# My weak student can beat your weak student!

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### Dave Rusin

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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.

### Rainer Rosenthal

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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)

### John R Ramsden

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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}"]

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
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 unread, 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: > >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 unread, 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: <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 unread, 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...] 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 unread, 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 :-) ### Dave Rusin unread, 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 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 unread, Jul 25, 2000, 3:00:00 AM7/25/00 to Dave Rusin <ru...@vesuvius.math.niu.edu> wrote in message news:8lkj4t$41h$1...@gannett.math.niu.edu... > > Some posters decided my comments said more about me than about the student. > > Dave is not a teacher by profession. > Hi Dave, please excuse my faulty impression. I am from Germany, so I am one of the Overseas reader you mentioned. I really did not want to offend you. It was pure ignorance as to the word TUTOR . In Germany a "Tutor" is some elder student helping those students who are in the first or second semester. You may have noticed, I read your story carefully - as well as I could with my limited knowledge of the circum- stances. ( By the way I liked that "steam" converting to "water"). For me as a foreigner it was really interesting to learn something about education in the USA. We do have a much more homogenous population - but things are changing. It is really interesting to hear in the newsgroup things related to mathematics in a broader sense. Thank you very much for your detailled information. May I wish you good humor all the time. Greeting to the student and my best wishes for her future. Regards, ### Oriana unread, Jul 25, 2000, 3:00:00 AM7/25/00 to In article <8lka1p$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. > 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 unread, 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."] 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 unread, 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. ### denis-feldmann unread, 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! 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 unread, 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? ### Anonymous unread, 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: >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 unread, Jul 26, 2000, 3:00:00 AM7/26/00 to In article <8lkshf$5sl$1...@nnrp1.deja.com>, 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 unread, 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? 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 unread, 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 ### Bill Taylor unread, 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 ------------------------------------------------------------------------------- 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 unread, 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. [*] [*] 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 unread, 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 ### David C. Ullrich unread, 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: >|> 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 unread, 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?" ### n_f_be...@my-deja.com unread, 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

### Zdislav V. Kovarik

unread,
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?" 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 unread, 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:

>|> 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