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
Jason Lee
> ...
> 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.
It most certainly can. Most math students are not fully aware of their
mathematical weaknesses. It is incumbent upon me to point out the places
where their knowledge is strong, and where it is weak. When I point out
a place where the students are weak, I do so insisting that the students
spend some more time brushing up on that material.
Granted, you don't want to turn it into, as you say, "an interrogation
session", but failing to point out where a student is weak is not good.
> 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.
Sorry, but that's BS. When I teach a course that relies heavily on a
previous course (for example a second term calculus class), I absolutely
refuse to allow the students to "get away" with not knowing the material
they should know, and I tell them this from day one. How can you
possibly understand much about integration if you don't understand how to
factor a quadratic polynomial?
JLee
--
Jason Lee jl...@math.ucsd.edu http://math.ucsd.edu/~jlee/
"Pay attention to the details and the big things will take care of themselves."
-- Ritch Price
Cal Poly SLO head baseball coach
Jason Lee
>
> 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.
Hear hear!
I have many times run into basically the same situation Dave ran into, so
I sympathize with him 100%. It sounds to me that perhaps he become more
exasperated than he needed to be, but the general reaction is right on
right on target
Rainer Rosenthal
Zdislav V. Kovarik <kov...@mcmail.cis.McMaster.CA> wrote in message
news:8lq9pc$n...@mcmail.cis.McMaster.CA...
>
> 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).
>
Ausnahmen bestätigen die Regel
This sounds paradoxical and is often used to legalize silly
statements or injust practices.
Normally used by the mightier part in a discussion.
Wonderful perversion of sound logic ! Works as follows:
"First of all: I am the Boss and A is true. If you give me
examples where A fails, then I simply declare them to
be exeptions from my rule. And together with that Latin-
based holy law of 'exeption proves the rule' we have a
rule, some exeptions and alas ! we have the rule proven.
So I AM RIGHT (as all the time)".
It's a lever of might, outright cynical and useful. So this
'good old' law ( or is it a rule itself ?) will survive and stink.
In German ther is quite a large distance between 'Gesetz'
and 'Regel'. So it is even more difficult to oppose, as 'Regel'
has by far not that strictness of 'Gesetz'.
Every 'Regel' is supposed to have 'Ausnahmen'. So the
very sentence reads simply as a definition for 'Regel':
If you have an exeption, this proves your statement to be
merely a rule, not a law.
May be this is the correct translation ?
- Rainer
Rainer Rosenthal
Zdislav V. Kovarik <kov...@mcmail.cis.McMaster.CA> wrote in message
news:8lq9pc$n...@mcmail.cis.McMaster.CA...
>
> 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).
>
as my last posting was a little too long, here I state the
conclusion.
"exeption tests the rule" is true, if read as
"exeption tests the ruleness".
It's the old game of pointer/reference/name !
Let R(x) = true iff x is a rule.
Then "x has an exeption" implies "R(x) is true"
But NOT: "x is true".
I think, this is not so bad an interpretation !?
- Rainer
--------------------------------
I am sick of sigs, sic !
Torkel Franzen
> 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.
It's a common misconception that "proves" in "the exception proves
the rules" has the meaning of "tests". The actual origin of the
phrase is well known. From the OED:
The legal maxim, 'Exception proves (or confirms) the rule in
the cases not excepted', which is in its original form an
example of sense 1 [the action of excepting (a person or
thing, a particular case) from the scope of a proposition,
rule, etc], is commonly quoted as 'The exception proves the
rule', the sb. being interpreted in sense 2 [something that
is excepted; a particular case which comes within the terms
of a rule, but to which the rule is not applicable].
John Savard
wrote, in part:
>I can't believe that so many of you are taking Dave to task here.
In my post, I did note that a professor indeed shouldn't be expected
to do what the schools should have.
One can hope that sometimes one can overcome a 'mental block', though,
so that the student can then study the material on his own and get
somewhere. But I tried not to be overly critical - instead, my intent
was to be helpful - the problem is not that he failed to have the
patience of a saint, which is indeed excusable, but merely that he
might not have thought of what to do next.
John Savard (teneerf <-)
Now Available! The Secret of the Web's Most Overused Style of Frames!
http://home.ecn.ab.ca/~jsavard/frhome.htm
Allan Adler
> It's a common misconception that "proves" in "the exception proves
> the rules" has the meaning of "tests". The actual origin of the
> phrase is well known. From the OED:
>
> The legal maxim, 'Exception proves (or confirms) the rule in
> the cases not excepted', which is in its original form an
> example of sense 1 [the action of excepting (a person or
> thing, a particular case) from the scope of a proposition,
> rule, etc], is commonly quoted as 'The exception proves the
> rule', the sb. being interpreted in sense 2 [something that
> is excepted; a particular case which comes within the terms
> of a rule, but to which the rule is not applicable].
I have a very vague recollection of the German word "probieren"
being relevant to this saying. One might guess that probieren
means prove but it actually means test. I don't know where I
remember it from; maybe an earlier discussion of this on
sci.math, maybe something on NPR, maybe gossip overheard
on a subway...
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. *
* *
****************************************************************************
Chas F Brown
Torkel Franzen wrote:
>
> kov...@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) writes:
>
> > 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.
>
> It's a common misconception that "proves" in "the exception proves
> the rules" has the meaning of "tests". The actual origin of the
> phrase is well known. From the OED:
>
> The legal maxim, 'Exception proves (or confirms) the rule in
> the cases not excepted', which is in its original form an
> example of sense 1 [the action of excepting (a person or
> thing, a particular case) from the scope of a proposition,
> rule, etc], is commonly quoted as 'The exception proves the
> rule', the sb. being interpreted in sense 2 [something that
> is excepted; a particular case which comes within the terms
> of a rule, but to which the rule is not applicable].
Just to be ultra-clear (and over-bearing ;)), my recollection is that
this comes up in legal cases where a law or rule is not clearly stated
(possibly due to sloppy bookkeeping in more ancient times). If someone
can, however, demonstrate that there is a case on the books where an
exception was granted to the law, then that by itself demonstrates that
the law was in effect and has some precedence.
If I remember correctly, the "exception" was often the king overruling
some local law, and that a reference to that overruling was taken as
evidence ("proof") that such a law existed.
---------------------------------------------------
C Brown Systems Designs
Multimedia Environments for Museums and Theme Parks
---------------------------------------------------
labuser
> > 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.
>
> It most certainly can.
It's a waste of time. Instructors tend to operate on an automotive
troubleshooting paradigm, where to fix the car you poke around
until at last you locate the Source Of Problem and fix it. This
leads to an office hours charade similar to what D.Rusin posted,
where it's clear within seconds that a student is hopelessly lost, but
the instructor feels bound by honor (or total denial) to lead them on
a march through whatever specific technical point in problem 3.27
is baffling them at the moment.
I'm suggesting that a psychotherapeutic paradigm is much closer
to the actual situation. Students build up deep cumulative
problems over years of math courses where they've had to fake
competence to get by, and pointing out some specific errors at
office hours is usually beside the point.
> Most math students are not fully aware of their
> mathematical weaknesses. It is incumbent upon me to point out the places
> where their knowledge is strong, and where it is weak.
The type of diagnosis you're talking about is so limited as to be
useless, except maybe for helping them fake it through another
semester. Their actual problems are deeper and pretending to the
contrary is a disservice. Realistically, 95 to 99 percent of students
in US college calculus courses understand almost nothing, but there
is institutional pressure to certify the opposite.
David Besser
math is elementary education, so you have children getting their
introduction to math from people that hate it. Elementary students have
art, music, and physical education teachers, so why not a math specialist?
