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Feb 3, 2000, 3:00:00 AM2/3/00

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On Fri, 4 Feb 2000, Michael Leary wrote:

:Date: Fri, 04 Feb 2000 03:00:52 GMT

:From: Michael Leary <le...@nwlink.s.p.a.m.com>

:Newsgroups: sci.math

:Subject: Order of study?

:

:I'm setting out to refresh my memory of all the math I've mostly forgotten

:(algebra, calculus), and to learn some new stuff as well (discrete math). I'm

:mostly interested in computer science applications.

:

:I got a recommendation to learn (about 2 semesters worth of) calculus first,

:and then to learn both linear algebra and discrete math at the same time,

:followed by abstract algebra, and then possibly advanced calculus, mathematical

:logic, and/or set theory.

:

:Does this sound about the right order?

:

There is no reason why linear algebra or abstract algebra should come

after calculus- except possibly for the purposes of examples. Logic and/or

set theory could be done at any time. It always confuses me when students

(here) seem to think that courses must be done in a particular sequence

when they don't require knowledge of the other. Certainly do more advanced

calculus after earlier calculus but as for the rest do them in whatever

order you please.

:Any book recommendations? (particularly for calculus -- or for whichever topic

For calculus, get a book that does "epsilons and deltas"- it is better

that way. Unfortunately, to the best of my knowledge, students here

usually don't get the "epsilon-delta" definitions in calculus courses but

only in "Modern analysis" courses etc. I think this is rather sad.

Read many books- go to the library and pick up several. If and when you

find one which is at a suitable level then you could consider buying

it. Otherwise, you'll probably get a whole load of different suggestions

and you probably can't/shouldn't buy them all.

:I ought, IYO, to study first) It's important to me that I get a solid

:conceptual understanding of things, so I can "see how it all fits together" and

Again, I recommend books with epsilons and deltas.

:be able to apply my learning in more creative ways than just solving a given

:equation.

:

:Thanks,

:

:Mike

:

:

Feb 4, 2000, 3:00:00 AM2/4/00

to

I'm setting out to refresh my memory of all the math I've mostly forgotten

(algebra, calculus), and to learn some new stuff as well (discrete math). I'm

mostly interested in computer science applications.

(algebra, calculus), and to learn some new stuff as well (discrete math). I'm

mostly interested in computer science applications.

I got a recommendation to learn (about 2 semesters worth of) calculus first,

and then to learn both linear algebra and discrete math at the same time,

followed by abstract algebra, and then possibly advanced calculus, mathematical

logic, and/or set theory.

Does this sound about the right order?

Any book recommendations? (particularly for calculus -- or for whichever topic

I ought, IYO, to study first) It's important to me that I get a solid

conceptual understanding of things, so I can "see how it all fits together" and

Feb 4, 2000, 3:00:00 AM2/4/00

to

In article <389a3ff8...@news.pacbell.net>,

Michael Leary <le...@nwlink.s.p.a.m.com> wrote:

>I'm setting out to refresh my memory of all the math I've mostly forgotten

>(algebra, calculus), and to learn some new stuff as well (discrete math). I'm

>mostly interested in computer science applications.

>

>I got a recommendation to learn (about 2 semesters worth of) calculus first,

>and then to learn both linear algebra and discrete math at the same time,

>followed by abstract algebra, and then possibly advanced calculus, mathematical

>logic, and/or set theory.

>

>Does this sound about the right order?

Michael Leary <le...@nwlink.s.p.a.m.com> wrote:

>I'm setting out to refresh my memory of all the math I've mostly forgotten

>(algebra, calculus), and to learn some new stuff as well (discrete math). I'm

>mostly interested in computer science applications.

>

>I got a recommendation to learn (about 2 semesters worth of) calculus first,

>and then to learn both linear algebra and discrete math at the same time,

>followed by abstract algebra, and then possibly advanced calculus, mathematical

>logic, and/or set theory.

>

>Does this sound about the right order?

It depends. The reasons why students are usually taught calculus first

has to do with th eneeds of other programs, and to give them a certain

amount of "mathematical maturity". However, there is no dependency in

terms of the math between calculus, linear algebra, discrete math,

math logic and set theory.

If you feel you have been out of the game for a while, going for

calculus first might be a good idea. Linear algebra will be your first

real meeting with abstraction, and can be a shock. Set Thoeory (at the

undergraduate level, anyway) is also a good introeuction to

abstractions and proofs.

I would suggest Calculus and set theory; then linear algebra and

discrete math; then advanced calculus (by which I assume you mean

analysis). Within calculus, you should do 1 variable first, then

mulitple variables. Mathematical logic would be good with advanced

calculus.

>Any book recommendations? (particularly for calculus -- or for whichever topic

>I ought, IYO, to study first) It's important to me that I get a solid

>conceptual understanding of things, so I can "see how it all fits together" and

>be able to apply my learning in more creative ways than just solving a given

>equation.

You won't get that froma book. You will get it as a result of study

and practice.

======================================================================

"It's not denial. I'm just very selective about

what I accept as reality."

