Usenet is a bulletin board? (was Re: ETHICS VIOLATION....)
Gerry Myerson
(Hugh LaMaster) wrote:
=>
=> Usenet is a bulletin-board. No one makes you read it. If
=> it is of no use to you, don't read it.
=>
=> Do you think that Stanford or Dartmouth or whomever or whatever
=> are responsible for all the paper gets stapled to kiosk
=> bulletin boards on campus?
Usenet is *like* a bulletin board. The analogy is useful, but not perfect.
It's not clear to me whether it is useful in the case at hand. Anyway,
I'd like to see something more convincing than this argument by analogy.
Gerry Myerson (ge...@mpce.mq.edu.au)
Centre for Number Theory Research (E7A)
Macquarie University, NSW 2109, Australia
Lee Lady
>
> I read your recent post in sci.psychology where you commented that in
>your experience math students fall into 2 groups. I've also noticed your NLP
>background in previous posts. If I remember, some students seemed to be able to
>do proofs and the other group could only learn to do them by rote.
> I seem to fall in the group that lacks ability. That's frustrating because
>I'm interested in some problems in computer science that are becoming more and
>more mathematical(Category theory and the Domain theory underlying denotational
>semantics) every year.I can use the results, no problem, but a lot of the newer
>methods say in type theory and inference require a little deeper insight than
>I've been able to acheive.
> So, all that said I'm interested if you've ever tried modeling the sucessful
>students and the unsucessful students to find the difference that makes the
>difference?
> I've done some thinking about this to try to find the smallest possible
>example of the places where my thinking breaks down. Properties of powersets,
>notions of continuity are possible candidates, I certainly have troubles in
>those areas. Based on your teaching experience do you have any thoughts on
>commonalities of those who have problems with mathematical creativity
>understanding?
Well, I have to say that I've eventually developed a sense of futility
about this. From the very beginning of my teaching career, one of my
main driving forces has been to find ways of teaching students to do
proofs. Learning NLP [NeuroLinguistic Programming] intensified my
belief that it should be possible to each any person to understand
mathematical thinking. But in practical terms, I have to say that my
results have been zilch --- and not for want of effort.
I have taken a lot of time in class doing things that I believe most
mathematicians never do in their teaching, showing students that many
proofs are not as creative as students assume but instead follow a basic
template that can be learned. I have tried to show them that the
perenniel question "How do you start?" is often not difficult to
answer. And yet at the same time I have stressed that they shouldn't get
discouraged, because things that seem very simple in retrospect are very
often arrived at, even by good thinkers, only after much futile trial
and error. (There's a story that when John von Neumann first arrived in
Princeton he hired a maid, and someone asked the maid what the famous
mathematician was like. "Oh, he seems like an okay person, except for
being a little strange in some ways. All day he sits and his desk and
scribbles, scribbles, scribbles. Then, at the end of the day, he takes
the sheets of paper he's scribbled on, scrunges them all up, and
throws them in the trash can.")
I have given students printed answers to homework problems, I have
given them lists of incorrect statements frequently found in student
proofs, along with a list of corresponding correct statements. In
addition to class time, I have spent enormous amounts of time in my
office with certain students who clearly have a sincere willingness to
learn.
And the results, as I have said, for practical purposes seem to be about
zilch.
On the other hand, a couple of years ago I had a student in Linear
Algebra who wrote beautiful proofs, although certainly she made her share
of mistakes. So one day I said to her, "Obviously you've had experience
in writing proofs before." And she answered, "No, I never had to do that
in any of my previous courses. Of course I do have a reasonably good
ability to express myself in writing, but otherwise this is all new to
me."
This is somewhat like my own experience learning mathematics. Nobody
ever taught me how to write a proof. I simply imitated what was in the
book and what my professors did. When I was very young, maybe before I
even took calculus, I was reading some book and the author wanted to
prove that a certain statement was true for all natural numbers n. He
showed that it was true for n=1 and that if it was true for a
particular value of n then it would also be true for n+1. And I
thought "Wow! What a clever way of proving something!" It was only much
later that I learned that there was a name for this type of proof:
mathematical induction. To me, it just seemed like common sense. And
yet most students seem to be absolutely incapable of understanding
induction, even when it is explained in great detail. I have juniors
and seniors still turning in proofs that say "The statement is true
for n=1 and for n=2, and so by induction it holds for all n."
It's now starting to seem to me that trying to teach students
theorem-proving in a systematic way is not only futile but probably also
pointless. It's like trying to teach a dog to play checkers. Even if he
can learn, the dog will still never be a good player.
There are a few fundamental hang-ups I can identify, but I don't know how
to get students past them. One is that students don't realize the
importance of knowing the definitions. In many calculus courses (and I
have to admit, this is true in the ones I teach myself) a student can get
through without ever having to state a formal definition. Then all of a
sudden they come into Linear Algebra or some upper level course, and
everything depends on being extremely familiar with the formal
definitions, without needing to stop and think. To most students, this
is not "mathematics" (it's not a recipe that produces an answer) and
it's not *fair*.
A student comes into my office and says, "I don't know how to do
problem 27. It says to prove that w is in the subspace spanned by
v1, ..., vn." So I ask him, "What does it mean to to say that v is
in the subspace etc?" And he says, "I don't know." This blows me away
and it happens over and over again with the same students. How can a
student not realize, even after being told repeatedly, that if you want
to prove something the place to start is by identifying the
definitions of the concepts involved?
Another thing that is a sense of absolute frustration for me is that
students cannot understand the idea of proving an "If...then" statement.
This is why they can't learn to do proofs by induction, they can't prove
that functions are one-to-one, and in Linear Algebra they can't learn
to prove vectors are linearly independent. I tell them again and
again, "When you want to prove 'If X, then Y,' you suppose that X is
true and then show that you can prove Y." They absolutely refuse to do
this. It clearly makes no sense to them, and even when a few of them
eventually agree to turn in correct proofs, it seems clear to me that
they are doing it under protest and do not believe that it makes
sense. Furthermore, as soon as they learn that the word "suppose" can
occur in a proof, all hell breaks loose. Asked to prove the statement,
"If a triangle is equilateral then it is equiangular," their proof will
begin: "Suppose the triangle is equiangular." This is a very
consistent pattern. Over and over again, regardless of the context, when
a student if asked to prove "If X then Y," he will begin by saying
"Suppose Y." At first, the teacher will think "The student can't seem
to understand the concept of a one-to-one function, or linear
independence" or whatever, but what students don't seem to understand
is the structure of hypothetical statements.
Another thing (which may be relevant to your difficulties with category
theory and the like) is that students have enormous difficulty in
thinking of a *set* as being an entity and as something one can label with
a symbol and make statements about. Over and over again students will
make statements like "A subspace is a vector that can be added and
multiplied by constants." When I stop them and point out that a subspace
is always a whole set of vectors, they listen patiently but without
paying much attention, as though I were correcting some minor point of
grammar. And they continue to turn in proofs using the same kind of
statement and get very angry about my "nitpicking" when I tell them that
I can't understand what their statements mean.
Another thing is that students don't seem to get the idea that proofs are
not mere word games but involve statements, albeit often in quite
abstract form, about quite concrete entities. (As concrete, anyway, as
integers and real numbers.) If I were to say, for instance, "Prove that
all dogs have wheels," anyone would look at me as if I were crazy. And
yet I can say in class, "One of the interesting things about doing
algebra with matrices is that it is possible that AB = 0, even though
neither A nor B is 0." And then I give a very simple example with
two by two matrices. And then in the homework, problem 3 says, "Give
an example of matrices A and B such that A != 0 and B != 0 but
AB = 0," and the student will obligingly cook up some rather
complicated example (forgetting about the fact that I inadvertently
already did the problem for them in class). But then in problem 4, he
will be writing a proof and say, "Since AB = 0, we conclude that
either A = 0 or B = 0," oblivious to the counter-example he has just
constructed himself.
It doesn't occur to the student to ask, "Wait a minute. What does this
statement actually say about concrete calculations? And does that make
any sense?"
Of course I have to admit that this last mistake is one that I, as
a professional mathematician, have committed many times myself, busting
my ass for several weeks trying to prove a certain statement because I
fervently *wanted* it to be true, only to finally realize when I looked
at very simple examples that it was clearly often not valid.
--Lee
--
Unlike past American intellectuals, who saw the educated nonacademic
public as their main audience, today's leftist intellectuals feel no
need to write for a larger audience; colleagues, departments, and
conferences have come to constitute their world. -- Russell Jacoby
Bhalchandra Thatte
Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:
[...]
>induction, even when it is explained in great detail. I have juniors
>and seniors still turning in proofs that say "The statement is true
>for n=1 and for n=2, and so by induction it holds for all n."
