Calculus success rate
James F. Epperson
Is there a =documented= national figure for passing calculus courses?
Jim Epperson
The right half of the brain controls the left half of the body.
This means that only left handed people are in their right mind.
Robert J. Pease
Such a figure would be meaningless because minimum standards for
"passing" vary enormously from different professors and schools.
Maybe the number of people getting a "3" or better on the AP Calculus
exam would be interesting....They have reasonable standards for this.
James F. Epperson
> In article <Pine.SUN.3.91.981006131045.10164F-100000@zonker> you write:
> >
> >Is there a =documented= national figure for passing calculus courses?
>
> My high school calculus teacher told me that no one had ever failed
> calculus - they just failed algebra while taking calculus.
I tell my ODE students a minor variation of this -- no one flunks
differential equations, they flunk algebra and calculus while taking
diff. eq.
Jim Epperson http://members.aol.com/jfepperson/causes.html
Duct tape is like the force. It has a light side, and a dark side, and
it holds the universe together ...
-- Carl Zwanzig
Dr. Michael Albert
> "passing" vary enormously from different professors and schools.
I beg to differ. As you point out, such a figure would say little
about the mathematical ability of the students, but something about
the way in which we choose to run our society.
-Mike
James F. Epperson
> James F. Epperson wrote:
> >
> > Is there a =documented= national figure for passing calculus courses?
>
> Such a figure would be meaningless because minimum standards for
> "passing" vary enormously from different professors and schools.
True enough, but when one's department is being badgered by the
administration for a "low" success rate, it would be nice to have
something to compare ourselves to, even if it is flawed.
William L. Bahn
choose to run our society? What would be different if the rate were 50%? How
about 80%?
Dr. Michael Albert wrote in message ...
>> > Is there a =documented= national figure for passing calculus courses?
>> Such a figure would be meaningless because minimum standards for
>> "passing" vary enormously from different professors and schools.
>
Nathan C Burnett
: Maybe the number of people getting a "3" or better on the AP Calculus
: exam would be interesting....They have reasonable standards for this.
Yes, but is the subset of students who take the AP test representative
of college students in general. I would say no. If a student doesn't
already have above average mathematical ability they generally don't
take the AP Calculus exam.
Nate
--
---
Nathan C. Burnett "It is not certain that everything
na...@acm.org is uncertain."
http://www.cse.msu.edu/~nate -Blaise Pascal
( ( (((In Stereo Where Available))) ) )
Dr. Michael Albert
> choose to run our society? What would be different if the rate were 50%? How
> about 80%?
> >about the mathematical ability of the students, but something about
> >the way in which we choose to run our society.
It has been my experience that in most calculus classes, the majority
of students do not "master" the material, and in many cases the majority
of them are reasonably clueless at the end of the semester. Nevertheless,
you can't flunk 80% of the class. Not only would one get into trouble
with the administration, but truthfully most people doing this would
think of themselves as unduly cruel. After all, calculus is important
but there are other things in life. So, people grade "on a curve". But
the weird thing about grading on a curve is that it guarantees
that someone looses. In principle, if you grade "on a curve", you
have decided what fracition of the class will fail *before* grading
the exams. Of course, in practice, you would fail a lot more under
almost any reasonable standard of "mastery". Note that even though
giving 80% of the class a "D/E/F" would be considered cruel, generally
people can live with giving 5% of the class a "D/E/F". In general this
is a 5% which didn't even show up to class. Thus "mastery" is
replaced with "effort".
Please note that I am not saying this is good or bad. I must admit
that in the non-science courses I took I suspect I got more
credit for effort than mastery--especially English literature, where
I almost always managed to "miss the point", though I suspect you
would have guessed that from reading my posts :-).
Best wishes,
Mike
C. Hillman
On Tue, 6 Oct 1998, James F. Epperson wrote:
> Is there a =documented= national figure for passing calculus courses?
Sort of. Look up back issues of the Notices of the American Mathematical
Society. Most math departments at large universities which can withstand
political and economic pressure from the consumer (students and their
parents and their state senators) are reluctant to release hard data.
What hard statistics are available (and you will find some in the Notices)
suggest that the pass rate in a good college calculus course is only about
60%, as I recall (if you count drops as failures) or 75-80% (if you
don't). I know the UW math department has carefully tracked these numbers
(but forget the percentages) and I assume most other departments do too.
I surveyed my own calculus students at the beginning of each course for a
few years and found a very simple and revealing rule of thumb. These are
bright students (UW is only somewhat selective, however, with an
acceptance rate of about 75%, much higher than its peer institutions) who
have often had high school calculus courses. I found that incoming
students consistently came into the course with the expectation that they
would do one grade point -better- (3.5 versus 2.5 on a scale of 0 to 4.0)
in the college course than they had in high school, and I consistently
found that in fact they did one grade point -worse- (1.5 versus 2.5), on
average. I am confident that the reason for the discrepancy was that we
demanded true mastery of the material, including an ability to interpret
the results of computations, rather than asking simply for the ability to
plug and chug using memorized formulae. I.e. our grades, but not the HS
grades, were an accurate reflection of the students level of mastery. If
this rule of thumb holds across the board, it could explain the
unpopularity of calculus courses among students :-/
Finally, check out Tolstoy's autobiographical novel, Childhood, Boyhood,
Youth (his first published work) which should give you some valuable
perspective on the whole problem of teaching calculus to engineering
students. (Tolstoy, like his hero in the novel, flunked out of the
engineering school at the University of Kazan in his first year, although
I am not sure if this was soley as a result of failing calculus.)
Chris Hillman
Please DO NOT email me at opti...@u.washington.edu. I post from this account
to fool the spambots; human correspondents should write to me at the email
address you can obtain by making the obvious deletions, transpositions,
and insertion (of @) in the url of my home page:
http://www.math.washington.edu/~hillman/personal.html
Thanks!
C. Hillman
On 6 Oct 1998, Nathan C Burnett wrote:
> Yes, but is the subset of students who take the AP test representative
> of college students in general. I would say no. If a student doesn't
> already have above average mathematical ability they generally don't
> take the AP Calculus exam.
And my experience very strongly suggests that doing well on the AP exam is
a poor predictor of success in a college calculus course.
Brian M. Scott
<alb...@esther.rad.tju.edu> wrote:
[snip]
>It has been my experience that in most calculus classes, the majority
>of students do not "master" the material, and in many cases the majority
>of them are reasonably clueless at the end of the semester. Nevertheless,
>you can't flunk 80% of the class. Not only would one get into trouble
>with the administration, but truthfully most people doing this would
>think of themselves as unduly cruel. After all, calculus is important
>but there are other things in life. So, people grade "on a curve". But
>the weird thing about grading on a curve is that it guarantees
>that someone looses. In principle, if you grade "on a curve", you
>have decided what fracition of the class will fail *before* grading
>the exams.
For which reason genuine grading on a curve is an abomination. If one
can't stomach giving the grades that have really been earned, a much
better approach is simply to relax one's standards at the lower end
until one's stomach quiets down sufficiently.
>Note that even though
>giving 80% of the class a "D/E/F" would be considered cruel, generally
>people can live with giving 5% of the class a "D/E/F". In general this
>is a 5% which didn't even show up to class.
I don't think that I've ever reached 80%, but I'm quite certain that
on a number of occasions I've recorded grades of F, D, or W (withdrew
during first 60% of the term) for at least 50% of a calculus class.
I'd be pleasantly surprised if the rate has ever been as low as 5%,
unfortunately. (Come to think of it, I'm pretty sure that I once had
a calculus class in which 2/3 of the students earned and received
grades of D or F.) And I consider a fair number of my D and C grades
to be distressingly generous.
Brian M. Scott
bo...@rsa.com
sc...@math.csuohio.edu (Brian M. Scott) wrote:
> On Tue, 6 Oct 1998 22:08:38 -0400, "Dr. Michael Albert"
> <alb...@esther.rad.tju.edu> wrote:
>
> [snip]
>
> >It has been my experience that in most calculus classes, the majority
> >of students do not "master" the material, and in many cases the majority
> >of them are reasonably clueless at the end of the semester.
I can recall a class I took at Harvard as an undergraduate. It was taught by
D.G.M. Anderson. It was advanced Calculus with tensor analysis, some Fourier
analysis, some Calc of variations, etc. All taken by either math or applied
math or physics majors. It was considered an upper level course. Out of a
class of 80 there was one A and one A-. 25 flunked. One does not take an
upper division math class at Harvard without a considerable amount of
mathematical maturity.
The T.F. for the course said he could not have finished the (3 hour) final
is less than 8 hours. The median grade out of 350 points was in the 60 point
range. I got about 155 on the exam and got a B+. The highest grade was
190ish.
As a teacher, Prof. Anderson was the worst I have ever seen. I remember
one class on non-cartesian tensors. He filled two blackboards with material,
then said: "No, this is wrong." He erased it and started over. And damn it,
he did the same thing again!
The second half of the course was on advanced methods in diffeq's, including
partial diffeq's etc. Now, 2/3 of the class got A's. It was taught by G.
Carrier
It is therefore possible to mis-teach even the very best math students.
Comments?
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Ronald Bruck
C. Hillman <opti...@u.washington.edu> wrote:
:
:On 6 Oct 1998, Nathan C Burnett wrote:
:
:> Yes, but is the subset of students who take the AP test representative
:> of college students in general. I would say no. If a student doesn't
:> already have above average mathematical ability they generally don't
:> take the AP Calculus exam.
:
:And my experience very strongly suggests that doing well on the AP exam is
:a poor predictor of success in a college calculus course.
This may be a self-selecting phenomenon. Students who do well on the AP
(4 or better) are permitted to opt out of the first semester of calculus
at the University of Southern California, and if they get such a score on the
BC version of the test, they can opt out of the SECOND semester.
So the students we see in calculus are not generally the very best. Every
class has its exceptions, of course, which makes life tenable, but that's
generally the case here at USC.
When we try value-add-ons, such as Ohio State's Calculus&Mathematica course,
or a special calculus-for-superior-engineers course, the good students STILL
don't opt for them.
Well, I can sympathize. When I went to the University of Chicago as a first-
year student, I placed out of almost two full years of undergraduate require-
ments. I was delighted (I could take more MATH classes!), although now I
wish I'd taken more of those "unnecessary" courses when I had the chance.
--Ron Bruck
Mike McCarty
)> choose to run our society? What would be different if the rate were 50%? How
)> about 80%?
)> >I beg to differ. As you point out, such a figure would say little
)> >about the mathematical ability of the students, but something about
)> >the way in which we choose to run our society.
)
)It has been my experience that in most calculus classes, the majority
)of students do not "master" the material, and in many cases the majority
)of them are reasonably clueless at the end of the semester. Nevertheless,
)you can't flunk 80% of the class. Not only would one get into trouble
)with the administration, but truthfully most people doing this would
)think of themselves as unduly cruel. After all, calculus is important
)but there are other things in life. So, people grade "on a curve". But
)the weird thing about grading on a curve is that it guarantees
)that someone looses. In principle, if you grade "on a curve", you
)have decided what fracition of the class will fail *before* grading
)the exams. Of course, in practice, you would fail a lot more under
This is decidedly not my experience. I agree that most calculus students
understand very little of what they learn. But I found it impossible to
"grade on a curve". I found that I had a strongly bimodal distribution
of grades. The students who studied made As and Bs. The ones who did not
study made Fs. Some principles I used in making my exams:
make the problems similar to, but not identitical with, problems
considered in the text
make the problems check for actual ability to apply what is
learned, so that memorization will not suffice
put at least one problem on the test that *any* student who
studied can work it
put at least one problem on the test that likely *no* student
who just learned what was covered can work it
give 50% credit for any problem on which the student showed he
at least understood the problem, whether he could work it or not
include an "extra credit" problem which is not difficult, but
which can only be worked by someone who *understands* the
material (this goes beyond just "application", and into
"research")
I found that
very few who were failing even tried to work problems - scores
like 5-10 of 100 were common among failing students
very few even tried the extra credit, but those who did
*learned* from the exam, and thanked me afterward
I had no problem with failing half of a class, generally between
10 and 20%
most students complained that I was the "hardest teacher I have
ever had"
a few of them *thanked* me for being the "hardest teacher I have
ever had"
none of them accused me of being unfair
)almost any reasonable standard of "mastery". Note that even though
)giving 80% of the class a "D/E/F" would be considered cruel, generally
)people can live with giving 5% of the class a "D/E/F". In general this
)is a 5% which didn't even show up to class. Thus "mastery" is
)replaced with "effort".
)
)Please note that I am not saying this is good or bad. I must admit
)that in the non-science courses I took I suspect I got more
)credit for effort than mastery--especially English literature, where
)I almost always managed to "miss the point", though I suspect you
)would have guessed that from reading my posts :-).
)
) Best wishes,
) Mike
)
--
----
char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
I don't speak for DSC. <- They make me say that.
Fabio Rojas
>> >It has been my experience that in most calculus classes, the majority
>
>D.G.M. Anderson. It was advanced Calculus with tensor analysis, some Fourier
>class of 80 there was one A and one A-. 25 flunked. One does not take an
>upper division math class at Harvard without a considerable amount of
>mathematical maturity.
I think every math department has a professor like this. At Berkeley
where I did undergrad and some grad work, one professor, who shall
remain nameless, flunked half of a *linear algebra* class. These
were upper division kids, too. Not the freshman, but math majors,
engineering students, physics and the occasional econ student.
>As a teacher, Prof. Anderson was the worst I have ever seen. I remember
>one class on non-cartesian tensors. He filled two blackboards with material,
>then said: "No, this is wrong." He erased it and started over. And damn it,
>he did the same thing again!
As long as we are talking horror stories, I once attanded a course
which didn't even have lectures or a book. the professor, who also
shall remain nameless, would hand out worksheets on logic
which no one understood becuase he seemed to have his own
notational system. The class would then be dedicated to people's
valiant attempts at solving problems in class.
>
>
>The second half of the course was on advanced methods in diffeq's, including
>partial diffeq's etc. Now, 2/3 of the class got A's. It was taught by G.
>Carrier
>
>It is therefore possible to mis-teach even the very best math students.
>
>Comments?
I agree. There are always a few kids who are "wunderkinds" for lack
of a better term. Even at elite schools, most students at most schools
need some input from faculty. The wunderkinds seem to be born with
a knowledge of eigenvalues, but the rest of us need to study the material.
So the teacher makes a big difference.
Since this thread is about grading, I think there is a simple
solution which few people use but I think is fairly decent.
I call it "conditional curving". At the beginning of a course,
the instructor decides on an objective standard. 80%= A, 60% = B,
etc. This should correspond to an assesment of what a student should
learn in the class. At the end of the course, the instructor looks
at the distribution of grades according to this "ideal system".
If no body gets A's or B's, as in the example above, then the
instructore should curve the course. Why? Because given a standard
nobody has acheived, eithr the students or at fault or teh faculty
is at fault. Or even both. It is the responsibility of instructors
to assign material that the students can master in a reasonable amount
of time and effort. Faculty can tell through home work grades if
students are actually trying hard or are being lazy.
This system has the benefit of being fairly easy to use, rewarding
acheivement and not punishing the student for taking TG Anderson's class.
In the above example, the fact that 2/80 = 2.5% of the class got A's
and 20/80 = 25% got F's at one of the most elite schools is probaboly
an indiciator that it was the prof's fault. Thus, in my system
the "conditional curve" would be invoked: some people would be bumped
to A's and B's from C's and the F's would probably turn into C's
while the no-shows and the truly clueless would stay in the F category.
Excpet for those bumped from B to A, no one gets off scott free.
It is unlikely that grades would be inflated with this sort of
grading. A consolatioon grade of C or B isn't to bad.
-fabio
Steve Monson
>Since this thread is about grading, I think there is a simple
>solution which few people use but I think is fairly decent.
>I call it "conditional curving". At the beginning of a course,
>the instructor decides on an objective standard. 80%= A, 60% = B,
>etc. This should correspond to an assesment of what a student should
>learn in the class. At the end of the course, the instructor looks
>at the distribution of grades according to this "ideal system".
>If no body gets A's or B's, as in the example above, then the
>instructore should curve the course.
My high school chemistry/physics teacher used a variant of this approach,
and it seemed to work fairly well. He had a numeric basis for grades, say
90%=A, 80%=B, etc., but for each test, he averaged the top three
grades and called that 100%. Thus, rather than a strict curve, where
a certain percentage of the class was guaranteed to get a given grade,
the perfect score was based on how the best students did. The others
were then graded by that standard. If the top scores were not right up
around 90-100%, he figured maybe the tests was too difficult, or covered
things he had not explained well enough.