--- Calvin ("Calvin and Hobbes")

======================================================================

Arturo Magidin

mag...@math.berkeley.edu

Feb 4, 2000, 3:00:00 AM2/4/00

to Arturo Magidin

On 4 Feb 2000, Arturo Magidin wrote:

> It depends. The reasons why students are usually taught calculus first

> has to do with th eneeds of other programs, and to give them a certain

> amount of "mathematical maturity". However, there is no dependency in

> terms of the math between calculus, linear algebra, discrete math,

> math logic and set theory.

There is no necessary dependency in the material itself but it is very

common for linear algebra to be taught with calculus as a prerequisite,

using differentiation as an example of a linear function(and some more

interesting ways too). Also taking logic before set theory is not

necessary but might be a good idea, I am currently taking a set theory

course and so far I have found my familiarity with formal logic to be very

useful.

Feb 4, 2000, 3:00:00 AM2/4/00

to

On 4 Feb 2000, Lieven Marchand wrote:

:Date: 04 Feb 2000 20:15:22 +0100

:From: Lieven Marchand <m...@bewoner.dma.be>

:Newsgroups: sci.math

:Subject: Re: Order of study?

:

:Richard Carr <ca...@math.columbia.edu> writes:

:

:> For calculus, get a book that does "epsilons and deltas"- it is better

:> that way.

:

Calculus is done here without epsilons and deltas because "it would be too

hard for the students to understand".

:I'm probably going to regret asking but what other undergraduate way

:to calculus is there without epsilons and deltas? I know about non

:standard analysis with infinitesimals but I don't suppose you mean

:that.

:

:For comparison, in a mathematically strong curriculum here in Belgium

:17 year old high school pupils get a first course on 1 variable

:calculus starting with a few chapters on topology (roughly up to

:definition of trace topology, Hausdorff topology and continuity) after

:which the topological definitions get translated into epsilons and

:deltas for the case of real functions.

:

:--

:Lieven Marchand <m...@bewoner.dma.be>

:If there are aliens, they play Go. -- Lasker

:

Feb 4, 2000, 3:00:00 AM2/4/00

to

Richard Carr <ca...@math.columbia.edu> writes:

> For calculus, get a book that does "epsilons and deltas"- it is better

> that way.

I'm probably going to regret asking but what other undergraduate way

Feb 4, 2000, 3:00:00 AM2/4/00

to

In article <389a3ff8...@news.pacbell.net>, le...@nwlink.s.p.a.m.com

(Michael Leary) wrote:

(Michael Leary) wrote:

> I'm setting out to refresh my memory of all the math I've mostly forgotten

> (algebra, calculus), and to learn some new stuff as well (discrete math). I'm

> mostly interested in computer science applications.

>

> I got a recommendation to learn (about 2 semesters worth of) calculus first,

> and then to learn both linear algebra and discrete math at the same time,

> followed by abstract algebra, and then possibly advanced calculus,

mathematical

> logic, and/or set theory.

>

> Does this sound about the right order?

>

That's the standard order. Unfortunately calculus is often taught so

badly that it gets in the way of understanding math. Try to find a book

or class that teaches rigorous calculus, stressing the concepts of limits,

continuity, etc., rather than making you memorize a bunch of formulas like

"pull down the exponent and subtract 1."

In particular, try not to take a calculus class that makes you use a

calculator. Trust me on this, the least important thing in calculus is to

compute some number. What is important to further study of mathematics is

the concepts.

Steve L

Feb 4, 2000, 3:00:00 AM2/4/00

to

In article <Pine.BSF.4.10.100020...@soda.csua.Berkeley.edu>,

Nicolas Bray <br...@soda.csua.Berkeley.edu> wrote:

>

>

>

>On 4 Feb 2000, Arturo Magidin wrote:

>

>> It depends. The reasons why students are usually taught calculus first

>> has to do with th eneeds of other programs, and to give them a certain

>> amount of "mathematical maturity". However, there is no dependency in

>> terms of the math between calculus, linear algebra, discrete math,

>> math logic and set theory.

>

>There is no necessary dependency in the material itself but it is very

>common for linear algebra to be taught with calculus as a prerequisite,

>using differentiation as an example of a linear function(and some more

>interesting ways too).

Nicolas Bray <br...@soda.csua.Berkeley.edu> wrote:

>

>

>

>On 4 Feb 2000, Arturo Magidin wrote:

>

>> It depends. The reasons why students are usually taught calculus first

>> has to do with th eneeds of other programs, and to give them a certain

>> amount of "mathematical maturity". However, there is no dependency in

>> terms of the math between calculus, linear algebra, discrete math,

>> math logic and set theory.

>

>There is no necessary dependency in the material itself but it is very

>common for linear algebra to be taught with calculus as a prerequisite,

>using differentiation as an example of a linear function(and some more

>interesting ways too).

Which falls under "giv[ing] them a certain amount of 'mathematical

maturity'", which includes access to examples.

>Also taking logic before set theory is not

>necessary but might be a good idea, I am currently taking a set theory

>course and so far I have found my familiarity with formal logic to be very

>useful.

Fair enough. Then again, I took set theory well before I took formal

logic, and I found my familiarity with set theory to be very useful

when I was taking the course in formal logic.