I have observed that even some students who can use induction
with some practice believe that induction proves the statements only for
some finitely many values of n.
>
>It's now starting to seem to me that trying to teach students
>theorem-proving in a systematic way is not only futile but probably also
>pointless. It's like trying to teach a dog to play checkers. Even if he
>can learn, the dog will still never be a good player.
>
>There are a few fundamental hang-ups I can identify, but I don't know how
>to get students past them. One is that students don't realize the
>importance of knowing the definitions. In many calculus courses (and I
>have to admit, this is true in the ones I teach myself) a student can get
>through without ever having to state a formal definition. Then all of a
>sudden they come into Linear Algebra or some upper level course, and
>everything depends on being extremely familiar with the formal
>definitions, without needing to stop and think. To most students, this
>is not "mathematics" (it's not a recipe that produces an answer) and
>it's not *fair*.
I think all this is because most students here have not enough background
for any math course that they do. They always get away without learning
anything. And of course it is not *fair* if you flunk them! You are not
supposed to assume that they know what they did in previous math classes.
They do not have enough manipulation skills. So when you teach anything,
they try to understand each and every addition subtraction that you do on
the board, it takes lot of time for them to do that, and as a result they
find the stuff very complicated. For example, in a vector calculus course
there are students who have done some math courses - you can teach them
distance/time = velocity, but if you use distance / velocity = time
anywhere without explaining, they do not grasp it easily. If they get stuck
at every intermediate step like that, they have no time to understand
the subject matter of this course. And proving theorems also involves
manipulations - may be more abstract manipulations - manipulations of
definitions in the simplest proofs.
I believe that what needs to be stressed in the schools is manipulation
skills. That can be done by teaching algebra and plane geometry.
There is a growing conception among many people that concepts
and manipulations are distinct, and manipulations are a waste of time,
since computers can do them. This is a false conception. One can't do math
without good manipulation skills.
Bhalchandra Thatte
Archimedes Plutonium
ge...@macadam.mpce.mq.edu.au (Gerry Myerson) writes:
> Usenet is *like* a bulletin board. The analogy is useful, but not perfect.
> It's not clear to me whether it is useful in the case at hand. Anyway,
> I'd like to see something more convincing than this argument by analogy.
In the beginning, the Atom created other atoms, .. .. much later on,
. . it created intelligent life in order to create more atoms by hands
on experience. This is called nucleosynthesis in polite circles.
But, along with creation, there is both growth and decay, mostly
decay. In polite circles it is called decay but to the average person
(bloke in Aussie) it is called sin. And we are awash in sin. For every
great growth or creation is accompanied by a 100 sins or decay.
For every 7 Wonders of this World, are accompanied by 700 spray
painting graffiti.
The Journal system of our time for math and physics is the gargantum
and Lord Humungus (Road Warrior movies) Sin of our times. This system
is a "scratch my back, I scratch your back"; "ladder of promotion for
uncreative professors"; "barr and bann all outsiders from publishing,
even if the outsider is a Galois". Our present math and physics journal
system is a sin and a disgrace. And the faster it is destroyed, the
better we will be.
In the year 1993, the Atom had created for us the worldwide Internet
system. The Atom looked up at Earth from the Nucleus and saw the decay
and sin that was the Journal system, and gave us the Internet. The
Internet must flourish and grow, and concomitantly, the Journal System
must diminish and be replaced.
Timothy Y. Chow
Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:
>There are a few fundamental hang-ups I can identify, but I don't know how
>to get students past them. One is that students don't realize the
>importance of knowing the definitions.
I don't know if this will help, but perhaps you can try assigning problems
where you make up a bunch of nonsense words and ask them to prove something
that is stated in terms of the nonsense words. The subject matter can be
taken from everyday life, i.e., the reasoning should be trivial once the
definitions are entangled. This might circumvent the problem suggested by
Bhalchandra Thatte that the students don't have enough background.
>How can a student not realize, even after being told repeatedly, that
>if you want to prove something the place to start is by identifying the
>definitions of the concepts involved?
I spend a lot of time in my classes telling students certain facts
repeatedly. However, I realize that in many cases telling a student
a fact n times accomplishes about the same amount of communication as
telling the student the fact zero times. If they aren't on the brink
of understanding it already, then it doesn't help to simply present
them with a new concept. It's sometimes necessary to backtrack to
something that they *do* understand and spoonfeed them one tiny
mouthful at a time.
The reason that I don't do this in class is that it only really works
in a one-on-one situation when both of you have plenty of time and
motivation on your hands. However, from your comments about office
hours, it seems that this is the case at least occasionally for you.
Again, I think that the only hope is to try to make a connection with
things that they *already* understand, whether it be mathematics that
they already understand or everyday situations. Otherwise there is no
chance of progress. This applies for your descriptions of the problems
with if-then statements and sets as well.
--
Tim Chow tyc...@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons. ---Popular Mechanics, March 1949
Prem Sobel
Y. Chow) writes:
>In article <CzrL8...@news.hawaii.edu>,
>Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:
>>There are a few fundamental hang-ups I can identify, but I don't know
how
>>to get students past them. One is that students don't realize the
>>importance of knowing the definitions.
>I don't know if this will help, but perhaps you can try assigning
problems
>where you make up a bunch of nonsense words and ask them to prove
something
>that is stated in terms of the nonsense words.
While what you both say is valid, definitions are important, some
of the key words/concepts are not defined but are rather axiomatically
related. One can take either Peano's axioms or the axioms of geometry
and substitute nonesense words for the stand ones, for example:
in Peano's axioms: zero -> foo
successor -> sky
equal -> fusion
etc.
in gemoetry: point -> ding
line -> expansion
etc.
According to the ease with which one makes a model or multiple models
one gets a hand at working with the system. It may allow new models to
be formed which are useful. In this case real words rather than
nonsense words would be used, as shown.
Prem
Mike Russo
>mathematicians never do in their teaching, showing students that many
>proofs are not as creative as students assume but instead follow a basic
>template that can be learned. I have tried to show them that the
>perenniel question "How do you start?" is often not difficult to
>answer. And yet at the same time I have stressed that they shouldn't get
>discouraged, because things that seem very simple in retrospect are very
>often arrived at, even by good thinkers, only after much futile trial
>and error.
As a high school student who has taken but three years of mathematics, and
has not done a good deal of proofs in his life, I can tell you first-hand I
never know where to start! I think I've finally realized that the key is to
play around with what you're trying to prove by turning subscripts and Greek
letters into their definitions, and then play with those using simple algebra,
moving them around until you get what you want to prove.
>And the results, as I have said, for practical purposes seem to be about
>zilch.
After having proofs shown to me a hundred times and reading history of math
books for some insight (and because I'm interested, of course), I still only
barely have an idea of how to begin (which I stated above). Students should
really be exposed to this kind of thinking at a younger age; later when
they're asked to prove something and they just don't understand, it might be
because of underexposure in their pre-teen years.
>On the other hand, a couple of years ago I had a student in Linear
>Algebra who wrote beautiful proofs, although certainly she made her share
>of mistakes. So one day I said to her, "Obviously you've had experience
>in writing proofs before." And she answered, "No, I never had to do that
>in any of my previous courses. Of course I do have a reasonably good
>ability to express myself in writing, but otherwise this is all new to
>me."
Of course, I could be wrong...
>This is somewhat like my own experience learning mathematics. Nobody
>ever taught me how to write a proof. I simply imitated what was in the
>book and what my professors did. When I was very young, maybe before I
>even took calculus, I was reading some book and the author wanted to
>prove that a certain statement was true for all natural numbers n. He
>showed that it was true for n=1 and that if it was true for a
>particular value of n then it would also be true for n+1. And I
>thought "Wow! What a clever way of proving something!" It was only much
>later that I learned that there was a name for this type of proof:
>mathematical induction. To me, it just seemed like common sense. And
>yet most students seem to be absolutely incapable of understanding
>induction, even when it is explained in great detail. I have juniors
>and seniors still turning in proofs that say "The statement is true
>for n=1 and for n=2, and so by induction it holds for all n."
My Statistics teacher just taught the class how to do mathematical
induction, and it was the first time I, or anyone in the class, had ever heard
of it. Half the class didn't know what the heck was going on, and the other
half, though they knew what the teacher was doing, were still going over in
their heads whether it would work or not. At first, I thought it was like
circular logic; you mean you're assuming what you want to prove (when you
assuming statement x holds for all n's) and then using that to prove it? Wait
a second.... is that guy in front of the class doing mumbo-jumbo with that
piece of chalk tryin' to pull a fast one on us?
But if you think about it hard enough it DOES make sense, because even if I
assumed it in order to prove n+1, once I've proved it, I don't have to
'assume' for n anymore, since if it works for n+1, it must work for n (if we
let n=n-1, we have n-1 +1, which is n.)