When i run into (a + b)^2 = a^2 + b^2, I draw the well known picture.
David Besser
Anton Sherwood
: > My calculus professor would have killed this guy.
Rainer Rosenthal <r.ros...@ngi.de> writes
: How cruel these professors are in the USA :-)
Well, maybe they go for euthanasia in Belgium,
but here the authorities frown on it.
--
Anton Sherwood -- br0...@p0b0x.com -- http://ogre.nu/
Brian Borchers
Dave Rusin wrote:
>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.
Now we begin to see an important problem: The courses that come before
calculus are not actually preparing the students for calculus. This
is a problem with the curriculum that should be addressed by changing
the precaclulus courses so that students who pass these courses
actually will have the skills and knowledge needed to succeed in the
calculus course.
By giving a student a passing grade in a prerequisite course, we
implicitly promise that the student is prepared for the next course. It's
misleading and unfair to students to pass them on when they really
aren't ready for the next course in the sequence.
Of course, even if Dave can see to it that the precalculus courses at
his institution are working well, he'll have to deal with students who
transfer into his university with credit from other institutions that
aren't doing such a good job.
Unfortunately, my institution faces the same problem, although things
aren't quite as bad. We have a placement test, and the results of the
placement test are mandatory- students are simply not allowed to take
courses that they haven't placed into. The placement test is
effective in the sense that about 90% of the students who place
directly into calculus pass on the first try.
On the other hand, students who place into the various levels of
precalculus courses do very poorly. The failure rate in the precalc
courses is about 50%. (The students that have to pass through two or
more semesters of precalculus almost never make it!) Of those who do
pass these courses and start calculus, only about 60% pass calculus on
the first try.
We're faced with two challenges. First, we have to ensure that a
passing grade in the precalculus course means that a student really is
prepared for calculus. Second, we need to increase the percentage of
students who actually make it through the precalculus courses.
To me, the first challenge is the easier one- figure out what skills
and knowledge the students are lacking and make sure that students
don't pass the precalculus courses until they have it. Fixing this
will require changing the curriculum, not just "raising standards."
However, it's likely in the short run that more students will fail
these courses.
The second challenge is far harder. When I look at why students fail
in our precalculus courses, I see lots of different issues:
- Some students are so poorly prepared that even our lowest level
courses are too advanced for them. We could offer even lower level
courses, but what's the point of spending two years in college
studying precalculus mathematics before starting calculus (as well
as calculus based courses in physics, engineering, etc.)?
Our students are all engineering and science majors. Most of them
have to take several more courses beyond calculus. This would
stretch out the undergraduate program to far more than the typical
four to five years. In my opinion, students who plan to major in
science and engineering simply have to learn some mathematics
in high school before they come to college.
- Many students aren't motivated. When students won't even show up
for class and work on homework outside of class, they're not going
to learn much. I don't have a clue as to how to deal with this
problem. (I'd be very interested to hear of any approaches that
have been shown to work.)
- There are many problems that are particular to individual students.
For example, I've dealt with a dyslexic student who couldn't copy
equations from one line to the next. I've seen other students with
severe test anxiety. By their very nature, these problems require
individual attention.
This has been a long rambling post. Let's see if I can summarize it
in a few sentences:
- It's important to recognize that their are problems with a system
which puts students into classes for which they aren't prepared. The
system needs to be fixed so that this doesn't happen.
- Ultimately, we still have to teach students. Blaming the student
for not being prepared isn't an answer. Setting up appropriate
classes and getting students the help they need is the right way
to go.
--
Brian Borchers borc...@nmt.edu
Department of Mathematics http://www.nmt.edu/~borchers/
New Mexico Tech Phone: 505-835-5813
Socorro, NM 87801 FAX: 505-835-5366
Anonymous
wrote:
>When i run into (a + b)^2 = a^2 + b^2, I draw the well known picture.
When a=4 and b=3, is this the picture you had in mind?
....bbb
....bbb
....bbb
aaaa...
aaaa...
aaaa...
aaaa...
Bob Silverman
borc...@rainbow.nmt.edu (Brian Borchers) wrote:
<snip>
> On the other hand, students who place into the various levels of
> precalculus courses do very poorly. The failure rate in the precalc
> courses is about 50%. (The students that have to pass through two or
> more semesters of precalculus almost never make it!)
May I ask: What happens to these students? Are they dismissed
from the college?
They certainly should not be granted a *college* degree, if they
have not yet learned *high school level* math.
> We're faced with two challenges. First, we have to ensure that a
> passing grade in the precalculus course means that a student really is
> prepared for calculus. Second, we need to increase the percentage of
> students who actually make it through the precalculus courses.
Which would probably require not admitting many of the students
who do get admitted. I think that many schools are too willing to
compromise academic integrity for the tuition they gain.
> The second challenge is far harder. When I look at why students fail
> in our precalculus courses, I see lots of different issues:
>
> - Some students are so poorly prepared that even our lowest level
> courses are too advanced for them.
Yes. So don't admit such students. Send them to the local
community college for a couple of years so that they may fill
in the holes.
>
> - Many students aren't motivated. When students won't even show up
> for class and work on homework outside of class, they're not
going
> to learn much.
Bingo! So tell them to take a few years off from school until they
ARE ready to study. This is really the only cure.
> For example, I've dealt with a dyslexic student who couldn't copy
> equations from one line to the next. I've seen other students
with
> severe test anxiety.
My experience is that "test anxiety" is a synonym for "afraid to
take the test because they haven't done the work". Students who
know the material are not afraid to take the test.
In life and in our jobs we are tested all the time. Are these people
going to go through life being afraid all the time?
>
> - It's important to recognize that their (sic) are problems with a
system
> which puts students into classes for which they aren't prepared.
The
> system needs to be fixed so that this doesn't happen.
>
> - Ultimately, we still have to teach students. Blaming the student
> for not being prepared isn't an answer. Setting up appropriate
> classes and getting students the help they need is the right way
> to go.
Which sometimes means getting remedial help BEFORE entering college.
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
Allan Adler
> My experience is that "test anxiety" is a synonym for "afraid to
> take the test because they haven't done the work". Students who
> know the material are not afraid to take the test. In life and
> in our jobs we are tested all the time. Are these people going
> to go through life being afraid all the time?
A friend of mine is a very smart fellow who simply had trouble
with high pressure tests. I don't know exactly what it was about
it that bothered him but I know that it intimidated him in ways
that caused him to prepare for the test in unproductive ways.
For example, his copy of Greenberg's algebraic topology book had
very little white space left in it because he had annotated in the
margin every picky little detail that they could possibly use to
trip him up on an exam. He wound up doing very badly on the exam
and left mathematics. However, before he left he happened to tell
me something he had been thinking about in connection with categories
and I realized that he had invented 2-categories on his own. He
was perfectly capable of thinking creatively about mathematics and
enjoying it for its own sake as long as he wasn't distracted by
the bureaucratic procedures designed to test him.
I have seen a similar phenomenon in which a professor at some godforsaken
school in the middle of nowhere, desperately trying to get a raise or
promotion by doing some research, wound up spending all his time proving
that he was really working and instead accomplished nothing. He was so
afraid of failure that he put all his effort into proving that he had
really tried, because at some level he could only accept failure if he
knew he had done his best and failed in spite of it. What he never
realized that this need to prove to himself that he was really trying
made it impossible for him to do any real work.
In this case, there was more doubt about whether he could have done
creative mathematics under more relaxed conditions. However, the
bureaucracies that tested him seemed to be satisfied with his ability
to take them seriously and retained him permanently. He was just what
they wanted.
Dennis Yelle
[...]
