a point by point system. Each
paper is out of 10, with 7 being average. (There has been one 10 so
far, and before awarding it I had two other grad students read it and
agree with me that it should be a 10.) To get a 10 you must do the
problems stated, explain things well, show that you understand, and go
further to explain things that are beyond the problems given. This
could include explaining the theory behind whatever the topic is,
taking the topic further, or noticing things about the problems that
were not pointed out. What is interesting is how obvious people trying
to bluff is. IMO you cannot get a good mark w/o a good understanding of
what is going on. In any case after the marking is done, for the rest
of the week we run our 1 hr clinic for general questions, and then we
have times posted on our doors that the students can choose from for
their meetings, which we do what I described earlier. Oh, and one of
the grad students has to keep track of all of the marks, but in return
has fewer groups.
Over the course of the semester the students will have some time to get
settled in, five 1-week projects, and a larger group project that takes
3.5 weeks. The final project is on a topic of their choice (although we
do give a list of suggestions). Topics can include such things as
learning about the history of calculus, presenting an alternate
formulation of calculus, there is one group that may do a critique of
how math 3 could be improved... As long as the topic is substantial and
involves math 3 material it is allowed. As always, those projects will
be discussed with the grad students, this time several times, and it
will be marked by the grad students.
Now here is what it is like in practice. For each grad student, and for
the students, we estimate 12-15 hrs a week. I do not know what it is
for the profs or for the graders. Not having the homework for marks has
resulted in a lot less complaints by the students, and a better
atmosphere between students and graders (who are more willing to
explain why they marked the way that they did). However we would like
to make the homework count in some way. One suggestion is that we could
always have quick open book quizzes in class with a selection of
problems from the homework. All that they have to write down are the
final answers. These quizzes would not add a lot of work, but they
would be very hard for those who did not do the homework. Secondly the
exams are multiple choice to cut down the work marking. However there
is some pressure to have a non-multiple choice question. Next we are
always trying to get more project ideas, and we do change the questions
on the assignments. (One suggestion of mine is the following. Consider
the function f(x) = x - x^2. Around 0 consider its best constant
approximation and its tangent line. (a) Which is a better
approximation? (b) Which is closer for x = 0.7? (c) Explain why (a) and
(b) do not contradict each other.)
The attitude of the students towards the projects seems to be that they
like them. (I have had a comment that the problem with the projects are
that they would like to spend more time on them, but they have other
things to do. :-) They generally do a lot worse on the first one, but
by the second they have a better idea of what we want. I do not know
what the effect is on going on in math and science, or how it affects
retention, but I have reason to believe that it is better than the
straight lecture method. Furthermore there are suprisingly few
problems. One of the main things that helps is the fact that the groups
change. Therefore if someone is not contributing then they can (and do)
get kicked out of the group. This tends to be a motivating influence.
There are other dynamics, but that freedom to switch, although
discouraged, certainly does help. In addition it seems that they form
friends within the groups and the groups turn into study groups. (IMO a
good thing!)
Are there any comments or suggestions out there about this? How about
other stories at other places?
(BTW, as you might expect, there aer a number of people here who are
very interested in education. I can name at least a half-dozen profs in
math here who regularly follow research on math education. :-)
Ben Tilly
Ron Maimon
|> In article <2bgssd$8...@scunix2.harvard.edu> rma...@husc9.Harvard.EDU (Ron Maimon) writes:
|>
|> >for example, say I wanted to prove that the sum of the integers from 1 to n is
|> >n(n+1)/2. I could do it by induction. This requires, of course, that someone
|> >smarter then me derive the formula first by some "inspiration" and then give
|> >it to me to prove by induction. As a matter of fact, some books actually do
|> >urge students to try to come up with "inspirations" and try to prove them by
|> >induction.
|>
|> Two things have been confused here. Following a proof requires knowing what
|> a proof is, and NOT how to do the original thinking to produce one. In some
|> cases, as the proof by induction of the sum of successive integers, or of a
|> geometric series, the carrying out of the proof should give the student little
|> problem. This is quite different from discovering the theorem; students should
|> be encouraged to find their own proofs, but inspiration cannot be taught.
I think that this distinction makes mathematics very boring to a lot of people,
and that youre last statement is just wrong.
You can teach student's what looks like inspiration. As a matter of fact, most
of "inspiration" is just intensive thought directed at a particular problem
from a point of view that works in another field.
For example, I will give you an anecdote.
I learned calculus inbetween my algebra and my precalculus class. When I got to
precalculus, we spent some time summing geometric and arithmetic series. I was
upset, because we did both those sums by an inspired trick, and this trick
doesn't work for any other sum. So I thought about the problem.
Well, the way I thought about it was to consider the sums of the odd numbers,
which I knew was the perfect squares. I then remembered that the difference
between two consecutive perfect squares is an odd number. This gave me a little
shock, because it suggested that summing and differencing were inverse operations
much like differentiation and integration, and I went on to derive all the
formulas for sums I had seen from that principle alone.