>There are a few fundamental hang-ups I can identify, but I don't know how
>to get students past them. One is that students don't realize the
>importance of knowing the definitions. In many calculus courses (and I
I agree there. The easiest way for me to even start a proof is to break what
I'm dealing with down into its simpler definitions.
>again, "When you want to prove 'If X, then Y,' you suppose that X is
>true and then show that you can prove Y." They absolutely refuse to do
>this. It clearly makes no sense to them, and even when a few of them
>eventually agree to turn in correct proofs, it seems clear to me that
>they are doing it under protest and do not believe that it makes
>sense.
Again, it is thought of as circular, because they can't understand the
statement. They don't understand that I HAVE TO assume X because the statement
above says 'If X'! So, if X is true, then if I want to show the above
statement true, I have to prove Y is true. It's only circular if you try to
prove Y is true because the statement says that if X is true then Y is true,
so then Y must be true. It's NOT circular if you can prove Y true by some
other means.
>Furthermore, as soon as they learn that the word "suppose" can
>occur in a proof, all hell breaks loose. Asked to prove the statement,
>"If a triangle is equilateral then it is equiangular," their proof will
>begin: "Suppose the triangle is equiangular." This is a very
>consistent pattern. Over and over again, regardless of the context, when
>a student if asked to prove "If X then Y," he will begin by saying
>"Suppose Y."
Now THAT'S circular!
>At first, the teacher will think "The student can't seem
>to understand the concept of a one-to-one function, or linear
>independence" or whatever, but what students don't seem to understand
>is the structure of hypothetical statements.
As I said before, it could be because the student hasn't had much exposure to
this logical way of thinking. Logic just wasn't their subject because they
didn't understand how it was done.
>It doesn't occur to the student to ask, "Wait a minute. What does this
>statement actually say about concrete calculations? And does that make
>any sense?"
If the teacher said it, it's true. If the textbook says it, it's true. That's
why I wish teachers (especially in high-school level mathematics) would PROVE
EVERYTHING THEY SAY. I went through all of Course II (which here in New York
State is mostly Euclidean geometry) learning constructions and no teacher
(except the head of the math department, who is also very angry of this
situation) ever proved a single construction to me. Here's how you construct a
perpendicular. Do this, move the compass like this, and they would do this
mumbo-jumbo with a compass and say 'NOW REMEMBER THAT FOR THE TEST!' How am I
supposed to remember something I'm not even sure is true?!
>Of course I have to admit that this last mistake is one that I, as
>a professional mathematician, have committed many times myself, busting
>my ass for several weeks trying to prove a certain statement because I
>fervently *wanted* it to be true, only to finally realize when I looked
>at very simple examples that it was clearly often not valid.
I want x/0 to be equal to something, and I still don't understand this
undefined business! I'll have to do some research on how many mathematicians
it gave nightmares to...
*** Mike Russo, living in luxury in fabulous... Brooklyn, New York!! ***
*** star...@dorsai.org Writing from Windows for Workgroups 3.11! ***
*** djt...@prodigy.com <Quotes suck! Huh heh huh huh...> ***
Timothy Y. Chow
Mike Russo <star...@dorsai.org> wrote:
>If the teacher said it, it's true. If the textbook says it, it's true. That's
>why I wish teachers (especially in high-school level mathematics) would PROVE
>EVERYTHING THEY SAY.
Unfortunately, that's impossible. It would take too long and it wouldn't
actually improve understanding in many cases. However, I do agree that a
strong effort should be made to combat the notion that mathematical truth
is determined by authority, and that *some* proofs should be done. In the
class I'm teaching now I try to make it clear when I make a statement that
requires proof but whose proof I am going to omit.
Herman Rubin
>>mathematicians never do in their teaching, showing students that many
>>proofs are not as creative as students assume but instead follow a basic
>>template that can be learned. I have tried to show them that the
>>perenniel question "How do you start?" is often not difficult to
>>answer. And yet at the same time I have stressed that they shouldn't get
>>discouraged, because things that seem very simple in retrospect are very
>>often arrived at, even by good thinkers, only after much futile trial
>>and error.
> As a high school student who has taken but three years of mathematics, and
>has not done a good deal of proofs in his life, I can tell you first-hand I
>never know where to start! I think I've finally realized that the key is to
>play around with what you're trying to prove by turning subscripts and Greek
>letters into their definitions, and then play with those using simple algebra,
>moving them around until you get what you want to prove.
You are quite perceptive. When one gets more experience, one develops
intuition about what things are likely to word. And when there is even
more experience, one realizes that this, or any other kind of intuition,
can result in following blind alleys.
BTW, this applies to all branches of mathematics. When one is asked to
prove a theorem, one is given the information that a proof exists.
Often the proof can be discerned by looking at special cases; sometimes
this is the worst that one can do. Theorms and proofs are not found
by producing directly the polished forms in the textbooks.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hru...@stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Paul N. Hilfinger
> I have taken a lot of time in class doing things that I believe most
> mathematicians never do in their teaching, showing students that many
> proofs are not as creative as students assume but instead follow a basic
> template that can be learned. I have tried to show them that the
> perenniel question "How do you start?" is often not difficult to
> answer. And yet at the same time I have stressed that they shouldn't get
> discouraged, because things that seem very simple in retrospect are very
> often arrived at, even by good thinkers, only after much futile trial
> and error.
The discussion reminds me of an interesting book:
Gasteren, A. J. M. (Antonetta J. M.) van, 1952-
On the shape of mathematical arguments
Berlin ; New York : Springer-Verlag, c1990.
Series title: Lecture notes in computer science ; 445.
(Van Gasteren was a student of Edsger Dijkstra, who wrote the forward
to this book). She has some interesting things to say about the
importance of notation, of how theorems are formulated, and of how the
form (shape) of an assertion can guide its proof ("suggest how to
start"). Like much of Dijkstra's work, it can both intrigue and
infuriate.
Paul Hilfinger
Gerhard Niklasch
In article <1994Nov25.1...@galois.mit.edu>,
|> where you make up a bunch of nonsense words and ask them to prove something
|> that is stated in terms of the nonsense words. The subject matter can be
|> taken from everyday life, i.e., the reasoning should be trivial once the
Raymond Smullyan's books are good at precisely this.
(It takes a bit of patience and discipline on the reader's part, though,
to use them in such a way that you actually learn something. This is
why I wouldn't recommend them as compulsory accompanying literature for
our undergraduates...;-)
Enjoy, Gerhard
--
+------------------------------------+----------------------------------------+
| Gerhard Niklasch | All opinions are mine --- I even doubt |
| <ni...@mathematik.tu-muenchen.de> | whether this Institute HAS opinions:-] |
+------------------------------------+----------------------------------------+
Craig John Douglas Webster
Herman Rubin <hru...@b.stat.purdue.edu> wrote:
>In article <stardate.7...@dorsai.org>,
>Mike Russo <star...@dorsai.org> wrote:
>> As a high school student who has taken but three years of mathematics, and
>>has not done a good deal of proofs in his life, I can tell you first-hand I
>>never know where to start! I think I've finally realized that the key is to
>>play around with what you're trying to prove by turning subscripts and Greek
>>letters into their definitions, and then play with those using simple algebra,
>>moving them around until you get what you want to prove.
>You are quite perceptive. When one gets more experience, one develops
>intuition about what things are likely to word. And when there is even
>more experience, one realizes that this, or any other kind of intuition,
>can result in following blind alleys.
It is true that one can often obtain a proof this way, but I always feel
cheated, somehow. Generally I tend to try and convince myself that the
result *is* true, and then ask myself what exactly about my assumptions
*makes* it true, and then try to formalise that. Of course, my little
brain often can't manage ... .. :(
--
Craig "Lemming" Webster | (609)258-9877 | "Why deny the
94 Blair Hall, Princeton U. | cweb...@princeton.edu | obvious, child?"
Princeton, NJ 08544, USA | N8LDA | -- Paul Simon
Dana Albrecht
In article 99...@galois.mit.edu, tyc...@math.mit.edu (Timothy Y. Chow) writes:
> In article <stardate.7...@dorsai.org>,
> Mike Russo <star...@dorsai.org> wrote:
> >If the teacher said it, it's true. If the textbook says it, it's true. That's
> >why I wish teachers (especially in high-school level mathematics) would PROVE
> >EVERYTHING THEY SAY.
> ^^^^^^^^^^
>
> Unfortunately, that's impossible. It would take too long and it wouldn't
> actually improve understanding in many cases. However, I do agree that a
> strong effort should be made to combat the notion that mathematical truth
> is determined by authority, and that *some* proofs should be done. In the
> class I'm teaching now I try to make it clear when I make a statement that
> requires proof but whose proof I am going to omit.