> We have a placement test, and the results of the
> placement test are mandatory- students are simply not allowed to take
> courses that they haven't placed into. The placement test is
> effective in the sense that about 90% of the students who place
> directly into calculus pass on the first try.
>
> On the other hand, students who place into the various levels of
> precalculus courses do very poorly. The failure rate in the precalc
> courses is about 50%. (The students that have to pass through two or
> more semesters of precalculus almost never make it!) Of those who do
> pass these courses and start calculus, only about 60% pass calculus on
> the first try.
Well then it looks like you should be using the placement
test as the final test for the last precalculus class.
It also indicates that many of these students are NOT simply
handicapped by bad previous schools, perhaps there really are things that
some people simply cannot learn. (Or unlearn as the case may be.)
Dennis Yelle
--
I am a computer programmer and I am looking for a job.
There is a link to my resume here: http://table.jps.net/~vert
Lee Rudolph
>Brian Borchers wrote:
>[...]
...
>> On the other hand, students who place into the various levels of
>> precalculus courses do very poorly. The failure rate in the precalc
>> courses is about 50%. (The students that have to pass through two or
>> more semesters of precalculus almost never make it!) Of those who do
>> pass these courses and start calculus, only about 60% pass calculus on
>> the first try.
>
>Well then it looks like you should be using the placement
>test as the final test for the last precalculus class.
This is (in my experience, which is not at Brian Borchers's
institution) politically infeasible.
>It also indicates that many of these students are NOT simply
>handicapped by bad previous schools,
Well...it might indicate that the nature of the handicap
provided by previous schooling is not that they were badly
taught the right things, but rather that they were successfully
taught the wrong things.
>perhaps there really are things that
>some people simply cannot learn.
I certainly believe that, in any given area, some people are
(born) stupid; but I also believe that, in any given area, many
people have been stupified (made stupid), often--alas--by the
schools (and other social institutions including the family).
For example, I am stupid in various areas, and I have good reason
to believe that in at least some of them I got that way through
assiduous training by the schools. In a few of these areas I
have somewhat reversed my stupefaction after much effort over many
years.
>(Or unlearn as the case may be.)
Unlearning is, I think, harder than learning right the first time.
Lee Rudolph
Keith Ramsay
Bob Silverman <bo...@my-deja.com> writes:
|My experience is that "test anxiety" is a synonym for "afraid to
|take the test because they haven't done the work". Students who
|know the material are not afraid to take the test.
I don't think this is always true, although it's tricky to claim a
clear counterexample, since hardly ever has a student known everything
on the test... and then done badly. But I've had students who've been
frightened far out of proportion to the actual difficulties they've
had with the material (as judged by how they dealt with problems
they've solved in office hours).
The situation is complicated by the plague of students' not knowing
when they actually know things.
Keith Ramsay
Dave Rusin
David Besser <dbe...@io.frii.com> wrote:
>My theory: One of the college majors that requires the least amount of
>math is elementary education,
This is subject to quite a lot of local variation. Whatever the
university-wide minimum for mathematics is (typical examples seem to
read, "one course from the following list:..."), that minimum is
probably achieved by students in arts and humanities, and possibly
some "professional certification" programs. (Is there a school which
offers a degree in Cosmetology?) It's hard to argue that painters
and analysts of Moliere need any math beyond what is necessary for
them to survive the General Education experience.
By comparison, students intending to teach are subjected to quite an
array of required courses, at least in states in which teaching is
a regulated profession. In Illinois, for example, I believe a minimum
of three math courses are required (one of which may be specific to
mathematical pedagogy). Other states may even require graduate degrees,
a requirement of dubious merit but one which at least in principle
can require more mathematics of those future teachers.
But certainly the math requirements for elementary teachers are
less demanding than those for secondary teachers. (Rightly so, I'd say.)
So, restricting our observations to future teachers only, we can
still agree with this:
>so you have children getting their
>introduction to math from people that hate it.
Also:
>Elementary students have
>art, music, and physical education teachers, so why not a math specialist?
The NCTM agrees. This may be one of the best-followed and most useful of
its recent recommendations. (Those interested in math education in the US
in grades K-12 should read the Standards document promulgated this year;
see nctm.org).
Of course, one issue is money. Consider making contact with your local
board of education...
dave
Chris Thompson
>
>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...
>>
>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.
I've always assumed that Littlewood was having a little fun here at the
expense of "profound philosophy" in general, and Wittgenstein in particular...
Chris Thompson
Email: cet1 [at] cam.ac.uk
Herman Rubin
>>Brian Borchers wrote:
>>> On the other hand, students who place into the various levels of
>>> precalculus courses do very poorly. The failure rate in the precalc
>>> courses is about 50%. (The students that have to pass through two or
>>> more semesters of precalculus almost never make it!) Of those who do
>>> pass these courses and start calculus, only about 60% pass calculus on
>>> the first try.
I remember that at the university I was then, one of the
winners of the prize for best undergraduate was a woman who
started at the university with high school algebra and
worked up.
>>Well then it looks like you should be using the placement
>>test as the final test for the last precalculus class.
>This is (in my experience, which is not at Brian Borchers's
>institution) politically infeasible.
>>It also indicates that many of these students are NOT simply
>>handicapped by bad previous schools,
>Well...it might indicate that the nature of the handicap
>provided by previous schooling is not that they were badly
>taught the right things, but rather that they were successfully
>taught the wrong things.
I have been maintaining that this is the case for quite
a long time. Part of the evidence here is that the
teachers could not learn the "new math"; the present
elementary school teacher candidates still cannot.
Nor can most of the prospective high school teachers
learn the structure of the integers, or the importance
of proofs, or that explanations should not just consist
of memorizing formulas. It is the idea of learning by
memorization and drill which, I believe, HAS done an
excellent job of weakening, or even destroying, the
minds of most whe are mistaught this way.
>>perhaps there really are things that
>>some people simply cannot learn.
>I certainly believe that, in any given area, some people are
>(born) stupid; but I also believe that, in any given area, many
>people have been stupified (made stupid), often--alas--by the
>schools (and other social institutions including the family).
>For example, I am stupid in various areas, and I have good reason
>to believe that in at least some of them I got that way through
>assiduous training by the schools. In a few of these areas I
>have somewhat reversed my stupefaction after much effort over many
>years.
>>(Or unlearn as the case may be.)
>Unlearning is, I think, harder than learning right the first time.
VERY much so. Those who go through years of arithmetic
drill have difficulty in learning that there is more to
the integers than strings of decimal digits.
>Lee Rudolph
--
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
Herman Rubin
Dave Rusin <ru...@vesuvius.math.niu.edu> wrote:
>In article <KVDg5.18$P3.170...@news.frii.net>,
>David Besser <dbe...@io.frii.com> wrote:
>>My theory: One of the college majors that requires the least amount of
>>math is elementary education,
>This is subject to quite a lot of local variation. Whatever the
>university-wide minimum for mathematics is (typical examples seem to
>read, "one course from the following list:..."), that minimum is
>probably achieved by students in arts and humanities, and possibly
>some "professional certification" programs. (Is there a school which
>offers a degree in Cosmetology?) It's hard to argue that painters
>and analysts of Moliere need any math beyond what is necessary for
>them to survive the General Education experience.
>By comparison, students intending to teach are subjected to quite an
>array of required courses, at least in states in which teaching is
>a regulated profession. In Illinois, for example, I believe a minimum
>of three math courses are required (one of which may be specific to
>mathematical pedagogy). Other states may even require graduate degrees,
>a requirement of dubious merit but one which at least in principle
>can require more mathematics of those future teachers.
They are subject to a variety of required names of courses.
This does not mean that they learn any real mathematics
whatever. Any school which tried to make the courses of
the level of mathematical rigor which has been shown to be
attainable by many, if not most, elementary school children
would find that it is no longer turning out teachers.