We students always thought it seemed fair enough.
Steve Monson
--
Little Willie was a chemist.
Little Willie is no more.
What he thought was H2O
Was H2SO4.
Brian M. Scott
(Fabio Rojas) wrote:
[snip]
>Since this thread is about grading, I think there is a simple
>solution which few people use but I think is fairly decent.
>I call it "conditional curving". At the beginning of a course,
>the instructor decides on an objective standard. 80%= A, 60% = B,
>etc.
This is not an objective standard. Depending on the examinations and
the details of scoring, 60% could be achievable only by an
extraordinarily good student, achievable by all but the very weakest
students, or anything in between.
> This should correspond to an assesment of what a student should
>learn in the class. At the end of the course, the instructor looks
>at the distribution of grades according to this "ideal system".
>If no body gets A's or B's, as in the example above, then the
>instructore should curve the course. Why? Because given a standard
>nobody has acheived, eithr the students or at fault or teh faculty
>is at fault.
And if (for the sake of argument) the students are at fault, why
should the grades be adjusted?
> Or even both. It is the responsibility of instructors
>to assign material that the students can master in a reasonable amount
>of time and effort.
And it is the responsibility of the students to master it.
> Faculty can tell through home work grades if
>students are actually trying hard or are being lazy.
This is simply not true. (Moreover, effort is pointless if it's
ill-directed.)
[snip]
Brian M. Scott
Herman Rubin
Fabio Rojas <aap...@mazel.spc.uchicago.edu> wrote:
>In article <6vg0ec$9fn$1...@nnrp1.dejanews.com>, <bo...@rsa.com> wrote:
>>> >It has been my experience that in most calculus classes, the majority
>>> >of students do not "master" the material, and in many cases the majority
>>> >of them are reasonably clueless at the end of the semester.
>>I can recall a class I took at Harvard as an undergraduate. It was taught by
>>D.G.M. Anderson. It was advanced Calculus with tensor analysis, some Fourier
>[snip]
>>class of 80 there was one A and one A-. 25 flunked. One does not take an
>>upper division math class at Harvard without a considerable amount of
>>mathematical maturity.
This is very UNlikely to be the case. They are much more likely to
have an ability to do computational manipulations, which, while they
may be useful, have nothing to do with any understanding. It would
be far better if they had done little computing, but had acquired
understanding. The concepts in the upper division courses can be
taught to someone who has had a GOOD high school algebra course, and
a GOOD high school geometry course, emphasizing proofs. Unfortunately,
these courses are rare, and at this time I doubt that the mathematics
departments could POLITICALLY get away with requiring calculus
students to understand the integers including induction, and the
basic structure of the real numbers. I doubt if they could have an
"upper division" algebra course, which really has not college
prerequisites, which teachers abstract algebra as a tool in
understanding linear algebra.
No, every computational course we put the students through is likely
to add to their difficulty to understand mathematical concepts.
...................
>I agree. There are always a few kids who are "wunderkinds" for lack
>of a better term. Even at elite schools, most students at most schools
>need some input from faculty. The wunderkinds seem to be born with
>a knowledge of eigenvalues, but the rest of us need to study the material.
>So the teacher makes a big difference.
Anyone who teaches these as the roots of |xI - A| = 0 already makes it
hard to understand. The ideas are easy, IF one presents them abstractly
in the first place.
>Since this thread is about grading, I think there is a simple
>solution which few people use but I think is fairly decent.
>I call it "conditional curving". At the beginning of a course,
>the instructor decides on an objective standard. 80%= A, 60% = B,
>etc. This should correspond to an assesment of what a student should
>learn in the class. At the end of the course, the instructor looks
>at the distribution of grades according to this "ideal system".
>If no body gets A's or B's, as in the example above, then the
>instructore should curve the course. Why? Because given a standard
>nobody has acheived, eithr the students or at fault or teh faculty
>is at fault. Or even both. It is the responsibility of instructors
>to assign material that the students can master in a reasonable amount
>of time and effort. Faculty can tell through home work grades if
>students are actually trying hard or are being lazy.
But what does the grade mean? If it means anything other than an
absolute measure of what the student knows and can do, the transcript
is meaningless. The present "system" is too much as you describe it;
it is unfortunate that we have anyone in any educational system who
would consider adjusting a given course to the backgrounds and
abilities of those who happen to enrol, unless they have what is
needed for the course. In many cases, one has no way of even
assessing this; no amount of computational mathematics is any
indication of the ability to understand mathematical concepts, let
alone proofs.
>This system has the benefit of being fairly easy to use, rewarding
>acheivement and not punishing the student for taking TG Anderson's class.
It does not reward achievement at all. Relative positions in a class
convey no information about how well a student has learned a subject.
>In the above example, the fact that 2/80 = 2.5% of the class got A's
>and 20/80 = 25% got F's at one of the most elite schools is probaboly
>an indiciator that it was the prof's fault. Thus, in my system
>the "conditional curve" would be invoked: some people would be bumped
>to A's and B's from C's and the F's would probably turn into C's
>while the no-shows and the truly clueless would stay in the F category.
>Excpet for those bumped from B to A, no one gets off scott free.
>It is unlikely that grades would be inflated with this sort of
>grading. A consolatioon grade of C or B isn't to bad.
This is the mockery which the educational system has foisted on us.
If everyone in the class knows the material which is supposed to be
taught in that course, they should all get good grades. If they do
not, they should get bad grades. Whether the instructor did a poor
job of teaching should not affect the absolute meaning of the grades.
I have seen many letters of recommendation from people who have
supposedly taught students the foundations of analysis saying that
this is a good student, and yet the student does not have a clue
as to what constitutes an open set in Euclidean space.
If grades are to have any value, they must be absolute. Courses
should be, as well. Even Podunk can teach good students good
courses, or if it cannot, it should make arrangements for the
students to learn them otherwise.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
William L. Bahn
assumes that your class of thirty students is a random sampling of all
students and that the mean and standard deviations for your class is
sufficiently close to the rest of the country/world that assigning
individual grades that have impact outside of the school (getting into
college, getting scholarships, etc) based on the group performance is
fundamentally flawed.
But the issue that Steve points out is a good one. You need something to
distinguish a poor exam from a poor examinee. I would probably take the top
three (or five or whatever) and call that something like 90% or 95% - since
you would not expect the average score of the best three students to be
perfect.
What I have noticed is that if you put the scores on a histogram - then
there are usually groupings of scores and the breaks between the groupings
are good places to put the break between grades. This, of course, is grading
on a curve and has the problems mentioned in the first paragraph. So the
second thing that must be done is to ask if the students are actually
getting approximately the grades that you subjectively feel they have
earned - neither too much lower nor too much higher than that.
What I would like to see is a nationally standardized test made available
(in each course) that teachers could use to get a feel for where their
students are on a national scale. In other words, give them information
about what the mean and std deviation of their students is compared to the
actual mean and std deviations for the entire nation. Then they can use that
information to curve the grades they give for a given class with a bias that
keeps the final grades more in line with a nationally applicable
interpretation. In other words, that a student from Mississippi that gets a
B in calculus is at least somewhere close to a student from California that
gets a B. The problem with this is that there are too many lazy teachers out
there that will simply use the national test as their sole means of
assigning their grades and I don't think that any such test would be good
enough to use it for that.
Part of the problem is that we have to decide what an grade is supposed to
mean. Is it an indicator of how the student did according to their immediate
peers, perhaps without regard to how well they did compared to a larger
group or without regard to prior groups? Or is it supposed to be an
indicator of how well they learned the material, irrespective of how many
others might have learned it better or learned it worse? These are two very
different measures and they provide you with quite different information. I
always liked scoring reports that gave you both the raw score and the
percentile score so that you had at least some measure of both.
Steve Monson wrote in message <6vsu8n$5...@euphony.tri.sbc.com>...
>See ye here Fabio Rojas's writings:
>>Since this thread is about grading, I think there is a simple
>>solution which few people use but I think is fairly decent.
>>I call it "conditional curving". At the beginning of a course,
>>the instructor decides on an objective standard. 80%= A, 60% = B,
>>etc. This should correspond to an assesment of what a student should
>>learn in the class. At the end of the course, the instructor looks
>>at the distribution of grades according to this "ideal system".
>>If no body gets A's or B's, as in the example above, then the
>>instructore should curve the course.
>
Brian M. Scott
wrote:
[snip]
>But the issue that Steve points out is a good one. You need something to
>distinguish a poor exam from a poor examinee. I would probably take the top
>three (or five or whatever) and call that something like 90% or 95% - since
>you would not expect the average score of the best three students to be
>perfect.
An exam on which the scores range from 20% to 80% is not necessarily a
poor exam. Indeed, if the class is a fairly normal cross-section, and
the scores are spread out through that range, the exam is quite likely
to have been a fairly good one. (One must also be on guard for
outliers.)
[snip paragraph with which I agree]
>What I would like to see is a nationally standardized test made available
>(in each course) that teachers could use to get a feel for where their
>students are on a national scale.
Why? (1) Any standardized test that's likely to be feasible on that
scale isn't worth the effort. (2) What is 'the same course' at
another school?
[snip]
Brian M. Scott
fgr...@my-dejanews.com
Send responses to aaprana" at" mazel.uchicago.edu
hru...@b.stat.purdue.edu (Herman Rubin) wrote:
> In article <F0p2t...@midway.uchicago.edu>,
> Fabio Rojas <aap...@mazel.spc.uchicago.edu> wrote:
> >>I can recall a class I took at Harvard as an undergraduate. It was taught by
> >>D.G.M. Anderson. It was advanced Calculus with tensor analysis, some Fourier
> >[snip]
> >>class of 80 there was one A and one A-. 25 flunked. One does not take an
> >>upper division math class at Harvard without a considerable amount of
> >>mathematical maturity.
>
> This is very UNlikely to be the case. They are much more likely to
Why is this so unlikely? I think some very high quality students need
decent instruction. Incompetent instructors can easily confuse good
students if it is the first time they have ever encountered proofs
as done in a real analysis course. Here is an example: at UC Berkeley
there is a professor - e-mail me personally if you want names and dates -
who taught an honors vector calculus class about banach spaces. Granted that
the class of Banach spaces includes R^3 and the vector spaces studied
in such a course, but it is a wholly inappropriate way to teach sophmore
level calculus. As a result, about a third of the class failed with only
one or two A's out of a class of 80. Isn't the responsibility of the professor
to
a) teach relevant material at the appropriate level of difficulty
b) communicate it effectively
c) administer exams that test the students on the material that was
clearly communicated ?
I know lots of instructors who screw up these simple premises. Including
myself. Why should students suffer for the mistakes of teachers?
Don't get the idea I'm a softy either - I've given out a fair share
of D's and F's in my time. But teaching involves two parties: students
and teachers. I have never seen a class with a perfect professor
or perfect students.
> have an ability to do computational manipulations, which, while they
> may be useful, have nothing to do with any understanding. It would
> be far better if they had done little computing, but had acquired
> understanding. The concepts in the upper division courses can be
> taught to someone who has had a GOOD high school algebra course, and
> a GOOD high school geometry course, emphasizing proofs. Unfortunately,
I don't think so. I think you have a fairly limited understanding of how
human beings learn. I think what you have listed are needed but not usually
enough except for the John Milnor's of the world. Most students need
*experience* as well as *knowledge*. Knowing how to write a geometry proof is
needed but most students would benefit immensly by having exposure to
sequences, integrals, derivatives, trigonometric functions and some physics.
It's just like learning a language. You need to know how to conjugate verbs
but it's not enough. A student of French will benefit from studying books,
talking with people, reading newspapers, watching French TV, etc. What you
are claiming above is like saying that a student who only knows how to
conjugate verbs and has a dictionary will be ready to translate Sartre's
"Being and Nothingness". Of course, some kids will always be able to, but
most will be stuck on the first page. Most people need much more experience
to help them out.
> these courses are rare, and at this time I doubt that the mathematics
> departments could POLITICALLY get away with requiring calculus
> students to understand the integers including induction, and the
Why should they? Have you ever heard of the division of labor? Why must all
people who walk into a calculus class be required to write proofs? Why is it
not good enough for people just to know how to apply math ? Do we require
people who learn English to learn syntax and phonology?
This may sound like a heresy but here's my reasoning. People take
calculus for at least two reasons: 1) applications and 2) beauty
of the subject. I took calculus for #2. Most people I know took
it for #1. Why should they be forced to learn various things
that they will never use? It doesn't mean you can't be challenging
but why must we think that all calculus must revolve around
real analysis? Isn't the fact that you can use physics to solve
all sorts of engineering problems amazing enough? I think any student
who can use calculus for these ends deserves a decent grade
from me.
>
> No, every computational course we put the students through is likely
> to add to their difficulty to understand mathematical concepts.
So would you not teach arithmetic to children? That's computational. Why not
go to your local kindergarten and start with the von Neumann construction of
the ordinals? Long division is out and division rings are in pre-school !!
That's crazy. Abstraction is important but most math students probably
base their understanding on some well chosen computational examples.
Computational courses with the right emphasis can serve as a good
introduction to more abstract concepts.
From my own experience, I can say that I learned about determinants by
manually computing a few examples years before I ever Iearned about
Pfaffians.
> > The wunderkinds seem to be born with
> >a knowledge of eigenvalues, but the rest of us need to study the material.
> >So the teacher makes a big difference.
>
> Anyone who teaches these as the roots of |xI - A| = 0 already makes it
> hard to understand. The ideas are easy, IF one presents them abstractly
> in the first place.
This is true. I can't disagree but this is always harder in practice than it
is in theory. It has been about at least 10-12 years since I first learned
about eigenvalues and they seem clear to me, but every time I explain it to
someone, I always run into new problems. There is a common thread to these
learning problems: most students need a concrete computational example to
help them see the general picture. I think abstraction almost always has a
very specific underpinning in specific cases.
Hilbert once said that he proved some fact about matrices by doing the 2x2
case first. If he needed a good concrete example, then most others probably
need it as well. ( can someone jog my memory? was this a version of the
spectral theorem for matrices? polar decomposition?)
>
> >Since this thread is about grading, I think there is a simple
> >solution which few people use but I think is fairly decent.
> >I call it "conditional curving". At the beginning of a course,
> >the instructor decides on an objective standard. 80%= A, 60% = B,
> >etc. This should correspond to an assesment of what a student should
> >learn in the class. At the end of the course, the instructor looks
> >at the distribution of grades according to this "ideal system".
> >If no body gets A's or B's, as in the example above, then the
> >instructore should curve the course. Why? Because given a standard
> >nobody has acheived, eithr the students or at fault or teh faculty
> >is at fault. Or even both. It is the responsibility of instructors
> >to assign material that the students can master in a reasonable amount
> >of time and effort. Faculty can tell through home work grades if
> >students are actually trying hard or are being lazy.
>
> But what does the grade mean? If it means anything other than an
> absolute measure of what the student knows and can do, the transcript
> is meaningless.
Meaningless? That's pretty strong stuff. Real world example: grade inflation
exists even at elite schools (I've been to a few, I know). Yet, most people
are convinced that the grades and degree mean something. I don't
think people take them that seriously, but they seem to mean something
or else why would people fret about them so much?
Here is a more radical example: What if I decided that in my class
that any student who is not at the level of C.F. Gauss would get an F?
That's an objective standard. Yet the grade conveys *no information*
at all since almost every student who enters the class is guaranteed
an F.
Here is another point. Why aren't grades considered reflections of
teachers and not students? Some students are beyond help, but
if there is a high failure rate why is that not the teacher's fault?
There is no a priori reason not to consider this as a possibility
without further information about the quality of the students.
>The present "system" is too much as you describe it;
Not in the places I have been in. Maybe Purdue has lax grades
in your opinion, but getting an "A" grade in most math and physics
departments I have been to is well deserved. I've seen some
graduate algebaric geometry courses where the grades are about 50% F's
(e-mail for details if you don't believe me).
The lax grading seems to be rampant in non-quantitative
social science and humanities departments rather than math
departments.
> it is unfortunate that we have anyone in any educational system who
> would consider adjusting a given course to the backgrounds and
> abilities of those who happen to enrol, unless they have what is
> needed for the course.
I don't think you have understood my argument correctly. You seem to think
that my argument is "curve the grades so we feel better" or "relative
standing in the class is the important thing."