I would be happy if teachers (especially in high school) were CAPABLE of proving
everything they say, proved some of the more important statements, and were
willing to furnish proofs of others outside of class time to interested students.
I would also be much happier if they fostered an environment where students
were encouraged to demand proofs of statements that ought to require them.
Dana W. Albrecht
d...@mirage.svl.trw.com
Gordon McLean Jr.
: >I have taken a lot of time in class doing things that I believe most
: >mathematicians never do in their teaching, showing students that many
: >proofs are not as creative as students assume but instead follow a basic
: >template that can be learned. I have tried to show them that the
: >perenniel question "How do you start?" is often not difficult to
: >answer. And yet at the same time I have stressed that they shouldn't get
: >discouraged, because things that seem very simple in retrospect are very
: >often arrived at, even by good thinkers, only after much futile trial
: >and error.
: As a high school student who has taken but three years of mathematics, and
: has not done a good deal of proofs in his life, I can tell you first-hand I
: never know where to start! I think I've finally realized that the key is to
: play around with what you're trying to prove by turning subscripts and Greek
: letters into their definitions, and then play with those using simple algebra,
: moving them around until you get what you want to prove.
<snip>
: If the teacher said it, it's true. If the textbook says it, it's true. That's
: why I wish teachers (especially in high-school level mathematics) would PROVE
: EVERYTHING THEY SAY.
Something better to wish for, is that at some point your teachers will
have given you enough tools so that *you* can prove anything that they
say!
When this happens, it is very heady stuff indeed. You feel so, well,
"empowered".
For me, this did not really come until some time during college
calculus, possibly even "advanced calculus" (early "real analysis").
I began to notice that assertions which, in high school, the teacher
would preface with, "Someday they will teach you that ... is true, but
for now we'll just accept it", I could actually prove to myself, in as
much gory detail as I desired. And not just stuff that had been
"accepted without proof" in high school. By the time you're taking
(college) calculus, you have stopped being told that you have to
accept stuff without proof because the proof would involve machinery
that is "beyond the scope of this course". You finally have enough
tools at hand to prove just about everything they tell you, whether or
not the teacher (or the textbook) takes the time to actually do the
proof in any given instance.
I was so drunk with this newfound ability in college that at one point
a math professor had to take me aside and tell me that the problem
sets I was turning in were just too damn long! I was proving
everything down to the nth level of detail, like a child who has
learned how to add and just can't stop adding numbers because it's so
exciting to be able to do it. He said that part of "mathematical
maturity" is knowing *when* it's important to prove some assertion
you've made, as opposed to just asserting it with some remark like "It
is clear that so-and-so". Otherwise the simplest theorems have
unbearably long proofs. You have to know when something "needs
proving" and when it's so clear that the details can be "left to the
reader".
My main message is twofold: (1) do not lose your desire to see things
proved, but (2) realize that it will not be until you have reached a
certain level in your education that you will stop hearing the phrase
"The proof of this is beyond the scope of this course".
Susan Schwartz Wildstrom
It is my belief that students frequently cannot appreciate the power of
mathematical induction until the third or fourth time that they are
exposed to it. But, I contend, that at some point, the understanding
comes like a bolt of lightning! I am usually the one who teaches them
induction for the first time, so I spend a lot of time looking at
beautifully constructed convincing proofs that my students have written,
but that they don't really "buy into." It's fun to be around when the
light bulb goes on.
Susan Schwartz Wildstrom
Lee Lady
opportunity to say that I'm sorry about all the mail I haven't been able
to answer.
In article <1994Nov25.1...@galois.mit.edu> tyc...@math.mit.edu
(Timothy Y. Chow) writes:
>In article <CzrL8...@news.hawaii.edu>,
>Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:
>>There are a few fundamental hang-ups I can identify, but I don't know how
>>to get students past them. One is that students don't realize the
>>importance of knowing the definitions.
>
>I don't know if this will help, but perhaps you can try assigning problems
>where you make up a bunch of nonsense words and ask them to prove something
>that is stated in terms of the nonsense words. The subject matter can be
>taken from everyday life, i.e., the reasoning should be trivial once the
>definitions are entangled. This might circumvent the problem suggested by
>Bhalchandra Thatte that the students don't have enough background.
It seems to me that this is actually a good description of what already
exists in many mathematics courses and is precisely one of the things
that makes mathematics, as done in the end of the 20th century,
inaccessible to almost everyone except the elite who have graduate
training in the subject. There are lots of students in physics and other
sciences who would really like to know about mathematical subjects such
as differential geometry, but after one week in a graduate math class
they disappear because they simply can' t cope with the rather arcane
style of communication we use.
We have reduced a lot of mathematics to elaborate word games. If one
takes the typical proof in linear algebra that my students have so much
trouble with, usually by the time you unwrap the elaborate layers of
definitions, what's left --- the real mathematical content --- is some
almost trivial equation or other statement.
Now as a mathematician, I know that there is value in this style of
communication in terms of the increased generality. There's an elegance
to the fact that using the concept of linear transformations one can
prove theorems which apply both to linear differential equations and the
solutions of systems of linear equations. But when I see the
difficulties which statements at this level of abstraction cause my
students, I wonder whether it would really be that great an evil to
simply prove (what I see as) the same theorem twice in two different
courses, stated in much more concrete terms.
I have a hard time justifying to my students that there is real value in
all the word games I am requiring them to play.
For instance, a statement such as "The kernel of a linear transformation
is a subspace" is totally impenetrable to most of my students. And when
I explain to them that this is simply another way of saying that the set
of solutions to a homogeneous linear system is closed under addition and
multiplication by scalars, they look at me as if I'm trying to recruit
them into the looney bin and convince them that that's normality. I can
see them thinking, "If that's what you mean, then why don't you just say
that? Is mathematics just an exercise in stating things in as obscure a
way as possible?"
For my students, even the statement "The set of solutions to a
homogeneous linear system is closed under addition" is extremely
abstract. When I ask a student, "What does it mean to be a solution to
the system?" I get a blank look in return, even though certainly
on some pre-verbal level the student does know. And the phrase
"closed under addition" just causes her eyes to glaze over. For me, it's
a major success if I can get the student to figure out for herself,
without my flat-out just telling her, that the statement in question
simply says that "If Ax = 0 and Ay = 0 then A(x+y) = 0," and thus
realize that the real mathematical content of the statement that's
causing her so much pain is virtually trivial.
The truth of the matter is that in some ways the courses I most enjoy
teaching and get the most satisfaction from are courses for liberal arts
majors (trying to satisfy their core requirement without taking calculus)
and for prospective elementary teachers (most of whom are terrified of
mathematics). In these courses I can go slowly and really talk about
ideas and try to give students some real understanding instead of just
teaching manipulations, whether of equations or of words.
When I first started teaching these courses for non-math majors, there
were a lot of ideas that I consider very exciting that I tried to
present, and I did it in the way I'd been taught to do mathematics,
presenting proofs full of lots of little symbols. But eventually I
started realizing that my communication didn't make much sense to my
students.
Finally I learned that, instead of doing what I would call a "proof," I
could present the same logic in terms of a particular example, always
emphasizing that there's a difference between merely seeing that an
example works because the calculation gives the right answer and being
able to see the *logic* that makes that example work, so that it's really
convincing that in fact every example would work the same way.
The last time I taught a class for prospective elementary school
teachers, for instance, I actually got them through the proof that the
period of the decimal expansion for 1/p, with p prime, is a divisor
of p-1, explaining all the logic in terms of concrete examples. They
had a hard time with it, of course, but some of them were able to
following the reasoning all the way through and really thought it was
neat. (Unfortunately, here at Hawaii there are technical considerations
that prevent me from teaching such courses any more.)
And to some extent I still do the same thing in courses like linear
algebra. But in these cases, I'm faced with a dilemma. If I present
reasoning in an informal way, then more students will be able to follow
it. But it's harder for them to take good notes so that they can go home
and still remember the reasoning. And --- probably a more important
consideration --- if I explain things informally in class, then I'm not
giving them good models to follow when they have to write proofs
themselves.
Ideally, of course, I would first explain ideas informally in terms of
concrete examples and then show how to formalize this reasoning by
writing an abstract proof. And, in fact, I do this some. But the pace
of a college course didn't doesn't give time for much of this. Already,
I'm acutely embarrassed by how little material I'm covering in my linear
algebra course --- well below the acceptable minimum. I've skipped inner
product spaces completely and am reduced to giving them a one-day quick
expository lecture on determinants, a three-day quick tour of
eigenvectors and eigenvalues. Orthogonal matrices and unitary matrices
won't even be mentioned. So how can these students claim to know
anything about linear algebra?