>But certainly the math requirements for elementary teachers are
>less demanding than those for secondary teachers. (Rightly so, I'd say.)
>So, restricting our observations to future teachers only, we can
>still agree with this:
>>so you have children getting their
>>introduction to math from people that hate it.
>Also:
>>Elementary students have
>>art, music, and physical education teachers, so why not a math specialist?
>The NCTM agrees. This may be one of the best-followed and most useful of
>its recent recommendations. (Those interested in math education in the US
>in grades K-12 should read the Standards document promulgated this year;
>see nctm.org).
I have read one of the earlier ones; I do not see how it
can achieve an understanding of mathematics.
>Of course, one issue is money. Consider making contact with your local
>board of education...
>dave
Money is not the main issue. It is AN issue. A nonobvious
issue is the mass of education courses required, which will
rapidly drive out the good students, together with the low
level of those in the mathematics education courses. Also,
the educationist idea of heterogeneous classes and keeping
with one's age group must be utterly destroyed if we are to
ever have even fair education.
patrick...@my-deja.com
degree. She was diagnosed by a psychologist as unable to learn basic
mathematics and her case was that she had a disability and was being
discriminated against. The court however ruled against her.
I once taught a freshman course in statistics. The curriculum heavily
emphasized the memorization of procedures of calculation, which struck
me as obsolete knowledge in the computer age. I think the explanation
was that there had been no significant change in statistics instruction
for fifty years because the consumer/student is powerless and
instructors have no motive to make any effort to change. Also some
students simply cannot learn anything mathematically abstract or
conceptual, while almost all in this class could learn to memorize
procedures. The effort this requires gives an excuse to reward the
student with a passing grade: failing a student is unpleasant for
everyone involved. Most of the students hated the course. I didn't
blame them.
So I feel some sympathy with the litigant student. Education has
evolved into something resembling a very lengthy, involved hazing
ritual. If one makes it through, he/she is a Member.
patrick...@my-deja.com
ach...@hotmail.com wrote:
> In article <8li1a4$duu$1...@gannett.math.niu.edu>,
> 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
It is good sport to make fun of this sort of thing, but almost everyone
is just as ignorant in many areas of life, be it auto repair, wine,
foreign languages, etc. Were I to visit France or be forced to play
semipro football I doubt I would have a better mastery of the basics
than this student.
So you can think of the students as tourists and try to ensure that
they have a pleasant stay. Then perhaps they will be equally welcoming
should we visit their territory. On the other hand, such stories can
be very good fun when you're certain that They can't overhear.
patrick...@my-deja.com
"denis-feldmann" <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?
>
>
I dunno, I don't think it is all that dumb. I had the same trouble in
junior high school(what do they mean "what is x"? Didn't they just say
x could be any number?) It wasn't until I took a graduate course in
logic that I really understood it. It shows that the student is
thinking and questioning, not just trying to do what the teacher wants.
The <> sign is somewhat ambiguous. You are using as shorthand for
NOT( for all a,b in reals : f(a) = g(b) ). One might naturally suppose
that the meaning is ( for all a,b in reals: f(a)<>g(b) ).
Lee Rudolph
>It is good sport to make fun of this sort of thing, but almost everyone
>is just as ignorant in many areas of life, be it auto repair, wine,
>foreign languages, etc. Were I to visit France or be forced to play
>semipro football I doubt I would have a better mastery of the basics
>than this student.
>
>So you can think of the students as tourists and try to ensure that
>they have a pleasant stay.
I suppose we can (try to) think such thoughts. Some of us, though,
might have thought--many years ago, when we got into this sorry
business--that we were going to be earning our livings by teaching
mathematics, not by being genial hosts at the buffet table, or
ticket-takers at the amusement park. Ah, well, that was before
Universities of Hospitality were thick on the ground.
>Then perhaps they will be equally welcoming
>should we visit their territory.
And we might not have planned an extended visit to, much less a
permanent exile in, Philistia.
Lee Rudolph
david_...@my-deja.com
patrick...@my-deja.com wrote:
[...]
>
> It is good sport to make fun of this sort of thing, but almost
everyone
> is just as ignorant in many areas of life, be it auto repair, wine,
> foreign languages, etc. Were I to visit France or be forced to play
> semipro football I doubt I would have a better mastery of the basics
> than this student.
But if you were a student taking a class in semi-pro football
and had no better mastery of the basics would you expect to pass?
(Um...)
> So you can think of the students as tourists and try to ensure that
> they have a pleasant stay. Then perhaps they will be equally
welcoming
> should we visit their territory. On the other hand, such stories can
> be very good fun when you're certain that They can't overhear.
>
Herman Rubin
>A failed junior college student in Aptos, California sued for a
>degree. She was diagnosed by a psychologist as unable to learn basic
>mathematics and her case was that she had a disability and was being
>discriminated against. The court however ruled against her.
>I once taught a freshman course in statistics. The curriculum heavily
>emphasized the memorization of procedures of calculation, which struck
>me as obsolete knowledge in the computer age.
This might not have been obsolete 50 years ago, but it was
just as bad a course. Knowing how to calculate does not
teach when. Knowing when is adequate to get help on the how.
Ronald Bruck
wrote:
:In article <8lm768$2dn$1...@wanadoo.fr>,
: "denis-feldmann" <denis-f...@wanadoo.fr> wrote:
:>
:
:> Me: You must realize that, sadly, (a+b)^2<>a^2+b^2. For instance, if
:you
:> take a=2, b=3, you get (a+b)^2=25 <>a^2+b^2=13.
:> Student: Agreed. But what happens if you dont take any example?
:>
:>
:
:I dunno, I don't think it is all that dumb. I had the same trouble in
:junior high school(what do they mean "what is x"? Didn't they just say
:x could be any number?)...
When I was seven or eight years old I asked my father, "What is
Algebra?" (Don't remember where I'd read the word.) He explained that
it was the subject where "Letters stood for numbers."
So the rest of that day I went around doing algebra: "A = 1, B = 2, C =
3, D = 4, ..."
Of course, I was eight, not fourteen. By fourteen you should be able to
get it.
--Ron Bruck
--
Due to University fiscal constraints, .sigs may not be exceed one
line.
Ronald Bruck
hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
:In article <8m6q4i$uss$1...@nnrp1.deja.com>, <patrick...@my-deja.com>
:wrote:
:>A failed junior college student in Aptos, California sued for a
:>degree. She was diagnosed by a psychologist as unable to learn basic
:>mathematics and her case was that she had a disability and was being
:>discriminated against. The court however ruled against her.
:
:>I once taught a freshman course in statistics. The curriculum heavily
:>emphasized the memorization of procedures of calculation, which struck
:>me as obsolete knowledge in the computer age.
:
:This might not have been obsolete 50 years ago, but it was
:just as bad a course. Knowing how to calculate does not
:teach when. Knowing when is adequate to get help on the how.
Ahh--I must remember this phrase: "Knowing when is adequate to get help
on the how." It summarizes the core of good mathematics instruction!
patrick...@my-deja.com
Ronald Bruck <br...@math.usc.edu> wrote:
>
> Ahh--I must remember this phrase: "Knowing when is adequate to get
help
> on the how." It summarizes the core of good mathematics instruction!
>
> --Ron Bruck
>
Playing devil's advocate, there is much to be said for purely technical
mindless repetitious algebra drill for the future mathematician. E.T.
