learning calculus through mathematica
j l
I am new to Mathematica and I would like to use it as a way to learn calculus (alongside some books that I have). Does anyone have a good reference page or some suggestions about how to go about doing this?
Sorry for the n00bish question; I'm mostly just trying to teach myself more math.
Thanks
Murray Eisenberg
MathEverywhere. This is a complete calculus course delivered via
Mathematica notebook files (along with some pdf files that are the
equivalent of printed materials).
http://www.matheverywhere.com/mei/courseware/
--
Murray Eisenberg mur...@math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
David Park
undertaking any significant math. Otherwise you will waste a lot of time.
Study most the Core Language Section of Help. (Also some of the items on the
bottom of the page.) Look at the Functional Programming material. It's VERY
useful, especially Function and pure functions. Do a lot of typing of the
examples in your own notebooks, with your own variations.
2) Learn the basic structure of notebooks so you can organize material in
Sections. Learn how to enter Text cells, and even Inline cells within Text
cells, so you can give textual description to your work.
3) Then "fly solo" with your calculus books. Or you might want to first try
even simpler books where you know the math and only have to worry about the
Mathematica. When it gets boring go on to new math material.
4) Get in the habit of writing definitions with arguments for objects that
you use. For example, don't write:
a=3.5
b=6.22
f= a Sin[b x]
Plot[f,{x,0,6}]
Write:
f[a_, b_][x_]:= a Sin[b x]
Plot[f[3.5,6.22][x],{x,0,6}]
Have fun, and ask questions on MathGroup, with specific examples if
possible. There are no such things as silly questions because silly people
never use Mathematica in the first place.
David Park
djm...@comcast.net
http://home.comcast.net/~djmpark/
ingol...@telia.com
http://www.mma-users.org/webMathematica/wiki/wiki.jsp?pageName=Links#_Educat
ional_tools_based_on_Mathematica
Please everyone, add more links there if you know other good links!
Best regard
Ingolf Dahl
Sweden
Canopus56
There a number of college texts of that can be located using Amazon.com and the keyword search "Calculus mathematica". Not suprisingly, one of the uses of Mathematica (and other systems) is in college analytic geometry and calculus courses.
I have:
1) Kevin O'Conner, Calculus Labs for Mathematica (2008). This is a short 120 page phamplet that integrates with the course textbook:
David William Cohen (Author), James M. Henle (Author), Calculus: The Language of Change (2004)
The contents of both can be previewed on Amazon.com.
2) Bruce F. Torrence, The Student's Introduction to MATHEMATICA : A Handbook for Precalculus, Calculus, and Linear Algebra. (2009)
O'Conner was simplier and more direct than Torrence.
Also look at the Wolfram Library Archive for sample notebooks.
http://library.wolfram.com/
Using "calculus" as a keyword, returns an overwhelming 962 cites to books and sample notebooks. Alot of the references are outdated, that is they have deprecated urls or are buit for older versions of Mathematica.
There is also a calculus index on the same Wolfram Library with 182 entries. It is easier to read and review.
http://library.wolfram.com/infocenter/BySubject/Mathematics/CalculusAnalysis/Calculus/
Also courseware - 25 entries
http://library.wolfram.com/infocenter/Courseware/Mathematics/CalculusAnalysis/Calculus/
See also the demos library - 15 entries
http://library.wolfram.com/infocenter/Demos/Mathematics/CalculusAnalysis/
- Kurt
David Bailey
> There are no such things as silly questions because silly people
> never use Mathematica in the first place.
>
>
I think there may be the odd exception to that rule!
David Bailey
http://www.dbaileyconsultancy.co.uk
Murray Eisenberg
for calculus. In short, "add-ons" to a standard calculus course.
MathEverywhere's "Calculus&Mathematica", on the other hand, represents
a rethinking of what calculus is about and how to teach it. And its
presentation THROUGH Mathematica makes it especially amenable to use for
self-study.