There's a real judgement issue here about what to teach at universities
such as UH. One can give an honest course, covering a reasonable amount
of material and simply taking it for granted that students will be able
to follow mathematical reasoning; usually what one requires from students
on tests is considerably below the level of the lectures, so that students
can pass the course, and sometimes even get an A, simply by learning to
do routine calculations even if they don't understand the theory that's
been presented at all. This, in fact, is pretty much what I do in
calculus.
In the extreme, this sort of teaching becomes a kind of intellectual
masturbation for the professor. I've sat through some courses like that
in graduate school, where the professor had a good time presenting some
material he was really fascinated by and the students simply physically
showed up and had very little idea what was being said and all got
A's. In fact, I've occasionally taught that sort of course myself.
The first time I taught the graduate algebra course was pretty much like
that (although the students didn't all get A's).
But that sort of thing doesn't interest me much any more. To me, it's
much more interesting to take a student who seems to be incapable of
learning a certain thing and teaching her how to learn it, how to think.
But it's a constant exercise in frustration. And sometimes I have to ask
myself if this is not just another type of mental masturbation. Because
what I'm doing is trying to force my students to learn something that
they have no real desire to know. And for most of them, I'm not even
sure that it really does have value, because they're never going to go on
to take high level courses where they need the skills I'm attempting to
teach them.
And then once in a while --- rarely --- a really competent student shows
up who would be capable of dealing with an honest course, and in
fact wants one, and I have to cringe at the fact that I'm not giving him
one. When I taught the undergraduate topology course, for instance, I
had a student like this and at the end of the semester he said to me
rather gently, "When you give a proof, you never just give it.
It's like you circle around the periphery of it and then slowly spiral
in, until you finally get to the actual proof. For me, it would be
better if you'd skip all the spiraling and just present the proof."
Well, I never planned to devote my whole life to one interest and I've
been involved in mathematics for much too long now anyway. Legally I can
retire at this point (although not on very favorable financial terms) and
it's time for me to move on to some other activity.
--
The best thing about being an artist, instead of a madman or someone who
writes letters to the editor, is that you get to engage in satisfying
work. --- Anne Lamott, BIRD BY BIRD
Mike Russo
>would preface with, "Someday they will teach you that ... is true, but
>for now we'll just accept it", I could actually prove to myself, in as
>much gory detail as I desired. And not just stuff that had been
>"accepted without proof" in high school. By the time you're taking
>(college) calculus, you have stopped being told that you have to
>accept stuff without proof because the proof would involve machinery
>that is "beyond the scope of this course". You finally have enough
>tools at hand to prove just about everything they tell you, whether or
>not the teacher (or the textbook) takes the time to actually do the
>proof in any given instance.
What I was getting at was the fact that the teacher would, while teaching
constructions, simply tell you what to do with the compass and straightedge,
and leave it at that. A proof of WHY the construction works is not beyond the
scope of the course; to prove that what I just did with the compass actually
DOES bisect an angle requires the addition of a couple of triangles and
triangle congruence theorems we learned in our freshman year. But we never got
those proofs -- the gem was lost.
>My main message is twofold: (1) do not lose your desire to see things
>proved, but (2) realize that it will not be until you have reached a
>certain level in your education that you will stop hearing the phrase
>"The proof of this is beyond the scope of this course".
I hope so! Think I'll even take the 2-period advanced placement calculus next
year... =)
Lee Lady
> .......
>
>The last time I taught a class for prospective elementary school
>teachers, for instance, I actually got them through the proof that the
>period of the decimal expansion for 1/p, with p prime, is a divisor
>of p-1, explaining all the logic in terms of concrete examples. They
>had a hard time with it, of course, but some of them were able to
>following the reasoning all the way through and really thought it was
>neat.
Let me go through this, because it will illustrate my point that often
the elaborate conceptual baggage we carry around in modern mathematics
obscures ideas rather than clarifying them.
It's easy to see that the period of the decimal expansion for 1/p is
nothing except the smallest exponent such that 10^p is congruent to 1
modulo 10. (This depends on the fact that neither 2 nor 5 divide p
but otherwise does not depend on the fact that p is prime.) Now as
mathematicians, we now quickly realize that the fact that the period
divides p-1 is a simple consequence of Fermat's Little Theorem and the
fact that the order of an element in a group divides the order of the
group.
Thus the result seems extremely simple. But when you start thinking of
how to explain the whole thing from scratch to someone who's never heard
of Fermat or a group, you realize that this proof is actually extremely
complicated. I certainly am not about to explain things that way to a
class of prospective elementary school teachers. In particular, I'm
not going to try to give this class any sort of reasoning that involves
cosets.
So instead... look at an example. For instance, 1/13 = .067923...
where the decimal repeats at the point where the ... is given. Thus
the period is 6, which divides 12, and 12 = 13-1. But I'm not
satisfied just to see that the arithmetic works out, I want to know WHY
this happens.
So think about the circular pattern: (067923). (Sorry, I can't put
these numbers in a circle on your computer screen the way I do for my
students.) If we circle through this pattern, we get six different
patterns, all of which give repeating decimals with a denominator of
13. For instance, .792306... = 9/13.
But again, I ask: WHY is this so? Why could .792306... not have
turned out to have a completely different denominator? Why couldn't it
be 10/17, for instance?
Well, if 1/13 = .067923... then it's pretty obvious that 10/13 =
.679230... (Actually, even this is not obvious to my students, but
they can figure out why given time.) And likewise 100/13 =
6.792306... So from this we see that .792306... is just the fractional
part of 100/13, namely (after a quick calculation) 9/13. For
essentially the same reason, .923067... and .230679... and
.306792... will all represent proper fractions with denominators of
13. Thus from the one circular pattern (067923) we get six
different(!) proper fractions with 13 as denominators.
But how many proper fractions exist with 13 as denominator? (A lot
of blank looks, but a few students will see the answer). 12. So there
must be some fractions with a different circular pattern.
For instance, 2/13 = .153846... with a circular pattern (153846). This
pattern also has length 6. Is this just a coincidence that the two
patterns have the same length, or is there a reason for it? (Basically,
I'm now about to show that any two cosets of a subgroup have the same
size.) Well, if we look at the patterns long enough we notice something
interesting. In fact (153846) is twice (067923), if we take into
consideration carrying during the arithmetic. That seems just too
strange to be a coincidence!
Oh! But that's just saying that 2/13 is twice 1/13. So since we're
just doubling the pattern, it's obvious why the new pattern has the same
length. (Actually, this reasoning is flawed, as we will see later when
we consider tripling the pattern for 1/27 = .037...)
USING THE FACT THAT 13 IS A PRIME, it's not that hard to find some
unflawed reasoning showing that the pattern for 1/13 and 2/13 would have
the same length.
Now we can cycle the new pattern (153846) just as before to get six
different fractions a/13 having this circular pattern and these will
all be different than the ones with the old pattern. (Why?) (I'm
repeating the standard proof of Lagrange's Theorem.)
So these two circular patterns give a total of 12 proper fractions with
13 as the denominator. But that's all there are! So there are only two
circular patterns for fractions a/12.
Notice the general principle: If p is a prime, then all the repeating
patterns for the decimal expansions for a/p will have the same length.
Furthermore, the length of the repeating (circular) pattern times the
number of circular patterns equals the total possible number of proper
decimals with denominator p. But this is p-1.
For instance, think about p = 23. Now it would be a drag to actually
compute 1/23, and it's probably too long to fit on my calculator. But
let's ask, just hypothetically, could the period for 1/23 possibly
be 7? (The students all shrug.) Well, think about it, if the length of
the circular pattern for 1/23 is 7, then how many different circular
patterns would there have to be? One certainly wouldn't be enough,
because we need to get 22 fractions a/23 and a circular pattern of
length 7 only produces 6 fractions. Two wouldn't be enough either,
because that would only give 14 fractions, and three would give 21
different fractions, which is just one too few. But four would give 28
fractions, which is too many.
So a period of 7 couldn't work for 23. So what would work? Could five
be okay? (More shrugs from students. Finally, somebody says, "5
couldn't work either because 5 doesn't divide 22.") So I give her a big
smile (reinforcement) and say, "Why would it have to divide 22?"
"Well, because etc. etc," probably rather inarticulate, but I don't want
the answer to come too quickly because I want the other students to have
a chance to figure it out too.
Finally, once every one accepts the fact that the length of the period
times the number of circular patterns equals p-1 and THEREFORE (which
my students see as an absolutely spectacular piece of reasoning) the
length of the pattern has to divide p-1, I ask "Why does p have to
be a prime in order for this to work?" And then we see where the
reasoning breaks down if we take p = 27.
It takes time to go through all this and not all students can follow the
complete chain of reasoning. But I think in the end, my class of
elementary education majors actually understands the result more clearly
than my class in number theory will.