Bell wrote that if one is not expert in algebra by age 18, it will
never happen. Someone else (Stanislaw Ulam?) wrote that purely
mechanical manipulation of symbols was the highest level of
understanding, though he most likely meant knowing some abstruse
operations so well that manipulations are completely routine. I've
noticed that some very good algebraicists have very little
"understanding"
of what they are doing: it seems helpful in that they have less baggage
to carry about and hence have more flexibility of thought. As a
graduate student in mathematics there was a good deal of emphasis on
memorization of definitions and theorems, as it is very helpful to have
this churning away in one's subconcious when attempting a proof.
On the other hand, none of the students under discussion in this thread
has any possibility of becoming a mathematician, so this observation
is irrelevant to the matter under discussion.
Neil Koblitz sat in on a Russian 9th grade class in Moscow in which the
students were developing the properties of a logarithm from its
definition as an integral. The main concern in the United States seems
to be that the primary and secondary educational systems not give any
student an advantage over any other, a goal in which they are fairly
successful. It is no wonder that Ayn Rand novels are perennially
popular in the high schools of America.
patrick...@my-deja.com
Hilbert was very literal-minded. He would visit undergraduate classes
and would sometimes be baffled by some subtle gap that all the students
could leap. It was no accident that he was the motive force behind the
mathematical foundations movement.
So the student straining at a gnat might be the next... Oh forget
about it.
J. Antonio Ramirez R.
> In article <bruck-72837F....@news.supernews.com>,
> Ronald Bruck <br...@math.usc.edu> wrote:
> > By fourteen you should be able to
> > get it.
> >
>
> Hilbert was very literal-minded. He would visit undergraduate classes
> and would sometimes be baffled by some subtle gap that all the students
> could leap.
Hmm. I'm curious, where did you read/learn this?
> [...]
--
Antonio
Herman Rubin
>:In article <8lm768$2dn$1...@wanadoo.fr>,
>: "denis-feldmann" <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?
>:I dunno, I don't think it is all that dumb. I had the same trouble in
>:junior high school(what do they mean "what is x"? Didn't they just say
>:x could be any number?)...
>When I was seven or eight years old I asked my father, "What is
>Algebra?" (Don't remember where I'd read the word.) He explained that
>it was the subject where "Letters stood for numbers."
>So the rest of that day I went around doing algebra: "A = 1, B = 2, C =
>3, D = 4, ..."
>Of course, I was eight, not fourteen. By fourteen you should be able to
>get it.
Even the limited use of letters for numbers can be
explained quite easily to first graders; it was standard in
some of the "new math" books. The use of variables for not
only mathematical purposes belongs with beginning reading.
patrick...@my-deja.com
ram...@theory.lcs.mit.edu (J. Antonio Ramirez R.) wrote:
> patrick...@my-deja.com writes:
>
> > In article <bruck-72837F....@news.supernews.com>,
> > Ronald Bruck <br...@math.usc.edu> wrote:
> > > By fourteen you should be able to
> > > get it.
> > >
> >
classes
> > and would sometimes be baffled by some subtle gap that all the
students
> > could leap.
>
> Hmm. I'm curious, where did you read/learn this?
>
> > [...]
>
> --
> Antonio
>
I believe it was a biography of Hilbert. There can't be very many.
Hilbert never prepared for class, he prefered to derive the results on
the spot with occasional dead ends and backtracking. Some students
were stimulated by this, others were baffled. His associate Klein's
lectures were at the opposite end of the spectrum and were models of
organization. Klein went temporarily insane from overwork when trying
to beat Hilbert to a result. Once he recovered he never returned to
research and devoted his career to administration and teaching. Such
incidents are not unusual amongst mathematicians, including such
notables as Isaac Newton, Jacobi, and Georg Cantor.
patrick...@my-deja.com
hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>
> Even the limited use of letters for numbers can be
> explained quite easily to first graders; it was standard in
> some of the "new math" books. The use of variables for not
> only mathematical purposes belongs with beginning reading.
Our first grade class in Norman, Oklahoma I was taught linguistics:
dipthong blends and the like. It was very effective. Unfortunately
our educational system has chosen the other path and is attempting to
abandon phonetics entirely.
My point is that first graders can learn much more of supposedly
"advanced" topics than is commonly supposed.
patrick...@my-deja.com
david_...@my-deja.com wrote:
> In article <8m6qqq$vd5$1...@nnrp1.deja.com>,
> patrick...@my-deja.com wrote:
> [...]
> >
> > semipro football I doubt I would have a better mastery of the basics
> > than this student.
>
> But if you were a student taking a class in semi-pro football
> and had no better mastery of the basics would you expect to pass?
> (Um...)
>
As in the United States a command of the very basics of mathematics is
required for admittance into a college football program, surely it
makes no less sense that an equally basic mastery of the skills of
American football be a prerequisite for entry into a course of
mathematical study. A ruling that the candidate be able to run 100
yards within 60 seconds would have challenged Sophus Lie, and a bench
press of 30 pounds might have proved overwhelming for Bernhard Riemann.
Lends new meaning to "My weak student can beat your weak student."
Aristarchus Opprobrium
> Neil Koblitz sat in on a Russian 9th grade class in Moscow in which the
> students were developing the properties of a logarithm from its
> definition as an integral.
The Russian system is well known for teaching math and physics
at a high level, but do they really teach the average 9th grader how to
develop calculus?
First, do you mean 9th *form*? Russian school starts one year later and
ends one year earlier than the 12-year system in the US and most of
Europe (10 years total). So 9th form would correspond to something
like 11th grade. Also, in the European systems that take 12 years,
calculus is usually taught during the last 2 years (or 3 semesters),
so teaching some form of calculus in 9th form would not be too
unusual in that light.
Also, were these students really using the machinery of integrals
and substitutions? Or a geometric approach using areas instead
of actual integrals. There is a popularization in Russian on hyperbolic
geometry, which contains this approach to the logarithm and also
the ordinary and hyperbolic trig functions. It's intended for
extracurricular math circles (seminars for bright pupils led by
grad students) rather than regular math classes.
patrick...@my-deja.com
Aristarchus Opprobrium <san...@bibimus.edu> wrote:
> patrick...@my-deja.com wrote:
>
> > Neil Koblitz sat in on a Russian 9th grade class in Moscow in which
the
> > students were developing the properties of a logarithm from its
> > definition as an integral.
>
> The Russian system is well known for teaching math and physics
> at a high level, but do they really teach the average 9th grader how
to
> develop calculus?
>
Certainly not. Talented pupils are separated very early, and quality
education is available mainly in Moscow (or at least this was the case
in 1977). The average level is lower than the US: it is not an
egalitarian system. Similarly, German students are tracked into trade
schools as early as the fourth grade.
Lieven Marchand
> As in the United States a command of the very basics of mathematics is
> required for admittance into a college football program, surely it
> makes no less sense that an equally basic mastery of the skills of
> American football be a prerequisite for entry into a course of
> mathematical study.
Why do American universities have (semi-professional) sports teams
anyway? European ones generally don't have sport teams and nobody
seems to miss them.
--
Lieven Marchand <m...@bewoner.dma.be>
Lambda calculus - Call us a mad club
Virgil
<m...@bewoner.dma.be> wrote:
>patrick...@my-deja.com writes:
>
>> As in the United States a command of the very basics of mathematics is
>> required for admittance into a college football program, surely it
>> makes no less sense that an equally basic mastery of the skills of
>> American football be a prerequisite for entry into a course of
>> mathematical study.
>
>Why do American universities have (semi-professional) sports teams
>anyway? European ones generally don't have sport teams and nobody
>seems to miss them.
The University of Chicago has done away with them, to its own benefit.
--
Virgil
vm...@frii.com
Lynn Killingbeck
>
> patrick...@my-deja.com writes:
>
> > As in the United States a command of the very basics of mathematics is
> > required for admittance into a college football program, surely it
> > makes no less sense that an equally basic mastery of the skills of
> > American football be a prerequisite for entry into a course of
> > mathematical study.