Then I got mad because the derivation in class was trickier, more complicated,
and harder to follow then the presentation for the calculus of finite
differences, which is natural, more general, and has that "I could have
come up with that myself"ness that the other methods lack.
|>
|> One could get the sum of the first n integers otherwise, as the legend claims
|> Gauss did at age 7. But that method will not work on the sum of the first n
|> squares or cubes. There is a cute method for cubes, but this was only done
|> after the formula was known. And the simplification by using factorial
|> moments did not arise until the binomial theorem, and Pascal's Triangle,
|> become known.
|>
|> ......................
|>
|> >if someone ever states the binomial theorem, a legitemate question I can ask is
|> >"how did you know that it's true" and if he says "well, it's clear by induction"
|> >I get really mad, because that is only a proof that he didn't derive it himself
|> >but read it in a book somewhere, memorized it, and is giving a heuristic
|> >last-ditch proof. If he proves other, more obvious things, by induction - like
|> >deriving the calculus of finite differences, and then deriving the binomial
|> >formula, I would be much more satisfied at the end of the proof.
|>
|> Given the statement of the binomial theorem, proving it by induction is not
|> at all difficult. Given the statement of Pascal's Triangle, likewise. Even
|> if we knew how, we do not have the time for a student to derive everything
|> without being told some of them.
|>
I had time to do that.
I want to emphasize that my mathematical education was completely standard, and
I had time to only see derivations of proofs that either I came up with myself,
or I had the feeling that I could have come up with them myself.
There were a few sticking points- Law of cosines I felt I couldn't have come
up with, but when I learned the dot product and vectors, it all fell into place.
The way the 180-degrees-in-a-triangle theorem is proven was too tricky for me,
I felt I couldn't have come up with that, so I wasn't satisfied until my
freshman year of college when I derived it another way. Finally, the derivation
of gamma(-.5)=sqrt(pi) is done in the most abhorrent way I could ever imagine,
a way I not only wouldn't have thought up, but a way I can hardly believe
_anyone_ thought up. it took me months of fiddeling arond with the properties
of the gamma function before I got it to pop out of stirling's (natural) formula
for n! (in a natural way) using only theorems I had derived myself.
|> .................
|>
|> >People mistakenly think that the way these "neat expressions" were derived was
|> >that someone _really_ smart guessed the form and then proved it by induction.
|> >This is completely untrue. They were derived by some way that is all too often
|> >forgotten, and then the proof was made into a rote exercise. People don't just
|> >sit around and get "inspired" about math, even if that is the way the
|> >mathematical world likes to present its results.
|>
|> As a mathematician, who does derive things like this and others, I must
|> disagree. There are ways to go about guessing forms, and fiddling around
|> is not that difficult. Those who derive these formulas have a combination
|> of talent and luck, and putting apparently unrelated things together. It
|> is most often the case that others who think in terms of structure and
|> rigorous content will at least consider the presentation reasonable, and
|> use those ideas to develop new results.
I agree that new results can be done by sitting and waiting for inspiration, I
just think this is a bad way to go about it. I have a "law of mathematics"
The "I could have come up with that myself" hypothesis
Any formula, no matter how tricky, can be derived in a way where it's trickiness
is reduced to the point that it has the property that anyone could have come up
with it.
And the related "the way I would have come up with it is also the way the guy
who first derived it came up with it" hypothesis
Any complicated formula derived in a really slick way in a textbook was derived
in a much more natural way by it's original inventor.
I know that these statements don't mean anything mathematically. They just
represent some heuristic observations about how humans do mathematics. And since
I am human, and mathematicans are usually human, and math students are certainly
human, I think that it pays off to teach them only using ideas and tricks that
they feel they could have come up with themselves- I know that its not impossible,
because that's how I learned math.
Ron Maimon
David Stein
>the students, we estimate 12-15 hrs a week. I do not know what it is
>for the profs or for the graders. Not having the homework for marks has
>resulted in a lot less complaints by the students, and a better
>atmosphere between students and graders (who are more willing to
>explain why they marked the way that they did). However we would like
>to make the homework count in some way.
Why!?! As you just explained, not grading the HW makes it more
useful, since the "graders" can concentrate on what's important.
>One suggestion is that we could
>always have quick open book quizzes in class with a selection of
>problems from the homework. All that they have to write down are the
>final answers. These quizzes would not add a lot of work, but they
>would be very hard for those who did not do the homework.
How boring and artificial. And what if I do not have the time
to do HW before the quiz? As an undergrad I had to work to pay
for college, and so presumeably do many students in Dartmouth.
Some flexibility when it comes to assignments is crucial for these
students; they can plan for long-term assignments, but not for
short-term HW.
>Secondly the
>exams are multiple choice to cut down the work marking. However there
>is some pressure to have a non-multiple choice question.
I vote to shoot any prof who gives multiple-choice question tests.
Multiple-choice belongs to the pages of popular magazines.
- David
--
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I'm not a native speaker of English, so I'm not sure what I wrote.