Set theory, at the level of, say Halmos's _Naive Set Theory_ does not

require formal logic, andin fact I woudl suspect that a good course on

formal logic would give little to such a set theory course. A more

advanced set theory course would be difficult without atl east some

familiarity with formal logic, of course.

Feb 4, 2000, 3:00:00 AM2/4/00

to

Richard Carr wrote:

>

> On Fri, 4 Feb 2000, Michael Leary wrote:

>

> :Date: Fri, 04 Feb 2000 03:00:52 GMT

> :From: Michael Leary <le...@nwlink.s.p.a.m.com>

> :Newsgroups: sci.math

> :Subject: Order of study?

> :

> :I'm setting out to refresh my memory of all the math I've mostly forgotten>

> On Fri, 4 Feb 2000, Michael Leary wrote:

>

> :Date: Fri, 04 Feb 2000 03:00:52 GMT

> :From: Michael Leary <le...@nwlink.s.p.a.m.com>

> :Newsgroups: sci.math

> :Subject: Order of study?

> :

> :(algebra, calculus), and to learn some new stuff as well (discrete math). I'm

> :mostly interested in computer science applications.

> :

> :I got a recommendation to learn (about 2 semesters worth of) calculus first,

> :and then to learn both linear algebra and discrete math at the same time,

> :followed by abstract algebra, and then possibly advanced calculus, mathematical

> :logic, and/or set theory.

> :

> :Does this sound about the right order?

> :

>

> There is no reason why linear algebra or abstract algebra should come

> after calculus- except possibly for the purposes of examples. Logic and/or

> set theory could be done at any time. It always confuses me when students

> (here) seem to think that courses must be done in a particular sequence

> when they don't require knowledge of the other. Certainly do more advanced

> calculus after earlier calculus but as for the rest do them in whatever

> order you please.

>

> :Any book recommendations? (particularly for calculus -- or for whichever topic>

> There is no reason why linear algebra or abstract algebra should come

> after calculus- except possibly for the purposes of examples. Logic and/or

> set theory could be done at any time. It always confuses me when students

> (here) seem to think that courses must be done in a particular sequence

> when they don't require knowledge of the other. Certainly do more advanced

> calculus after earlier calculus but as for the rest do them in whatever

> order you please.

>

>

> For calculus, get a book that does "epsilons and deltas"- it is better

> that way. Unfortunately, to the best of my knowledge, students here

> usually don't get the "epsilon-delta" definitions in calculus courses but

> only in "Modern analysis" courses etc. I think this is rather sad.

> Read many books- go to the library and pick up several. If and when you

> find one which is at a suitable level then you could consider buying

> it. Otherwise, you'll probably get a whole load of different suggestions

> and you probably can't/shouldn't buy them all.

>

> :I ought, IYO, to study first) It's important to me that I get a solid> usually don't get the "epsilon-delta" definitions in calculus courses but

> only in "Modern analysis" courses etc. I think this is rather sad.

> Read many books- go to the library and pick up several. If and when you

> find one which is at a suitable level then you could consider buying

> it. Otherwise, you'll probably get a whole load of different suggestions

> and you probably can't/shouldn't buy them all.

>

> :conceptual understanding of things, so I can "see how it all fits together" and

>

> Again, I recommend books with epsilons and deltas.

>

> :be able to apply my learning in more creative ways than just solving a given>

> :equation.

> :

> :Thanks,

> :

> :Mike

> :

> :

Ghosts of departed quantities.

--

John O'Brien

If replying by e-mail, please remove "nospam." from address.

Feb 4, 2000, 3:00:00 AM2/4/00

to

Michael Leary <le...@nwlink.s.p.a.m.com>

[sci.math Fri, 04 Feb 2000 03:00:52 GMT]

<http://forum.swarthmore.edu/epigone/sci.math/laflolclix>

[sci.math Fri, 04 Feb 2000 03:00:52 GMT]

<http://forum.swarthmore.edu/epigone/sci.math/laflolclix>

wrote

>I'm setting out to refresh my memory of all the math I've

>mostly forgotten (algebra, calculus), and to learn some new

>stuff as well (discrete math). I'm mostly interested in

>computer science applications.

>

>I got a recommendation to learn (about 2 semesters worth of)

>calculus first, and then to learn both linear algebra and

>discrete math at the same time, followed by abstract algebra,

>and then possibly advanced calculus, mathematical logic,

>and/or set theory.

>

>Does this sound about the right order?

>

>Any book recommendations? (particularly for calculus -- or

>for whichever topic I ought, IYO, to study first) It's

>important to me that I get a solid conceptual understanding

>of things, so I can "see how it all fits together" and be able

>to apply my learning in more creative ways than just solving

>a given equation.

>

>Thanks,

>

>Mike

Given the comments you made in your last paragraph, I'd recommend

that you begin with Spivak and Courant/John as primary texts and

use Courant/Robbins and Hardy for supplementary reading.

Michael Spivak, CALCULUS, 3'rd edition, Publish or Perish, 1994.

[Amazon.com has 16 reviews of this book, with an average rating

of 5 stars (out of a maximum of 5).]