>
> Why do American universities have (semi-professional) sports teams
> anyway? European ones generally don't have sport teams and nobody
> seems to miss them.
>
Just a guess, but...
(1) Money
(2) Money
(3) Money
(...)
Lynn Killingbeck
Tim274
>> Why do American universities have (semi-professional) sports teams
>> anyway? European ones generally don't have sport teams and nobody
>> seems to miss them.
>Just a guess, but...
>(1) Money
>(2) Money
>(3) Money
Except that most universities and colleges lose a lot of money on athletic
programs, which are then subsidized by student fees. My feeling is that
tradition plays more of a role, plus the rationalization that more people
donate money to schools with winning records.
Tim274
Randy Poe
wrote:
>patrick...@my-deja.com writes:
>
>> As in the United States a command of the very basics of mathematics is
>> required for admittance into a college football program, surely it
>> makes no less sense that an equally basic mastery of the skills of
>> American football be a prerequisite for entry into a course of
>> mathematical study.
>
>Why do American universities have (semi-professional) sports teams
>anyway? European ones generally don't have sport teams and nobody
>seems to miss them.
And American baseball uses a completely different system of
professional minor leagues to develop players for the top-level teams.
Why wouldn't a "farm team" system work for football and basketball?
Probably it would, but I guess the answer is money. College basketball
and football are very popular on TV, and that means that the TV
stations get a lot of money by showing them. Which in turn means that
they can afford to pay obscene fees to the colleges for the rights.
Which is more than most colleges can afford to turn down. Hence scenes
like the one I remember at my university, where I watched them raise a
sidewalk in such a way that rain and snow runoff would be dumped into
a lake at the entrance of the Physics Building. Why? So that the
sidewalks to the new stadium would be dry.
- Randy
Bob Silverman
Virgil <vm...@frii.com> wrote:
> The University of Chicago has done away with them, to its own benefit.
>
Almost true. Organized sports still exist, but they are *clubs*.
This allows grad students to participate as well as undergrads.
(speaking from first hand experience)
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
Bob Silverman
Bob Silverman <bo...@my-deja.com> wrote:
> In article <vmhjr-553909....@news.frii.com>,
> Virgil <vm...@frii.com> wrote:
> > The University of Chicago has done away with them, to its own
benefit.
> >
>
> Almost true. Organized sports still exist, but they are *clubs*.
> This allows grad students to participate as well as undergrads.
>
> (speaking from first hand experience)
Also, it is not true that European universities have no sports.
Look at crew at Oxford & Cambridge.
(which is what I did as a grad student at U. of C)
Ronald Bruck
<bo...@my-deja.com> wrote:
:In article <8megej$ml1$1...@nnrp1.deja.com>,
: Bob Silverman <bo...@my-deja.com> wrote:
:> In article <vmhjr-553909....@news.frii.com>,
:> Virgil <vm...@frii.com> wrote:
:> > The University of Chicago has done away with them, to its own
:benefit.
:> >
:>
:> Almost true. Organized sports still exist, but they are *clubs*.
:> This allows grad students to participate as well as undergrads.
:>
:> (speaking from first hand experience)
:
:
:Also, it is not true that European universities have no sports.
:Look at crew at Oxford & Cambridge.
:
:(which is what I did as a grad student at U. of C)
Just curious--where? On the Midway, after a rainstorm? ;-)
Of course, you don't HAVE to have a river near campus to enjoy this
sport. USC and UCLA practice in Ballona Creek, probably ten miles from
the UCLA campus and fifteen from USC. And much closer to the sewage
treatment plant, to Ballona's great detriment.
Lieven Marchand
> Also, it is not true that European universities have no sports.
> Look at crew at Oxford & Cambridge.
I said semi-professional sport teams. Off course, in probably almost
all the universities, different faculties have teams that compete in
friendly games, but certainly not on the scale of college
(base|foot)ball, nor is it taken as a factor in enrolling students.
Fred Galvin
> Bob Silverman <bo...@my-deja.com> writes:
>
> > Also, it is not true that European universities have no sports.
> > Look at crew at Oxford & Cambridge.
>
> I said semi-professional sport teams. Off course, in probably almost
> all the universities, different faculties have teams that compete in
> friendly games, but certainly not on the scale of college
> (base|foot)ball, nor is it taken as a factor in enrolling students.
You mean (basket|foot)ball, don't you?
Aristarchus Opprobrium
> > > Neil Koblitz sat in on a Russian 9th grade class in Moscow in
> > > which the students were developing the properties of a logarithm
> > > from its definition as an integral.
Koblitz and his wife wrote several articles about their experience
in mathematics classrooms around the world. Is the story about
integrals in Russian 9th grade written in more detail somewhere?
> > The Russian system is well known for teaching math and physics
> > at a high level, but do they really teach the average 9th grader how
> > to develop calculus?
>
> Certainly not. Talented pupils are separated very early,
They are mostly identified, not separated.
Math/physics high schools do exist in various Russian cities as do
extracurricular math circles and olympiads. Some schools such
as #57 (?) in Moscow are famous for producing
scientists, and there was a similar school in Leningrad that produced a
huge number of olympiad champions. However, most mathematicians
and physicists did not go to specialized high schools.
BTW, China, which also identifies talent early, does not have
specialized high schools.
> and quality
> education is available mainly in Moscow (or at least this was the case
> in 1977).
Mainly in big cities. Moscow, St Petersburg, Kiev, also some
academic centers such as Novosibirsk (Akademgorok).
> The average level is lower than the US: it is not an
> egalitarian system.
I doubt that the average level is lower. The US with
its property tax-driven school system is not
egalitarian either, except maybe in the Procrustean
sense of giving everyone a shot at ignorance.
> Similarly, German students are tracked into trade
> schools as early as the fourth grade.
Not true. It may have been true several
decades ago. There have been many
reforms in the last 50 years.
First of all, the German systems'
vocational schools in some subjects
(such as engineering) run longer than the
US 4-year high school, cover a good deal
of US university-level material, and lead to
professional licensing. So "trade school"
is a somewhat misleading description.
Second, it is possible to move
upward from the vocational tracks
to gymnasium.
Third, external matriculation is available as a
direct track to university.
Finally, US students are also stratified,
also as early as fourth grade and continuing
through high school and university. The
main difference is simply that students from
all strata are taught in the same school building.
But the academic segregation is, by high school
level, comparable in rigidity to any current European
school system.
Andrey....@get-lost-spammer.uni-ulm.de
> patrick...@my-deja.com wrote:
>> > > Neil Koblitz sat in on a Russian 9th grade class in Moscow in
>> > > which the students were developing the properties of a logarithm
>> > > from its definition as an integral.
> Koblitz and his wife wrote several articles about their experience
> in mathematics classrooms around the world. Is the story about
> integrals in Russian 9th grade written in more detail somewhere?
>> > The Russian system is well known for teaching math and physics
>> > at a high level, but do they really teach the average 9th grader how
>> > to develop calculus?
Yeah, "they" do. Hehe. Unbelivable? These stupid Soviets taught kids
integrals instead of "consumer lessons".
>> Certainly not. Talented pupils are separated very early,
Certainly yes.
[..]
>> The average level is lower than the US: it is not an
>> egalitarian system.
No shit? :)
> I doubt that the average level is lower. The US with
> its property tax-driven school system is not
> egalitarian either, except maybe in the Procrustean
> sense of giving everyone a shot at ignorance.
--
Andrey Nikolaev Ulm university,
Department of Biophysics. Germany.