My argument is:
a) instructors should set objective and appropriate standards (not dependent
on students) b) instructors should help students acheive this standard (give
clear lectures, write exams based on the material covered in the course, show
up for office hours, etc) c) examine the distribution of the final scores d)
if this distribution is suspicious (it usually is not, but sometimes it is)
then consider the case that the teacher *may* have screwed up e) if you think
you gave an unusually hard exam or otherwise messed up (violate point a. or
b. ) then curve the class such that f) the grades are not distorted too much
(few people move into the A/B category and thus retain the value of working
hard, but most would move from D/F to D/C) g) if after reflection, you think
that your exams were fair (which is probably the case if 10%-20% of the class
"gets it"=A each time you teach) but people still did horribly, then stick
with the original grades h) this is a reasonable system since: 1) it is
based on the idea that grading is a question of *measurement*. Instructors
have a duty to try to accurately measure students acheivement and report
there measurements to the public and to the students through transcripts 2)
Sometimes measurement errors occur. Since there is little research on the
accuracy of college level math exams, these errors probably go uncorrected
and are unnoticed. It is the duty of the instructor to seriously investigate
whether he or she made such an error. Sometimes it may be as simple as
asking the T.A.'s or colleagues "was this too hard?" or comparing to how
students did the last time you asked a question like this. 3) Since errors
occur, there is no a priori reason to assume that the students must bear the
full cost of such mistakes. 4) Since retesting students is an unfeasible
solution and so is giving everybody an A, we make a guess about how bad our
measurement was and correct for this error. Thus my scheme above does not
reward slackers or those who didn't learn. People at the bottom stay at the
bottom. Only a few people would move into the high grade category: probably
in such a system, we move from 0-5% A's to 5-15%. The main effect would be
to move the mean from D/F range to the C range. Since the A's would still be
in the small minority and C's don't get you anywhere, it doesn't debase hard
work. 5) It uses relative position as a proxy for ability only if there is
sufficient reason to believe measurement error has occured. But I argue that
you should stick with a class of all F's if you think the test was fair.
The point is that when measurement error has occured, you have no other
information about the student except the relative class position. I
conjecture that this is not such a bad estimator of ability on the average
in the *absence of other information*.
That may have seemed wordy, but I think it is good to lay all my
cards on the table so that an incorrect oversimplification of my
argument does not pass as the real one.
>In many cases, one has no way of even
> assessing this; no amount of computational mathematics is any
> indication of the ability to understand mathematical concepts, let
> alone proofs.
This is quite wrong. Do you know of *any* mathematicians who learned
Galois theory before knowing the quadratic formula? If you had
a student who couldn't integrate x^2, would you recommend that
he take real analysis because his inability to do a high school
calculus problem is no indication of his mathematical ability?
Would you recommend that a person who had difficulty with SAT level algebra
problems jump to a graduate course on commutative algebra?
Hog wash!! We use computational ability as a judge of math ability
all the time. Of course, there are always *some* kids who are horrid
at freshman calculus but stunning at real analysis, but for most
there is a correlation.
Real example: A kid gets a D in vector calculus. He asks Herman Rubin
whether he should take real analysis. I would hope that Herman Rubin would
recommend against it but let the kid give it a shot anyway. I seriously
doubt he would say, "I think there is a real good chance you would
be great at it.That 'D' grade is just your computational math grade.
Don't worry." If he did say that, I would sue for advisor malpractice!
> >This system has the benefit of being fairly easy to use, rewarding
> >acheivement and not punishing the student for taking TG Anderson's class.
>
> It does not reward achievement at all. Relative positions in a class
> convey no information about how well a student has learned a subject.
>
> >grading. A consolatioon grade of C or B isn't to bad.
>
> This is the mockery which the educational system has foisted on us.
Having at least half the class get a C or less (the curving which is
present in many math departments) is not a "mockery". A grade
of C is not good at all. It usually means you can't get into grad school
or professional schools. A single C can blow your chances for
high honors, etc. Getting a C is in no way getting off easy. just
because a some people got C's, does not pull down the A's.
Mockery is grading in the humanities.You virtually have to be dead to get a
low grade in a humanities class. You can get away without knowing anything or
having done the reading in many classes. From my experience, the A's in
curved math classes (usually less than 20%) are kids that actually know
something. I have rarely seen a graduate of an American math department with
a high GPA who actually didn't know something about math.
Grading in math, chemistry and physics may be the last refuge of
remotely accurate grading in American education.
You should go hang out in a humanities department to get
some perspective.
> If everyone in the class knows the material which is supposed to be
> taught in that course, they should all get good grades.
Why? Is it not possible that the text may be confusing? or the
professor is confusing? Or the exam was not good?
Why do you insist on assuming that teachers are always perfect
and students are always at fault? Before I get flamed, let me say
once again that students often deserve D's and F's. I once
gave 50% F's in a "college algebra" class. But sometimes I
have given a bad exam and I don't want to punish the students
for my mistake.
> If they do
> not, they should get bad grades. Whether the instructor did a poor
> job of teaching should not affect the absolute meaning of the grades.
Why not? If your doctor screws up, you get to sue. If your lawyer
screws up, you get to appeal. If the waiter brings the wrong food, you
get to send it back.
Why are math teachers magically exempt from this rule?
If the teacher implied that you should learn A but tested on B (which
has happened to me), how can the grade not be misleading?
the message I get from you is that : "it is the responsibility of the
student to read the mind of the teacher. the actions of the
teacher in class are irrelevant. No matter how grave the teahcer's
error is, students are always to blame. Teachers are not
responsible at all for the quality of teaching. Even if the teacher
speaks in Sanskrit and writes in Chineese, the teacher is not
to blame for the poor grade."
>
> I have seen many letters of recommendation from people who have
> supposedly taught students the foundations of analysis saying that
> this is a good student, and yet the student does not have a clue
> as to what constitutes an open set in Euclidean space.
Actually, I think this is a different problem completely. In many schools,
vector calculus and other topics go under the name of "real analysis" and the
courses never teach a single thing about metric spaces. Foundations of
analysis can mean Fourier series, some numerical analysis, basics of proof
writing, etc. At Berkeley, applicants were required to write down what texts
were used for their advanced math class. If Rudin or Herrstein (for algebra)
was listed, then that was an indicator of quality. Most faculty I knew had a
good sense of which schools really taught analysis and which taught glorified
calculus and judged applicant accordingly.
The problem is about the label "foundations of analysis" and not
about standards. I think there are some decent standards
but a lot confusion about what happens in classrooms after
the end of the calculus sequence.
> > If grades are to have any value, they must be absolute. Courses
> should be, as well. Even Podunk can teach good students good
> courses, or if it cannot, it should make arrangements for the
> students to learn them otherwise.
I agree with you here. I would prefer that students transfer to
a better school than get a second rate education at Podunk.
I think the main difference between you and I is that we have different
opinions on what happens in classrooms. I bet that we would probably agree on
grades about 90% or more of the time in the real world.
I don't really believe that classroom grades can be absolute in a very
meaningful way because of all the noise and error that gets into measuring
students' abilities. As a teacher of statistics, you must have some
appreciation of this issue
I do believe students have a true level of ability. I do believe that
teachers can asses this most of the time. But there is a significant but not
gigantic amount of error and it just seems crazy to me to assume this doesn't
happen and that we instructors have no duty to mitigate the results of our
errors. I do not advocate the abandonment of standards, just the idea that
when we screw up we should have back up plans that are fair. "Let them eat
cake" is not a good grading policy as far as I am concerned.
I hope I haven't been too offensive.
> Herman Rubin,
> hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Fabio Rojas
fgrojas at harper.uchicago.edu
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Brian M. Scott
> hru...@b.stat.purdue.edu (Herman Rubin) wrote:
[most snipped]
> >In many cases, one has no way of even
> > assessing this; no amount of computational mathematics is any
> > indication of the ability to understand mathematical concepts, let
> > alone proofs.
> This is quite wrong. Do you know of *any* mathematicians who learned
> Galois theory before knowing the quadratic formula? If you had
> a student who couldn't integrate x^2, would you recommend that
> he take real analysis because his inability to do a high school
> calculus problem is no indication of his mathematical ability?
> Would you recommend that a person who had difficulty with SAT level algebra
> problems jump to a graduate course on commutative algebra?
Your response is a logical non sequitur. Complete lack of computational
skill *may* be an indicator of inability to understand the concepts, but
that wasn't Herman's point. His point - and in my experience he's
basically right - is that computational skill is at best a very poor
indicator of ability to understand concepts.
> Hog wash!! We use computational ability as a judge of math ability
> all the time.
The point is that we shouldn't, because it's a lousy indicator.
> Having at least half the class get a C or less (the curving which is
> present in many math departments) is not a "mockery". A grade
> of C is not good at all.
It *should* be perfectly satisfactory, however.
> Mockery is grading in the humanities.You virtually have to be dead to get a
> low grade in a humanities class.
That depends entirely on the class. I know quite a few humanities
instructors whose standards are higher than those of some of my
colleagues in the math department.
> > If everyone in the class knows the material which is supposed to be
> > taught in that course, they should all get good grades.
> Why? Is it not possible that the text may be confusing? or the
> professor is confusing? Or the exam was not good?
Eh? What does this have to do with Herman's statement?
> > If they do
> > not, they should get bad grades. Whether the instructor did a poor
> > job of teaching should not affect the absolute meaning of the grades.
> Why not? If your doctor screws up, you get to sue. If your lawyer
> screws up, you get to appeal. If the waiter brings the wrong food, you
> get to send it back.
So take the grade off the record. But don't make it dishonest.
Brian M. Scott
Fabio Rojas
>fgr...@my-dejanews.com wrote:
>> hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>> >In many cases, one has no way of even
>> > assessing this; no amount of computational mathematics is any
>> > indication of the ability to understand mathematical concepts, let
>> > alone proofs.
>> problems jump to a graduate course on commutative algebra?
>
>skill *may* be an indicator of inability to understand the concepts, but
>that wasn't Herman's point. His point - and in my experience he's
>basically right - is that computational skill is at best a very poor
>indicator of ability to understand concepts.
It is not a non sequitor. Herman claimed that computational
ability is no indicator of mathematical ability (writing
proofs, understanding concepts, etc.). He made a pretty
strong claim. He said "no amount of computational
mathematics is any indiciation of the ability to understand
concepts". These are his words, not mine. My response
is that if you have math ability, you may probably
have computational ability, thus, computational ability
is not such a bad way to guess that a student has
math ability *if you have no other information*.
Think about it: why do we have *any* prerequisites
for advanced math courses? Why not have all freshman
take real analysis? There's a simple reason. Instructors
want to see if the student can handle taking derivatives
and solving word problems. This is "computational" rather
than "mathematical". If they can understand how to integrate
by parts and calculate radii of convergence, then there
is a good chance that they could handle proof writing.
It's not certain. It's not perfect. In fact, in the real
world of teaching we actually use this as a method for
selecting students. If a student wants to major in math or
statistics, the first questions are usually "what were
your calculus grades?" OF course, most of the kids who take
real analysis do badly because the do not have math. ability,
but it is completely reasonable to use computational ability
as a *first guess* about the potential to do math.
I think this is an empirical matter that can't be settled
in a forum like this. What one would have to do is to
figure out the computational abilities of sucessful
math students and compare them with proof writing skills.
If bet that most proof writers can do copmutations, but most
"copmuters" can't write proofs. Which is consistent with
what I claim, but not with what Herman claims.
>
>> Hog wash!! We use computational ability as a judge of math ability
>> all the time.
>
>The point is that we shouldn't, because it's a lousy indicator.
>
I think in the lack of empirical evidence, it is now a moot point.
>> Having at least half the class get a C or less (the curving which is
>> present in many math departments) is not a "mockery". A grade
>> of C is not good at all.
>
>It *should* be perfectly satisfactory, however.
I argued that it will make students unhappy, but that it is
satisfactory. No disagreement.
>
>> Mockery is grading in the humanities.You virtually have to be dead to get a
>> low grade in a humanities class.
>
>That depends entirely on the class. I know quite a few humanities
>instructors whose standards are higher than those of some of my
>colleagues in the math department.
You know I am talking about long term trends and not specific
cases. At UC Berkeley, the average GPA in almost all humanities
and social sciences is much higher than almost all the sciences.
Of course, there will always be *some* people who are
honest garders in the humanities, but many are taking the
path of least resistance.
Here is some empirical evidence: Bok's recent book on affirmative
action (no, this is not an AA thread but just listen for a sec)
he compiles extensive statistics on the effects of SAT, socio-economic
class, etc. on percentile rank. Being a science major
usually has a negative coefficient in the regression.
>
>> > If everyone in the class knows the material which is supposed to be
>> > taught in that course, they should all get good grades.
>
>> Why? Is it not possible that the text may be confusing? or the
>> professor is confusing? Or the exam was not good?
>
>Eh? What does this have to do with Herman's statement?
Herman said earlier in the argument that even if the professor
screws up, that the students still have a respobnsibility
to know the material. I was responding to that statement
and arguing, not too persuasively, that this is not
a plausible. I think it came off the wrong way when I wrote it.
>
>> > If they do
>> > not, they should get bad grades. Whether the instructor did a poor
>> > job of teaching should not affect the absolute meaning of the grades.
>
>> Why not? If your doctor screws up, you get to sue. If your lawyer
>> screws up, you get to appeal. If the waiter brings the wrong food, you
>> get to send it back.
>
>So take the grade off the record. But don't make it dishonest.
I agree but this is easier said than done. Grades are often removed
if there was a serious interpersonal conflict between student
and teacher. Maybe the student had a disciplinary problem or
the faculty harrassed the student in some way. But I have *never*
heard of a case where the administration said, "Gee whiz, Prof. X
is a horrid teacher. It's a shame you got a D. We'll remove it
from the transcript. " Ok, Stanford's now defunct policy
of allowing students to drop courses on the last day of the class,
including final exam day comes close but this is by far the
exception.
In most cases, incompetent teachers rarely suffer any consequences
of bad teaching. Even if the university does reprimand a professor,
it is often years after students have left. "Conditional curving"
is a low cost way to protect students from the occasional
pedagogical screw up. It distorts grades some what but not too
much.
>
>Brian M. Scott
Fabio
Colin Richard Day
aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
No, the reason we don't have freshmen take real analysis
is that most calculus students are not math majors and
have little interest in doing proofs.
>want to see if the student can handle taking derivatives
>and solving word problems. This is "computational" rather
>than "mathematical". If they can understand how to integrate
>by parts and calculate radii of convergence, then there
>is a good chance that they could handle proof writing.
Shouldn't students learn proof-writing much earlier,
as part of a liberal education? Maybe not deep proofs
in analysis, but proofs using sentential and
predicate logic.
Also, whereas some computational background may be
helpful, are students losing the forest for the trees?
>It's not certain. It's not perfect. In fact, in the real
>world of teaching we actually use this as a method for
>selecting students. If a student wants to major in math or
>statistics, the first questions are usually "what were
>your calculus grades?" OF course, most of the kids who take
>real analysis do badly because the do not have math. ability,
>but it is completely reasonable to use computational ability
>as a *first guess* about the potential to do math.
So only math and statistics majors should know about
proofs? Does that mean that everyone else has to take
stuff on faith?
>
>I think this is an empirical matter that can't be settled
>in a forum like this. What one would have to do is to
>figure out the computational abilities of sucessful
>math students and compare them with proof writing skills.
>If bet that most proof writers can do computations, but most
>"copmuters" can't write proofs. Which is consistent with
>what I claim, but not with what Herman claims.
Your claim doesn't mean much. Suppose that 70% of the population
can compute, 10% of the population can do proofs and 7% can do both.
We have that 70% of the proof writers can do computations and that only
10% of the computers can write proofs. Does this allow us to use
computational skill as a basis for judging proof-writing ability?
Absolutely not, for the two are statistically independent. What you
need to show is that a higher percentage of computers than noncomputers
can write proofs.
>Fabio
>
Fabio Rojas
>In article <F0u1w...@midway.uchicago.edu>,
>>strong claim. He said "no amount of computational
>>mathematics is any indiciation of the ability to understand
>>concepts". These are his words, not mine. My response
>>is that if you have math ability, you may probably
>>have computational ability, thus, computational ability
>>is not such a bad way to guess that a student has
>>math ability *if you have no other information*.
>>
>>Think about it: why do we have *any* prerequisites
>>for advanced math courses? Why not have all freshman
>>take real analysis? There's a simple reason. Instructors
>
>is that most calculus students are not math majors and
>have little interest in doing proofs.
This is not so clear to me. A lot of people try to major in math
because they enjoy problem solving of a rather abstract sort.
they often do not know what a proof is when they start in the
math major. If we only encouraged people to major in math
who had an explicit interest in proofs , the classes
would be mostly empty.