Richard Courant and Fritz John, INTRODUCTION TO CALCULUS AND

ANALYSIS, Volume 1, Springer-Verlag, 1999. [Reprint of the 1965

(I think) edition. Amazon.com has 3 reviews of this book, with

an average rating of 5 stars.]

Richard Courant and Herbert Robbins, WHAT IS MATHEMATICS?,

2'nd edition (revised with the assistance of Ian Stewart), 1996.

[Amazon.com has 7 reviews of this book, with an average rating

of 5 stars.]

G. H. Hardy, A COURSE IN PURE MATHEMATICS, Cambridge Univ.

Press, 1947. [This may be out of print, as I didn't find it at

amazon.com. However, most libraries will have a copy, and you

can find it listed under QA 303 .H24 1947.]

Dave L. Renfro

Feb 5, 2000, 3:00:00 AM2/5/00

to

In article <stevel-0402...@192.168.100.2>,

Steve Leibel <ste...@coastside.net> wrote:

Steve Leibel <ste...@coastside.net> wrote:

[.snip.]

> Unfortunately calculus is often taught so

>badly that it gets in the way of understanding math. Try to find a book

>or class that teaches rigorous calculus, stressing the concepts of limits,

>continuity, etc., rather than making you memorize a bunch of formulas like

>"pull down the exponent and subtract 1."

There's a calculus textbook, long out of print but recently reissued.

It is highly recommended by Martin Gardner, who helped edit the new

version. I've heard very good things about it, but I haven'ta ctually

read it myself, so take this with a grain of salt (unless you trust

Martin Gardner implicitly). The book is:

"Calculus made easy" by Silvanus P. Thompson and Martin Gardner

St. Martin's Press.

Amazon has it for $15.37, at

www.amazon.com/exec/obidos/ASIN/0312185480/qid%3D949709061/002-1368031-6767434

Feb 6, 2000, 3:00:00 AM2/6/00

to

In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

Lieven Marchand <m...@bewoner.dma.be> writes:

|I'm probably going to regret asking but what other undergraduate way

|to calculus is there without epsilons and deltas?

Lieven Marchand <m...@bewoner.dma.be> writes:

|I'm probably going to regret asking but what other undergraduate way

|to calculus is there without epsilons and deltas?

In the United States, many calculus classes (and I am fairly sure

most calculus classes) teach the concept of "limit" giving only

informal descriptions of what it means and giving examples.

I've seen some students who were taught the definition having quite a

bit of difficulty in applying it. I'm not sure why this is. The

quantifier alternation (one of the quantifiers being "for all" and the

other one being "there exists") seems to make it more difficult for

them. I remember tutoring a student who appeared to be completely

stumped by the following question: given an arbitrary epsilon>0, how

can you give me a number between 0 and epsilon? For specific values of

epsilon he could answer the question, but he seemed unprepared for the

idea of dealing with such a question with a parameter in it. It seems

many American students expect not to get anything so un-algorithmic in

their math classes.

It's commonplace for professors to say that applying the definition

is simply too difficult a task to put into most calculus courses. If

anything, the trend seems to be toward NOT teaching that definition,

except to mathematics majors.

I don't know about other people, but *I* always appreciated having

gotten the "real" definition.

Keith Ramsay

Feb 6, 2000, 3:00:00 AM2/6/00

to

kra...@aol.commangled (Keith Ramsay) writes:

>It's commonplace for professors to say that applying the definition

>is simply too difficult a task to put into most calculus courses. If

>anything, the trend seems to be toward NOT teaching that definition,

>except to mathematics majors.

The fact is that applying the definition (of limit) in a calculus

course has inevitable bad effects on the instructor's teaching

evaluations, which in turn lead to bad effects on the instructor's

future employment prospects (if not tenured) or future salary

(if tenured). (There are no doubt *some* colleges and universities

where this is not [yet] the case.)

This situation is, as we say, enough to make a cat sick. But

apparently nothing can be done.

Lee Rudolph

Feb 6, 2000, 3:00:00 AM2/6/00

to

you nailed quite well:)

In article <87kdca$2sf$1...@panix2.panix.com>,

--

-signature-

maky m. atheist #Ln(2)

chair of the eac theist bashing dept

http://members.tripod.com/~mmanch01/

Sent via Deja.com http://www.deja.com/

Before you buy.

Feb 7, 2000, 3:00:00 AM2/7/00

to

"maky m." wrote:

> you nailed quite well:)

Except that "enough to make a cat sick" was far too polite.

Feb 7, 2000, 3:00:00 AM2/7/00

to

Lieven Marchand wrote:

> Richard Carr <ca...@math.columbia.edu> writes:

>

> > For calculus, get a book that does "epsilons and deltas"- it is better

> > that way.

>

> I'm probably going to regret asking but what other undergraduate way

> to calculus is there without epsilons and deltas? I know about non

> standard analysis with infinitesimals but I don't suppose you mean

> that.

You're thinking of calculus as a mathematics course. I don't know

about how it was in the old days, or how it is today where you are,

but here it's not what you and I would think of as "mathematics".

If you ask what the word "derivative" means you're lucky if

someone says the definition is "n*x^(n-1)" .

