My weak student can beat your weak student!
Dave Rusin
A student just left from an hour of private office help. Since I
really oughn't go out for a drink at mid-day, I hope venting some steam
in public will have a purgative effect instead. This is for real.
[Student has come in with the weak background that leaves calculus
instructors scratching their heads. We talk about computing derivatives
as limits of difference quotients -- student had flubbed the derivative of
3x+(4/x) by virtue of failing miserably to subtract the necessary fractions
using common denominators. We discuss the need for algebra skills, then
move on. That, perhaps, was my fatal error...]
Student: "I'm having some real problems in this class [Calculus 1].
Can you show me how to do problems like this in case we have some on
the next test?" [Student points to a problem of the form, "Sketch a
function which satisfies f'(x)>0 for x in [0,1], etc."]
Me [somewhat puzzled, since there's nothing to 'do' on those problems]:
"OK, let's try one to see if you were with me in class today. Sketch a
function which has f' positive everywhere but f''(x) > 0 for x < 2 and
f''(x) < 0 for x > 2."
S: "So I need f' to be increasing on the first part."
[Sketches something like y=log(x) ] "Like this?"
M: "You were right -- you needed f' to be increasing. Now, f' measures
the slope of the tangent line, right? So you mean to say the slope of the
tangent line is getting greater as you move from left to right?"
[S is silent, appears to have thought so.]
M: "What would you say the slope is here?" [Points to left edge of the graph,
slope is around 4. Hard to tell -- this is freehand, no grid.]
S: "Around 1?"
M [puzzled]: "Hm, well the picture's a little unclear but OK. And here?"
[Points to rightmost point, where the slope is really a little less than 1.]
S: "Maybe 4?"
M [pause]: "You're saying the slope at this second point is four times as
great as the slope at the first point?" [Effort to mask incredulity is
probably wasted.] "What does 'slope' mean?"
S: "See, I can't describe it so well. I know the formula..."
M: "If you understand it, you should be able to describe it in half a dozen
words, tops. Look, let's do this accurately" [Produce graph paper with
half-inch grid, recreate general shape, mark two interior points on graph
near left and right edges.] "OK, here are the two points on the graph.
Show me the tangent lines" [Provides a ruler.]
[Student draws the line segments well.] "OK, now what's the slope
of this first line?" [The points (1,2) and (3,6) are conveniently near the
endpoints of student's line segment.]
S: "I'd have to write it down and I..."
M: "Sure go ahead. Write anything you need to."
S: [Carefully writes, correctly, "(y1-y0)/(x1-x0)=". Counts coordinates
1, 2, 3, 4, 5, 6; 1, 2; 1, 2, 3; 1. Writes "(6-2)/(3-1)=4/2=2".]
"The slope is 2".
M: "Right. But you made it much too hard for yourself." [Thinking ahead to
the other point, whose coordinates are around (10,18)...] "See, all you
needed to do is to count the _difference_ between the y-coordinates and
then between the x-coordinates" [Draws the little triangle]. "Most students
just remember 'rise over run'. See, that's what the '6-2' measures is
the _difference_ between the y-coordinates -- what we usually just call
the 'rise'." [Small lecture followed.] "So now what's the slope of the
second line?"
S: [Draws the small triangle this time. Coordinates are not lattice points.]
"Well, the rise is..." [counts off] "It's more than two; could it be
two and a half?"
M: [Surprised that this could be a question, though probably 2+1/3 is closer.]
"Yes, sure, that's close enough. And the run is...?"
S: [Counts carefully.] "3" [Looks for confirmation] "So the slope is
two-and-a-half over 3".
M: [Sensing that we have yet to hit bottom] "Right. Two-and-a-half thirds.
As a simple fraction that would be..."
S: [Unsure] "You mean this?" [Writes "{2.5}\over{3}"]
M: "Yes, but you can write that as a simple fraction, you know, a ratio of
two whole numbers. Here, you've written that numerator as a decimal, which
is fine but you can write it as a fraction, too. What's two and a half as
a fraction?"
S: [Writes "2 {1\over 2}"] "You mean like this?"
M: "Well, that's a mixed number. You can write it as a simple fraction.
What is it -- eleven ninths? seven fourths? What is two and a half as a
fraction?"
S: [Draws parentheses around the "2" and the "1/2"] "One?"
M: [Losing patience now] "No! If I give you two and a half bucks, have I
just given you one dollar? What's two and a half? It's two AND a half.
That means two PLUS a half"
S: "Oh so it's back to common denominators". [Starts to fumble with halves.
Perhaps recognizing this is supposed to be the kind of thing one should
be able to do mentally, announces:] "Three halves."
M: [Barely resisting the impulse to be insulting, reaches for some coins.]
"Look, I haven't got any half-dollars, so pretend these other coins are all
half-dollars. [Throw in pairs] Here's one dollar, here's another, here's
an extra half-dollar. I've just given you two and a half dollars, and it's
what? Five half-dollars. That's five halves of a dollar. Two and a half
is five halves." [Small diatribe about the need for students to actually
_learn_ the material they deal with in math classes. Small concession
thrown in for student's emotional well-being, recognizing that the student
probably got shafted by lousy teachers early on. I can't change that now...]
M: "OK, great. So this numerator is five halves. Now that slope is
five halves over 3. Now that's a compound fraction. Do you remember how
to simplify those?"
S: [tired and embarassed now] "No."
M: "Here, let me remind you". [Big fraction bars used to show the
invert-and-multiply routine.] "So you get five-sixths for this slope.
Now, is that bigger than the slope we had over here? Is it even bigger
than one?"
S: "I don't know."
M: [Well-practiced skills of teachers' patience clearly wearing thin]
"What is five sixths? Can you describe what that is in some other way?
What is that number?"
S: "Is it, um, negative..."
I didn't let the student finish describing the number.
Meet you in the bar in five minutes.
Rainer Rosenthal
Dave Rusin <ru...@vesuvius.math.niu.edu> wrote in message
news:8li1a4$duu$1...@gannett.math.niu.edu...
>
> A student just left from an hour of private office help. Since I
> really oughn't go out for a drink at mid-day, I hope venting some steam
> in public will have a purgative effect instead. This is for real.
>
Hi Dave,
I believe you fully. My opinion regarding this sad
little story is as follows: This guy is not so bad at all,
since he (1) knows, he needs help and (2) does what
you ask him to do - as well as he is able to.