Incidentally, it's much easy to see why if a/n has a decimal period of k,
then n must divide 99...9 = 10^k-1 (assuming that n is prime to 10).
Because consider .1234... for instance, with a period of 4. Now
.1234... is 1234 times .0001..., and it's easy to establish that
.0001... = 1/9999. Therefore .1234... = 1234/9999, and from this we see
that if a/n = .1234... then a/n = 1234/9999 and it follows (fairly
easily) that n must divide 9999 = 10^4-1.
From these two facts, we can now figure out fairly easily the answer to
questions such as: What fractions 1/p have periods of length 2?
(Only 1/11). Of length 3? (1/37). Of length 4? (1/101). Of length 5?
(1/41 and 1/271). Of length 6? (1/7 and 1/13). This is a convenient
thing for a sixth grade teacher to know.
Gordon McLean Jr.
<snip>
: >>>When I was very young, maybe before I
: >>>even took calculus, I was reading some book and the author wanted to
: >>>prove that a certain statement was true for all natural numbers n. He
: >>>showed that it was true for n=1 and that if it was true for a
: >>>particular value of n then it would also be true for n+1. And I
: >>>thought "Wow! What a clever way of proving something!" It was only much
: >>>later that I learned that there was a name for this type of proof:
: >>>mathematical induction. To me, it just seemed like common sense.
: My Statistics teacher just taught the class how to do mathematical
: of it. Half the class didn't know what the heck was going on, and the other
: half, though they knew what the teacher was doing, were still going over in
: their heads whether it would work or not. At first, I thought it was like
: circular logic;
I'm glad you responded to this. I personally think that of all the
proof techniques one encounters in high school, mathematical induction
is the hardest to grok. It's *complicated*! It requires that you
understand how to prove "If X then Y" (see your discussion below), and
also why, when proving that an assertion of the form "If X then Y"
holds for all n, you get to say "Let n be an integer", and why after
proving it for this hypothetical n, you get to say "Since n was
arbitrary, it's true for all n", and why if you've proven A and "If A
then B", you've proven B, and so on and so on.
Basically you have to understand *everything* to be able to do
mathematical induction!
: you mean you're assuming what you want to prove (when you
: assuming statement x holds for all n's) and then using that to prove it?
Be careful! You don't assume your statement holds for *all* n's and
then use that to prove it. You assume it holds for *one* (arbitrary)
n, and then show that this implies it must also hold for that same n
plus 1.
Question: But why do you get to assume it for even *one* n?
The reason is that you're trying to prove an "If X then Y" kind of
thing.
Mathematical induction requires (among other things) that you prove,
"If the statement holds for some given n, then it holds for that same
n, plus 1". As you know, to prove an "If X then Y" sort of thing, you
get to say "Suppose X is true", then use that to deduce Y, and finally
sum up by saying, "Therefore, *if* X, *then* Y". (This last summing up
step is where you have _stop_ assuming X.)
For the case at hand, the X part is, "The statement holds for n".
So of course you see the proof saying, "Suppose the statement holds
for n"!
: Wait
: a second.... is that guy in front of the class doing mumbo-jumbo with that
: piece of chalk tryin' to pull a fast one on us?
: But if you think about it hard enough it DOES make sense, because even if I
: assumed it in order to prove n+1, once I've proved it, I don't have to
: 'assume' for n anymore, since if it works for n+1, it must work for n (if we
: let n=n-1, we have n-1 +1, which is n.)
You're trying hard to justify the step of assuming it for all n's, but
you can't succceed because it's *not* correct to assume it for all
n's, as we discussed above. So the thing you're worried about, you
should be worried about!
: >There are a few fundamental hang-ups I can identify, but I don't know how
: >to get students past them. One is that students don't realize the
: >importance of knowing the definitions. In many calculus courses (and I
: I agree there. The easiest way for me to even start a proof is to break what
: I'm dealing with down into its simpler definitions.
: >again, "When you want to prove 'If X, then Y,' you suppose that X is
: >true and then show that you can prove Y." They absolutely refuse to do
: >this. It clearly makes no sense to them, and even when a few of them
: >eventually agree to turn in correct proofs, it seems clear to me that
: >they are doing it under protest and do not believe that it makes
: >sense.
: Again, it is thought of as circular, because they can't understand the
: statement. They don't understand that I HAVE TO assume X because the statement
: above says 'If X'!
You got it!
: So, if X is true, then if I want to show the above
Timothy Y. Chow
Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:
[Re: math as manipulation of nonsense words]
>It seems to me that this is actually a good description of what already
>exists in many mathematics courses and is precisely one of the things
>that makes mathematics, as done in the end of the 20th century,
>inaccessible to almost everyone except the elite who have graduate
>training in the subject. There are lots of students in physics and other
>sciences who would really like to know about mathematical subjects such
>as differential geometry, but after one week in a graduate math class
>they disappear because they simply can' t cope with the rather arcane
>style of communication we use.
However, I don't believe that the solution is to avoid using the arcane
language entirely; rather, show them that the arcane language, despite
appearances, is nothing to be afraid of. When the students hear you
talk about a kernel, their eyes glaze over because they haven't learned
how to deal with hearing a word that they don't understand. What we
need to do is to get them to understand that learning "higher" math is
a lot like learning a foreign language. If we hear a word in foreign
language that we don't understand, we instinctively seek to find out
what it means. We need to try to cultivate this instinct in students.
And a first step in this direction is to emphasize the word game aspect,
e.g., by stripping away the mathematical content entirely, leaving a
*pure* word game.
>I have a hard time justifying to my students that there is real value in
>all the word games I am requiring them to play.
One way is to treat math terms as abbreviations. Your students hopefully
understand how "x^2 + 2x - 1 = 0" is shorthand for something that would
otherwise take a lot of words to say, and that the shorthand also leads
to new ideas because of its simplicity and clarity. By analogy they may
be able to appreciate how math jargon has similar benefits.
I do appreciate the difficulties, though; trying to motivate all the
abstract verbiage surrounding linear transformations is practically
impossible unless you can show them a tangible payoff (like the one
you mentioned---the ability to transfer the reasoning verbatim to a
new context such as differential equations), and quite often this is
extremely difficult.
Furthermore, as you say, one really needs to deal with concepts as
well as mere manipulation of words, and together these can swallow up
massive amounts of time. This goes back to the debate about the Moore
method; done well, the Moore method can really excite students and
teach them to think mathematically and prove theorems. The cost is
that only a microscopic amount of material can be covered.
Lee Lady
>Mike Russo (star...@dorsai.org) wrote:
><snip>
>
>proof techniques one encounters in high school, mathematical induction
>is the hardest to grok. It's *complicated*! It requires that you
>understand how to prove "If X then Y" (see your discussion below), and
>also why, when proving that an assertion of the form "If X then Y"
>holds for all n, you get to say "Let n be an integer", and why after
>proving it for this hypothetical n, you get to say "Since n was
>arbitrary, it's true for all n", and why if you've proven A and "If A
>then B", you've proven B, and so on and so on.
I think that the first time I saw a proof by induction, which was in a
quite old-fashioned book, the author, without ever using the using the
word induction, proved the theorem for n=1 and then said, "Now suppose
that we have already proved the theorem for a certain integer n." This
made complete sense to me.
Although I could not have articulated it at the time, I think what was so
marvelous to me about that proof is that one proved the theorem by
essentially backing off to a meta-position. Essentially what one was
doing was proving that a proof existed by giving an algorithm for
constructing the proof for any given n. Later, when I studied logic and
model theory, I encountered other examples of this sort of "meta-proof"
and they always seemed delightful to me.
After many years of teaching, I finally started using this phrase "Now
suppose that we have already proved the theorem..." with my own students.
This seems to make more sense to them than saying "Now suppose the
theorem is true for a certain n."
I think the course where I was most successful in teaching induction was
Discrete Mathematics. In that course, the students had already learned
about recursive algorithms and I said, "A proof by induction is just a
type of recursive algorithm, except that instead of computing a numerical
value you're computing the truth value of a [quantified] statement."
Of course from a rigorous point of view this is circular, since one needs
induction in order to prove the validity of recursive algorithms.
John S. McGowan
> >Mike Russo (star...@dorsai.org) wrote:
> ><snip>
> >
> >I'm glad you responded to this. I personally think that of all the
> >proof techniques one encounters in high school, mathematical induction
> >is the hardest to grok.
Does it help to rewrite it in terms of well ordering? "If it is not
alwayss true, let n be the smallest integer for which it is false... is
n=1? ANS=NO (separate proof for n=1). Then it is true for n-1... and
then...(induction) it is true for n... contradiction"
Regards,
--
John S. McGowan | jmcg...@bigcat.missouri.edu [COIN] (preferred)
| j.mcg...@genie.geis.com [GEnie]
| jom...@eis.calstate.edu [CORE]
----------------------------------------------------------------------
Gordon McLean Jr.