Email: Andrey.Nikolaev@!get-lost-spammer!.uni-ulm.de
Substitute physik instead of !*! .
patrick...@my-deja.com
Aristarchus Opprobrium <san...@bibimus.edu> wrote:
> patrick...@my-deja.com wrote:
>
> > > > Neil Koblitz sat in on a Russian 9th grade class in Moscow in
> > > > which the students were developing the properties of a logarithm
> > > > from its definition as an integral.
>
> Koblitz and his wife wrote several articles about their experience
> in mathematics classrooms around the world. Is the story about
> integrals in Russian 9th grade written in more detail somewhere?
>
He told me this at dinner in the late 70s.
You're right, me knowledge of German education is circa 1963. And I'm
no expert on European secondary education.
Any Germans or Russians out there with the straight scoop?
Paul Burridge
<san...@bibimus.edu> opined thusly:
>BTW, China, which also identifies talent early, does not have
>specialized high schools.
An interesting dichotomy in China. talent for athletics, ping-pong and
swimming is widely applauded and promoted; WHEREAS a talent for
democratic politics and human rights gets one a bullet in the back of
the head - for which one's family have to pay.
Yes, they do identify talent early - for all the wrong reasons.
--
Noverint universi per presentes et futuri...
jmfb...@aol.com
Have you ever been to a children's palace in China? Perhaps
you should visit one before you judge their approaches.
/BAH
Subtract a hundred and four for e-mail.
Rob Pratt
<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?
In one of the Abstract Algebra textbooks (either Artin or
Dummit/Foote), I remember an exercise that asked the student to prove
that in a field of characteristic p, (x + y)^p = x^p + y^p. (You just
need to show that p divides the binomial coefficients, so all but two
terms of the binomial expansion drop out.) Anyway, the exercise was
labeled "The Freshman's Dream".
Paul Burridge
>Have you ever been to a children's palace in China? Perhaps
>you should visit one before you judge their approaches.
I couldn't possibly do that, having boycotted them over their actions
in Tibet.
Igor V Litvinyuk
: On Tue, 08 Aug 00 10:16:29 GMT, jmfb...@aol.com opined thusly:
: >Have you ever been to a children's palace in China? Perhaps
: >you should visit one before you judge their approaches.
: I couldn't possibly do that, having boycotted them over their actions
: in Tibet.
: --
That a boy. That will teach them stinking commies. Now if you also
could do us all a favour and boycott soc.culture.russia over, say,
Russia's action in Chechnia, we would appreciate it even more.
I.
Ian Dickson
>On Sat, 05 Aug 2000 16:57:00 -0400, Aristarchus Opprobrium
><san...@bibimus.edu> opined thusly:
>
>>BTW, China, which also identifies talent early, does not have
>>specialized high schools.
>
My wife was in school under the Soviet system and she started at 6,
which would make 9th grade age 14.
I did calculus in a British comp at age 16 and was not any kind of
program. Age 14 would be perfectly fine given a system that perhaps
allowed people who were bright (as opposed to gifted, they are doing
degrees at 14) to move forward faster. Certainly my understanding of
that system is that there were a lot of specialist schools and say the
top 10-15% in any field (sports, science, art etc) got sent to
specialist schools and an education that concentrated on those areas.
--
Ian Dickson Moneyweb - http://www.moneyweb.co.uk
Find your Local IFA. 01452 862637 (fax 862670)
"probably the UK's most comprehensive Personal Finance site" - The FT
"lots of useful information"- Which? "packed to bursting"- WWW Directory
UK FinServ Professional?, join Finservuk-list via Moneyweb.
jmfb...@aol.com
Paul Burridge <p...@osiris1.co.uk> wrote:
>On Tue, 08 Aug 00 10:16:29 GMT, jmfb...@aol.com opined thusly:
>
>>Have you ever been to a children's palace in China? Perhaps
>>you should visit one before you judge their approaches.
>
>I couldn't possibly do that, having boycotted them over their actions
>in Tibet.
Sigh! So you've decided to not learn from people with whom
you disagree. That narrows your opportunities to nothing.
Those children's palaces are really neat places. The kids
go there after school is done for the day. I saw music
rooms, art rooms, computer rooms, and that's where we
got stuck because the instructor wanted to talk with
us about computers..even though we didn't have a speaking
language in common. Those people were so interested in
learning and, as long as we kept our politics out of the
conversation, they showed very little prejudice because
of those differences. Quite refreshing.
Andrey....@get-lost-spammer.uni-ulm.de
> I did calculus in a British comp at age 16 and was not any kind of
> program. Age 14 would be perfectly fine given a system that perhaps
> allowed people who were bright (as opposed to gifted, they are doing
> degrees at 14) to move forward faster. Certainly my understanding of
> that system is that there were a lot of specialist schools and say the
> top 10-15% in any field (sports, science, art etc) got sent to
> specialist schools and an education that concentrated on those areas.
There were special schools, no doubt there. F.e. I myself is graduate
of one of those special schools (#145, Kiev), but calculus was in
cirriculum of _ordinary_ school in USSR. In our school we effectively
were taught not only calculus but differential geometry on 10th grade
and linear algebra. That was already beyond program of ordinary school.
Paul Burridge
>Sigh! So you've decided to not learn from people with whom
>you disagree. That narrows your opportunities to nothing.
See below.
>Those children's palaces are really neat places.
You might be silly enough to fall for this propaganda but thankfully
some of us look beyond it.
>The kids
>go there after school is done for the day. I saw music
>rooms, art rooms, computer rooms, and that's where we
>got stuck because the instructor wanted to talk with
>us about computers..even though we didn't have a speaking
>language in common.
That 'conversation' didn't get very far then.
>Those people were so interested in
>learning and, as long as we kept our politics out of the
>conversation, they showed very little prejudice because
>of those differences.
You said it: "as long as we kept our politics out of the conversation"
So the Chinese decline to learn from someone with whom *they*
disagree. That narrows *their* opportunities to nothing, wouldn't you
say.
>Quite refreshing.
About as refreshing as inhaling the stench of stale vomit.
jmfb...@aol.com
Paul Burridge <p...@osiris1.co.uk> wrote:
>On Wed, 09 Aug 00 10:45:53 GMT, jmfb...@aol.com opined thusly:
>
>>Sigh! So you've decided to not learn from people with whom
>>you disagree. That narrows your opportunities to nothing.
>
>See below.
>
>>Those children's palaces are really neat places.
>
>You might be silly enough to fall for this propaganda but thankfully
>some of us look beyond it.
Propoganda? How do you figure that a building where
kids go is propoganda? BTW, for the others reading this
thread, there were also math rooms and science rooms but
I didn't get to see them :-( since we did spend most of
our tourist time in the computer room.
>
>>The kids
>>go there after school is done for the day. I saw music
>>rooms, art rooms, computer rooms, and that's where we
>>got stuck because the instructor wanted to talk with
>>us about computers..even though we didn't have a speaking
>>language in common.
>
>That 'conversation' didn't get very far then.
Oh, quite the contrary, we got very far which really woke
me up. I had not considered BASIC to be a vehicle
for spoken lanaguage before then; and I didn't believe
it were possible to discuss hardware and software with
gestures :-). I learned a lot in that hour's visit.
>
>>Those people were so interested in
>>learning and, as long as we kept our politics out of the
>>conversation, they showed very little prejudice because
>>of those differences.
>
>You said it: "as long as we kept our politics out of the conversation"
>So the Chinese decline to learn from someone with whom *they*
>disagree. That narrows *their* opportunities to nothing, wouldn't you
>say.
No. It was called agreeing to disagree and get on with other
topics that were more productive and of benefit to both.
>
>>Quite refreshing.
>
>About as refreshing as inhaling the stench of stale vomit.