> For comparison, in a mathematically strong curriculum here in Belgium

> 17 year old high school pupils get a first course on 1 variable

> calculus starting with a few chapters on topology (roughly up to

> definition of trace topology, Hausdorff topology and continuity) after

> which the topological definitions get translated into epsilons and

> deltas for the case of real functions.

Wow. Where do I sign up?

Feb 7, 2000, 3:00:00 AM2/7/00

to

Keith Ramsay wrote:

> In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

> Lieven Marchand <m...@bewoner.dma.be> writes:

> |I'm probably going to regret asking but what other undergraduate way

> |to calculus is there without epsilons and deltas?

>

> In the United States, many calculus classes (and I am fairly sure

> most calculus classes) teach the concept of "limit" giving only

> informal descriptions of what it means and giving examples.

>

> I've seen some students who were taught the definition having quite a

> bit of difficulty in applying it. I'm not sure why this is. The

> quantifier alternation (one of the quantifiers being "for all" and the

> other one being "there exists") seems to make it more difficult for

> them.

The quantifiers is exactly the problem. They can't understand the

definition because they simply do not know what the string

"for every A there is a B such that..." really means. Presumably because

they've never been required to actually make sense of anything

that subtle. So they can't make sense of it, so they complain if

they're expected to, so people give up because they get in trouble

when the kids complain, _so_ they're still not required to

understand it, so they don't...

Which of course means they really can't understand any mathematics

that they read (hence comments about how the book is unreadable,

etc). IMO the most important reason to try to teach them how the

definition actually works is so that they _will_ understand the

significance of "for every A there is a B such that..." in other

contexts. But this is not a very fashionable view - if they still

haven't got straight what the construction means in some other

context we need to develop some interactive multimedia

instructional materials - the idea that basic technical literacy

is a good thing is sort of reactionary.

> I remember tutoring a student who appeared to be completely

> stumped by the following question: given an arbitrary epsilon>0, how

> can you give me a number between 0 and epsilon? For specific values of

> epsilon he could answer the question, but he seemed unprepared for the

> idea of dealing with such a question with a parameter in it. It seems

> many American students expect not to get anything so un-algorithmic in

> their math classes.

>

> It's commonplace for professors to say that applying the definition

> is simply too difficult a task to put into most calculus courses. If

> anything, the trend seems to be toward NOT teaching that definition,

> except to mathematics majors.

>

Feb 7, 2000, 3:00:00 AM2/7/00

to

"David C. Ullrich" wrote:

> You're thinking of calculus as a mathematics course. I don't know

> about how it was in the old days, or how it is today where you are,

> but here it's not what you and I would think of as "mathematics".

>

> If you ask what the word "derivative" means you're lucky if

> someone says the definition is "n*x^(n-1)" .

Well, at least that is better than "x*e^(x-1)".

Feb 7, 2000, 3:00:00 AM2/7/00

to

Pertti Lounesto wrote:

Heh-heh. Yes, things could be worse.

Feb 8, 2000, 3:00:00 AM2/8/00

to

On Mon, 7 Feb 2000, David C. Ullrich wrote:

:Date: Mon, 07 Feb 2000 13:52:23 -0600

:From: David C. Ullrich <ull...@math.okstate.edu>

:Newsgroups: sci.math

:Subject: Re: Order of study?

:

:

:

:

Also, why do people have trouble integrating by substitution with

variables other than u? Too much emphasis on using the same variable all

the time for substituting; it's as though substituting with a variable

like theta or w etc. is illegal.

Similarly, integration by parts. Say, you had to integrate ve^v

(with respect to v) (instead of xe^x with respect to x).

The first question would be "What do I make u and what do I make dv?" (or

possibly dv/dx). You'd get students wanting to put u=v and dv/dx=e^v or

u=e^v and dv/dx=v (even though there is no x in the probelm).

Thus in case 1, we get xve^v-int(xe^v dv/dx) (as they rarely put a dv or a

dx or whatever on the end) =xve^v-int(x e^v e^v)=xve^v-1/2 x^2e^v e^v

(with +C, if you're lucky). There'd even be no attempt to put e^v

e^v=e^{2v}. It is rather unfortunate that the majority (but not, usually,

all) of students are not learning mathematics but rather following

'algorithms' and following them with such rigidity that if different letters

appear they can not cope.

Feb 8, 2000, 3:00:00 AM2/8/00

to

"David C. Ullrich" <ull...@math.okstate.edu> writes:

> Lieven Marchand wrote:

>

> > For comparison, in a mathematically strong curriculum here in Belgium

> > 17 year old high school pupils get a first course on 1 variable

> > calculus starting with a few chapters on topology (roughly up to

> > definition of trace topology, Hausdorff topology and continuity) after

> > which the topological definitions get translated into epsilons and

> > deltas for the case of real functions.

>

> Wow. Where do I sign up?

>

I think you're a bit too old ;-)

That year we also did 3 dimensional synthetic geometry, matrices,

combinatorics and probability theory. And 6 hours a week reading

Virgil and Horace in the original.

The next year the program was integrals, analytic geometry especially

applied to conics and projective geometry.