: >I began to notice that assertions which, in high school, the teacher
: >would preface with, "Someday they will teach you that ... is true, but
: >for now we'll just accept it", I could actually prove to myself, in as
: >much gory detail as I desired. And not just stuff that had been
: >"accepted without proof" in high school. By the time you're taking
: >(college) calculus, you have stopped being told that you have to
: >accept stuff without proof because the proof would involve machinery
: >that is "beyond the scope of this course". You finally have enough
: >tools at hand to prove just about everything they tell you, whether or
: >not the teacher (or the textbook) takes the time to actually do the
: >proof in any given instance.
: What I was getting at was the fact that the teacher would, while teaching
: constructions, simply tell you what to do with the compass and straightedge,
: and leave it at that. A proof of WHY the construction works is not beyond the
: scope of the course; to prove that what I just did with the compass actually
: DOES bisect an angle requires the addition of a couple of triangles and
: triangle congruence theorems we learned in our freshman year. But we never got
: those proofs -- the gem was lost.
Well, it sounds like you're already doing what I was sort of getting
at, namely proving things for yourself. You may also be trying to say
that in your opinion the teacher (or textbook) should have been
proving it for *everyone*, because it was so important.
Bear in mind, however, that there will always be things you want to
prove to yourself, that the teacher/textbook will *rightly* (and
perhaps implicitly) "leave to the reader". I agree that your current
example is not perhaps one of these, that a geometry teacher should be
saying *why* the mysterious construction works, if only to get
students thinking that by default there should be no mystery/magic,
that unless something is really clear without further elaboration, a
proof is in order (at some level).
: >My main message is twofold: (1) do not lose your desire to see things
: >proved, but (2) realize that it will not be until you have reached a
: >certain level in your education that you will stop hearing the phrase
: >"The proof of this is beyond the scope of this course".
: I hope so! Think I'll even take the 2-period advanced placement calculus next
: year... =)
Sounds exciting. Calculus is one of the most beautiful things ever
invented/discovered. I hope you will find it as dazzling as I did. I
can't resist a word of warning however. I'm a little suspicious of
high school calculus courses. I've seen students take "lightweight"
calculus in high school, and then when they get to college, they try
to place out of calculus because "I already had that stuff in high
school". The net effect is that they never see full-bore *rigorous*
calculus, with no hand-waving.
By all means take high school calculus. But then take "real" calculus
in college, perhaps in an honors track. That way by the time they're
slinging epsilons and deltas and Riemann sums at you in college,
you'll have enough familiarity with the subject matter to appreciate
what all the rigor is for.
: *** Mike Russo, living in luxury in fabulous... Brooklyn, New York!! ***
Herman Rubin
>Mike Russo (star...@dorsai.org) wrote:
>: >I began to notice that assertions which, in high school, the teacher
>: >would preface with, "Someday they will teach you that ... is true, but
>: >for now we'll just accept it", I could actually prove to myself, in as
>: >much gory detail as I desired.
There are some places where this approach is justified, but few. Most
"mathematics" courses, at all except the higher levels, are cookbook
courses, without understanding. Even where things are not proved, the
ideas should be presented, and an indication of WHY.
Quite a few high schools do not even have a course where proofs are
of any importance. The teachers themselves often see the theorems
as items to be memorized, and do not themselves have the understanding
to see the logic behind them.
It is not so much teaching students to prove theorems, but suggesting
it, encouraging it, and testing it. One can teach what a proof is, but
one cannot teach how to prove. At best, a few hints can be given.
It is not true that "geometric intuition" is the only way to find proofs.
Although this was widely stated when I was a student, I was quite aware
that I had algebraic and logical intuition, and could use them better in
very many cases.
........................
>Sounds exciting. Calculus is one of the most beautiful things ever
>invented/discovered. I hope you will find it as dazzling as I did. I
>can't resist a word of warning however. I'm a little suspicious of
>high school calculus courses. I've seen students take "lightweight"
>calculus in high school, and then when they get to college, they try
>to place out of calculus because "I already had that stuff in high
>school". The net effect is that they never see full-bore *rigorous*
>calculus, with no hand-waving.
Do not assume that college calculus is any better. There are few
schools which provide rigorous calculus below the junior level,
and many do not even do it then.
>By all means take high school calculus. But then take "real" calculus
>in college, perhaps in an honors track. That way by the time they're
>slinging epsilons and deltas and Riemann sums at you in college,
>you'll have enough familiarity with the subject matter to appreciate
>what all the rigor is for.
I would suggest that, if you take high school calculus, or the usual
college calculus, that you look at it as a cookbook course with little
mathematical content. You would be better served by learning what
proofs are, taking an abstract algebra course with as little before
as you can, then a rigorous analysis course, and then practicing
calculus and linear algebra manipulations. I personally would teach
the abstract mathematics in high school, and let the student pick up
the manipulations; there is no unlearning to be done this way, and
any student who cannot handle the rigorous mathematics early is very
unlikely to be able to get it after more years of manipulation.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hru...@stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!a.stat!hrubin(UUCP)
Herman Rubin
Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:
>In article <D074z...@atria.com> gor...@atria.com (Gordon McLean Jr.) writes:
>>Mike Russo (star...@dorsai.org) wrote:
>><snip>
>>I'm glad you responded to this. I personally think that of all the
>>proof techniques one encounters in high school, mathematical induction
>>is the hardest to grok. It's *complicated*! It requires that you
>>understand how to prove "If X then Y" (see your discussion below), and
>>also why, when proving that an assertion of the form "If X then Y"
>>holds for all n, you get to say "Let n be an integer", and why after
>>proving it for this hypothetical n, you get to say "Since n was
>>arbitrary, it's true for all n", and why if you've proven A and "If A
>>then B", you've proven B, and so on and so on.
>I think that the first time I saw a proof by induction, which was in a
>quite old-fashioned book, the author, without ever using the using the
>word induction, proved the theorem for n=1 and then said, "Now suppose
>that we have already proved the theorem for a certain integer n." This
>made complete sense to me.
..............................
I would not be surprised if both your books were almost designed to
confuse. JUSTIFYING proof by induction in a formal system can be a
little difficult, but proofs, and calculations, by induction, are
nothing more than the "domino theory", and should be made that simple.
I had it the really old-fashioned way; the students were expected to
have learned what a proof is from a "Euclid"-type geometry course, and
few mathematicians had done much more on foundations at the time. But
it was expected that any student going on to calculus would be able to
use induction.
I believe that arguments by induction can, and should, be used in the
primary grades. It seems much harder to teach it later.
Gordon McLean Jr.
: In article <D074z...@atria.com> gor...@atria.com (Gordon McLean Jr.) writes:
: >Mike Russo (star...@dorsai.org) wrote:
: ><snip>
: >
: >I'm glad you responded to this. I personally think that of all the
: >proof techniques one encounters in high school, mathematical induction
: >is the hardest to grok. It's *complicated*! It requires that you
: >understand how to prove "If X then Y" (see your discussion below), and
: >also why, when proving that an assertion of the form "If X then Y"
: >holds for all n, you get to say "Let n be an integer", and why after
: >proving it for this hypothetical n, you get to say "Since n was
: >arbitrary, it's true for all n", and why if you've proven A and "If A
: >then B", you've proven B, and so on and so on.
: I think that the first time I saw a proof by induction, which was in a
: quite old-fashioned book, the author, without ever using the using the
: word induction, proved the theorem for n=1 and then said, "Now suppose
: that we have already proved the theorem for a certain integer n." This
: made complete sense to me.
I'm a little worried by this.
How would you feel about the following analogous heuristic?
"When trying to prove a statement of the form 'If X then Y', start by
saying, 'Suppose we have already proved X'".
This heuristic seems misleading. Suppose S is the set of axioms that
can be used in a proof of "If X then Y". Then the heuristic actually
proves something like "If S |- X, then S |- Y", i.e. "If X is a
theorem, then Y is a theorem". This is very different from proving
"S |- 'If X then Y'", i.e. "'If X then Y' is a theorem".
As an example, let S be Peano Arithmetic, let Y be "0 = 1", and let X
be some formally undecidable sentence of PA, e.g. the Goedel sentence
for PA. In this case X is not a theorem of S, hence if X is a
theorem, Y is a theorem. But 'If X then Y' cannot be a theorem,
because then, since "not Y" is a theorem of S, S could prove "not X".
And this it cannot do, X being formally undecidable.
So (meta)proving "If X is a theorem, then Y is a theorem" doesn't
prove "'If X then Y' is a theorem".
Now in your case of mathematical induction, X and Y are not (closed)
sentences; they usually have at least one free variable, namely "n".