>
> Student: "I'm having some real problems in this class [Calculus 1].
>
Fine, does not pretend to know.
> M: "Sure go ahead. Write anything you need to."
>
> S: [Carefully writes, correctly, "(y1-y0)/(x1-x0)=". Counts coordinates
> 1, 2, 3, 4, 5, 6; 1, 2; 1, 2, 3; 1. Writes "(6-2)/(3-1)=4/2=2".]
> "The slope is 2".
>
> M: "Right. But you made it much too hard for yourself."
Letting him count the difficult way is a good technique. So
he's got something to do and is happy with your confirmation.
It was meant friendly by you, helping him optimize this step.
But blows up the discussion and brings more information
into it. ONE bit of "Right" confirmed knowledge in this
DESERT of non-knowledge is VERY MUCH.
Like a little plant it should be cared for and cultivated.
Don't expect it to become a tree within seconds.
(Well this is my feeling far from the place where this all
happened. And it is easy to make wise remarks. But why
not collecting the steam which you mentioned in the beginning
of your mail and converting it into water for plants like this ?)
>
> S: [Draws parentheses around the "2" and the "1/2"] "One?"
>
This really makes one take a deep breath. You look into an abyss.
But without exercizing and practicing, all these conventions on
math formalism cannot be grasped. So the only thing that can
be concluded here is: STOP - we need basic help for this guy.
> M: [Losing patience now] "No! If I give you two and a half bucks, have I
> just given you one dollar? What's two and a half? It's two AND a half.
> That means two PLUS a half"
>
This is an ad hoc trial for "basic help". But two things are lacking:
(1) Patience and (2) a plan as to which goal is to be achieved next.
> S: "Oh so it's back to common denominators". [Starts to fumble with
halves.
> Perhaps recognizing this is supposed to be the kind of thing one should
> be able to do mentally, announces:] "Three halves."
>
Applause :-)
>
> M: "OK, great. So this numerator is five halves. Now that slope is
> five halves over 3. Now that's a compound fraction. Do you remember how
> to simplify those?"
>
> S: [tired and embarassed now] "No."
>
Another lesson in "basic help".
> "What is five sixths? Can you describe what that is in some other way?
> What is that number?"
>
> S: "Is it, um, negative..."
>
Yes that beats all. But I cannot laugh (even though you told
the story livingly).
From my own experience I can assure you, that helping
people in such a situation with basic information ( and not
too much optimization and 'overhelp') and getting their
math brain working - is very very satisfying.
You soon get AHA's and surprise and even fun. I know that.
And it's so cheap: because you KNOW.
But for 'knowers' the 'not-knowers' are often like aliens.
They are not. They are like you and me (more or less).
With friendly regards,
Rainer
----------------
Conversation between programs:
"Do you believe in programmers ?" (Rainer)
John R Ramsden
>
> M: [Surprised that this could be a question, though probably 2+1/3 is closer.]
> "Yes, sure, that's close enough. And the run is...?"
>
> S: [Counts carefully.] "3" [Looks for confirmation] "So the slope is
> two-and-a-half over 3".
>
> M: [Sensing that we have yet to hit bottom] "Right. Two-and-a-half thirds.
> As a simple fraction that would be..."
>
> S: [Unsure] "You mean this?" [Writes "{2.5}\over{3}"]
I think at this point it might have been simpler to say "You can multiply
top and bottom by the any non-zero number without changing the value of
the fraction. So what whole number can you multiply 2.5 by to turn the
result into a whole number? Hint, how many lots of $2.50 must I have to
end up with a whole number of dollars?"
> M: "Yes, but you can write that as a simple fraction, you know, a ratio of
> two whole numbers. Here, you've written that numerator as a decimal, which
> is fine but you can write it as a fraction, too. What's two and a half as
> a fraction?"
>
> S: [Writes "2 {1\over 2}"] "You mean like this?"
>
> M: "Well, that's a mixed number. You can write it as a simple fraction.
> What is it -- eleven ninths? seven fourths? What is two and a half as a
> fraction?"
>
> S: [Draws parentheses around the "2" and the "1/2"] "One?"
>
> M: [Losing patience now] "No! If I give you two and a half bucks, have I
> just given you one dollar? What's two and a half? It's two AND a half.
> That means two PLUS a half"
>
> S: "Oh so it's back to common denominators". [Starts to fumble with halves.
> Perhaps recognizing this is supposed to be the kind of thing one should
> be able to do mentally, announces:] "Three halves."
>
> [...]
> [Small diatribe about the need for students to actually
> _learn_ the material they deal with in math classes. Small concession
> thrown in for student's emotional well-being, recognizing that the student
> probably got shafted by lousy teachers early on. I can't change that now...]
It sounds like you're in a practically impossible position having
to teach this student calculus, and so are they being expected to
learn it with such a tenuous grasp of elementary arithmetic.
> M: "OK, great. So this numerator is five halves. Now that slope is
> five halves over 3. Now that's a compound fraction. Do you remember
> how to simplify those?"
>
> S: [tired and embarassed now] "No."
>
> M: "Here, let me remind you". [Big fraction bars used to show the
> invert-and-multiply routine.] "So you get five-sixths for this slope.
> Now, is that bigger than the slope we had over here? Is it even bigger
> than one?"
>
> S: "I don't know."
It may seem a bit eccentric but I often think children, and students
like this, would be able to learn arithmetic far more thoroughly and
with much less effort and stress if they played some shoot-em-up
computer game such as Quake with an add-on feature that would pop
up simple arithmetic questions whose correct answer had to be typed
to open a door or collect ammo or health boosts etc, especially if
pupils were competing with each other. I know there are games for
younger children, to help them learn to read (collecting letters
and so forth), but I'm sure that adapted games could be made more
rewarding, and not boringly worthy, for older pupils as well.
Cheers
---------------------------------------------------------------------------
John R Ramsden (j...@redmink.demon.co.uk)
---------------------------------------------------------------------------
The new is in the old concealed, the old is in the new revealed.
St Augustine.
---------------------------------------------------------------------------
Allan Adler
who, we all hope, did not read Dave Rusin's posting.
In one of his educational essays, Andre Weil said that rigor does
not consist in proving everything but in maintaining a clear distinction
between what has and what has not been proved. I think such a point
of view would have made it possible for both the student and Dave
to accomplish their goals. This is a case where both parties basically
want the same thing, so there is no need for it to degenerate into
the rout that Dave described. To quote from some commercials on a
simliar subject, grab hold of yourself before grabbing hold of your
child.
