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Nov 24, 1994, 12:21:34 AM11/24/94

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In article <3atld6$e...@onramp.arc.nasa.gov>, lama...@viking.arc.nasa.gov

(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?

(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

Nov 24, 1994, 4:04:48 AM11/24/94

to

In email (quoted with permission), John Heron<her...@smtplink.NGC.COM> writes:

>

> 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?

>

> 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

Nov 24, 1994, 10:08:48 PM11/24/94

to

In article <CzrL8...@news.Hawaii.Edu>,

Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:

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

Nov 25, 1994, 12:05:49 PM11/25/94

to

In article <941124051...@macadam.mpce.mq.edu.au>

ge...@macadam.mpce.mq.edu.au (Gerry Myerson) writes:

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.

Nov 25, 1994, 1:45:04 PM11/25/94

to

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.

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

Nov 26, 1994, 7:45:49 AM11/26/94

to

In <1994Nov25.1...@galois.mit.edu> tyc...@math.mit.edu (Timothy

Y. Chow) writes:

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

Nov 27, 1994, 7:39:07 PM11/27/94

to

>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.

>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...> ***

Nov 27, 1994, 10:33:43 PM11/27/94

to

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.

^^^^^^^^^^

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.

Nov 28, 1994, 9:17:21 AM11/28/94

to

>>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.

>>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)

Nov 28, 1994, 2:09:30 PM11/28/94

to

> 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

Nov 29, 1994, 10:23:20 AM11/29/94

to

In article <1994Nov25.1...@galois.mit.edu>,

|> 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 disentangled.|> 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:-] |

+------------------------------------+----------------------------------------+

Nov 28, 1994, 5:30:45 PM11/28/94

to

In article <3bcoph$28...@b.stat.purdue.edu>,

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.

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

Nov 29, 1994, 6:43:03 PM11/29/94

to

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

Nov 30, 1994, 9:19:39 AM11/30/94

to

Mike Russo (star...@dorsai.org) wrote:

: >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.

: >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".

Dec 1, 1994, 1:33:29 PM12/1/94

to

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

Dec 1, 1994, 4:31:55 PM12/1/94

to

I certainly seem to have hit a nerve with my article. I want to take the

opportunity to say that I'm sorry about all the mail I haven't been able

to answer.

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

Dec 1, 1994, 6:55:46 PM12/1/94

to

>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.

>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... =)

Dec 1, 1994, 6:47:00 PM12/1/94

to

In article <D05I...@news.hawaii.edu> I write:

> .......

>

>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.

> .......

>

>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.

Dec 2, 1994, 1:35:32 PM12/2/94

to

Mike Russo (star...@dorsai.org) wrote:

<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.

<snip>

: My Statistics teacher just taught the class how to do mathematical

<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

: 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;

: 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

Dec 2, 1994, 3:53:56 PM12/2/94

to

In article <D05I...@news.hawaii.edu>,

Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:

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.

Dec 2, 1994, 5:22:11 PM12/2/94

to

In article <D074z...@atria.com> gor...@atria.com (Gordon McLean Jr.) writes:

>Mike Russo (star...@dorsai.org) wrote:

><snip>

>

>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.

>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.

Dec 2, 1994, 9:44:54 PM12/2/94

to

> 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.

> >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]

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

Dec 5, 1994, 12:15:02 PM12/5/94

to

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. 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 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!! ***

Dec 5, 1994, 1:28:11 PM12/5/94

to

In article <D0CL9...@atria.com>, Gordon McLean Jr. <gor...@atria.com> wrote:

>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.

>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)

Dec 5, 1994, 1:09:32 PM12/5/94

to

In article <D07FG...@news.hawaii.edu>,

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>

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.

Dec 5, 1994, 1:32:04 PM12/5/94

to

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.

: 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

Dec 9, 1994, 3:16:27 PM12/9/94

to

Lee Lady <la...@uhunix3.uhcc.Hawaii.Edu> wrote:

>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.

Dec 9, 1994, 6:27:46 PM12/9/94

to

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.

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?

Dec 10, 1994, 8:32:16 AM12/10/94

to

Timothy Y. Chow (tyc...@math.mit.edu) wrote:

: 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.

: 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.

Dec 10, 1994, 10:02:59 AM12/10/94