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Oct 6, 1998, 3:00:00 AM10/6/98

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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.

Oct 6, 1998, 3:00:00 AM10/6/98

to

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.

Oct 6, 1998, 3:00:00 AM10/6/98

to

On Tue, 6 Oct 1998, Alan Morgan wrote:

> 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

Oct 6, 1998, 3:00:00 AM10/6/98

to

> > 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.

> "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

Oct 6, 1998, 3:00:00 AM10/6/98

to

On 6 Oct 1998, Robert J. Pease wrote:

> 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.

Oct 6, 1998, 3:00:00 AM10/6/98

to

Let's suppose the rate is 20% - what does that say about the way in which we

choose to run our society? What would be different if the rate were 50%? How

about 80%?

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.

>

Oct 6, 1998, 3:00:00 AM10/6/98

to

Robert J. Pease <bobp...@pop3.concentric.net> wrote:

: 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))) ) )

Oct 6, 1998, 3:00:00 AM10/6/98

to

> Let's suppose the rate is 20% - what does that say about the way in which we

> choose to run our society? What would be different if the rate were 50%? How

> about 80%?

> 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.

> >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

Oct 6, 1998, 3:00:00 AM10/6/98

to

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!

Oct 6, 1998, 3:00:00 AM10/6/98

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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.

Oct 7, 1998, 3:00:00 AM10/7/98

to

On Tue, 6 Oct 1998 22:08:38 -0400, "Dr. Michael Albert"

<alb...@esther.rad.tju.edu> wrote:

<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

Oct 7, 1998, 3:00:00 AM10/7/98

to

In article <361ae886...@news.csuohio.edu>,

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.

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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Oct 7, 1998, 3:00:00 AM10/7/98

to

In article <Pine.A41.4.05.981006...@dante23.u.washington.edu>,

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.

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

Oct 7, 1998, 3:00:00 AM10/7/98

to

In article <Pine.GSO.3.95.981006...@esther.rad.tju.edu>,)> Let's suppose the rate is 20% - what does that say about the way in which we

)> 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

)> 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.

Oct 12, 1998, 3:00:00 AM10/12/98

to

In article <6vg0ec$9fn$1...@nnrp1.dejanews.com>, <bo...@rsa.com> wrote:

>> >It has been my experience that in most calculus classes, the majority

>> >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]>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

Oct 12, 1998, 3:00:00 AM10/12/98

to

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.

>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.

Oct 12, 1998, 3:00:00 AM10/12/98

to

On Mon, 12 Oct 1998 03:25:06 GMT, aap...@mazel.spc.uchicago.edu

(Fabio Rojas) wrote:

(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

Oct 12, 1998, 3:00:00 AM10/12/98

to

In article <F0p2t...@midway.uchicago.edu>,

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.

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

Oct 12, 1998, 3:00:00 AM10/12/98

to

This is the problem that I have, in general, with grading on a curve - it

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.

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.

>

Oct 13, 1998, 3:00:00 AM10/13/98

to

On Mon, 12 Oct 1998 21:28:20 -0600, "William L. Bahn" <ba...@bfe.com>

wrote:

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

Oct 14, 1998, 3:00:00 AM10/14/98

to

I've snipped and added a lot...

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

-----== Posted via Deja News, The Leader in Internet Discussion ==-----

http://www.dejanews.com/rg_mkgrp.xp Create Your Own Free Member Forum

Oct 14, 1998, 3:00:00 AM10/14/98

to

fgr...@my-dejanews.com wrote:

> 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

Oct 14, 1998, 3:00:00 AM10/14/98

to

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.

>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?

>

>> 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.

>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

Oct 15, 1998, 3:00:00 AM10/15/98

to

In article <F0u1w...@midway.uchicago.edu>,

aap...@mazel.spc.uchicago.edu (Fabio Rojas) wrote:

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

>

Oct 15, 1998, 3:00:00 AM10/15/98

to

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

>

>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.

>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!

Oct 15, 1998, 3:00:00 AM10/15/98

to

>>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

Oct 16, 1998, 3:00:00 AM10/16/98

to

In article <F0u1w...@midway.uchicago.edu>,

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?

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