There were a lot of ways to handle such a student. This was not
teaching, it was a game of "I got you now, you sonofabitch".
That is all that the endless backtracking really amounted to.
I won't say that a specific alternate approach would have worked
from the beginning, since I don't know enough about that student.
But it does seem to me that this student had it in him to learn
what he was asking Dave to explain to him.
Having addressed some generalities, let me discuss some details.
> Student: "I'm having some real problems in this class [Calculus 1].
> Can you show me how to do problems like this in case we have some on
> the next test?" [Student points to a problem of the form, "Sketch a
> function which satisfies f'(x)>0 for x in [0,1], etc."]
>
> Me [somewhat puzzled, since there's nothing to 'do' on those problems]:
Of course there is something to do. One thing to be done is to find the
function and another is to sketch it. Which thing one does first is a
matter of taste. The approach Dave tried to follow seemed to be based
on the ability to sketch asketching a grap and make a visual determination
of a function's suitability, and to use that ability to decide what kind
of graph one needs to draw. There are clearly other approaches.
One thing Dave might have done was ask him to write down the formula
for some function and see whether it met the condition. If he wrote
down f(x)=2, it wouldn't, but with a little prodding, he might have
come up with f(x)=x, which does work. The student could have done
this entirely computationally and then graphed the function.
If the student eventually demonstrated consistent skill in finding
functions, by whatever method, that answered the questions, then it
would have made sense to go back over these examples to try to impart
some deeper understanding.
> "OK, let's try one to see if you were with me in class today. Sketch a
> function which has f' positive everywhere but f''(x) > 0 for x < 2 and
> f''(x) < 0 for x > 2."
> S: "So I need f' to be increasing on the first part."
> [Sketches something like y=log(x) ] "Like this?"
> M: "You were right -- you needed f' to be increasing. Now, f' measures
> the slope of the tangent line, right? So you mean to say the slope of the
> tangent line is getting greater as you move from left to right?"
>
> [S is silent, appears to have thought so.]
>
> M: "What would you say the slope is here?" [Points to left edge of the graph,
> slope is around 4. Hard to tell -- this is freehand, no grid.]
Why did Dave ask for numbers when the visual information required
to appreciate an increasing slope is qualitative? Presumably, in
order to test the student, since the student wasn't saying anything.
This was the begining of the game of "I got you now, you sonofabitch."
Perhaps saying "the slope of the tangent line is getting greater" is
not as visual or intuitive as saying that the tangent line seems to be
tilting upwards to the right more and more, or something like that.
Who needs numbers?
> S: "Around 1?"
> M [puzzled]: "Hm, well the picture's a little unclear but OK. And here?"
> [Points to rightmost point, where the slope is really a little less than 1.]
> S: "Maybe 4?"
> M [pause]: "You're saying the slope at this second point is four times as
> great as the slope at the first point?" [Effort to mask incredulity is
> probably wasted.] "What does 'slope' mean?"
Given that we are using numbers, this is a reasonable question at this point.
However:
> S: "See, I can't describe it so well. I know the formula..."
at this point, it would have been better simply to accept that this
how this student deals with the estimation of slopes. Getting him
to feel comfortable with other methods is of course desirable, but
it is not what Dave needed to accomplish at that moment. Let him
use the stupid formula if he likes and redraw the graph so that
he can find numbers to calculate with. (Actually, Dave eventually
did so.)
> M: "If you understand it, you should be able to describe it in half a dozen
> words, tops.
If he really understands it, he should be able to explain it to the
next person he meets on the street, but that wouldn't justify sending
him out into the street to find someone to explain it to. The same
applies here, since the point was entirely peripheral. Moreover, if
one really wanted to teach the student how to think about slopes,
there were better approaches.
The time available for helping the student was limited. There is a certain
kind of panic or rage that can strike a teacher when he/she realizes that
the explanation he knows he has just enough time to give will not do the
job of fixing a much deeper problem. In my opinion, that is what really
happened.
I think the student deserves an apology. A teacher should always seek
reasons for optimism and in this case there were reasons for it. Emphasize
the positive in dealing with a student and be his/her partner in pursuit
of a common goal. There are a lot of perfectly good reasons why a teacher
might not feel up to it on a particular occasion and, when they arise,
it is better to defer to another time.
One reason for optimism is that the student showed up at all. You would
not believe how few students bother to ask for help at many places.
Treating him so judgementally and throwing him out might guarantee
that he never asks you for help again.
On bended knee, apologize.
Allan Adler
a...@zurich.ai.mit.edu
****************************************************************************
* *
* Disclaimer: I am a guest and *not* a member of the MIT Artificial *
* Intelligence Lab. My actions and comments do not reflect *
* in any way on MIT. Morever, I am nowhere near the Boston *
* metropolitan area. *
* *
****************************************************************************
Lieven Marchand
> Dave Rusin <ru...@vesuvius.math.niu.edu> wrote in message
> news:8li1a4$duu$1...@gannett.math.niu.edu...
> >
> > A student just left from an hour of private office help. Since I
> > really oughn't go out for a drink at mid-day, I hope venting some steam
> > in public will have a purgative effect instead. This is for real.
> >
>
> Hi Dave,
> I believe you fully. My opinion regarding this sad
> little story is as follows: This guy is not so bad at all,
> since he (1) knows, he needs help and (2) does what
> you ask him to do - as well as he is able to.
>
At university level?
My calculus professor would have killed this guy. Actually, he
wouldn't have made it past the entrance exams.
This is stuff they don't even repeat at introductory calculus on a
university. You're supposed to know this from when you're 14.
--
Lieven Marchand <m...@bewoner.dma.be>
Lambda calculus - Call us a mad club
Torkel Franzen
> But it does seem to me that this student had it in him to learn
> what he was asking Dave to explain to him.
Yes, but it is not necessarily the case that anything Dave might
have said could have helped him learn, except fortuitously.
> One reason for optimism is that the student showed up at all.
True. I don't have the optimism and dedication of teachers who are
prepared to spend any amount of time on going through things with
students, but it is unconditionally incumbent even on us slackers
not to put anybody down who comes to us for help.
labuser
> A student just left from an hour of private office help. Since I
> really oughn't go out for a drink at mid-day, I hope venting some steam
> in public will have a purgative effect instead. This is for real.