I have read some biographies of mathematicians and math related
types (physicists, statisticians, engineers, etc). In most cases,
calculus and problem solving is what started the journey.
I might be factually wrong, but that is my impression.
>
>>want to see if the student can handle taking derivatives
>>and solving word problems. This is "computational" rather
>>than "mathematical". If they can understand how to integrate
>>by parts and calculate radii of convergence, then there
>>is a good chance that they could handle proof writing.
>
>Shouldn't students learn proof-writing much earlier,
>as part of a liberal education? Maybe not deep proofs
>in analysis, but proofs using sentential and
>predicate logic.
I think we might be in complete agreement. Instead of torturing
liberal arts students with watered down calculus, why not
have them take a solid course in euclidean geometry?
Sometimes, students' only encounter with proofs is through
the symbolic logic course in the philosophy dept.
>
>Also, whereas some computational background may be
>helpful, are students losing the forest for the trees?
That is why they need to go to school!! By interacting
with experienced mathematicians and students, they can start
to gain perspective. To much computation can be bad,
but we shouldn't discourage people from learning it.
I remember an algebraic topology class I took with Robin
hartshorne. He was shocked to find out that almost nobody
could write out the formula for a homotopty he verbally
described. It was because students had almost no practice
in computation at Berkeley. Loosing sight of the importance
of computation is as bad thinking computation is all there
is to mathematics.
>>as a *first guess* about the potential to do math.
>
>So only math and statistics majors should know about
>proofs? Does that mean that everyone else has to take
>stuff on faith?
Well, my comment was a response to Herman Rubin who suggested
that you can *Never* judge a students' math ability by
his computational ability. My response is simply that
unless you have tested the individual's proof writing
ability beforehand, your only evidence of any sort of
mathematical ability comes from performance in computationally
oriented classes.
Should only math majors only know about proofs? Maybe. I can
enjoy novels without knowing about Bakhtin or DErrida. I
can enjoy history without knowing about methodological debates
amongst historians. I think most science students should
know what a proof is, but I think it is no crime if
your average sociology or history student does not know.
It would benefit them intellectually, but they haven't been
robbed if they don't learn it.
>
>>
>>I think this is an empirical matter that can't be settled
>>in a forum like this. What one would have to do is to
>>figure out the computational abilities of sucessful
>>math students and compare them with proof writing skills.
>>If bet that most proof writers can do computations, but most
>>"copmuters" can't write proofs. Which is consistent with
>>what I claim, but not with what Herman claims.
>
>Your claim doesn't mean much. Suppose that 70% of the population
>can compute, 10% of the population can do proofs and 7% can do both.
>We have that 70% of the proof writers can do computations and that only
>10% of the computers can write proofs. Does this allow us to use
>computational skill as a basis for judging proof-writing ability?
Yes - if you don't have any other information about the student.
My claim is not: "taking derivatives = proof writing ability".
That's crazy. My claim is : " if all you know is a kid's calculus
grades, put your money on the kids with the high grades."
I also make the claim that this is how math instructors operate
in the class.
>Absolutely not, for the two are statistically independent. What you
Is it? I have not seen evidence that the distribution of computational
ability is *statistically independent* of mathematical ability.
This is the point of debate not an assumption. I could be wrong.
For the most part, proof writers are almost a proper subset
of "computers" but a rather small subset. I have met only one student
in my small number of years teaching who genuinely could do advanced
proofs but could not do an integral so save his life. The poor
guy just struggled through every ODE/numerical that he was forced
to take. But damn, he was quite the logician.
>need to show is that a higher percentage of computers than noncomputers
>can write proofs.
No. I only need to show that "proof writer --> computer" and
that "not computer --> not proof writer" for some similarly high
number of students.
My claim is that the compliment of the set
[ (proof writers) intersect (computers) ]
is fairly small.
If you buy that, which I think is reasonable, then encouraging
students on the basis of calculus grades and copmutational
ability is a good way to make a first guess about a students'
ability/.
Perhaps you think I believe that "computer --> proof writer".
I do not and most math teachers don't either.
>>Fabio
Fabio, again!
Brian M. Scott
>>fgr...@my-dejanews.com wrote:
>>> hru...@b.stat.purdue.edu (Herman Rubin) wrote:
You've just repeated the error. Herman said that A does not imply B.
You've observed that B generally implies A and used this to argue that
Herman was wrong. This is a non sequitur. If you still don't see
this, take A to be 'is black' and B to be 'is a crow'. Obviously it
would be foolish to guess that something is a crow if all you knew
about it was that it was black. The fact that all(?) crows are black
is irrelevant.
>Think about it: why do we have *any* prerequisites
>for advanced math courses? Why not have all freshman
>take real analysis? There's a simple reason. Instructors
>want to see if the student can handle taking derivatives
>and solving word problems.
Not this instructor. I can find make a better estimate of a student's
chances in real analysis after 15 minutes of conversation than I can
get from his grades in calculus.
> This is "computational" rather
>than "mathematical". If they can understand how to integrate
>by parts and calculate radii of convergence, then there
>is a good chance that they could handle proof writing.
How much teaching have you done?! There most certainly is *not* a
good chance that they can handle proof writing.
>It's not certain. It's not perfect. In fact, in the real
>world of teaching we actually use this as a method for
>selecting students. If a student wants to major in math or
>statistics, the first questions are usually "what were
>your calculus grades?"
I don't. If the calculus grades are poor, of course, odds are that
the student won't do well in the real math courses, though I've seen a
fair number of exceptions to that principle. But good calculus grades
are meaningless in this context. Of the math courses that our
students normally take in their first two years the only one that
might be considered any kind of positive indicator is linear algebra,
and even it is too computational to show much.
> OF course, most of the kids who take
>real analysis do badly because the do not have math. ability,
>but it is completely reasonable to use computational ability
>as a *first guess* about the potential to do math.
Only if you recognize that it's almost worthless except perhaps as a
negative predictor.
>I think this is an empirical matter that can't be settled
>in a forum like this. What one would have to do is to
>figure out the computational abilities of sucessful
>math students and compare them with proof writing skills.
>If bet that most proof writers can do copmutations, but most
>"copmuters" can't write proofs. Which is consistent with
>what I claim, but not with what Herman claims.
You're wrong: it's consistent with Herman's statement and inconsistent
with yours. This is the same logical error that you made up at the
top. If most 'computers' can't write proofs, then the ability to do
computations is a very poor indicator of ability write proofs - which
is exactly what Herman and I have been saying.
>>> Hog wash!! We use computational ability as a judge of math ability
>>> all the time.
>>The point is that we shouldn't, because it's a lousy indicator.
>I think in the lack of empirical evidence, it is now a moot point.
You may lack empirical evidence; after a quarter-century or so of
teaching I have a great deal. I have no doubt that Herman has even
more.
[snip]
>>> Mockery is grading in the humanities.You virtually have to be dead to get a
>>> low grade in a humanities class.
>>That depends entirely on the class. I know quite a few humanities
>>instructors whose standards are higher than those of some of my
>>colleagues in the math department.
>You know I am talking about long term trends and not specific
>cases.
No, I don't. You made a flat assertion that is flatly wrong.
> At UC Berkeley, the average GPA in almost all humanities
>and social sciences is much higher than almost all the sciences.
>Of course, there will always be *some* people who are
>honest garders in the humanities, but many are taking the
>path of least resistance.
And there are a great many people taking the path of least resistance
elsewhere, too. In my experience the difference in GPA has less to do
with the intentions of the instructors than with the nature of the
subject. In math and science and performing arts it's easier to agree
on what isn't good performance.
[snip]
>>> > If everyone in the class knows the material which is supposed to be
>>> > taught in that course, they should all get good grades.
>>> Why? Is it not possible that the text may be confusing? or the
>>> professor is confusing? Or the exam was not good?
>>Eh? What does this have to do with Herman's statement?
>Herman said earlier in the argument that even if the professor
>screws up, that the students still have a respobnsibility
>to know the material. I was responding to that statement
>and arguing, not too persuasively, that this is not
>a plausible.
If you were responding to that statement, why did you place your
response after a completely different statement to which it was a non
sequitur?
>>> > If they do
>>> > not, they should get bad grades. Whether the instructor did a poor
>>> > job of teaching should not affect the absolute meaning of the grades.
>>> Why not? If your doctor screws up, you get to sue. If your lawyer
>>> screws up, you get to appeal. If the waiter brings the wrong food, you
>>> get to send it back.
>>So take the grade off the record. But don't make it dishonest.
>I agree but this is easier said than done.
That's not sufficient justification for advocating dishonest grading.
[snip]
Brian M. Scott
Herman Rubin
Fabio Rojas <aap...@mazel.spc.uchicago.edu> wrote:
>Brian M. Scott <sc...@math.csuohio.edu> wrote:
>>fgr...@my-dejanews.com wrote:
>>> hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>>> >In many cases, one has no way of even
>>> > assessing this; no amount of computational mathematics is any
>>> > indication of the ability to understand mathematical concepts, let
>>> > alone proofs.
>>> Would you recommend that a person who had difficulty with SAT level algebra
>>> problems jump to a graduate course on commutative algebra?
>>Your response is a logical non sequitur. Complete lack of computational
>>skill *may* be an indicator of inability to understand the concepts, but
>>that wasn't Herman's point. His point - and in my experience he's
>>basically right - is that computational skill is at best a very poor
>>indicator of ability to understand concepts.
>It is not a non sequitor. Herman claimed that computational
>ability is no indicator of mathematical ability (writing
>proofs, understanding concepts, etc.).
There are DISTINCT abilities involved here. The ability to produce
proofs, which is not important for the non-mathematician in general,
is different from the understanding of concepts, and in fact, either
can occur without the other. The test of the understanding of a
concept is the ability to apply it. There are even those who can
often produce enough of a proof that others can complete it without
understanding that a proof is not "a convincing argument".
He made a pretty
>strong claim. He said "no amount of computational
>mathematics is any indiciation of the ability to understand
>concepts". These are his words, not mine. My response
>is that if you have math ability, you may probably
>have computational ability, thus, computational ability
>is not such a bad way to guess that a student has
>math ability *if you have no other information*.
There is general mental ability, so this statement is quite
likely to be correct. But using it as a criterion to teach
mathematics or to structure the curriculum is a major error.
It would not surprise me if the ability to play a musical
instrument has a positive correlation; should we make this
a requirement?
>Think about it: why do we have *any* prerequisites
>for advanced math courses?
There is a difference between having SOME prerequisites and
having IRRELEVANT prerequisites. Before the educationists
go their hands on the high school curriculum, the "Euclid"
type geometry course was standard preparation for college.
This course really has no mathematical prerequisites whatever;
algebra was not invented until centuries after Euclid. Were
the ancient Greek students weeded out on the basis of not
having strong arithmetic abilities? I suspect not.
Why not have all freshman
>take real analysis?
I would suggest abstract algebra first, as most of them do not
know what formal proofs are. Certainly, they should understand
the properties of the integers, including induction. But it
would not be that difficult to give a conceptual real variables
course before calculus to most of them who will ever by able to
learn it, and low grades in manipulative calculus could drive
many out. Going from the years of almost complete computational
mathematics to learning concepts is a MAJOR shock.
There's a simple reason. Instructors
>want to see if the student can handle taking derivatives
>and solving word problems.
They should be able to FORMULATE word problems, but this should
be completely separated from the computational part. And why
should someone memorize how to act like a machine, instead of
understanding what is being done, and why?
...................
>>> > If everyone in the class knows the material which is supposed to be
>>> > taught in that course, they should all get good grades.
>>> Why? Is it not possible that the text may be confusing? or the
>>> professor is confusing? Or the exam was not good?
>>Eh? What does this have to do with Herman's statement?
>Herman said earlier in the argument that even if the professor
>screws up, that the students still have a respobnsibility
>to know the material. I was responding to that statement
>and arguing, not too persuasively, that this is not
>a plausible. I think it came off the wrong way when I wrote it.
>>> > If they do
>>> > not, they should get bad grades. Whether the instructor did a poor
>>> > job of teaching should not affect the absolute meaning of the grades.
>>> Why not? If your doctor screws up, you get to sue. If your lawyer
>>> screws up, you get to appeal. If the waiter brings the wrong food, you
>>> get to send it back.
>>So take the grade off the record. But don't make it dishonest.
Why are we operating on a grade-credit system? It leads to very bad
consequences, at best. It is the cause of teachers drilling children
and preparing them to take examinations on trivial pursuit, instead
of having them learn for the distant future. It leads to students
taking unnecessary preparatory courses to raise their averages. Why
should a student get any more credit for taking such courses? If a
student learns a given amount of linear algebra in one 3-credit course,
or in 3 5-credit courses, why should there be any different indication
on the record? Grades should be given to the student for information
and guidance, but a grade for easily forgotten material memorized for
examinations is of little value, as is giving credit for such.
>In most cases, incompetent teachers rarely suffer any consequences
>of bad teaching. Even if the university does reprimand a professor,
> it is often years after students have left. "Conditional curving"
>is a low cost way to protect students from the occasional
>pedagogical screw up. It distorts grades some what but not too
>much.
Curving protects the students from getting poor grades for not having
learned the material, because they were not taught it, or because
they were not prepared for the course, or other similar reasons. It
does not protect them, or other students, from the consequences of
being put into higher level courses because of having "passed" the
prerequisites, and then having the higher level classes trashed by
attempts to cater to them. It damages the other students in those
higher level courses.
It is precisely the attitude of teaching to whomever is in the class
and grading on a curve which is responsible for much of our educational
woes. Our credentials should indicate knowledge and ability; an "A"
in a given course from Podunk should indicate nothing less than one
from Harvard.
>>Brian M. Scott
>Fabio
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Fabio Rojas
Herman Rubin <hru...@b.stat.purdue.edu> wrote:
>>It is not a non sequitor. Herman claimed that computational
>>ability is no indicator of mathematical ability (writing
>>proofs, understanding concepts, etc.).
>
>There are DISTINCT abilities involved here. The ability to produce
>proofs, which is not important for the non-mathematician in general,
>is different from the understanding of concepts, and in fact, either
>can occur without the other. The test of the understanding of a
>concept is the ability to apply it. There are even those who can
>often produce enough of a proof that others can complete it without
>understanding that a proof is not "a convincing argument".
I think I would agree that "math" and "computation" are distinct
abilities. But my theory is that is that students with strong computation
abilities have math ability often enough so that it is not
unreasonable to say that you should encourage the "computers"
to continue and discourage those who have little computational
ability.
>
>He made a pretty
>>strong claim. He said "no amount of computational
>>mathematics is any indiciation of the ability to understand
>>concepts". These are his words, not mine. My response
>>is that if you have math ability, you may probably
>>have computational ability, thus, computational ability
>>is not such a bad way to guess that a student has
>>math ability *if you have no other information*.
>
>There is general mental ability, so this statement is quite
>likely to be correct. But using it as a criterion to teach
>mathematics or to structure the curriculum is a major error.
>It would not surprise me if the ability to play a musical
>instrument has a positive correlation; should we make this
>a requirement?
Oh dear. I never meant to imply such a thing. The original
poster said that in class that he once took at Harvard,
some very high percentage of students got F's. He said that
this was probably an unreasonable distribution of grades since
it was a class of upper division students at Harvard tended
to have a decent amount of ability (I buy this although
I have never attended or taught at Harvard). You response
was that the grade distribution was probably ok because
a) computational ability, which the original poster emphasized
in about Harvard student, was no substitute for math ability
and b) even if the professor was incomprehensible or incompetent
the students were still to blame for their poor performance.
I was responding to statement a) and b) - not recommending
some radical overhaul of the curriculum ( which is ok
as far as I am concerned for the most part ). I was very
skeptical of your idea that all those kids really were
so bad that they deserved F's. I don't think you should
read more into it than that.
>
>>Think about it: why do we have *any* prerequisites
>>for advanced math courses?
Oh boy, let's debate these one at a time:
>
>There is a difference between having SOME prerequisites and
>having IRRELEVANT prerequisites.
I don't think computational abilities are irrelevant. For
example, say you want to teach an analysis class about
the Reimann integral. It is useful to calculate
an example using the anti-derivative and then to show that
the Reimann sum converges to the same asnwer. Both of these
involve doing calculations of the freshman calculus type.
Of course, there is more at work than this, but you can't
do without calculation abilities.
> Before the educationists
>go their hands on the high school curriculum, the "Euclid"
>type geometry course was standard preparation for college.