When you start university after that year, all the stuff you see in

high school gets repeated in the first month at a fairly high speed.

It helps to have a school system that gives courses from 9 to 4 with

practically no time spent on non academic pursuits (1 hour of sports a

week, 1 hour of drawing and that's about it). The general idea was

that you should do your hobbies in your spare time. They also didn't

let the lunatics run the asylum like your professor evaluations

sound. Of course, the system worked so well that the

idiots^Wpoliticians are going to change it and are talking about

cutting down the "modern mathematics" and putting more effort in

"relating the mathematics to real life".

Feb 8, 2000, 3:00:00 AM2/8/00

to

Lieven Marchand <m...@bewoner.dma.be> writes:

>They also didn't

>let the lunatics run the asylum like your professor evaluations

>sound.

Please. We are letting the purchasers run the marketplace. You

have a problem with that, Bud? .be--that's one of them Communist

countries, right?

Lee Rudolph, just another cornflake who's too old and too mean

to develop a yummy sugar frosting

Feb 8, 2000, 3:00:00 AM2/8/00

to

Lieven Marchand wrote:

> [...]

>

> It helps to have a school system that gives courses from 9 to 4 with

> practically no time spent on non academic pursuits (1 hour of sports a

> week, 1 hour of drawing and that's about it). The general idea was

> that you should do your hobbies in your spare time. They also didn't

> let the lunatics run the asylum like your professor evaluations

> sound. Of course, the system worked so well that the

> idiots^Wpoliticians are going to change it and are talking about

> cutting down the "modern mathematics" and putting more effort in

> "relating the mathematics to real life".

Here in Oklahoma (where education has never been all that

popular) there are politicians who are talking about less math

and science period. Even "real-life math" - we don't need so

much of that because most of our students won't be using

math in their careers anyway. If everybody know all that

stuff where will we get people to wait on tables in the

next century?

Not making it up, actual politicians are talking

that way.

Feb 8, 2000, 3:00:00 AM2/8/00

to

On 8 Feb 2000, Lee Rudolph wrote:

:Date: 8 Feb 2000 16:26:12 -0500

:From: Lee Rudolph <lrud...@panix.com>

:Newsgroups: sci.math

:Subject: Re: Order of study?

:

:Lieven Marchand <m...@bewoner.dma.be> writes:

:

:>They also didn't

:>let the lunatics run the asylum like your professor evaluations

:>sound.

:

:Please. We are letting the purchasers run the marketplace. You

:have a problem with that, Bud? .be--that's one of them Communist

:countries, right?

I hope you are kidding.

:

:Lee Rudolph, just another cornflake who's too old and too mean

:

Feb 9, 2000, 3:00:00 AM2/9/00

to

In article <Pine.BSF.4.10.100020...@soda.csua.Berkeley.edu>,

Nicolas Bray <br...@soda.csua.Berkeley.edu> wrote:

Nicolas Bray <br...@soda.csua.Berkeley.edu> wrote:

>On 4 Feb 2000, Arturo Magidin wrote:

>> It depends. The reasons why students are usually taught calculus first

>> has to do with th eneeds of other programs, and to give them a certain

>> amount of "mathematical maturity". However, there is no dependency in

>> terms of the math between calculus, linear algebra, discrete math,

>> math logic and set theory.

>There is no necessary dependency in the material itself but it is very

>common for linear algebra to be taught with calculus as a prerequisite,

>using differentiation as an example of a linear function(and some more

>interesting ways too). Also taking logic before set theory is not

>necessary but might be a good idea, I am currently taking a set theory

>course and so far I have found my familiarity with formal logic to be very

>useful.

The "claimed" reason to have calculus before much of anything at

all is to get "mathematical maturity"; I believe that it only

succeeds in driving students out, and weakening the courses.

In fact, linear algebra, essentially as taught, uses nothing

more than decent high school algebra. But one can ask why

linear algebra is taught before, or as a part of, abstract

algebra? Most of the current abstract algebra courses use

examples from linear algebra, and there are theorems in

algebra, usually not reached in the first course, which use

linear algebra as a model for representations.

Logic is used in all of mathematics. The advantage of a

good formal logic course is that it is efficiently gathered

and clearly stated, instead of being introduced piecemeal,

and sometimes even incorrectly stated. Most of the solution

procedure of high school algebra comes from applying the

one rule, the SAME operation on equal entities yields equal

results. This is often done as several rules for numbers,

but this is unnecessary. However, be sure it is the same

operation; many of the paradoxes come from failing to observe

this restriction.

Full basic mathematical logic is accessible in elementary

school with present materials. Basic formal set theory,

likewise. The more computation done without understanding,

the harder it gets to understand.

--

This address is for information only. I do not claim that these views

are those of the Statistics Department or of Purdue University.

Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399

hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

Feb 9, 2000, 3:00:00 AM2/9/00

to

In article <20000206114850...@nso-fp.aol.com>,

Keith Ramsay <kra...@aol.commangled> wrote:

>In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

>Lieven Marchand <m...@bewoner.dma.be> writes:

>|I'm probably going to regret asking but what other undergraduate way

>|to calculus is there without epsilons and deltas?