Your position at office hours is not that of math tutor. Instead,
it's closely analogous to *psychotherapist*.
Office hour visitors know they have problems, because there is
some proximate difficulty they can't surmount, such as solving
exercises 3 and 7 of the homework. However, other than not
being able to solve it, they can't put their finger on what
it is they're missing. Most likely it's a whole host of problems
dating back to childhood --- gaps and misunderstandings of math
they've learned ever since elementary school. You (the teacher)
find this out as they reveal their thoughts to you, exposing their
inadequacies to an authority figure. You asking further probing
questions to locate the root of the disorder: they don't understand
variables, or fractions. Maybe actual psychological complications
are involved, such as fear of the humiliations associated with math classes.
The point is, as in psychotherapy, you will not cure or even
identify all their problems in one session, though general symptoms
may become apparent. Probably it will take a lot of time and
work on their part (let them know this), with expert guidance
being a helpful thing (offer some or let them know where to get it).
Also, getting help can be humiliating from the student's point of
view, and turning the visit into an interrogation session to root out
(and implicitly judge as "bad") all their weaknesses, as described
in D.Rusin's posting, does not help. Finally, for many students
it will be simply necessary for them and the teacher to accept
most of their mathematical weaknesses and see what else can
be gained from the course. (One possibility is to reduce their
contempt for math and its practitioners, with revenge for
years of awful schooling to be exacted when scientists come
asking for public funds.)
I don't suggest to explicitly deal in psychology or
other non-mathematics in dealing with students.
I do suggest that misunderstanding one's position as
"math tutor" in office hours or similar situations, can
lead to a lot of frustration and wasted time during
those conversations.
Rainer Rosenthal
Allan Adler <a...@nestle.ai.mit.edu> wrote in message
news:y93bszn...@nestle.ai.mit.edu...
>
> There were a lot of ways to handle such a student. This was not
> teaching, it was a game of "I got you now, you sonofabitch".
>
are you citing Eric Berne ? "99 games for adults" or so ? Nice book,
interesting author.
> The time available for helping the student was limited. There is a certain
> kind of panic or rage that can strike a teacher when he/she realizes that
> the explanation he knows he has just enough time to give will not do the
> job of fixing a much deeper problem. In my opinion, that is what really
> happened.
>
So think I.
>
> I think the student deserves an apology. A teacher should always seek
> reasons for optimism and in this case there were reasons for it
>
Right.
>
> One reason for optimism is that the student showed up at all. You would
> not believe how few students bother to ask for help at many places.
> Treating him so judgementally and throwing him out might guarantee
> that he never asks you for help again.
>
Right.
>
> On bended knee, apologize.
>
This is too harsh either: be patient with Dave, don't play this
game of "I gotcha..." yourself please.
Dave is not a teacher by profession.
- Rainer
John Savard
wrote, in part:
>S: [Unsure] "You mean this?" [Writes "{2.5}\over{3}"]
At this point, you made things a bit too hard for him.
Since the student doesn't know how to manipulate fractions, that
concept will have to be explained in simple terms.
Thus: $2.50 is that fraction of $3.00. But we would like to have
something on the top and on the bottom that doesn't have decimal
points in it.
Suppose we think of the $2.50 as being all in quarters. And let's
think of the $3.00 the same way.
Then, we have that 10 quarters is the same fraction of 12 quarters;
they are both the same amount of money.
So we got rid of the decimal point. But both the numbers on the top
and on the bottom are even; the fraction isn't in lowest terms.
Dollar bills are too big; they make a decimal point. Quarters are too
small; both numbers are even. In this case, if we think of 50-cent
pieces, we get it just right.
But you are quite right that a student attending a university-level
mathematics course should have been paying attention during the study
of grade school math; while it is improper to expect that professors
somehow, in first-year courses, do what 12 years of schooling had
failed to do, it sometimes _is_ possible, in a few minutes, to
overcome a "mental block" that has kept someone from understanding
mathematics.
It is unfortunate, though, that this skill is not found in enough
elementary and high school teachers. But then, if it is not universal
among professors teaching first-year courses either, I suppose one
can't entirely fault the educational system; it may simply be a rare
skill.
John Savard (teneerf <-)
Now Available! The Secret of the Web's Most Overused Style of Frames!
http://home.ecn.ab.ca/~jsavard/frhome.htm
Bart Goddard
jsa...@tenMAPSONeerf.edmonton.ab.ca (John Savard) wrote:
> >S: [Unsure] "You mean this?" [Writes "{2.5}\over{3}"]
>
> At this point, you made things a bit too hard for him.
>
> Since the student doesn't know how to manipulate fractions, that
> concept will have to be explained in simple terms.
I can't believe that so many of you are taking Dave to task here.
Dave didn't make things too hard, rather, the student was
trying to take Calculus without having mastered 6th-grade math.
It's not slacking to refuse to waste time teaching a student
who's not ready for the class (by 7 years!) The right thing to
do here is not to spend googal office hours privately tutoring
a student who is not yet capable of the material, but to find
that student a course more suitable to his background. When so
many students start in remedial math (perhaps 4 courses below
Calc) why should this student get 4 courses worth of free,
one-on-one instruction? It's astounding and noble generosity
on Dave's part if he gives it, but I don't know of anyone who
could keep up that workload for long.
If Dave did anything wrong, it was to be too patient. As soon
as the student demonstrated no understanding for slope (let alone
fractions) he should have been sent to the registrar, (and perhaps
a nasty phone call to the student's advisor.)
Prepared students have to work very hard to keep up in Calc.
There is no justification for stealing money from an unprepared
student by putting him in a class he has no chance of passing.
My 3 cent,
Bart
Sent via Deja.com http://www.deja.com/
Before you buy.
Bob Silverman
Bart Goddard <godd...@my-deja.com> wrote:
<snip>
> I can't believe that so many of you are taking Dave to task here.
> Dave didn't make things too hard, rather, the student was
> trying to take Calculus without having mastered 6th-grade math.
I agree wholeheartedly.
Why was this student accepted to college???
>
> It's not slacking to refuse to waste time teaching a student
> who's not ready for the class (by 7 years!) The right thing to
> do here is not to spend googal office hours privately tutoring
> a student who is not yet capable of the material, but to find
> that student a course more suitable to his background.