It still is, Herman!! Some schools have abolished it, but
a great majority of high schools still teach it in some form.
Perhaps the quality of the course has declined in general.
The SAT still asks many elementary geometry questions of the
sort "If lines X and Y are parallel, what can you say about angles A,B ..."
It would be pointless to ask if no one ever took geometry
in high school.
But you do have a good point. Ed. researchers have often found
that what is advertised in courses is rarely taught. Here
in Chicago, it was found out that a large number of schools
had classes labled "Algebra" but when investigators showed up,
it was just another course in arithmetic.I don't think
you can blame the ed. school kids for this one (although
they are responsible for a lot of bad stuff in schools).
Teaching proofs is hard and a lot of teachers are just lazy.
Add to that the poor supervision most schools have.
>This course really has no mathematical prerequisites whatever;
>algebra was not invented until centuries after Euclid. Were
>the ancient Greek students weeded out on the basis of not
>having strong arithmetic abilities? I suspect not.
Are you so sure? Ancient greek geometry was often highly
connected to arithmetic. The Pythagorean theorem,
a geometrical fact, emerged from studying the arithmetical
relationships between the sides of triangles. A great
deal of geometry does not ever refer to arithmetic
computations, but a lot of it did. I think it is just
plain wrong to see computation and proofs as being
far apart from each other. They really feed each other
and they did for the Greeks.
I do not know about Greek education in general, by the
cult of the Pythagoreans were infatuated with all sorts
of arithmetic properties (like triangular numbers). They
were the "Research Establishment" in Greece. I bet if
a student couldn't play with ratios , then the Pythagoreans
probably gave the guy the boot. (If I remember correctlty,
the Pythagoreans allegedly killed someone for revealing to
the public that irrational numbers existed - that's
definitely a computational fact).
>
>Why not have all freshman
>>take real analysis?
>
>I would suggest abstract algebra first, as most of them do not
>know what formal proofs are. Certainly, they should understand
>the properties of the integers, including induction. But it
>would not be that difficult to give a conceptual real variables
>course before calculus to most of them who will ever by able to
>learn it, and low grades in manipulative calculus could drive
>many out. Going from the years of almost complete computational
>mathematics to learning concepts is a MAJOR shock.
I agree abstract algebra wouldn't be too bad. But real
variables? What motiviation for real analysis would students
have if they didn't understand calculus? Metric spaces only
made sense to me after thinking about limits in the context
of calculus. I could be wrong on this one.
But there is an empirical example: Reed College. For a long
time, they would teach real analysis in this fasion. Perhaps
someone familiar with the success of this program at Reed
could tell us about the results.
>
>There's a simple reason. Instructors
>>want to see if the student can handle taking derivatives
>>and solving word problems.
>
>They should be able to FORMULATE word problems, but this should
>be completely separated from the computational part. And why
>should someone memorize how to act like a machine, instead of
>understanding what is being done, and why?
I am not asking people to memorize like a machine, but that they
simply have to get the mechanical side of math done before
they get much insight from more advanced concepts. My theory
is simply that math ability is built on a foundation
of computation. Some kids, can skip it, but I think it
is better for most kids to get calculus down correctly
on the mechanical side before they can start worrying
about metric spaces.
>
>Why are we operating on a grade-credit system? It leads to very bad
>consequences, at best. It is the cause of teachers drilling children
>and preparing them to take examinations on trivial pursuit, instead
>of having them learn for the distant future. It leads to students
>taking unnecessary preparatory courses to raise their averages. Why
>should a student get any more credit for taking such courses? If a
>student learns a given amount of linear algebra in one 3-credit course,
>or in 3 5-credit courses, why should there be any different indication
>on the record? Grades should be given to the student for information
>and guidance, but a grade for easily forgotten material memorized for
>examinations is of little value, as is giving credit for such.
I think I share the same sentiment as you do, but life exists
outisde of the math department. The grade-credit system is demanded
by the outside world. People want to finish their degrees in
a reasonable amount of time. People need to demonstrate
knowledge to others. The "knowledge only" system you indicate
might cost too much money or time. Etc. Etc. I think it is
the best system, but might not survive. I remember UC Santa
Cruz abolished grades and had instructors write detailed
descriptions of what students learned in the class. How much
more honest can you get? Unfortunately, their graduates had problems
getting int o grad school so they changed back to the grade-credit
system.
>
>>In most cases, incompetent teachers rarely suffer any consequences
>>of bad teaching. Even if the university does reprimand a professor,
>> it is often years after students have left. "Conditional curving"
>>is a low cost way to protect students from the occasional
>>pedagogical screw up. It distorts grades some what but not too
>>much.
>
>Curving protects the students from getting poor grades for not having
>learned the material, because they were not taught it, or because
>they were not prepared for the course, or other similar reasons. It
>does not protect them, or other students, from the consequences of
>being put into higher level courses because of having "passed" the
>prerequisites, and then having the higher level classes trashed by
>attempts to cater to them. It damages the other students in those
>higher level courses.
You might not think so, but I think I agree with what is written
above. If you remember from my posting earlier, I only recommended
curving the course in unusual circumstances. Not as a general
practice. I also recommended a very weak version of curving
as well. I don't ever recommend giving 25% A's (or whatever)
every time. That would create the problems you cited above.
I am willing to make a trade off: every once in a while
I will curve a class if I believe I have given an unfair exam
and I have reason to believe that the students have the ability
and put out the effort in exchange for the chance that a few
students will undeservedly go to the next higher level.
I can live with the small amount of error. Remember that never
curving also excludes people from going to the next level
who got low grades because the professor made some sort
of error (like one Berkeley professor who taught ancient Chineese
math in a freshman calc class but testing on power series).
Using the example I just cited, it is unfair to ruin
somebody's academic record because this joker couldn't
teach the right material in freshman calculus. As long
as you realize that I rarely advocate the use of curving,
but do not exclude it, then I think you might see my side
a little better because it produces few mistakes of the sort
you described (which I can live with).
>Our credentials should indicate knowledge and ability; an "A"
>in a given course from Podunk should indicate nothing less than one
>from Harvard.
I agree with this but I should add that a C from Podunk with
a competent teacher is really a different C than Harvard
with an incompetent teacher. Why can't we agree on this?
>>Fabio
>
>hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Fabio, again!!
Brian M. Scott
[most snipped]
> (If I remember correctlty,
> the Pythagoreans allegedly killed someone for revealing to
> the public that irrational numbers existed - that's
> definitely a computational fact).
No, it's a theoretical fact. You can neither discover nor prove it by
computation.
> Metric spaces only
> made sense to me after thinking about limits in the context
> of calculus.
Neither one is the place to learn limits and continuity; the ideas are
much clearer in general topological spaces.
Brian M. Scott
Fabio Rojas
>
>[most snipped]
>> the Pythagoreans allegedly killed someone for revealing to
>> the public that irrational numbers existed - that's
>> definitely a computational fact).
>
>computation.
I have a hunch that the GReeks found out irrational numbers
by noticing that the length of the diagonal of the right iscoceles (sp?)
triangle whose two equal sides have length one can't be expressed
using a fraction by plugging in all sorts of candidate fractions
into formula for the Pythagorean theorem. I bet they didn't make
the hard and fast distinction between computation and theory
that we makde today. Of course, the modern proof of the existence
of irrationals is pureley theoretical, but the Greeks only
discovered teh fact through computation.
only after doing lost of computations did the Greeks decide
a proof was in order.
>
>> Metric spaces only
>> made sense to me after thinking about limits in the context
>> of calculus.
>
>Neither one is the place to learn limits and continuity; the ideas are
>much clearer in general topological spaces.
There is a difference between the logical context of a mathematical
concept and its pedagogical context. Sequences and what not
make the most sense from a topoligcal perspective but
maybe throwing the notion of topological space at someone
who has never seen a sequence in a calculus courser might be a bad idea.
For most people, abstract concepts are best learned with
concrete examples - even if it violates the most "correct"
logical order of things.
My theory is the one that underlies most teaching of advanced
math at the undergraduate level - computation provides a set
of concrete examples that people use to understand more abstract
structures. Computational examples often indicate why a more
general fact is true in a specific case. Heck, this is the
way much mathematics is discovered. It is my contention that
we should not reject computation as a foundation for mathematics.
>
>Brian M. Scott
Fabio
Brian M. Scott
(Fabio Rojas) wrote:
>Brian M. Scott <sc...@math.csuohio.edu> wrote:
>>Fabio Rojas wrote:
>>> (If I remember correctlty,
>>> the Pythagoreans allegedly killed someone for revealing to
>>> the public that irrational numbers existed - that's
>>> definitely a computational fact).
>>No, it's a theoretical fact. You can neither discover nor prove it by
>>computation.
>I have a hunch that the GReeks found out irrational numbers
>by noticing that the length of the diagonal of the right iscoceles (sp?)
>triangle whose two equal sides have length one can't be expressed
>using a fraction by plugging in all sorts of candidate fractions
>into formula for the Pythagorean theorem. I bet they didn't make
>the hard and fast distinction between computation and theory
>that we makde today. Of course, the modern proof of the existence
>of irrationals is pureley theoretical, but the Greeks only
>discovered teh fact through computation. [...]
How the Greeks *discovered* it is unknown. We know, however, that
they *proved* it, and this is distinctly not a matter of computation.
As for making a clear distinction between theory and computation, that
is precisely the Greeks' most significant mathematical contribution;
read any standard history of mathematics.
>>> Metric spaces only
>>> made sense to me after thinking about limits in the context
>>> of calculus.
>>Neither one is the place to learn limits and continuity; the ideas are
>>much clearer in general topological spaces.
>There is a difference between the logical context of a mathematical
>concept and its pedagogical context.
Sometimes.
> Sequences and what not
>make the most sense from a topoligcal perspective but
>maybe throwing the notion of topological space at someone
>who has never seen a sequence in a calculus courser might be a bad idea.
I didn't say anything about doing so. In fact you need some basic
notion of limit even to assign meaning to the symbol '0.333...', which
is normally encountered in grade school. The fact remains that
epsilons and deltas clutter up the basic idea of continuity something
awful.
>For most people, abstract concepts are best learned with
>concrete examples - even if it violates the most "correct"
>logical order of things.
But I suspect that the people most likely to end up doing abstract
mathematics generally aren't in that group.
>My theory is the one that underlies most teaching of advanced
>math at the undergraduate level - computation provides a set
>of concrete examples that people use to understand more abstract
>structures. [...]
The fact remains that some of us find it much easier to start with the
structure, learn its basic properties, and *then* look at concrete
examples. In my experience that includes a large fraction of the best
students. I also note (from experience) that weaker students are
often misled by concrete examples: they tend to assume that the
general object is more like the special cases that they know than it
really is.
There are of course areas in which this approach doesn't work very
well; I'm talking about primarily about such areas as topology,
abstract algebra, real analysis, and set theory.
Brian M. Scott
Dr. Michael Albert
> structure, learn its basic properties, and *then* look at concrete
> examples. In my experience that includes a large fraction of the best
Certaintly I can't say what goes on in other people's minds, nor say what
is best for them, and my experience with the "best" students is limited.
Nevertheless, I think the importance of examples and special cases is
underrated. I find it hard to imagine learning, say, Reimannian geometry
without first looking at 2-spaces embedded in three space.
There is a danger of students generalizing uncritically from the special
case to the general. For example, after one has worked in metric spaces
for a while it is a bit hard to remember that in a general topological
setting one can't count on first-countability[1], but I think the
solution is to show relevant examples where this fails--for example,
showing that one can think of point-wise convergence in the setting
of the weak-product topology[2], which certaintly isn't first-countable.
It is certaintly true that computational aspects are overemphasized
in the elementary course (say, grade school through 2nd year college in
the US system, or maybe more unfortunately). I would find it rare
that someone with real math ability didn't show at least a passing
ability at this. The problem is that we introduce computation for
computation sake. There are endless excercises after which the student
knows nothing more than when they began. For example, when studying
integration by parts, it doesn't hurt to introduce special cases of the
gamma function, and even point out to the students that something
interesting is going on.
NOTES(reminders for those to whom topology is a distant memory):
[1] First countability= at each point there is a countable basis, ie.,
at each point there is a countable collection of open sets such that
for any open set containing the point, one of the basis sets is in
that open set.
[2] Weak product. Let X be a set and Y a topological space. Consider
the space of all functions from X into Y. A basis for the topology is
given as follows: for any finite set collection of points in X, x_1, x_2,
x_3, . . ., x_n, and open sets U_1, U_2, U_3,...U_n in Y, then the set of
all function such that f(x_i) is in U_i form an open set.
Herman Rubin
Fabio Rojas <aap...@mazel.spc.uchicago.edu> wrote:
>I'd like to say that I find this debate stimulating..
>Herman Rubin <hru...@b.stat.purdue.edu> wrote:
>>>It is not a non sequitor. Herman claimed that computational
>>>ability is no indicator of mathematical ability (writing
>>>proofs, understanding concepts, etc.).
>>There are DISTINCT abilities involved here. The ability to produce
>>proofs, which is not important for the non-mathematician in general,
>>is different from the understanding of concepts, and in fact, either
>>can occur without the other. The test of the understanding of a
>>concept is the ability to apply it.
................
>I think I would agree that "math" and "computation" are distinct
>abilities. But my theory is that is that students with strong computation
>abilities have math ability often enough so that it is not
>unreasonable to say that you should encourage the "computers"
>to continue and discourage those who have little computational
>ability.
I find two errors, one empirical and one logical. The empirical
one is that those with strong computational abilities have
mathematical ability fairly often. It is CONCEIVABLE that
this is true, but attempting to teach mathematical concepts,
or even mathematical proofs, to those who have had a large amount
of successful computation shows that it does not work. The
logical one is that those who have little computational ability
should be discouraged from learning good mathematics.
In fact, those without computational ability NEED the concepts.
The one who can compute nothing needs to know how to formulate
the real problems in symbolic mathematics, and to interpret the
results obtained by machines or other people in real terms.
Someone who knows when to add, and what to add, can get others
to find the correct answer, while someone who knows how cannot.
>>He made a pretty
>>>strong claim. He said "no amount of computational
>>>mathematics is any indiciation of the ability to understand
>>>concepts". These are his words, not mine. My response
>>>is that if you have math ability, you may probably
>>>have computational ability,
I know enough counterexamples. My own computational abilities
are sufficiently high that I can see that they do not contribute
to understanding.
thus, computational ability
>>>is not such a bad way to guess that a student has
>>>math ability *if you have no other information*.
So let us get the other information! We can teach mathematical
concepts to those who can barely read, and proceed from that
point. We can find out much earlier than now. Computational
ability is essentially memorization without structure.
..............
>>>Think about it: why do we have *any* prerequisites
>>>for advanced math courses?
>Oh boy, let's debate these one at a time:
>>There is a difference between having SOME prerequisites and
>>having IRRELEVANT prerequisites.
>I don't think computational abilities are irrelevant. For
>example, say you want to teach an analysis class about
>the Reimann integral.
You have hit on a real peeve of mine. Why should we teach the
Riemann integral, with its unnecessary specialization, rather
than the general notion of integral with respect to measure?
The oldest calculation of such an integral is much older. An
integral is a sum of products, or a limit of such. This should
be taught using only high school algebra and a rudimentary
approach to limits, such as is needed to understand infinite
decimals.
It is useful to calculate
>an example using the anti-derivative and then to show that
>the Reimann sum converges to the same asnwer.
It is important for understanding that the student have lots
of examples of calculation of integrals from the definition
BEFORE using antiderivatives. Otherwise, they never get away
from antiderivatives. I have had this problem in teaching
students; they know too much about computation to be able
to get rid of it to understand the concepts.
Both of these
>involve doing calculations of the freshman calculus type.
>Of course, there is more at work than this, but you can't
>do without calculation abilities.
BTW, Fermat computed the integral of the q-th power, where
q is rational, without using anything more than the definition
and high school algebra. The needed algebra should be taught
much earlier. The rationality of q makes a rigorous computation
of limits easy.
>> Before the educationists
>>go their hands on the high school curriculum, the "Euclid"
>>type geometry course was standard preparation for college.
>It still is, Herman!! Some schools have abolished it, but
>a great majority of high schools still teach it in some form.
In some form is not good enough. The main "useful" purpose of
the course is to teach proof, not geometry.
>Perhaps the quality of the course has declined in general.
>The SAT still asks many elementary geometry questions of the
>sort "If lines X and Y are parallel, what can you say about angles A,B ..."
>It would be pointless to ask if no one ever took geometry
>in high school.
