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Jul 24, 2000, 3:00:00 AM7/24/00

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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.

Jul 24, 2000, 3:00:00 AM7/24/00

to

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)

Jul 24, 2000, 3:00:00 AM7/24/00

to

ru...@vesuvius.math.niu.edu (Dave Rusin) wrote:

>

> 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: [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.

---------------------------------------------------------------------------

Jul 24, 2000, 3:00:00 AM7/24/00

to

Dave Rusin summarizes a frustrating conversation with a student

who, we all hope, did not read Dave Rusin's posting.

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. *

* *

****************************************************************************

Jul 24, 2000, 3:00:00 AM7/24/00

to

"Rainer Rosenthal" <r.ros...@ngi.de> writes:

> 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

Jul 25, 2000, 3:00:00 AM7/25/00

to

Allan Adler <a...@nestle.ai.mit.edu> writes:

> 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.

Jul 25, 2000, 3:00:00 AM7/25/00

to

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.

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.

Jul 25, 2000, 3:00:00 AM7/25/00

to

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

Jul 25, 2000, 3:00:00 AM7/25/00

to

On 24 Jul 2000 18:18:44 GMT, ru...@vesuvius.math.niu.edu (Dave Rusin)

wrote, in part:

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

Jul 25, 2000, 3:00:00 AM7/25/00

to

In article <397d8e9...@news.ecn.ab.ca>,

jsa...@tenMAPSONeerf.edmonton.ab.ca (John Savard) wrote:

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.

Jul 25, 2000, 3:00:00 AM7/25/00

to

In article <8lk7nr$kqg$1...@nnrp1.deja.com>,

Bart Goddard <godd...@my-deja.com> wrote:

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"

Jul 25, 2000, 3:00:00 AM7/25/00

to

Dave Rusin (ru...@vesuvius.math.niu.edu) wrote:

: [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 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

Jul 25, 2000, 3:00:00 AM7/25/00

to

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 :-)

Jul 25, 2000, 3:00:00 AM7/25/00

to

Yesterday, in article <8li1a4$duu$1...@gannett.math.niu.edu>, I wrote:

> A student just left from an hour of private office help.

and described a tale of woe in which a Calculus student was found> 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

Jul 25, 2000, 3:00:00 AM7/25/00

to

news:8lkj4t$41h$1...@gannett.math.niu.edu...

>

> Some posters decided my comments said more about me than about the

student.

> > Dave is not a teacher by profession.

>

>

> 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,

Jul 25, 2000, 3:00:00 AM7/25/00

to

In article <8lka1p$mo5$1...@nnrp1.deja.com>,

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.

>

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

Jul 25, 2000, 3:00:00 AM7/25/00

to

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."]

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

Jul 25, 2000, 3:00:00 AM7/25/00

to

ach...@hotmail.com wrote:

[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.

Jul 26, 2000, 3:00:00 AM7/26/00

to

[cut]>

> 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!

> 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?

Jul 26, 2000, 3:00:00 AM7/26/00

to

"denis-feldmann" <denis-f...@wanadoo.fr> writes:

> 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?

Jul 26, 2000, 3:00:00 AM7/26/00

to

On Wed, 26 Jul 2000 10:23:56 +0200, "denis-feldmann"

<denis-f...@wanadoo.fr> wrote:

<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 ==----------

Jul 26, 2000, 3:00:00 AM7/26/00

to

In article <8lkshf$5sl$1...@nnrp1.deja.com>,

Oriana <oria...@my-deja.com> wrote:

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

Jul 26, 2000, 3:00:00 AM7/26/00

to

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?

> [...]

> 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

Jul 26, 2000, 3:00:00 AM7/26/00

to

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

Jul 27, 2000, 3:00:00 AM7/27/00

to

|> 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...

Geez!

-------------------------------------------------------------------------------

Bill Taylor W.Ta...@math.canterbury.ac.nz

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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...

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Jul 27, 2000, 3:00:00 AM7/27/00

to

denis-feldmann wrote:

>

> 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. [*]>

> 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]

Jul 27, 2000, 3:00:00 AM7/27/00

to

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

Jul 27, 2000, 3:00:00 AM7/27/00

to

On 27 Jul 2000 05:35:25 GMT, mat...@math.canterbury.ac.nz (Bill

Taylor) wrote:

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.

Jul 27, 2000, 3:00:00 AM7/27/00

to

>> > > 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?"

Jul 27, 2000, 3:00:00 AM7/27/00

to

In article <%D0g5.810$Di4.1...@news.uswest.net>,

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

Jul 27, 2000, 3:00:00 AM7/27/00

to

In article <%D0g5.810$Di4.1...@news.uswest.net>,

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?"

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).

Jul 27, 2000, 3:00:00 AM7/27/00

to

In article <8lohmt$5h6$3...@cantuc.canterbury.ac.nz>,

mat...@math.canterbury.ac.nz (Bill Taylor) wrote:

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