Look, I'm not even trying to convince you. All I'm doing is
using this opportunity to tell a story for the purpose of
pointing out a successful approach to teaching the kiddies.
/BAH
Paul Burridge
>Look, I'm not even trying to convince you. All I'm doing is
>using this opportunity to tell a story for the purpose of
>pointing out a successful approach to teaching the kiddies.
Well, even Cambodia under Pol Pot spared its children, I guess.
Nicolas Bray
On Thu, 10 Aug 2000, Paul Burridge wrote:
[snip]
> You said it: "as long as we kept our politics out of the conversation"
> So the Chinese decline to learn from someone with whom *they*
> disagree.
[snip]
Some Chinese people do, some Chinese people don't. It's ridiculous to
assign almost any characteristic to a group of 1 billion people. In any
case even if your statement was true that doesn't support your position.
There's something to be learned from almost everyone and if you have a
rational mind, you have nothing to fear from trying.
Vlad
Aristarchus Opprobrium <san...@bibimus.edu> wrote:
> patrick...@my-deja.com wrote:
>
> > Neil Koblitz sat in on a Russian 9th grade class in Moscow in which
the
> > students were developing the properties of a logarithm from its
> > definition as an integral.
>
> at a high level, but do they really teach the average 9th grader how
to
> develop calculus?
>
and
> ends one year earlier than the 12-year system in the US and most of
> Europe (10 years total).
No, Russian system was 11 years and, if I am not mistaken, it is
currently 12 years. And they always started at the same age as USA: 6 to
7.
< So 9th form would correspond to something
> like 11th grade. Also, in the European systems that take 12 years,
> calculus is usually taught during the last 2 years (or 3 semesters),
> so teaching some form of calculus in 9th form would not be too
> unusual in that light.
>
> Also, were these students really using the machinery of integrals
> and substitutions? Or a geometric approach using areas instead
> of actual integrals. There is a popularization in Russian on
hyperbolic
> geometry, which contains this approach to the logarithm and also
> the ordinary and hyperbolic trig functions. It's intended for
> extracurricular math circles (seminars for bright pupils led by
> grad students) rather than regular math classes.
Actually, I have never seen this approach. Trig was not very popular in
my times.
Vlad
patrick...@my-deja.com wrote:
> In article <3988FEC5...@bibimus.edu>,
> Aristarchus Opprobrium <san...@bibimus.edu> wrote:
> > patrick...@my-deja.com wrote:
> >
> > > Neil Koblitz sat in on a Russian 9th grade class in Moscow in
which
> the
> > > students were developing the properties of a logarithm from its
> > > definition as an integral.
> >
> > The Russian system is well known for teaching math and physics
> > at a high level, but do they really teach the average 9th grader how
> to
> > develop calculus?
> >
>
> education is available mainly in Moscow (or at least this was the case
> in 1977).
Rusisan authorities, your kid will stay in the same school all through
his school life. However, all medium and large cities have specialized
schools. specializing: most in foreign languages, some in math/physics,
some in biology, some in literature, some in music, some in art, and so
on. Math schools start at grade 7 or 8 but they take new students into
grades 9 and even 10. It is up to the student or his parent to apply to
such schools. These schools are sort of the inspiration for the modern
"magnet schools" that Americans are now trying.
> The average level is lower than the US:
That's a laugh! While education in non-specialized schools in some small
towns is pretty lousy, it certainly teaches you ten times more than does
a typical US public school. Especially in physics and math: what average
American kids learn in grade 10, average Russian kids finish learning in
grade 2. It is only in college that US education starts matching Russian
education.
> it is not an egalitarian system.
No. They don't sacrifice the needs of bright kids for the sake of not
offending lazy stupid ones, the way US education does.
> Similarly, German students are tracked into trade
> schools as early as the fourth grade.
I very much doubt if your knowledge of German education is any better
than of Russian one.
Vlad
Aristarchus Opprobrium <san...@bibimus.edu> wrote:
> patrick...@my-deja.com wrote:
>
> > > > Neil Koblitz sat in on a Russian 9th grade class in Moscow in
> > > > which the students were developing the properties of a logarithm
> > > > from its definition as an integral.
>
> in mathematics classrooms around the world. Is the story about
> integrals in Russian 9th grade written in more detail somewhere?
>
> > > at a high level, but do they really teach the average 9th grader
how
> > > to develop calculus?
> >
> > Certainly not. Talented pupils are separated very early,
>
> Math/physics high schools do exist in various Russian cities as do
> extracurricular math circles and olympiads. Some schools such
> as #57 (?) in Moscow are famous for producing
> scientists, and there was a similar school in Leningrad that produced
a
> huge number of olympiad champions. However, most mathematicians
> and physicists did not go to specialized high schools.
>
> specialized high schools.
>
> > education is available mainly in Moscow (or at least this was the
case
> > in 1977).
>
> academic centers such as Novosibirsk (Akademgorok).
>
> > egalitarian system.
>
> I doubt that the average level is lower. The US with
> its property tax-driven school system is not
> egalitarian either, except maybe in the Procrustean
> sense of giving everyone a shot at ignorance.
These two are "non-egalitarian" in different senses. US education
discriminates on the basis of parents' wealth, Russian - on the basis of
ability. I leave it up to you to decide what's more fair.
For some reason, the famous words of Tom Lehrer (said more than 30
years ago) come to mind:
"Army is the place that takes the idea of equality to its logical
conclusion. Not only does it try not to discriminate on th ebasis of
race or religion. It doesn't discriminate on the basis of ability
either."
> > Similarly, German students are tracked into trade
> > schools as early as the fourth grade.
>
> Not true. It may have been true several
> decades ago. There have been many
> reforms in the last 50 years.
I just wonder why American people hold these weird stereotypes about
other countries' education. I guess it helps them to deny the disastrous
state of American education (other than college, of course).
> First of all, the German systems'
> vocational schools in some subjects
> (such as engineering) run longer than the
> US 4-year high school, cover a good deal
> of US university-level material, and lead to
> professional licensing. So "trade school"
> is a somewhat misleading description.
>
> Second, it is possible to move
> upward from the vocational tracks
> to gymnasium.
>
> Third, external matriculation is available as a
> direct track to university.
>
> Finally, US students are also stratified,
> also as early as fourth grade and continuing
> through high school and university. The
> main difference is simply that students from
> all strata are taught in the same school building.
> But the academic segregation is, by high school
> level, comparable in rigidity to any current European
> school system.
>
>
Vlad
patrick...@my-deja.com wrote:
> In article <398C7F9C...@bibimus.edu>,
> Aristarchus Opprobrium <san...@bibimus.edu> wrote:
> > patrick...@my-deja.com wrote:
> >
> > > > > Neil Koblitz sat in on a Russian 9th grade class in Moscow in
> > > > > which the students were developing the properties of a
logarithm
> > > > > from its definition as an integral.
> >
> > Koblitz and his wife wrote several articles about their experience
> > in mathematics classrooms around the world. Is the story about
> > integrals in Russian 9th grade written in more detail somewhere?
> >
>
>
> You're right, me knowledge of German education is circa 1963. And I'm
> no expert on European secondary education.
>
> Any Germans or Russians out there with the straight scoop?
First of all, one would have to take Neil's views on USSR with a bit of
salt, given his political views.
I personally studied calculus in grade 8 but this was the most rigorous
math school in all of Russia. And integrals were studied in grade 9.
However, we learned more esoteric techniques for evaluating them than do
American math students upon getting their Ph.D. Way too much for my
taste! :-(
In mediocre Russian schools, integrals were not taught until 1980s. So,
I suspect Neil was talking about specialized math schools, not about
typical schools.