This has nothing to do with understanding of proof. Facts can
be looked up, and what the question asks for is facts. Drill
in memorization is far too great.
>But you do have a good point. Ed. researchers have often found
>that what is advertised in courses is rarely taught. Here
>in Chicago, it was found out that a large number of schools
>had classes labled "Algebra" but when investigators showed up,
>it was just another course in arithmetic.I don't think
>you can blame the ed. school kids for this one (although
>they are responsible for a lot of bad stuff in schools).
What makes you think that the teachers understand the purpose of
the course to be other than computation? One of my colleagues
was tutoring a student who was often doing difficult problems,
but failing to do simple ones. It turned out that he was missing
the KEY point of algebra, namely, using symbols to formulate.
This student was too bright to have forgotten it. This belongs
in first grade, but are the teachers capable of learning it?
>Teaching proofs is hard and a lot of teachers are just lazy.
>Add to that the poor supervision most schools have.
Teaching what proofs are requires knowing it first.
>>This course really has no mathematical prerequisites whatever;
>>algebra was not invented until centuries after Euclid. Were
>>the ancient Greek students weeded out on the basis of not
>>having strong arithmetic abilities? I suspect not.
>Are you so sure? Ancient greek geometry was often highly
>connected to arithmetic. The Pythagorean theorem,
>a geometrical fact, emerged from studying the arithmetical
>relationships between the sides of triangles.
That it may have been discovered from this has no relevance
whatever. The Babylonians knew about Pythagorean triples,
but does this mean that they knew that these were the sides
of right triangles? For all we know, it was an arithmetic
game to them.
But the contribution of the Greeks is that mathematics is
not based on observation and computation, but upon logical
deduction. The arithmetic in a good Euclid-type course
can be essentially ZERO. We do use algebra now for easier
communication, but Euclid could not.
..............
>>Why not have all freshman
>>>take real analysis?
>>I would suggest abstract algebra first, as most of them do not
>>know what formal proofs are. Certainly, they should understand
>>the properties of the integers, including induction. But it
>>would not be that difficult to give a conceptual real variables
>>course before calculus to most of them who will ever by able to
>>learn it, and low grades in manipulative calculus could drive
>>many out. Going from the years of almost complete computational
>>mathematics to learning concepts is a MAJOR shock.
>I agree abstract algebra wouldn't be too bad. But real
>variables? What motiviation for real analysis would students
>have if they didn't understand calculus?
We still call an important analytic property of the real
numbers the Archimedean property. Archimedes also made
use of the idea of limit, quite explicitly. Without this,
how does one know that the circle has a length or an area?
The Greeks knew that the directrix had its limit point on
the x axis (they did not have that terminology) was at 2/pi.
They explicitly DID use limits.
When Newton explained his results to his contemporaries,
he did not use calculus, but algebra, geometry, and limits.
Metric spaces only
>made sense to me after thinking about limits in the context
>of calculus. I could be wrong on this one.
I agree with Scott that general topology is much easier to
understand. In fact, in probabilistic situations, the
metrics are confusing and unnecessary.
>But there is an empirical example: Reed College. For a long
>time, they would teach real analysis in this fasion. Perhaps
>someone familiar with the success of this program at Reed
>could tell us about the results.
>>There's a simple reason. Instructors
>>>want to see if the student can handle taking derivatives
>>>and solving word problems.
>>They should be able to FORMULATE word problems, but this should
>>be completely separated from the computational part. And why
>>should someone memorize how to act like a machine, instead of
>>understanding what is being done, and why?
>I am not asking people to memorize like a machine, but that they
>simply have to get the mechanical side of math done before
>they get much insight from more advanced concepts.
What "advanced" concepts? I have provided examples to show
that the ancient Greeks used these concepts. Only the later
ones had any algebraic notation whatever, and their schools
were destroyed before they could develop enough.
My theory
>is simply that math ability is built on a foundation
>of computation. Some kids, can skip it, but I think it
>is better for most kids to get calculus down correctly
>on the mechanical side before they can start worrying
>about metric spaces.
I have taught students who have had computational calculus.
I wish they had not had any such material.
>>Why are we operating on a grade-credit system? It leads to very bad
>>consequences, at best. It is the cause of teachers drilling children
>>and preparing them to take examinations on trivial pursuit, instead
>>of having them learn for the distant future. It leads to students
>>taking unnecessary preparatory courses to raise their averages. Why
>>should a student get any more credit for taking such courses? If a
>>student learns a given amount of linear algebra in one 3-credit course,
>>or in 3 5-credit courses, why should there be any different indication
>>on the record? Grades should be given to the student for information
>>and guidance, but a grade for easily forgotten material memorized for
>>examinations is of little value, as is giving credit for such.
>I think I share the same sentiment as you do, but life exists
>outisde of the math department. The grade-credit system is demanded
>by the outside world.
Only because the educationists have claimed that it provides a
reasonable measure of knowledge and ability.
People want to finish their degrees in
>a reasonable amount of time. People need to demonstrate
>knowledge to others.
This is why the phony grade-credit system needs to go. This is
why education should teach for the distant future, not the
current term exam.
The "knowledge only" system you indicate
>might cost too much money or time.
I doubt it. The process is standard in India. Those who could
learn would get through faster. Courses would improve, as
students would not pressure their instructors to weaken the
material, knowing that they would later be expected to
demonstrate the knowledge. Especially, elementary and high
school, for those with college ability, could be compressed by
at least 1/3.
Etc. Etc. I think it is
>the best system, but might not survive. I remember UC Santa
>Cruz abolished grades and had instructors write detailed
>descriptions of what students learned in the class. How much
>more honest can you get?
Giving assessment of a course at the end of a course is
extremely poor. Good graduate schools insist on comprehensive
exams for their students, and largely ignore the grades they
give.
Unfortunately, their graduates had problems
>getting int o grad school so they changed back to the grade-credit
>system.
Having read thousands of applications for graduate admission
and support, I suspect that the problems came from the weaker
schools and the admissions people, not the faculty at the
good schools. If you want an administrator to think, never
make a supposedly comprehensive number available.
>>>In most cases, incompetent teachers rarely suffer any consequences
>>>of bad teaching. Even if the university does reprimand a professor,
>>> it is often years after students have left. "Conditional curving"
>>>is a low cost way to protect students from the occasional
>>>pedagogical screw up. It distorts grades some what but not too
>>>much.
The worst problems come from the teachers who will teach the
trivial pursuit now demanded by the students. The student taught
computational calculus and computational linear algebra usually
has been harmed by this. At this time, credentials from American
schools are essentially garbage.
>>>Fabio
>Fabio, again!!
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Brian M. Scott
<alb...@esther.rad.tju.edu> wrote:
[snips]
>Nevertheless, I think the importance of examples and special cases is
>underrated.
Then why is it almost impossible to find undergraduate topology texts
that start with the definition of a topological space? I certainly
don't deny that it's possible to go overboard, but the ship has two
sides.
>There is a danger of students generalizing uncritically from the special
>case to the general. For example, after one has worked in metric spaces
>for a while it is a bit hard to remember that in a general topological
>setting one can't count on first-countability[1], but I think the
>solution is to show relevant examples where this fails--for example,
>showing that one can think of point-wise convergence in the setting
>of the weak-product topology[2], which certaintly isn't first-countable.
If the students are really uncomfortable beginning with the simplest
spaces and adding structure (as in John Greever's excellent little
book of some years ago), I think that a better solution is to begin
with something familiar that is so obviously a special case that the
temptation to generalize will be much smaller. My choice would be
linearly ordered spaces.
> The problem is that we introduce computation for
>computation sake.
I've no quarrel with this at all. (Or with computation, for that
matter: this term I'm teaching a course from Graham, Knuth, and
Patashnik.)
Brian M. Scott
Herman Rubin
><alb...@esther.rad.tju.edu> wrote:
>[snips]
>>Nevertheless, I think the importance of examples and special cases is
>>underrated.
>Then why is it almost impossible to find undergraduate topology texts
>that start with the definition of a topological space? I certainly
>don't deny that it's possible to go overboard, but the ship has two
>sides.
One should definitely have examples and special cases. In general,
it is a good idea to present these AFTER the general concept has
been presented, so as to avoid confusion. In addition, the student
should not be limited by the special cases all having some properties
not in the general case; unlearning is the most difficult part.
>>There is a danger of students generalizing uncritically from the special
>>case to the general. For example, after one has worked in metric spaces
>>for a while it is a bit hard to remember that in a general topological
>>setting one can't count on first-countability[1], but I think the
>>solution is to show relevant examples where this fails--for example,
>>showing that one can think of point-wise convergence in the setting
>>of the weak-product topology[2], which certaintly isn't first-countable.
One cannot count on any separation properties. Upper semi-continuity
IS continuity with an appropriate topology on the image space which
is T0 but not T1. I have made use, in dealing with convergence
properties of probability measures and random variables, of finite
spaces without separation properties. While the FINAL results are
on "well-behaved" spaces, using these "nasty" spaces provides the
understanding lacking in the proofs which only use good separation.
Even for metric spaces, it is very often confusing to use some very
artificial metric which happens to provide the topology, rather than
to describe the topology in a manner which is not obviously metric.
>If the students are really uncomfortable beginning with the simplest
>spaces and adding structure (as in John Greever's excellent little
>book of some years ago), I think that a better solution is to begin
>with something familiar that is so obviously a special case that the
>temptation to generalize will be much smaller. My choice would be
>linearly ordered spaces.
Again, one should be careful of limiting the concept in this manner.
>> The problem is that we introduce computation for
>>computation sake.
>I've no quarrel with this at all. (Or with computation, for that
>matter: this term I'm teaching a course from Graham, Knuth, and
>Patashnik.)
I agree completely. While computation is occasionally helpful in
understanding, being able to do the computation is rarely so.
Computation is useful, and should be learned mainly for the purpose
of getting explicit answers.
The non-adept needs to be able to set up the problem so the computer,
or those who know how to compute, can do the computations without
knowing the original problem.
ull...@math.okstate.edu
> >
> >[most snipped]
> >> the Pythagoreans allegedly killed someone for revealing to
> >> the public that irrational numbers existed - that's
> >> definitely a computational fact).
> >
> >computation.
>
> I have a hunch that the GReeks found out irrational numbers
> by noticing that the length of the diagonal of the right iscoceles (sp?)
> triangle whose two equal sides have length one can't be expressed
> using a fraction by plugging in all sorts of candidate fractions
> into formula for the Pythagorean theorem. I bet they didn't make
> the hard and fast distinction between computation and theory
> that we makde today. Of course, the modern proof of the existence
> of irrationals is pureley theoretical, but the Greeks only
> discovered teh fact through computation.
This is nonsense. People have already pointed out it's
nonsense. Instead of just repeating yourself why not explain
_how_ the Greeks or anyone else can discover the irrationality
of sqrt(2) "through computation".
> only after doing lost of computations did the Greeks decide
> a proof was in order.
> >
> >> Metric spaces only
> >> made sense to me after thinking about limits in the context
> >> of calculus.
> >
> >Neither one is the place to learn limits and continuity; the ideas are
> >much clearer in general topological spaces.
>
> There is a difference between the logical context of a mathematical
> concept and its pedagogical context. Sequences and what not
> make the most sense from a topoligcal perspective but
> maybe throwing the notion of topological space at someone
> who has never seen a sequence in a calculus courser might be a bad idea.
>
> For most people, abstract concepts are best learned with
> concrete examples - even if it violates the most "correct"
> logical order of things.
>
> My theory is the one that underlies most teaching of advanced
> math at the undergraduate level - computation provides a set
> of concrete examples that people use to understand more abstract
> structures. Computational examples often indicate why a more
> general fact is true in a specific case. Heck, this is the
> way much mathematics is discovered. It is my contention that
> we should not reject computation as a foundation for mathematics.
I hope nobody has suggested that computation is not important.
Of course one discovers generalities through considering computations
regarding special cases, etc. Now explain how we discover the
irrationality of sqrt(2) through computation.
ull...@math.okstate.edu
sc...@math.csuohio.edu (Brian M. Scott) wrote:
> On Sat, 17 Oct 1998 05:05:24 GMT, aap...@mazel.spc.uchicago.edu
> How the Greeks *discovered* it is unknown. We know, however, that
> they *proved* it, and this is distinctly not a matter of computation.
> As for making a clear distinction between theory and computation, that
> is precisely the Greeks' most significant mathematical contribution;
That's what I thought, in fact that's what I would have thought
since about third grade when I learned about their inventing this
silly stuff about axioms and proofs. But he has a hunch to the
contrary - I don't think we can dismiss his hunches so easily.
> read any standard history of mathematics.
The authors of most standard histories of mathematics are
unaware of his hunches, so they really wouldn't be qualified to
say, would they now? (Wait, maybe I got something backwards there.)
Brian M. Scott
the math majors are generally weaker. <sigh> It's a new course for us,
and I'm not going to get nearly as far as I'd hoped, but I still like
the material. As you know, I agree that manipulative skill is
important; but I also agree with Herman that at the level Fabio was
talking about it's a very poor predictor of mathematical ability.
Brian M. Scott
- - - - - - - -
Alberto Moreira wrote:
>
> Also sprach sc...@math.csuohio.edu (Brian M. Scott) :
>
> >I've no quarrel with this at all. (Or with computation, for that
> >matter: this term I'm teaching a course from Graham, Knuth, and
> >Patashnik.)
>
> If you're teaching GKP - and maybe you'll have some computer science
> majors in your classroom - you may be able to see why I find
> computation and mathematical ability to be intimately intertwined,
> even though I do not disagree when some say that one may not
> necessarily imply the other.
>
> But when studying computer algorithms, the sort of manipulation
> prowess required by a text such as GKP may be well above many CS
> students. And that level of manipulation is required if one's going to
> understand some of the proofs of asymptotic time behavior for many
> computer algorithms.
>
> So, I believe that both are necessary, at least for computer science
> and engineering: a solid conceptual basis, and a strong manipulation
> skill.
>
> Alberto.
Colin Richard Day
aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
>Colin Richard Day <cd...@ix.netcom.com> wrote:
>>In article <F0u1w...@midway.uchicago.edu>,
>>>strong claim. He said "no amount of computational
>>>mathematics is any indiciation of the ability to understand
>>>concepts". These are his words, not mine. My response
>>>is that if you have math ability, you may probably
>>>have computational ability, thus, computational ability
>>>is not such a bad way to guess that a student has
>>>math ability *if you have no other information*.
>>>
>>>Think about it: why do we have *any* prerequisites
>>>for advanced math courses? Why not have all freshman
>>>take real analysis? There's a simple reason. Instructors
>>
>>No, the reason we don't have freshmen take real analysis
>>is that most calculus students are not math majors and
>>have little interest in doing proofs.
>
>This is not so clear to me. A lot of people try to major in math
>because they enjoy problem solving of a rather abstract sort.
>they often do not know what a proof is when they start in the
>math major. If we only encouraged people to major in math
>who had an explicit interest in proofs , the classes
>would be mostly empty.
But you warn potential math majors that they will have to do
proofs to advance in the subject. Also, I still don't believe
that more than 35% of Calc I students are math majors.
>
>I have read some biographies of mathematicians and math related
>types (physicists, statisticians, engineers, etc). In most cases,
>calculus and problem solving is what started the journey.
>
>I might be factually wrong, but that is my impression.
>>
I have also read of students who went into math because you
could prove things in it, such as Gauss' choice of math
over philology for just that reason
I was not claiming that students should know about proof
techniques for the sake of contemplating mathematical
objects, but rather as a general cognitive skill. One
could use logic to analyze the statements of politicains,
as well as the claims of physics and biology. Also, would
studying Derrida help one to understand literature? Just
wondering if I should start reading my copy of On
Grammatology.
>>
>>>
>>>I think this is an empirical matter that can't be settled
>>>in a forum like this. What one would have to do is to
>>>figure out the computational abilities of sucessful
>>>math students and compare them with proof writing skills.
>>>If bet that most proof writers can do computations, but most
>>>"copmuters" can't write proofs. Which is consistent with
>>>what I claim, but not with what Herman claims.
>>
>>Your claim doesn't mean much. Suppose that 70% of the population
>>can compute, 10% of the population can do proofs and 7% can do both.
>>We have that 70% of the proof writers can do computations and that only
>>10% of the computers can write proofs. Does this allow us to use
>>computational skill as a basis for judging proof-writing ability?
>
>Yes - if you don't have any other information about the student.
>My claim is not: "taking derivatives = proof writing ability".