Keith Ramsay <kra...@aol.commangled> wrote:

>In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

>Lieven Marchand <m...@bewoner.dma.be> writes:

>|I'm probably going to regret asking but what other undergraduate way

>|to calculus is there without epsilons and deltas?

I would not quite use epsilons and deltas, but neighborhoods.

But this is only a slight modification.

However, it is not even too difficult to develop algorithmic

differential calculus rigorously without any notion of limit.

One can have an operation of differentiation which maps

functions into functions, often with reduced domains, and

satisfies the key properties; this is also called differential

algebra, and is what is used in proving theorems about

integration in closed form.

>In the United States, many calculus classes (and I am fairly sure

>most calculus classes) teach the concept of "limit" giving only

>informal descriptions of what it means and giving examples.

>I've seen some students who were taught the definition having quite a

>bit of difficulty in applying it. I'm not sure why this is. The

>quantifier alternation (one of the quantifiers being "for all" and the

>other one being "there exists") seems to make it more difficult for

>them. I remember tutoring a student who appeared to be completely

>stumped by the following question: given an arbitrary epsilon>0, how

>can you give me a number between 0 and epsilon? For specific values of

>epsilon he could answer the question, but he seemed unprepared for the

>idea of dealing with such a question with a parameter in it. It seems

>many American students expect not to get anything so un-algorithmic in

>their math classes.

Learning algorithms does not give any understanding. Some

realize this and try to develop understanding despite not

being told, but too many do not. Also, the algorithmic

teaching of arithmetic, often with more than 10 times the

number of exercises needed, conveys the impression that this

is all of mathematics. And it continues after that.

The new math was introduced because it was observed that

being good at manipulation did not convey understanding,

but the teachers could only understand manipulation, and

could not then, and cannot now, teach understanding. The

old mathematics curriculum did have the old "Euclid" course,

which at least eliminated those who could not understand

what a proof is, even though that is also manipulative.

Most students do not take such a course now, and many do not

even have access to one, and many of those who do are only

asked to memorize them.

>It's commonplace for professors to say that applying the definition

>is simply too difficult a task to put into most calculus courses. If

>anything, the trend seems to be toward NOT teaching that definition,

>except to mathematics majors.

>I don't know about other people, but *I* always appreciated having

>gotten the "real" definition.

The ones who cannot understand the "real" definition also

cannot understand what a non-terminating decimal means.

The notion of limit is needed to understand that, and is

very definitely not taught.

>Keith Ramsay

Feb 9, 2000, 3:00:00 AM2/9/00

to

Herman Rubin wrote:

>

> In article <20000206114850...@nso-fp.aol.com>,

> Keith Ramsay <kra...@aol.commangled> wrote:

> >In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

> >Lieven Marchand <m...@bewoner.dma.be> writes:

> >|I'm probably going to regret asking but what other undergraduate way

> >|to calculus is there without epsilons and deltas?

>

> I would not quite use epsilons and deltas, but neighborhoods.

> But this is only a slight modification.

>

> In article <20000206114850...@nso-fp.aol.com>,

> Keith Ramsay <kra...@aol.commangled> wrote:

> >In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

> >Lieven Marchand <m...@bewoner.dma.be> writes:

> >|I'm probably going to regret asking but what other undergraduate way

> >|to calculus is there without epsilons and deltas?

>

> I would not quite use epsilons and deltas, but neighborhoods.

> But this is only a slight modification.

Coincidentally, I've just been looking at a first analysis

text, authors Moss and Roberts, which does just this. I don't

have the book handy, so I can't recall the title, I'm afraid.

Continuity is defined via neighbourhoods, and differentiability

via continuity (f is differentiable at a if there is a function

p continuous at a such that f(x)=f(a)+p(x)(x-a); then

f'(a)=p(a)).

Limits are introduced eventually, but it's surprising what you

can cover before doing that.

Feb 9, 2000, 3:00:00 AM2/9/00

to

In article

<Pine.LNX.4.21.000208...@cpw.math.columbia.edu>

Richard Carr <ca...@math.columbia.edu> writes:

<Pine.LNX.4.21.000208...@cpw.math.columbia.edu>

Richard Carr <ca...@math.columbia.edu> writes:

> Also, why do people have trouble integrating by substitution with

> variables other than u? Too much emphasis on using the same variable all

> the time for substituting; it's as though substituting with a variable

> like theta or w etc. is illegal.

One "theory" I recently heard which might address this issue is that

(typical American) students are very good at following instructions,

but that they are not so good at interpreting, or even thinking about,

what they are actually doing. As in, "what does this mean, and why

does it work". So they can easily "let u=x^2+1"; but if the problem

involves "t" rather than "x", they will "let u=x^2+1".

This habit can possibly be "broken" by the teacher using both examples

including "let u=x^2+1" and "let u=t^2+1", but doing so fails to

address the underlying issue (and probably won't help when the problem

involves "s" rather than "t" or "x").

Related to this, my suspicion is that when a student complains (about

an exam question) "there was nothing like this in the homework", their

(operational) definition of "like" means "at most one thing changed".

