Order of study?
Richard Carr
: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
:
:
Michael Leary
(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
Arturo Magidin
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
Nicolas Bray
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.
Richard Carr
: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
:
Lieven Marchand
> 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
Steve Leibel
(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
Arturo Magidin
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.
John O'Brien
>
> 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.
>
>
> For calculus, get a book that does "epsilons and deltas"- it is better
> 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
>
>
> :equation.
> :
> :Thanks,
> :
> :Mike
> :
> :
Ghosts of departed quantities.
--
John O'Brien
If replying by e-mail, please remove "nospam." from address.
Dave L. Renfro
[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
Arturo Magidin
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
Keith Ramsay
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
Lee Rudolph
>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
maky m.
In article <87kdca$2sf$1...@panix2.panix.com>,
--
-signature-
maky m. atheist #Ln(2)
chair of the eac theist bashing dept
http://members.tripod.com/~mmanch01/
Sent via Deja.com http://www.deja.com/
Before you buy.
David C. Ullrich
> you nailed quite well:)
Except that "enough to make a cat sick" was far too polite.
David C. Ullrich
Lieven Marchand wrote:
> Richard Carr <ca...@math.columbia.edu> writes:
>
> > For calculus, get a book that does "epsilons and deltas"- it is better
> > that way.
>
> I'm probably going to regret asking but what other undergraduate way
> to calculus is there without epsilons and deltas? I know about non
> standard analysis with infinitesimals but I don't suppose you mean
> that.
You're thinking of calculus as a mathematics course. I don't know
about how it was in the old days, or how it is today where you are,
but here it's not what you and I would think of as "mathematics".
If you ask what the word "derivative" means you're lucky if
someone says the definition is "n*x^(n-1)" .
> For comparison, in a mathematically strong curriculum here in Belgium
> 17 year old high school pupils get a first course on 1 variable
> calculus starting with a few chapters on topology (roughly up to
> definition of trace topology, Hausdorff topology and continuity) after
> which the topological definitions get translated into epsilons and
> deltas for the case of real functions.
Wow. Where do I sign up?
David C. Ullrich
Keith Ramsay wrote:
> In article <87fjsp$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.
>
Pertti Lounesto
> 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)".
David C. Ullrich
Pertti Lounesto wrote:
Heh-heh. Yes, things could be worse.
Richard Carr
: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.
Lieven Marchand
> 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".
Lee Rudolph
>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
David C. Ullrich
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.
Richard Carr
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
:
Herman Rubin
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
Herman Rubin
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
Robert Low
>
> 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.
Kevin Foltinek
<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.
Richard Carr
: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]
:
Lieven Marchand
> 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.
Peter Percival
>
> 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
Herman Rubin
<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.