Bingo
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
Miguel A. Lerma
: [Student has come in with the weak background that leaves calculus
: instructors scratching their heads. We talk about computing derivatives
: as limits of difference quotients -- student had flubbed the derivative of
: 3x+(4/x) by virtue of failing miserably to subtract the necessary fractions
: using common denominators. We discuss the need for algebra skills, then
: move on. That, perhaps, was my fatal error...]
In some universities there are special counselors and services
to help students to fill gaps in their knowledge. In the case
you describe it seems to me that there is no point in learning
derivatives until the student has acquired the basic skills for
dealing with fractions. I would make that a top priority before
going ahead with derivatives. Refer the student to the appropriate
counselor or service, or advise him to get a tutor who can help him
to fill the gap.
Miguel A. Lerma
Rainer Rosenthal
Lieven Marchand <m...@bewoner.dma.be> wrote in message
news:m3ittv4...@localhost.localdomain...
>
> At university level?
>
Well I wondered a little and thought I did not read
correctly.
With respect to THAT you - and some others who
made their comments - are right.
But generally what I was saying was not wrong.
The very moment, Dave went into the teaching
elementary stuff, he could as well have stopped,
saying: NO !
But once begun he should have known this was
not the sort of teaching for calculus but for somone
without any basis in math.
> My calculus professor would have killed this guy.
How cruel these professors are in the USA :-)
Dave Rusin
> A student just left from an hour of private office help.
to be very weak in basic mathematics skills.
I don't know that I was looking for any kind of response at all, but
since I seem to have gotten one, perhaps I ought to clarify a few points.
First there is the matter of pronoun: the student is "she".
That doesn't matter at all except that in my experience it makes
it less likely that this suggestion will help:
> I often think children, and students like this, would be able to learn
> arithmetic far more thoroughly and with much less effort and stress if
> they played some shoot-em-up computer game
Second, I can clarify the student's background. The answer to
> Why was this student accepted to college?
is in this particular case a bit of social engineering. Ours is a public
university which makes a special effort to serve typically under-served
portions of our state population. Students with some indication of promise
can sometimes be admitted even with serious gaps in their school records.
Typically I personally see few of these students: they don't often pursue
degrees requiring much math, or they drop out during a remediation period.
This particular student did go through our remedial program, which
extends to a peculiarly American subject called "College Algebra". I think
she earned a "B" in that one. In my opinion that could have been a
justified grade: that's a mechanical course, in which a student can
succeed by mimicking very specific manipulations. It's not always
easy for these students by any means, so we would have real problems
raising the bar on that course. On the other hand, the only real point
of the course is to prepare the student for later "real" courses,
and as many of us know, those later courses assume genuine facility
with algebra -- half-remembered mysterious procedures are useless.
The student followed with a "pre-calc" course, which discusses
the concept of functions, trigonometry, and the exponential and log functions.
It's not really a bad course, and significantly more substantial than
the corresponding course taught in U.S. secondary schools. (It runs a lot
faster, too.) Given the student's ability to parrot simple steps without
understanding very much of anything, perhaps you won't be surprised
to hear the student had a D grade in that course. Unfortunately for all
concerned, current school rules allow the student into calculus with
that performance. (As it happens, we're changing that this year.)
Note that repeating _that_ course would not necessarily improve the
student's algebra skills.
The student then took Calculus, and failed. This summer is a repeat for this
student and, as it turns out, for half of the class. Since our school
does not (usually) allow "three-peats" of a course, the students are
actually quite motivated in the summer: they can't fail it a second
time if passing this course is required for something else they want
to do. All things considered, it's an interesting group to teach.
(I also get students like one 15-year-old high school student looking
for a way to fill his summer. He's got one of the highest grades so far.)
So on the question of why the student is in college, and why she is taking
this class, the blame comes back in part to the school and in particular
to the department. I don't know where exactly we should have drawn the
line and said "we will not help you any more". (By the way, we are not
"taking the student's money" in this particular case, since there is
federal and state funding to include students like these in class. Other
students are paying their own way.)
Faculty at U.S. universities probably understand how we get put into
situations like this. Sorry if this seems incomprehensible to overseas readers.
Suffice it to say these things happen, even while most students do just fine.
Several people spoke to the possible ways the student and I ought to
proceed from where we now stand. Opinions differ, of course.
> Refer the student to the appropriate counselor or service, or advise
> him to get a tutor who can help him to fill the gap.
The student had one tutor, who later quit. She's getting another now.
> it is unconditionally incumbent even on us slackers
> not to put anybody down who comes to us for help.
I agree, and try to do so. Most of the class understands they have a lot
of work to do, and are trying hard to stay caught up. We work well together.
It is difficult to convey moods in USENET postings, so perhaps I ought
to clarify my teaching and tutoring styles. I am a demanding teacher
who in particular is known for giving exams the students always find
very challenging. On the other hand, I'm also well known locally for
maintaining a rather lively classroom setting (I even sing in class)
and, I think, a fairly welcoming office environment. So for example,
this analysis:
> This was not teaching, it was a game of "I got you now, you sonofabitch".
was rather wide of the mark. I was not trying to "trap" the student
so much as to lay bare the source of the student's difficulties.
I find it useful to help the students discover whether they are failing
because of (a) algebra skills (b) conceptual problems involving functions
(c) poor work habits (d) etc.
Some posters decided my comments said more about me than about the student.
> Dave is not a teacher by profession.
No comment :-)
dave
Rainer Rosenthal
>
> Some posters decided my comments said more about me than about the
student.
> > Dave is not a teacher by profession.
>
Hi Dave,
please excuse my faulty impression. I am from Germany,
so I am one of the Overseas reader you mentioned.
I really did not want to offend you. It was pure ignorance
as to the word TUTOR .
In Germany a "Tutor" is some elder student helping those
students who are in the first or second semester.
You may have noticed, I read your story carefully - as
well as I could with my limited knowledge of the circum-
stances. ( By the way I liked that "steam" converting to "water").
For me as a foreigner it was really interesting to learn
something about education in the USA. We do have a much
more homogenous population - but things are changing.
It is really interesting to hear in the newsgroup things related
to mathematics in a broader sense.
Thank you very much for your detailled information. May
I wish you good humor all the time. Greeting to the student
and my best wishes for her future.
Regards,
Oriana
Bob Silverman <bo...@my-deja.com> wrote:
> In article <8lk7nr$kqg$1...@nnrp1.deja.com>,
> Bart Goddard <godd...@my-deja.com> wrote:
>
> <snip>
>
> > I can't believe that so many of you are taking Dave to task here.