>That's crazy. My claim is : " if all you know is a kid's calculus
>grades, put your money on the kids with the high grades."
>I also make the claim that this is how math instructors operate
>in the class.
>
>>Absolutely not, for the two are statistically independent. What you
>
>Is it? I have not seen evidence that the distribution of computational
>ability is *statistically independent* of mathematical ability.
Perhaps is isn't, but your previous statement did not rule it out.
>This is the point of debate not an assumption. I could be wrong.
>For the most part, proof writers are almost a proper subset
>of "computers" but a rather small subset. I have met only one student
>in my small number of years teaching who genuinely could do advanced
>proofs but could not do an integral so save his life. The poor
>guy just struggled through every ODE/numerical that he was forced
>to take. But damn, he was quite the logician.
I don't know if you have the data, but if you do, you could look at
the *percentage* of students with good computation skills who can
also write proofs, and compare it to the percentage of students
without computation skills who can write proofs.
>
>
>>need to show is that a higher percentage of computers than noncomputers
>>can write proofs.
>
>No. I only need to show that "proof writer --> computer" and
>that "not computer --> not proof writer" for some similarly high
>number of students.
That's a contraposition! Of course they have the same probability!
What you need to show is that (using standard notation for
conditional probability, with 'p' for proof writer and 'c' for computer)
P(p | c)>P(p | ~c), with the difference in probabilities being
significant.
>
>My claim is that the compliment of the set
>[ (proof writers) intersect (computers) ]
>is fairly small.
The complement relative to what universe? And is it small
because proof writing and computational skill are
positively correlated, or because there are just so few
proof writers?
>
>If you buy that, which I think is reasonable, then encouraging
>students on the basis of calculus grades and copmutational
>ability is a good way to make a first guess about a students'
>ability/.
>
>Perhaps you think I believe that "computer --> proof writer".
>I do not and most math teachers don't either.
No, I wasn't saying that.
>>>Fabio
>
>Fabio, again!
>
>
Patricia Handwerk
Colin Richard Day <cd...@ix.netcom.com> wrote in article
<70g46e$m...@dfw-ixnews6.ix.netcom.com>...
> In article <F0uwq...@midway.uchicago.edu>,
> aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
>
> >Colin Richard Day <cd...@ix.netcom.com> wrote:
> But you warn potential math majors that they will have to do
> proofs to advance in the subject. Also, I still don't believe
> that more than 35% of Calc I students are math majors.
>
I agree with you on that one. Now I'm only using my college as a sample so
the results are statistically reliable. We may have approximately 75
students taking Calc I and II. By the time we get to Calc III we only have
approximately 10 students in the class. Of those 10 only 3 are math
majors. So that works out to approximately 4% of the students in Calc I
end up being math majors.
--
Patti Handwerk
Patt...@prodigy.net
Fabio Rojas
<ull...@math.okstate.edu> wrote:
>
> I hope nobody has suggested that computation is not important.
>Of course one discovers generalities through considering computations
>regarding special cases, etc. Now explain how we discover the
>irrationality of sqrt(2) through computation.
I definitely mis-stated myself in the beginning of the argument,
but people were saying that the Greeks did everything without
computation. They painted a picture of teh GReeks are purely
abstract theorem proovers who had no connection to run of the
mill computation. My point is that proof and intuition
emerge from doing computational examples. That is what
I meant. If you accept that, then the idea that we should
ditch computation as a way of teaching proof writing
or for evaluating the talent of students is off the mark.
-fabio
Fabio Rojas
>In article <36283e61...@news.csuohio.edu>,
> sc...@math.csuohio.edu (Brian M. Scott) wrote:
>> On Sat, 17 Oct 1998 05:05:24 GMT, aap...@mazel.spc.uchicago.edu
>> (Fabio Rojas) wrote:
>> >Brian M. Scott <sc...@math.csuohio.edu> wrote:
>> How the Greeks *discovered* it is unknown. We know, however, that
>> they *proved* it, and this is distinctly not a matter of computation.
>> As for making a clear distinction between theory and computation, that
>> is precisely the Greeks' most significant mathematical contribution;
>
> That's what I thought, in fact that's what I would have thought
>since about third grade when I learned about their inventing this
>silly stuff about axioms and proofs. But he has a hunch to the
>contrary - I don't think we can dismiss his hunches so easily.
>
>> read any standard history of mathematics.
>
> The authors of most standard histories of mathematics are
>unaware of his hunches, so they really wouldn't be qualified to
>say, would they now? (Wait, maybe I got something backwards there.)
Read the original post, ullrich.
Herman Rubin said that Greeks did not dismiss students from learning
proof writing because they couldn't do arithmetic.
I said that this was probably false. Why? Because the intuition for
Greek mathematics was derived from having done all sorts of
computations.
So:
computational patterns/anamolies ---> need for proofs ---> actual proofs
(All sorts of cultures) (Greek math)
This is completely compatible with standard histories of mathematics.
There is *no* indication that the Greeks completely abandoned
computational ability in the teaching of math or computational
examples as motiviations for proofs. If you buy into the
huerestic above, then Herman's statement is not true.
-fabio
Brian M. Scott
(Fabio Rojas) wrote:
[snip]
>I definitely mis-stated myself in the beginning of the argument,
>but people were saying that the Greeks did everything without
>computation. They painted a picture of teh GReeks are purely
>abstract theorem proovers who had no connection to run of the
>mill computation.
That's a gross misrepresentation. All Herman said was: ' Were the
ancient Greek students weeded out on the basis of not having strong
arithmetic abilities? I suspect not.' That's a far cry from the
straw man view that you describe.
> My point is that proof and intuition
>emerge from doing computational examples.
Sometimes, and to some extent.
> That is what
>I meant. If you accept that, then the idea that we should
>ditch computation as a way of teaching proof writing
>or for evaluating the talent of students is off the mark.
Computation as a way of teaching proof *writing*?! Don't be absurd!
It's bad enough as a measure of talent; it's downright
counterproductive when it comes to writing proofs, however helpful it
may sometimes be in getting the idea for one.
Brian M. Scott
Jesse
then I propose the student/victim would
be better served by stating that goal
from the beginning, and then teaching
that "skill" directly.
pretending that math translates directly
to other things is quite an assumption,
imho.
could be wrong :)
-Jesse
jcs.r...@asu.edu
Colin Richard Day wrote:
> In article <F0uwq...@midway.uchicago.edu>,
> aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
>
> >Colin Richard Day <cd...@ix.netcom.com> wrote:
> >>In article <F0u1w...@midway.uchicago.edu>,
[..]
> I was not claiming that students should know about proof
> techniques for the sake of contemplating mathematical
> objects, but rather as a general cognitive skill. One
> could use logic to analyze the statements of politicains,
> as well as the claims of physics and biology. Also, would
> studying Derrida help one to understand literature? Just
> wondering if I should start reading my copy of On
> Grammatology.
?signed colin richards?
[..]
ull...@math.okstate.edu
aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
> <ull...@math.okstate.edu> wrote:
> >In article <36283e61...@news.csuohio.edu>,
> > sc...@math.csuohio.edu (Brian M. Scott) wrote:
> >> (Fabio Rojas) wrote:
> >> >Brian M. Scott <sc...@math.csuohio.edu> wrote:
> >> How the Greeks *discovered* it is unknown. We know, however, that
> >> they *proved* it, and this is distinctly not a matter of computation.
> >> As for making a clear distinction between theory and computation, that
> >> is precisely the Greeks' most significant mathematical contribution;
> >
> > That's what I thought, in fact that's what I would have thought
> >since about third grade when I learned about their inventing this
> >silly stuff about axioms and proofs. But he has a hunch to the
> >contrary - I don't think we can dismiss his hunches so easily.
> >
> >> read any standard history of mathematics.
> >
> > The authors of most standard histories of mathematics are
> >unaware of his hunches, so they really wouldn't be qualified to
> >say, would they now? (Wait, maybe I got something backwards there.)
>
> Read the original post, ullrich.
>
> Herman Rubin said that Greeks did not dismiss students from learning
> proof writing because they couldn't do arithmetic.
>
> I said that this was probably false. Why? Because the intuition for
> Greek mathematics was derived from having done all sorts of
> computations.
"read the original post"??? I wasn't replying to the original
post, nor to anything Herman said, I was replying to your _assertions_
(conveniently missing here) that the Greeks discovered the irrationality
of sqrt(2) through computation, and that they did not make a clear
distinction between computation and proof.
> So:
>
> computational patterns/anamolies ---> need for proofs ---> actual proofs
> (All sorts of cultures) (Greek math)
>
> This is completely compatible with standard histories of mathematics.
Of course it is. Nobody said it wasn't.
> There is *no* indication that the Greeks completely abandoned
> computational ability in the teaching of math or computational
> examples as motiviations for proofs. If you buy into the
> huerestic above, then Herman's statement is not true.
What does the question of whether Herman's statement is
true have to do with what I said about what you said? (Or
rather here, what I said about what Brian said about what you
said.) Somehow you're getting from what I said to "everything
you've said is wrong, and everything everyone else said is
true". But I didn't say that.
I _was_ poking fun at your saying "I have a hunch that"
in a discussion like this, as though we should take your
hunches as some sort of evidence. _Regardless_ of whether
your hunches are correct or not you're going to find that
people prefer facts. If Herman were talking about his
hunches I'd find that amusing as well. But I have a hunch
that he might be able to support most of his assertions
with a bit more than just a hunch.
ull...@math.okstate.edu
> (Fabio Rojas) wrote:
>
> [snip]
>
> >I definitely mis-stated myself in the beginning of the argument,
> >but people were saying that the Greeks did everything without
> >computation. They painted a picture of teh GReeks are purely
> >abstract theorem proovers who had no connection to run of the
> >mill computation.
>
> That's a gross misrepresentation. All Herman said was: ' Were the
> ancient Greek students weeded out on the basis of not having strong
> arithmetic abilities? I suspect not.'
Thanks for clarifying that. I couldn't find Herman saying
what he said he said, didn't have time to search the whole thread.
> That's a far cry from the
> straw man view that you describe.
>
> > My point is that proof and intuition
> >emerge from doing computational examples.
>
> Sometimes, and to some extent.
>
> > That is what
> >I meant. If you accept that, then the idea that we should
> >ditch computation as a way of teaching proof writing
> >or for evaluating the talent of students is off the mark.
>
> Computation as a way of teaching proof *writing*?! Don't be absurd!
> It's bad enough as a measure of talent; it's downright
> counterproductive when it comes to writing proofs, however helpful it
> may sometimes be in getting the idea for one.
>
> Brian M. Scott
>
-----------== Posted via Deja News, The Discussion Network ==----------
Fabio Rojas
>
>> That is what
>>I meant. If you accept that, then the idea that we should
>>ditch computation as a way of teaching proof writing
>>or for evaluating the talent of students is off the mark.
>
>Computation as a way of teaching proof *writing*?! Don't be absurd!
>It's bad enough as a measure of talent; it's downright
>counterproductive when it comes to writing proofs, however helpful it
>may sometimes be in getting the idea for one.
>
>Brian M. Scott
By teaching proof writing, I mean using computation as a
away of giving motivation for more abstract concepts.
Also in the course of writing proofs, long and difficult
computations may be necessary. All I am arguing is that
in the course of teaching proofs, we use a lot of copmutational
examples to help us and that this is a good thing.
Fabio
Fabio Rojas
>In article <F13G3...@midway.uchicago.edu>,
> aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
>> <ull...@math.okstate.edu> wrote:
>> >> (Fabio Rojas) wrote:
>> >> >Brian M. Scott <sc...@math.csuohio.edu> wrote:
>> >> they *proved* it, and this is distinctly not a matter of computation.
>> >> As for making a clear distinction between theory and computation, that
>> >> is precisely the Greeks' most significant mathematical contribution;
>> >
>> > That's what I thought, in fact that's what I would have thought
>> >since about third grade when I learned about their inventing this
>> >silly stuff about axioms and proofs. But he has a hunch to the
>> >contrary - I don't think we can dismiss his hunches so easily.
>> >
>> >> read any standard history of mathematics.
>> >
>> > The authors of most standard histories of mathematics are
>> >unaware of his hunches, so they really wouldn't be qualified to
>> >say, would they now? (Wait, maybe I got something backwards there.)
>>
>> Read the original post, ullrich.
>>
>> Herman Rubin said that Greeks did not dismiss students from learning
>> proof writing because they couldn't do arithmetic.
>>
>> I said that this was probably false. Why? Because the intuition for
>> Greek mathematics was derived from having done all sorts of
>> computations.
>
> "read the original post"??? I wasn't replying to the original
>post, nor to anything Herman said, I was replying to your _assertions_
>(conveniently missing here) that the Greeks discovered the irrationality
>of sqrt(2) through computation, and that they did not make a clear
>distinction between computation and proof.
My assertions are included above and sumamrized in my responses but
you have to understand them in context - I was responding to what
Herman said. I think what he said was important and my comments
only make sense in the way that I meant them in reference
to what he said. I stand by my statement that the Greeks
did not completely divorce computation and proof from each other.
The idea that proofs and computation are radically different
is a fairly modern context. This is not to say that the GReeks
did not produce two distinct sets of objects: proofs and computations.
I think they just interpreted them a little differently than we
do.
>
>> So:
>>
>> computational patterns/anamolies ---> need for proofs ---> actual proofs
>> (All sorts of cultures) (Greek math)
>>
>> This is completely compatible with standard histories of mathematics.
>
> Of course it is. Nobody said it wasn't.
>
>> There is *no* indication that the Greeks completely abandoned
>> computational ability in the teaching of math or computational
>> examples as motiviations for proofs. If you buy into the
>> huerestic above, then Herman's statement is not true.
>
> What does the question of whether Herman's statement is
>true have to do with what I said about what you said? (Or
>rather here, what I said about what Brian said about what you
>said.) Somehow you're getting from what I said to "everything
>you've said is wrong, and everything everyone else said is
>true". But I didn't say that.
>
> I _was_ poking fun at your saying "I have a hunch that"
>in a discussion like this, as though we should take your
>hunches as some sort of evidence. _Regardless_ of whether
In the course of argument, you should present your reasons
for what they are. I was presenting a conclusion that
I had drawn from what I had read but that is not directly
made in what I read. It is totally ok to present
"hunches" if you present them as such.
>your hunches are correct or not you're going to find that
>people prefer facts. If Herman were talking about his
When talking about history, sometimes the "facts" are
missing and inference is all we have. I have not yet
read a book about *exactly* how the Pythagorean theorem
was prooved in Greece. Perhaps someone has written such
a book. SO in the absence of evidence, inference or "hunch"
are all we have. The record seems to be that previous
cultures had discovered the Pythagorean theorem empirically
but only the Greeks came up with a proof. How someone
came up with the proof is unkown (or is it? Could you
recommend a book?)
What the record is missing is how the Pythagoreans taught
thsi theorem to young people. Herman claimed that they
probably never turned away students because they couldn't
do computation. My *HUNCH* - because we have no evidence
eitehr way - is that they did turn students away from
learning proofs if they couldn't do arithmetic because
so much of Greeks math - but definitely not all by a long
shot - is inspired by computational example. If little
Euclid couldn't do some of these examples, I bet they
would have sent him on his way. I think this is
a pretty valid line of reasoning in the absence of
other evidence.
>hunches I'd find that amusing as well. But I have a hunch
>that he might be able to support most of his assertions
>with a bit more than just a hunch.
He and I don't disagree on specific facts at all, it's
the interpretation. We both agree theorem prooving
is different than computation, even back in the time
of the GReeks who invented theorem prooving. I was just
arguing a different point, about teaching.
>
>-----------== Posted via Deja News, The Discussion Network ==----------
>http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own
Fabio!!
Colin Richard Day
Jesse <jcs.r...@asu.edu> wrote:
>if the goal is general cognitive skill,
>then I propose the student/victim would
>be better served by stating that goal
>from the beginning, and then teaching
>that "skill" directly.
>
>pretending that math translates directly
>to other things is quite an assumption,
>imho.
>
>could be wrong :)
>
>-Jesse
>jcs.r...@asu.edu
>
>Colin Richard Day wrote:
>
>> In article <F0uwq...@midway.uchicago.edu>,
>> aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
>>
>> >Colin Richard Day <cd...@ix.netcom.com> wrote:
>> >>In article <F0u1w...@midway.uchicago.edu>,
>
>[..]
>
>> I was not claiming that students should know about proof
>> techniques for the sake of contemplating mathematical
>> objects, but rather as a general cognitive skill. One
>> could use logic to analyze the statements of politicains,
>> as well as the claims of physics and biology. Also, would
>> studying Derrida help one to understand literature? Just
>> wondering if I should start reading my copy of On
>> Grammatology.