Flames will be ignored unless you post them in perfect Czech.
================================ - David (the metamathician) - ===
Herman Rubin
>>Now here is what it is like in practice. For each grad student, and for
>>the students, we estimate 12-15 hrs a week. I do not know what it is
>>for the profs or for the graders. Not having the homework for marks has
>>resulted in a lot less complaints by the students, and a better
>>atmosphere between students and graders (who are more willing to
>>explain why they marked the way that they did). However we would like
>>to make the homework count in some way.
> Why!?! As you just explained, not grading the HW makes it more
> useful, since the "graders" can concentrate on what's important.
>>One suggestion is that we could
>>always have quick open book quizzes in class with a selection of
>>problems from the homework. All that they have to write down are the
>>final answers. These quizzes would not add a lot of work, but they
>>would be very hard for those who did not do the homework.
> How boring and artificial. And what if I do not have the time
> to do HW before the quiz? As an undergrad I had to work to pay
> for college, and so presumeably do many students in Dartmouth.
> Some flexibility when it comes to assignments is crucial for these
> students; they can plan for long-term assignments, but not for
> short-term HW.
I agree with this. My own quiz strategy (this is for service courses)
is to give problems which can be done easily IF the student understands
the ideas, but the details are relatively unimportant.
>>Secondly the
>>exams are multiple choice to cut down the work marking. However there
>>is some pressure to have a non-multiple choice question.
> I vote to shoot any prof who gives multiple-choice question tests.
> Multiple-choice belongs to the pages of popular magazines.
With this I agree completely. Multiple choice exams provide good
topics for problems for elementary probability courses, but are poor
methods for assessing knowledge and ability. It is far more important
for a student to be able to look at a word problem and decide how to
proceed rather than to grind out answers using routine procedures.
There is a possible modification of this which is quite difficult to
carry out; I do not know if it has been done to any extent, except in
oral exams. This is to allow the student to ask for help in deciding
how to proceed, with of course no credit for that part of the problem.
Lawrence R. Mead
: In article <2bfhtj$a...@tamsun.tamu.edu>
: kjd...@tamsun.tamu.edu (Kathryn Jean Drennan) writes:
: pushing the students to learn what we think is most important?
: Incidentally this *is* possible. If there is sufficient interest I will
: describe the steps that have been taken at Dartmouth to make people's
: grades reflect their ability to understand the concepts.
: Ben Tilly
I am much interested in this. If you have already posted the followup
i have missed it somehow. Would you then repost?
Thanks, Ben.
--
Lawrence R. Mead (lrm...@whale.st.usm.edu) | ESCHEW OBFUSCATION !
Associate Professor of Physics
Diane Palme x2617
:
: You will get no arguments from me on this. Now how do we implement this?
: In a low-level course, the cheating problem is bad enough on in-class
: tests; take-home is essentially impossible. This is not something
: which an individual teacher can do much about.
My suggestion is this: if the student cheats on a test in an
introductory course, do you think that they will be able to pass a higher
level course in the same subject? Example: student A cheats in Dynamics
and gets a good enough grade to go on to Mechanisms/Machine Design. They
are now in the position to do a project on gear design (or some such thing).
Will they be able to do it? I don't think so. Will they get help from
another student? Maybe, but generally the students who put the extra effort
in the lower level course will realise that student A has *NO* idea what is
going on and will be reluctant to help them. I was the same way. If it
was obvious that a person hadn't even *TRIED* to do the problem, or had no
idea what the fundamental theory was behind the problem, well, they were on
their own or they had to go see the prof. Worrying about cheating in the
lower classes is a moot point. These students will "wash out" once they
are forced to apply the knowldege later.
Just my $0.02,
D.
--
Diane Palme, EIT You really think that A-B would
Department Engineer, Central Inspection accept my opinions as their own?
Allen-Bradley Co. (414) 382-2617 <sheesh!>
_ /\ dsp...@mke.ab.com
_|_|/ J
|___| new .sig under construction!
O O *** Watch this space for details!! ***
Boucher David
#
#I use to think that it was the students responsibility for learning
#the material, but I've had similar incidents like this with several
#professors, and not just math. So, it makes me wonder just how much
#of it is students responsiblity or just how much it is the professor's
#responsibility for killing his student's spirit.
Try teaching for a few terms and you'll have your answer. The majority
of students are bored, lazy, and not very bright. More than a few of them
are dishonest to boot. That's what happens when you fill up the colleges
with people who are there for no other reason than to get a piece of
paper that they hope will qualify them for a reasonably high-paying job
that doesn't require too much manual labor. Dynamic and innovative
teaching methods are wasted on most of them, because all they care about
is how to get the grade they want with an absolute minimum of effort.
After a few years of teaching such characters, even the most dedicated
prof is liable to get burned out. That's probably why many profs prefer
higher-level courses, where most of the zeros have already been weeded
out.
- db
--
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"Come on down to the Big Dig. Can't get around the Big Dig."
- Don Van Vliet (Smithsonian Institute Blues)
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