> > Dave didn't make things too hard, rather, the student was
> > trying to take Calculus without having mastered 6th-grade math.
>
I don't think anyone here is trying to bash David. Speaking strictly
from experience as a student (i only just got my BS) its often
difficult to learn something as abstract as mathematics from
instructors since their way of looking at things influences their way
of explaining them. I'm applying to graduate school to study
mathematics, and i can't do 2 1/2 unless i write it out as 5/2, and
even that requres me to pause for a second. I would have never
understood an example using dollars and cents, but, heck write them out
to me this way, and i'm fine. Also, i, unlike (or maybe like) your
student would never have (when first seeing it) understood slope from
just a formula, only from the drawing, and counting out "the little
squares" thats simply a difference between people who have stronger
audio or visual comprehension, i (and possibly your student) am
(strongly) of the latter. When i was instructed by someone of the
former mentality, as some of my teachers were, i had no clue what they
meant!
Also, don't underestimate her. In highschool, i nearly flunked all my
math courses until i got to calculus. I don't know if it was poor
instruction, or poorly planned class content, but i couldn't even graph
a functio until i got to calc. And Calc i aced! Don't ask me how i
managed to get INTO calculus with out knowing that, probably the same
way i got through Algebra two not knowing basics of exponents!
The American education system does stink. It appalled me, when i came
here during fifth grade that my classmates only now learning long
division! I had been doing some basic algebra already. And i am yet to
meet and elementary shool teacher who actally KNEW what he/she was
teaching those students!
I guess i've ranted long enough, my point is that the system does
stink, but that only means that when those who teach recognize this,
they must endeavor iven harder to remedy the situation. If you're not
willing to, DON'T TEACH (god know i won't!!)
> I agree wholeheartedly.
>
i don't
> Why was this student accepted to college???
so that David could help her learn these things
>
> >
> > It's not slacking to refuse to waste time teaching a student
> > who's not ready for the class (by 7 years!) The right thing to
> > do here is not to spend googal office hours privately tutoring
> > a student who is not yet capable of the material, but to find
> > that student a course more suitable to his background.
>
Its this kind of irresponcibility that spawns students like David's!
> --
> Bob Silverman
> "You can lead a horse's ass to knowledge, but you can't make him
think"
>
> Sent via Deja.com http://www.deja.com/
> Before you buy.
>
--
Every time I close the door on reality, it comes in through the window
ach...@hotmail.com
ru...@vesuvius.math.niu.edu (Dave Rusin) wrote:
>
> A student just left from an hour of private office help. Since I
> really oughn't go out for a drink at mid-day, I hope venting some
steam
> in public will have a purgative effect instead. This is for real.
>
> [Student has come in with the weak background that leaves calculus
> instructors scratching their heads. We talk about computing
derivatives
> as limits of difference quotients -- student had flubbed the
derivative of
> 3x+(4/x) by virtue of failing miserably to subtract the necessary
fractions
> using common denominators. We discuss the need for algebra skills,
then
> move on. That, perhaps, was my fatal error...]
>
> Student: "I'm having some real problems in this class [Calculus 1].
> Can you show me how to do problems like this in case we have some on
> the next test?" [Student points to a problem of the form, "Sketch a
> function which satisfies f'(x)>0 for x in [0,1], etc."]
Dave,
Your tale of woe here reminded me of what happened to a friend of
mine when we were graduate students at Berkeley. She worked at some
sort of Math Learning Center where students could come in and get
tutored for free on any mathematical subject. She was getting ready to
go home at 5:00 when a student came in. I shall refer to her as M and
to the student as S.
M: I'm getting ready to leave, now. I hope your question is a short
one.
S: It's just one little thing. I understand this stuff pretty well,
but there is one point I am having a little trouble with.
M: OK. What's the problem.
S: You know, where they find the limit as x goes to 0 of sin(x)/x and
get 1?
M: Yeah.
S: Well, how come they don't just cancel the x's and get sin?
M: (with a straight face, even!) Well if you could do that, then
couldn't you take sqart(x) / x , cancel the x's and just get the square
root sign? (This would work better if there were a square root symbol
on my keyboard)
S: Wow, that's really neat! I never thought about that way before!
Back to the present. That's as much of the conversation as I heard, but
my friend told the same story to a lot of different people, and she
wasn't the sort to make this kind thing up. As I recall, she didn't get
any dinner until quite late that evening, and she wasn't very happy
about it either.
Regards,
Achava
Erik Max Francis
[teacher's anecdote]
> S: You know, where they find the limit as x goes to 0 of sin(x)/x and
> get 1?
>
> M: Yeah.
>
> S: Well, how come they don't just cancel the x's and get sin?
This sort of thing isn't all that uncommon; it's a sign of learning by
rote memorization and procedure, rather than getting at what's actually
going on. If sin, x, and a bar underneath are just meaningless symbols
and you're taught that you can cancel the same symbol when it appears on
both sides of a bar, then sin x/x = sin makes sense. Obviously it's
total nonsense, but if one's been inadequately taught or inadequately
learned (not laying the blame here), then such symbolic manipulation is
about the best one can manage.
I recall a calculus teacher who explicitly pointed out a gaffe like this
on one of his quizzes (obviously he was kind enough not to say who the
responsible party was). He was, needless to say, rather flabbergasted.
--
Erik Max Francis / m...@alcyone.com / http://www.alcyone.com/max/
__ San Jose, CA, US / 37 20 N 121 53 W / ICQ16063900 / &tSftDotIotE
/ \ Triumph cannot help being cruel.
\__/ Jose Ortega y Gasset
The laws list / http://www.alcyone.com/max/physics/laws/
Laws, rules, principles, effects, paradoxes, etc. in physics.
denis-feldmann
[cut]>
> S: Well, how come they don't just cancel the x's and get sin?
>
> couldn't you take sqart(x) / x , cancel the x's and just get the square
> root sign? (This would work better if there were a square root symbol
> on my keyboard)
>
> S: Wow, that's really neat! I never thought about that way before!
A few years ago, i got this in an exam paper (for 16 years old in France,
litterary options):
Solve a^3=b^3 (*)
Answer: taking square roots, (*) becomes aVa=bVb
(i use V for sqrt symbol, obviously)
then, let V equals 1
(yes, you read it right)
we get a*a=b*b=>a^2=b^2 ...