(You may substitute another small number for "one".) For example,

"f(x)=x^2-3x+1" is like "f(t)=t^2-3t+1" or "f(x)=x^2-2x+1", but not

like "f(t)=t^2-2t+1".

> [snip]

> It is rather unfortunate that the majority (but not, usually,

> all) of students are not learning mathematics but rather following

> 'algorithms' and following them with such rigidity that if different letters

> appear they can not cope.

Yes; presumably (according to the "theory" I mentioned above) this

rigidity comes from a (near-)complete lack of understanding of the

reasoning behind the algorithm, or even why each step is a valid

operation.

The problem originates long before the student enters calculus.

Kevin.

Feb 9, 2000, 3:00:00 AM2/9/00

to

On 9 Feb 2000, Kevin Foltinek wrote:

:Date: 09 Feb 2000 13:24:33 -0600

:From: Kevin Foltinek <folt...@math.utexas.edu>

:Newsgroups: sci.math

:Subject: Re: Order of study?

:

:In article

:<Pine.LNX.4.21.000208...@cpw.math.columbia.edu>

:Richard Carr <ca...@math.columbia.edu> writes:

:

:> Also, why do people have trouble integrating by substitution with

:> variables other than u? Too much emphasis on using the same variable all

:> the time for substituting; it's as though substituting with a variable

:> like theta or w etc. is illegal.

:

:One "theory" I recently heard which might address this issue is that

:(typical American) students are very good at following instructions,

:but that they are not so good at interpreting, or even thinking about,

:what they are actually doing. As in, "what does this mean, and why

:does it work". So they can easily "let u=x^2+1"; but if the problem

:involves "t" rather than "x", they will "let u=x^2+1".

:

Often, they have to be told what substitution to make also.

:This habit can possibly be "broken" by the teacher using both examples

:including "let u=x^2+1" and "let u=t^2+1", but doing so fails to

:address the underlying issue (and probably won't help when the problem

:involves "s" rather than "t" or "x").

:

:Related to this, my suspicion is that when a student complains (about

:an exam question) "there was nothing like this in the homework", their

:(operational) definition of "like" means "at most one thing changed".

:(You may substitute another small number for "one".) For example,

:"f(x)=x^2-3x+1" is like "f(t)=t^2-3t+1" or "f(x)=x^2-2x+1", but not

:like "f(t)=t^2-2t+1".

:

Or that the "exact same" question with the same wording and numbers

etc. has not been done in class.

:> [snip]

:

Feb 9, 2000, 3:00:00 AM2/9/00

to

lrud...@panix.com (Lee Rudolph) writes:

> Lieven Marchand <m...@bewoner.dma.be> writes:

>

> >They also didn't

> >let the lunatics run the asylum like your professor evaluations

> >sound.

>

> Please. We are letting the purchasers run the marketplace. You

> have a problem with that, Bud? .be--that's one of them Communist

> countries, right?

Definitely.

Our education system is totally corrupted by the French grammarians

who have banned visualisation completely and replaced it with abstract

nonsense no average guy needs.

Mar 5, 2000, 3:00:00 AM3/5/00

to

Herman Rubin wrote:

>

> In article <20000206114850...@nso-fp.aol.com>,

> Keith Ramsay <kra...@aol.commangled> wrote:

> >In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

> >Lieven Marchand <m...@bewoner.dma.be> writes:

> >|I'm probably going to regret asking but what other undergraduate way

> >|to calculus is there without epsilons and deltas?

>

> I would not quite use epsilons and deltas, but neighborhoods.

> But this is only a slight modification.

>

<snip>>

> In article <20000206114850...@nso-fp.aol.com>,

> Keith Ramsay <kra...@aol.commangled> wrote:

> >In article <87fjsp$d5s$1...@newnews1.news.nl.uu.net>,

> >Lieven Marchand <m...@bewoner.dma.be> writes:

> >|I'm probably going to regret asking but what other undergraduate way

> >|to calculus is there without epsilons and deltas?

>

> I would not quite use epsilons and deltas, but neighborhoods.

> But this is only a slight modification.

>

How do you feel about H J Keisler's "Elementary Calculus: An Approach

Using Infinitesimals". Has it been used much for teaching?

PP

Mar 5, 2000, 3:00:00 AM3/5/00

to

In article <38C2740E...@cwcom.net>,

<snip>

It has been used for teaching. It is not clear whether

this is better; I do not think that even the usual

treatment of decimal approximation makes sense without

the notion of limits. That non-standard analysis works

is due to the fact that proofs using infinitesimals and

non-standard integers or real numbers can be translated

into standard proofs; they may be easier to grasp. An

integer larger than all standard integers is NOT infinite.

The structure of the real number system and elementary

topological ideas, taught generally, should precede

calculus. I include power series and integration with

respect to arbitrary measures, taught so that at least

most of the generalizations come out easily.

BTW, in teaching topological ideas, even if not much

is done with "pathological" spaces, enough about those

spaces should be introduced so that the student does

not have to unlearn later. Adding conditions is easy;

removing them (generalizing) is hard. And the abstract

ideas are not at all difficult IF the student has not

been brainwashed into computation and rigid intuition.

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