And if X has a free variable, it's a confusion to talk of having
"already proved the theorem [i.e. X] for a certain integer n". You
can't prove an open sentence, unless you implicitly mean the universal
closure of the sentence, which is certainly not what's intended here.
I think you will agree that by saying, "Suppose that we have already
proved the theorem for a certain integer n", all one is really saying
is, "Suppose X" (where one fills in the X). If one is in fact saying
*more* than this, then it seems one is confused.
It's my opinion that the circumlocution you cited is really the result
of an attempt to save space. For example, suppose X is the formula
"the arctangent of n is a snarblatted Ferengi module over a
semi-closed pseudo-field of ADICS having charteristic 0". The
statement of the theorem would be, "For all natural numbers n, the
arctangent of n is ...". Now when it comes time for the induction
step in the body of the proof, it is much simpler just to say,
"Suppose the theorem is true for n", than it is to actually repeat X,
which you already took up half the page stating in the first place,
viz. "Suppose that the arctangent of n ... ." And of course one
typically sums up the inductive step with yet another cicumlocution to
save space, e.g. "So if the theorem is true for n, it is true for n +
1", to avoid expanding X with n + 1 substituted for n.
: Although I could not have articulated it at the time, I think what was so
: marvelous to me about that proof is that one proved the theorem by
: essentially backing off to a meta-position. Essentially what one was
: doing was proving that a proof existed by giving an algorithm for
: constructing the proof for any given n.
This worries me severely! One does not in mathematical induction give
a proof for constructing a proof for any given n, as this would be
fallacious!
It may well be that one can prove X(n) for each definite value of n,
i.e. one can prove X(0), and X(1), and X(2), and X(3), and so on, but
still not be able to prove "For all n, X(n)". There are formulas X(n)
such that "For all n, X(n)" does *not* follow from the fact that X(0),
X(1), X(2), X(3), ... . For such formulas, there are interpretations
of the language of the theory, under which X(0), X(1), X(2), X(3),
.. are all true, but "For all n, X(n)" is false. (Theories that
admit such strange examples are called "omega incomplete". Any
consistent axiomatizable theory containing elementary arithmetic
admits such examples.)
Note that it may well be possible to give an algorithm that will
produce a proof of X(n) for each value of n. The existence of this
algorithm does not constitute a proof of "For all n, X(n)", as we have
just seen.
Unfortunately, I think this is a good example of the kind of confusion
that can result in doing "proof by metaproof".
Certainly dumb old mathematical induction should not be presented as a
mysterious example of proof by metaproof, with allusions to algorithms
for constructing proofs and so forth.
Unfortunately, if one wishes to stick with the usual deductive rules,
one is forced to introduce the notion of an "axiom schema", so that
one can say from a metaposition, "Any sentence of the form
such-and-such is an axiom", where such-and-such reads like "'If A(0)
and (n)(A(n)->A(n+1)) then (n)A(n)', for some formula A(x)". This
axiom schema may be replaced by different more fundamental axiom
schemas if one is *proving* the principle of mathematical induction
in, say, set theory.
: Later, when I studied logic and
IMRE BOKOR
>I have a hard time justifying to my students that there is real value in
>all the word games I am requiring them to play.
The moment you see mathematics as merely a matter of "word games", you should
seriously consider another vocation.
I am sure most students do not balk at discussing computers. Yet all
*computers* really do is to manipulate formal symbolism. That is truly
but a word-game.
Similarly, long division and the multiplication taught in primary
schools is nothing more than a formal word game. After all, try to
multilpy fivehundredandsixtyseven by fourhundredandseventythree *without*
using formal manipulation of algebraic symbols (i.e. "abstract word game").
If you can do it in less than about twenty hours, you are doing very
well indeed. Of course if you need to check that you haven't erred,
than you will be very busy for a very long time.
d.A.
d.A.
Timothy Y. Chow
IMRE BOKOR <ibo...@metz.une.edu.au> wrote:
>The moment you see mathematics as merely a matter of "word games", you should
>seriously consider another vocation.
I think you miss the point. *Some* math consists of word gaming; math is
chock-full of jargon, and you must possess the skill (that some of us take
for granted) of being able to "chase definitions" or "follow your nose."
Some of his students are not capable of doing this. It seems alien to them
and they think that acquiring this skill is pointless. How do you convince
them otherwise?
IMRE BOKOR
: In article <3cadur$7...@grivel.une.edu.au>,
: IMRE BOKOR <ibo...@metz.une.edu.au> wrote:
: >The moment you see mathematics as merely a matter of "word games", you should
: >seriously consider another vocation.
: I think you miss the point. *Some* math consists of word gaming; math is
: chock-full of jargon,
I disagree. The point of mathematics is frequently to reduce computations
to mere "word-games", so that the actual process of calculating is purely
mechanical. But the *mathematics* of the situation is in the reduction to
the syntactic manipulation. The rest is accountancy, to put it in the
form of a crude caricature.
By "jargon" I presume you are referring to the technical language of
mathematics. That is no more jargon than speaking of "wing-nuts"
or "thrus bearings". The relationship between the concepts lurking
behind the words is the mathematics, not the manip[ulation of the words.
: and you must possess the skill (that some of us take
: for granted) of being able to "chase definitions" or "follow your nose."
: Some of his students are not capable of doing this. It seems alien to them
: and they think that acquiring this skill is pointless. How do you convince
: them otherwise?
Whatever sphere of expertise anyone aspires to master, there is always a
measure of "foot-slogging". A lawyer needs to be able to follow "legal
reasoning", which, at least on the face of it, often seems like nit-picking
about subtle linguistic differences. The same is true of carpentry,
automotiuve mechanics, gardening. One difference is that te difference
between a carburettor and a fuel-injection unit is visible to
anyone, whereas the objects of mathematics are not tangible and visual
in the same way. There is no mathematics without abstraction.
d.A.
Timothy Y. Chow
IMRE BOKOR <ibo...@metz.une.edu.au> wrote:
<Timothy Y. Chow (tyc...@math.mit.edu) wrote:
<
<: I think you miss the point. *Some* math consists of word gaming; math is
<: chock-full of jargon,
<
<I disagree. The point of mathematics is frequently to reduce computations
<to mere "word-games", so that the actual process of calculating is purely
<mechanical. But the *mathematics* of the situation is in the reduction to
<the syntactic manipulation. The rest is accountancy, to put it in the
<form of a crude caricature.
I'm willing to accept your definition of the word "mathematics" for the
sake of argument. The question, rephrased in your language, is how to
teach and motivate students to do accounting. Lady's students don't
know how to do accounting and don't see the point of learning how to
do it. How do you convince them otherwise?
<By "jargon" I presume you are referring to the technical language of
<mathematics. That is no more jargon than speaking of "wing-nuts"
<or "thrus bearings". The relationship between the concepts lurking
<behind the words is the mathematics, not the manip[ulation of the words.
Fine. But this doesn't address the question.
<Whatever sphere of expertise anyone aspires to master, there is always a
<measure of "foot-slogging". A lawyer needs to be able to follow "legal
<reasoning", which, at least on the face of it, often seems like nit-picking
<about subtle linguistic differences. The same is true of carpentry,
<automotiuve mechanics, gardening.
So we both agree that the accounting and foot-slogging is important (even
though you don't call it "mathematics"). So, how do you teach foot-slogging
to students who don't see the point of it? Do we just lecture them on the
indispensability of foot-slogging in general? That's not enough motivation
for the average student.
Herman Rubin
>>I have a hard time justifying to my students that there is real value in
>>all the word games I am requiring them to play.
>The moment you see mathematics as merely a matter of "word games", you should
>seriously consider another vocation.
Not merely, but it is a matter of symbolics. All communication between
different individuals is by the passing of symbols.
>I am sure most students do not balk at discussing computers. Yet all
>*computers* really do is to manipulate formal symbolism. That is truly
>but a word-game.
>Similarly, long division and the multiplication taught in primary
>schools is nothing more than a formal word game. After all, try to
>multilpy fivehundredandsixtyseven by fourhundredandseventythree *without*
>using formal manipulation of algebraic symbols (i.e. "abstract word game").
^^^^^^^^^
>If you can do it in less than about twenty hours, you are doing very
>well indeed. Of course if you need to check that you haven't erred,
>than you will be very busy for a very long time.
This is certainly wrong, as we know that multiplication was routinely
done by the Egyptians at the time of pyramid building, and by many other
cultures, while algebraic symbols seem to have been introduced by
Diophantus around 300 CE.
As to whether it could be done by someone who did not have some
knowledge of some kind of symbols, the statement itself would be
meaningless. Also, I do not know of a culture which would have
wanted to multiply (or even add)which did not already have a
notation for numbers which lended itself to formal manipulation.