I commented on the paper : it would have been easier to say "let 3 equals 2"
directly...
But the oral discussion was even worse, as the student said indignantly:
Why, you are always doing things like that in your teachings, like when you
say "let phi equals (1+sqrt 5)/2...."
At this stage, words fails.
One other example (in same class, not the same year):
Me: You must realize that, sadly, (a+b)^2<>a^2+b^2. For instance, if you
take a=2, b=3, you get (a+b)^2=25 <>a^2+b^2=13.
Student: Agreed. But what happens if you dont take any example?
Torkel Franzen
> Me: You must realize that, sadly, (a+b)^2<>a^2+b^2. For instance, if you
> take a=2, b=3, you get (a+b)^2=25 <>a^2+b^2=13.
> Student: Agreed. But what happens if you dont take any example?
A profound question, bringing to mind my old favorite: "Suppose X is
the number of sheep. -But sir, what if X is not the number of sheep?"
Maybe the student was a natural comedian?
Anonymous
<denis-f...@wanadoo.fr> wrote:
>Me: You must realize that, sadly, (a+b)^2<>a^2+b^2. For instance, if you
>take a=2, b=3, you get (a+b)^2=25 <>a^2+b^2=13.
>Student: Agreed. But what happens if you dont take any example?
Then you get Cantor's theory of transfinite numbers. :-)
Insignificant I
--------== Posted Anonymously via Newsfeeds.Com ==-------
Featuring the worlds only Anonymous Usenet Server
-----------== http://www.newsfeeds.com ==----------
Bart Goddard
Oriana <oria...@my-deja.com> wrote:
> thats simply a difference between people who have stronger
> audio or visual comprehension, i (and possibly your student) am
> (strongly) of the latter. When i was instructed by someone of the
> former mentality, as some of my teachers were, i had no clue what they
> meant!
This illustrates the two main schools of education. The
current one (the one that stinks) says that education is
to tailor facts to fit into the mind of a student, regardless
of the mind's state. The other one (the one that works)
says that education is the strengthening of the students
mind.
The students most emotionally taxing for me are the 50-year-
old mom's returning to school. They work SOOOO hard trying
to the the new concept into their brains, but fail so
miserably because they don't realize that the new concept
requires a change of their brains. 18-year-old are much more
malleable.
The point here is that if one is weak in visual or audio,
then the point of education is to strenghten that weakness,
not to cater to it, and thereby prolong it.
> I guess i've ranted long enough, my point is that the system does
> stink, but that only means that when those who teach recognize this,
> they must endeavor iven harder to remedy the situation. If you're not
> willing to, DON'T TEACH (god know i won't!!)
You point jumps the gun. Those of us who are successful
teachers DO work to remedy the situation. The issue here
is what that remedy is.
> > Why was this student accepted to college???
>
> so that David could help her learn these things
No, it was so that the college could bilk the government
and the student's parents out of 10 or 20 thousand dollars
before sending him on his way.
> > > It's not slacking to refuse to waste time teaching a student
> > > who's not ready for the class (by 7 years!) The right thing to
> > > do here is not to spend googal office hours privately tutoring
> > > a student who is not yet capable of the material, but to find
> > > that student a course more suitable to his background.
> >
>
> Its this kind of irresponcibility that spawns students like David's!
It would be irresponsible to let a student pay for Calculus
when he had no chance of passing. We both agree that the
student should be taught "where he is". For some reason,
you think that this means doing it in the context of
Calculus, rather than in a context already designed for a
student of his level. The same reasoning says that if a
student comes to me with a psychological problem, I should
attempt to treat him myself, rather than refer him to
the proper context of a shrink's office.
Indeed, the irresponsibility that spawns students like David's
is the first type of education, which seeks always to make
facts easier to swallow, and thereby leave the student
mentally weak.
If you're going to grad school in math, you'll need to
become competent in both audio and visual skills. Not
great, just competent.
Bart
Nico Benschop
> [...]
> One other example (in same class, not the same year):
>
> For instance, if you take a=2, b=3, you get (a+b)^2=25 <>a^2+b^2=13.
> Student: Agreed. But what happens if you dont take any example?
Brilliant, then you get algebra: (a+b)^2 - (a^2 + b^2) = 2ab
Showing that [1] for ab=0 is false,
and that [1] for ab>0 yields: (a+b)^2 > a^2+b^2.
It appears you missed a chance to show the power of algebra,
beyond working out specific examples;-)
--
Ciao, Nico Benschop
denis-feldmann
Nico Benschop <n.ben...@chello.nl> a écrit dans le message :
397EEA0B...@chello.nl...
> denis-feldmann wrote:
> > [...]
> > One other example (in same class, not the same year):
> >
> > Me: You must realize that, sadly, (a+b)^2 <> a^2+b^2 ...[1]
> > For instance, if you take a=2, b=3, you get (a+b)^2=25 <>a^2+b^2=13.
> > Student: Agreed. But what happens if you dont take any example?
>
> Brilliant, then you get algebra: (a+b)^2 - (a^2 + b^2) = 2ab
Not really... You might get characteristic 2 -fields (as 0=1+1);but i would
hate to see the immediate conclusion 2=0. Anyway, algebra has no sense
without examples
>
> Showing that [1] for ab=0 is false,
> and that [1] for ab>0 yields: (a+b)^2 > a^2+b^2.
>
> It appears you missed a chance to show the power of algebra,
> beyond working out specific examples;-)
Well, you can do a lot with (a+b)^2 =a^2+b^2+ 2ab (like getting to the
canonical form), but not with (a+b)^2=a^2+b^2 "because it is easier that
way"
> --
> Ciao, Nico Benschop
Bill Taylor
|> Maybe the student was a natural comedian?
Speaking of comedians, Littlewood was being an unintentional clown, I thought,
when he reported this amusing vignette, in "Mathematician's Miscellany".
He immediately followed it with the comment -
"I once asked Wittgenstein if this were not a profound philosophical joke,
and he said that it was."
I mean, *really*! To have to *ask* someone if something is a joke or not...
and then to *believe* them...
Geez!
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
I heard that Bertrand Russell once got a letter
from a lady saying that till recently she thought
she was the only solipsist in the universe...
-------------------------------------------------------------------------------
Nico Benschop
>
> Nico Benschop <n.ben...@chello.nl> a écrit dans le message :
> 397EEA0B...@chello.nl...