>
>?signed colin richards?
>[..]
>
I was not claiming that math courses were the best place to
teach logic; it could be done in a logic course. Also, I
do not compel my students to do proofs unless the topic is on
the syllabus.
Sorry for forgetting the sig.
Colin Day cd...@ix.netcom.com
Colin Richard Day
aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
> <ull...@math.okstate.edu> wrote:
>>In article <F13G3...@midway.uchicago.edu>,
>> aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
>>> <ull...@math.okstate.edu> wrote:
>>> >In article <36283e61...@news.csuohio.edu>,
>>> > sc...@math.csuohio.edu (Brian M. Scott) wrote:
>>> >> On Sat, 17 Oct 1998 05:05:24 GMT, aap...@mazel.spc.uchicago.edu
>>> >> (Fabio Rojas) wrote:
>>> >> >Brian M. Scott <sc...@math.csuohio.edu> wrote:
Much snipping
>
>What the record is missing is how the Pythagoreans taught
>thsi theorem to young people. Herman claimed that they
>probably never turned away students because they couldn't
>do computation. My *HUNCH* - because we have no evidence
>eitehr way - is that they did turn students away from
>learning proofs if they couldn't do arithmetic because
>so much of Greeks math - but definitely not all by a long
>shot - is inspired by computational example. If little
>Euclid couldn't do some of these examples, I bet they
>would have sent him on his way. I think this is
>a pretty valid line of reasoning in the absence of
>other evidence.
Many ancient Greeks actually taught for money, and might have
been as loath to reject students as modern American universities.
The Pythagoreans were probably not among such Greeks, but Thales
might have been patient with a wealthy but dumb student.
Would someone with a deeper knowledge of classical history care
to respond?
>
>>
>>-----------== Posted via Deja News, The Discussion Network ==----------
>>http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own
>
>Fabio!!
>
Colin Day cd...@ix.netcom.com
ull...@math.okstate.edu
aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:
[...]
> I stand by my statement that the Greeks
> did not completely divorce computation and proof from each other.
Well good for you. Now we have one more person who
stands by his statements in spite of the fact that he has no
idea what he's talking about, and in spite of the fact that
he's had the truth explained to him by various people. This
is great - we were running out of those.
> The idea that proofs and computation are radically different
> is a fairly modern context.
Assuming that "fairly modern" means on a geolgical
or astronomical sort of time scale (ie "in the last two or
three thousand years") I'd have to agree to this.
[...]
> When talking about history, sometimes the "facts" are
> missing and inference is all we have. I have not yet
> read a book about *exactly* how the Pythagorean theorem
> was prooved in Greece. Perhaps someone has written such
> a book.
Um, Euclid wrote such a book.
> SO in the absence of evidence, inference or "hunch"
> are all we have. The record seems to be that previous
> cultures had discovered the Pythagorean theorem empirically
> but only the Greeks came up with a proof. How someone
> came up with the proof is unkown (or is it? Could you
> recommend a book?)
I wouldn't know how someone _came up with_ the
proof. When I look at the math journals in the library
I have no idea how the authors came up with those proofs
either.
> What the record is missing is how the Pythagoreans taught
> thsi theorem to young people. Herman claimed that they
> probably never turned away students because they couldn't
> do computation. My *HUNCH* - because we have no evidence
> eitehr way - is that they did turn students away from
> learning proofs if they couldn't do arithmetic because
> so much of Greeks math - but definitely not all by a long
> shot - is inspired by computational example. If little
> Euclid couldn't do some of these examples, I bet they
> would have sent him on his way. I think this is
> a pretty valid line of reasoning in the absence of
> other evidence.
>
> >hunches I'd find that amusing as well. But I have a hunch
> >that he might be able to support most of his assertions
> >with a bit more than just a hunch.
>
> He and I don't disagree on specific facts at all, it's
> the interpretation. We both agree theorem prooving
> is different than computation, even back in the time
> of the GReeks who invented theorem prooving. I was just
> arguing a different point, about teaching.
Huh? There's nothing about teaching in the bit you
said that I said something about. One _more_ time: when
you say something and someone claims it was incorrect
talking about something _else_ you said doesn't make much
sense as a rebuttal.
Herman Rubin
Fabio Rojas <aap...@mazel.spc.uchicago.edu> wrote:
> <ull...@math.okstate.edu> wrote:
>>In article <36283e61...@news.csuohio.edu>,
>> sc...@math.csuohio.edu (Brian M. Scott) wrote:
>>> On Sat, 17 Oct 1998 05:05:24 GMT, aap...@mazel.spc.uchicago.edu
>>> (Fabio Rojas) wrote:
>>> >Brian M. Scott <sc...@math.csuohio.edu> wrote:
>>> they *proved* it, and this is distinctly not a matter of computation.
>>> As for making a clear distinction between theory and computation, that
>>> is precisely the Greeks' most significant mathematical contribution;
>> That's what I thought, in fact that's what I would have thought
>>since about third grade when I learned about their inventing this
>>silly stuff about axioms and proofs. But he has a hunch to the
>>contrary - I don't think we can dismiss his hunches so easily.
>>> read any standard history of mathematics.
..................
>Herman Rubin said that Greeks did not dismiss students from learning
>proof writing because they couldn't do arithmetic.
>I said that this was probably false. Why? Because the intuition for
>Greek mathematics was derived from having done all sorts of
>computations.
I do not believe that this was the case; they might have gotten
started that way, but one will find little need for computation
anywhere in Euclid's _Elements_, and essentially none in the
geometric section.
>So:
>computational patterns/anamolies ---> need for proofs ---> actual proofs
>(All sorts of cultures) (Greek math)
This is not correct. The classical geometric results may have used
observations on triangles, but still very little computation, even to
discover the results. One cannot be sure that a 20-21-29 triangle is
a right triangle by measurement, as the measurements are too crude.
The Egyptian value for pi, obtained by observation and measurement, is
far worse than 3.14 or 22/7. Even with the theory obtained by Archimedes,
he was not able to get too close. Any real improvement required coming
up with infinite series, or other infinite processes, and were not based
on any computational devices. Those who are now computing mathematical
constants and functions to high accuracy still do not need to be able to
do "hand" computations.
>This is completely compatible with standard histories of mathematics.
>There is *no* indication that the Greeks completely abandoned
>computational ability in the teaching of math or computational
>examples as motiviations for proofs. If you buy into the
>huerestic above, then Herman's statement is not true.
The use of numerical or other computational examples as motivation
still does not require that the person using it can do the actual
manipulations, and they have always been few.
Herman Rubin
>>Brian M. Scott
In writing proofs, and I have published quite a few of them in
various parts of mathematics, only occasionally has there been
a fair amount of computation, and most of that has been symbolic.
There was one where I needed to use a lot of computation to
achieve the result, but I made no attempt to do it by hand, but
rather put it on the computer.
There are cases in which I have done computation to get a good
idea of what was happening, but again, given good facilities, I
would have used electronic means.
It is rarely the case that a long computation by a human would
be trusted; the possibilities for error are far too great.
Bill Taylor
|> > I stand by my statement that the Greeks...
|> Well good for you. Now we have one more person who
|> stands by his statements in spite of the fact that he has no
|> idea what he's talking about, and in spite of the fact that
|> he's had the truth explained to him by various people.
Yes, quite.
Eggshelly, one can almost declare a uniform rule here, a classic "giveaway".
The mere use of the phrase "I stand by my statement that..."
============================
tends to be good enough to mark down the author as a bit of a crank, or
at least someone who's been caught out in a nonsense and is too embarrassed
and/or feather-ruffled to admit it and back down (or at least shut up).
It's usually an excellent clue that such a poster is not to be taken seriously.
Another similar giveaway is the use of "methinks", which almost always
signals a complete pillock. ========
Any others?
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Clever but long saying reduced to a microdot ----> "."
-------------------------------------------------------------------------------
P.S. Let's not have any responses about nincompoops
who use conversationalities like "eggshelly"...
Fabio Rojas
>ull...@math.okstate.edu writes:
>|> > I stand by my statement that the Greeks...
>
>|> Well good for you. Now we have one more person who
>
The crank is back!! Wow, have a disagreement and people call you
a crank. If someone says, "I disagree with your rebuttal and
stand by what I said" they get called a crank. That's a swell
standard.
Look, here's the argument: Herman claimed that we should *never*
use a student's computational ability as evidence that
they may have real math ability. In the course of debate, he
claimed that the Greeks did not turn away students who couldn't
do arithmetic. My response was that Greek math contained and
was inspired by many computational examples. As such, I found
it unlikely that if a Greek kid could not do arithmetic
that he was encouraged to do math. You responded that
many Greeks were teachers for hire who probably took many
kids who were bad at arithmetic. I had not thought of this
and I think this is right. I still stand by my statement
that Greek math was so based on arithmetic examples
that many students who were bad at arithmetic probably had
trouble with Greek math.
Here are some examples from eve's "An introduction to the
History of mathematics":
figurate numbers: numbers which could be arranged in a square, pentagon, etc.
ratios and proportions: who could a student who didn't do arithmetic
understand these central concepts?
the Greek (including archimedes) approximations of pi: how
could any person with a lack of arithmetic skills understand
the concept of approximation? of even of the number pi?
These may not be as famous as Euclid's axiomatic proofs
but they were very important to the Greeks .
Some sections of Greek math, most importantly the elements,
contain very little arithmetic. it is all proof. My argument
all along is that such proofs may seem pointless or difficult
to apply to specific cases if you have no arithmetic ability.
I also argued that Greeks didn't see proofs and computation
as all that different. This does sound odd, but the Pythagoreans
at least did consider proofs as explanations of computational
irregularities that had mystical qualities. I don't know anybody
who would consider modern number theory to be holy in the same
way. This is straight from the history texts. My point
is that proofs and computations were so tied in the minds of the
Greeks that the modern person would find it odd. The Elements
is the culmination of a long train of thought that was wrestling
with computation and trying to turn it into proof (succesfuly).
In modern times, we *start* with the elements or the axioms,
at least I did in high school and college, so we think of computation and proof
as being very far off or or somewhat unrelated activities.
Ok, here is my crank tip-off, so everyone can now ignore what
I said: I think this is not such a crazy argument and that previous
rebuttals were talking aboout a different thing. Previous people
seem to think I said something like "for GReeks computations
were the same as proofs". I did say one thing that was completely
wrong(I meant something else but I just wrote it very badly),
about irrational numbers. But the rest is factually correct.
My point is that we are accustomed to the definition - theorem -
proof style of math (at least I am) that we forget that
for all mathematicians (including the Greeks) proofs only
come from years of computational testing coupled with theoretical
("mathematical") insight. You really can't have one without the
other.
Here is one last question about computation and teaching,
which is really what this argument was originally about:
If a student got an F in calculus, would you recommend
that the student take real analysis or try to dissuade
them? When I used to teach, I would definitely say
retake calculus before analysis. Is this crazy advice?
If you really belive that math and computation are truly independent
abilities, then you would say that I gave the wrong advice. Why is
that ? Most teachers I have known would give the same advice.
-fabio
Herman Rubin
Fabio Rojas <aap...@mazel.spc.uchicago.edu> wrote:
>Bill Taylor <mat...@math.canterbury.ac.nz> wrote:
>>ull...@math.okstate.edu writes:
>>|> > I stand by my statement that the Greeks...
>>|> Well good for you. Now we have one more person who
................
>Look, here's the argument: Herman claimed that we should *never*
>use a student's computational ability as evidence that
>they may have real math ability. In the course of debate, he
>claimed that the Greeks did not turn away students who couldn't
>do arithmetic. My response was that Greek math contained and
>was inspired by many computational examples.
And I repeat that this was not the case. No lengthy list of
"Pythagorean triples" will give any indication that the
Pythagorean THEOREM is true. The Babylonians did have
constructing many of these as an exercise in arithmetic,
but this is all we know about it.
As such, I found
>it unlikely that if a Greek kid could not do arithmetic
>that he was encouraged to do math.
We do not know; but we do know that the Greeks stressed
approaches based on logic.
You responded that
>many Greeks were teachers for hire who probably took many
>kids who were bad at arithmetic. I had not thought of this
>and I think this is right. I still stand by my statement
>that Greek math was so based on arithmetic examples
>that many students who were bad at arithmetic probably had
>trouble with Greek math.
We have little evidence that the Greeks did much with
arithmetic.
>Here are some examples from eve's "An introduction to the
>History of mathematics":
>figurate numbers: numbers which could be arranged in a square, pentagon, etc.
>ratios and proportions: who could a student who didn't do arithmetic
>understand these central concepts?
There is no problem whatever with this. People can understand
concepts without doing manipulations. Also, the Greeks had problems
with these, as they were lacking algebraic notation until
Diophantus. With algebraic notation, it becomes very easy.
>the Greek (including archimedes) approximations of pi: how
>could any person with a lack of arithmetic skills understand
>the concept of approximation? of even of the number pi?
It would be impossible to COMPUTE an approximation to pi
without arithmetic skills in the days before computers.
But, to the person not brainwashed into believing that
mathematics is computation, understanding these are easier
than to the one who is.
>These may not be as famous as Euclid's axiomatic proofs
>but they were very important to the Greeks .
>Some sections of Greek math, most importantly the elements,
>contain very little arithmetic. it is all proof. My argument
>all along is that such proofs may seem pointless or difficult
>to apply to specific cases if you have no arithmetic ability.
In order to apply the geometry, it was usually necessary to
go to trigonometry. Before calculators, getting the answers
of course required computational ability. But understanding
did not.
>I also argued that Greeks didn't see proofs and computation
>as all that different. This does sound odd, but the Pythagoreans
>at least did consider proofs as explanations of computational
>irregularities that had mystical qualities. I don't know anybody
>who would consider modern number theory to be holy in the same
>way. This is straight from the history texts. My point
>is that proofs and computations were so tied in the minds of the
>Greeks that the modern person would find it odd.
The mystic use of numbers uses only quite small numbers.
The Elements
>is the culmination of a long train of thought that was wrestling
>with computation and trying to turn it into proof (succesfuly).
There is no wrestling with computation there.
>In modern times, we *start* with the elements or the axioms,
>at least I did in high school and college, so we think of computation and proof
>as being very far off or or somewhat unrelated activities.
................
>Here is one last question about computation and teaching,
>which is really what this argument was originally about:
>If a student got an F in calculus, would you recommend
>that the student take real analysis or try to dissuade
>them? When I used to teach, I would definitely say
>retake calculus before analysis. Is this crazy advice?
Retaking a course, unless the student just goofed off, is RARELY a
good action. It may be needed for stupid reasons, like graduation
requirements. If a student got an F in calculus, unless the other
problems were corrected, such as not having an understanding of
algebra or logic, retaking it would at best get a D or a lucky C,
and not have any better understanding.
There is nothing about memorizing rules for calculating derivatives
and antiderivatives, memorizing techniques for evaluation by
integration by parts and various kinds of substitutions, etc.,
which will provide any help in understanding what is going on.
The student would have a much better chance by taking analysis.
>If you really belive that math and computation are truly independent
>abilities, then you would say that I gave the wrong advice. Why is
>that ? Most teachers I have known would give the same advice.
I regret to have to say that most teachers make no attempt to teach
understanding, especially in courses where students will complain,
"Don't bother me with the `theory'. Just tell me how to do the
problems on the exam." Even those who would teach concepts have
to cater to these students, with the result that the substantial
minority who could learn concepts do not. Many of them become
teachers of mathematics.
Gerry Myerson
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
=> The mere use of the phrase "I stand by my statement that..."
=> ============================
=> tends to be good enough to mark down the author as a bit of a crank....
Will you stand by this statement?
GM
Bill Taylor
|> => ============================
|> => tends to be good enough to mark down the author as a bit of a crank....
|> Will you stand by this statement?
Eh? I don't remember saying that.
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Personally, I believe in solipsism, but that's just one man's opinion.
-------------------------------------------------------------------------------
david.g...@gmail.com
> Is there a =documented= national figure for passing calculus courses?
>
> Jim Epperson
>
> The right half of the brain controls the left half of the body.
> This means that only left handed people are in their right mind.
I am getting bent over in my math 151 01 course, calculus 1 sequence. I took calc 1 in high school and got a c, and now ill be lucky to finish this course with a d. They really are expecting you to be a master of calculus 1 if you want to excel into the higher math courses. Looks like ill be taking math 151 again next semester...smh.