> > denis-feldmann wrote:
> > > [...]
> > > One other example (in same class, not the same year):
> > >
> > > Me: You must realize that, sadly, (a+b)^2 <> a^2+b^2 ...[1]
> > > For instance, if you take a=2, b=3,
> > > you get (a+b)^2=25 <> a^2+b^2=13.
> > > Student: Agreed. But what happens if you dont take any example?
> >
> > Brilliant, then you get algebra: (a+b)^2 - (a^2 + b^2) = 2ab
>
> Not really... You might get characteristic 2 -fields (as 0=1+1);
> but i would hate to see the immediate conclusion 2=0.
[*] True;)
I only wanted to convey that the student's answer was far from
stupid, and hit the essence of algebra: it covers so much more than
examples. Like the <> sign in [1]: if inequality is claimed, it is
useful to look at the difference of the two sides, and check the
conditions for which it might be zero, yielding the exception ab=0.
Of course, also in the checking for zero difference, complete
inspection means to work through several cases (here: <0 =0 >0 )
but that wraps it up (completely, assuming normal arithmetic,
not residues or even wilder non-commutative algebra's;-(
BTW: (a+b)^p == a^p + b^p (in residues mod p^k, prime p>2, k>1)
does have interesting solutions, re FLT;-) ...[2]
> >
> > Showing that [1] for ab=0 is false,
> > and that [1] for ab>0 yields: (a+b)^2 > a^2+b^2.
> >
> > It appears you missed a chance to show the power of algebra,
> > beyond working out specific examples;-)
>
> Well, you can do a lot with (a+b)^2 = a^2+b^2 + 2ab (like getting to
> the canonical form), but not with (a+b)^2 = a^2+b^2 "because it is
> easier that way"
Ciao, Nico Benschop -- http://home.iae.nl/users/benschop/ferm.htm [2]
Clark
Bill Taylor wrote:
>
> |> A profound question, bringing to mind my old favorite: "Suppose X is
> |> the number of sheep. - But sir, what if X is not the number of sheep?"
>
> |> Maybe the student was a natural comedian?
>
> Speaking of comedians, Littlewood was being an unintentional clown, I thought,
> when he reported this amusing vignette, in "Mathematician's Miscellany".
> He immediately followed it with the comment -
>
> "I once asked Wittgenstein if this were not a profound philosophical joke,
> and he said that it was."
>
> I mean, *really*! To have to *ask* someone if something is a joke or not...
> and then to *believe* them...
>
Well, but asking whether 'this were not a profound philosophical joke',
he may not have been asking whether it was a joke, but rather whether
the (agreed-to-be) joke was profoundly philosophical. That seems more
likely, really, given that Wittgenstein's metier was more in the line of
profound philosophy than stand-up comedy.
Bob
David C. Ullrich
Taylor) wrote:
>|> A profound question, bringing to mind my old favorite: "Suppose X is
>|> the number of sheep. - But sir, what if X is not the number of sheep?"
>
>|> Maybe the student was a natural comedian?
>
>Speaking of comedians, Littlewood was being an unintentional clown, I thought,
>when he reported this amusing vignette, in "Mathematician's Miscellany".
>He immediately followed it with the comment -
>
>"I once asked Wittgenstein if this were not a profound philosophical joke,
> and he said that it was."
>
>I mean, *really*! To have to *ask* someone if something is a joke or not...
>and then to *believe* them...
Hey, I never realized that was funny before! Thanks.
David Petry
>> > > Me: You must realize that, sadly, (a+b)^2 <> a^2+b^2 ...[1]
>> > > For instance, if you take a=2, b=3,
>> > > you get (a+b)^2=25 <> a^2+b^2=13.
>> > > Student: Agreed. But what happens if you dont take any example?
Sometimes students don't ask what they mean to ask.
The student was probably asking "how could you know that equality
does not hold by some method other than trying an example?"
n_f_be...@my-deja.com
Precisely my point: a very good question - going to the heart of the
matter. While I had the impression that it was mentioned as example
of: my student is even more stupid than yours (re: the thread title;-)
Ciao, Nico Benschop
Zdislav V. Kovarik
David Petry <dpe...@uswest.net> wrote:
:
:>> > > Me: You must realize that, sadly, (a+b)^2 <> a^2+b^2 ...[1]
:>> > > For instance, if you take a=2, b=3,
:>> > > you get (a+b)^2=25 <> a^2+b^2=13.
:>> > > Student: Agreed. But what happens if you dont take any example?
:
:Sometimes students don't ask what they mean to ask.
:
:The student was probably asking "how could you know that equality
:does not hold by some method other than trying an example?"
Yes, and often in the background there is this plausible falsehood
"exception proves the rule", sometimes tacitly assumed, sometimes spelled
out in full.
This persisted in the past when functions just had to be differentiable
(except at isolated points), and I heard of an older (now long retired)
instructor of Calculus II who was convinced that "the limit of a
two-variable function at a point is always found along straight lines
passing through the point, except in isolated counterexamples concocted by
malicious mathematicians to create confusion".
I've read that the witticism is a barbaric translation from Latin, which
should read "exception tests the rule" or "exception improves the rule",
in the sense that it helps to look for possible exceptions while we
formulate the rule. After the rule is stated, the post factum exceptions
invalidate the rule (and in better cases, force re-formulation).
The mechanics of negating a quantified statement (De Morgan's Laws) are a
mystery to many students, and some professionals, too. (Add constructivism
into the picture, and you have endless debates...)
Greetings, ZVK(Slavek).
Virgil
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
>|> A profound question, bringing to mind my old favorite: "Suppose X is
>|> the number of sheep. - But sir, what if X is not the number of sheep?"
>
>|> Maybe the student was a natural comedian?
>
>Speaking of comedians, Littlewood was being an unintentional clown, I
>thought,
>when he reported this amusing vignette, in "Mathematician's Miscellany".
>He immediately followed it with the comment -
>
>"I once asked Wittgenstein if this were not a profound philosophical joke,
> and he said that it was."
>
>I mean, *really*! To have to *ask* someone if something is a joke or
>not...
>and then to *believe* them...
>
Having read "A Mathematician's Miscellany" several times, and enjoyed it
thoroughly, I very much doubt that Littlewood was unintentional about
any of it.
--
Virgil
vm...@frii.com