Another possibility I didn't mention in my original post is Keith
Stroyan's "Calculus: the Language of Change" which closely integrates
Mathematica into the development and makes substantial use of modeling
as motivation from the very start. (That's insteaad of the usual
calculus text approach of perhaps nominally mentioning some "real-world"
application, then going off doing all the relevant math, and finally
returning to applying that math.) Thus it introduces the derivative in
the context of modeling an epidemic by the SIR model.
--
Richard Fateman
There is a substantial list of links to calculus resources at
http://www.calculus.org/ This includes complete on-line courses.
I have searched in vain for objective evidence that students who learn
calculus with a computer algebra system at hand learn it better than
students without such a tool. (e.g. higher exam grades.)
This is disappointing to people who would like every student to learn
how to use a CAS at the earliest opportunity.
Historically, the big success for calc students was using computers
to plot functions. Handy to understand slopes and areas. Very easy to use.
Not so prone to arithmetic mistakes, though with problems of their own.
Murray Eisenberg
University of Illinois and Ohio State. They should be able to point you
to, or provide, the objective studies they did on performance in their
Calculus & Mathematica program vs. standard courses.
As I recall, one of their methods was to compare grades of the two
groups in subsequent courses.
Unless I recall incorrectly, I posted about that very project to this
group before. Whether or not I did, please do try to follow up on my
suggestion above.
--
David Park
possible age. But maybe those who are seriously interested in a technical
career and are motivated should. Maybe it wouldn't be a part of regular
secondary school education, but be done on their own, or in math clubs, or
via mentoring over the Internet.
Maybe it's true that CAS have not made a significant positive impact in
technical education. Does that mean people should give up? Maybe we haven't
properly learned how to use them yet. When new technologies come in they are
often used to just make the old approaches more efficient. Usually what is
needed is entirely new approaches. Instead of mass lectures and mass exams,
maybe there should be more self study, more mentoring and more mathematical
essay writing. As things stand now I have the sneaky suspicion that students
just don't know Mathematica well enough and it is another obstacle to
getting through the course. So, why should they do better?
Also, Mathematica off the shelf is not a great educational tool. It does too
much at a high level with commands like Solve, Integrate or Limit. That's
all fine, but students need something I call "hierarchical depth", the
ability to do mathematics at different levels and see how things work. It is
somewhat ironic that as a computer algebra system, Mathematica (and I
suspect most systems) are poor at providing the kind of algebraic
manipulations that students need to work with. They are hierarchically thin.
This all could be provided, but it takes more work.
Helen Read
> Certainly not every student should be learning Mathematica at the earliest
> possible age. But maybe those who are seriously interested in a technical
> career and are motivated should. Maybe it wouldn't be a part of regular
> secondary school education, but be done on their own, or in math clubs, or
> via mentoring over the Internet.
>
> Maybe it's true that CAS have not made a significant positive impact in
> technical education. Does that mean people should give up? Maybe we haven't
> properly learned how to use them yet. When new technologies come in they are
> often used to just make the old approaches more efficient. Usually what is
> needed is entirely new approaches. Instead of mass lectures and mass exams,
> maybe there should be more self study, more mentoring and more mathematical
> essay writing. As things stand now I have the sneaky suspicion that students
> just don't know Mathematica well enough and it is another obstacle to
> getting through the course.
I teach my calculus students Mathematica more or less by immersion,
while they are learning calculus. We use it routinely throughout the
course, and every one of them is competent at using it by the middle of
the semester; it really only takes a couple of weeks to get them up and
running. The very weakest students might think of Mathematica as an
obstacle, but even they are mostly OK with it, especially now that we
have the Classroom Assistant palette, which eases the learning curve a
great deal.
I have the good fortune of teaching in a classroom (we have two such
rooms) equipped with a computer for each student, however, as well as .
This means we can use Mathematica any time we want. Often times I will
introduce a topic, do some chalk-and-talk, then give the class some
examples to work on with the aid of Mathematica, and let them discover a
lot of things by doing the examples. Then we'll discuss the examples
afterward, and do some additional chalk-and-talk analysis. Or sometimes
I'll give them pencil-and-paper exercises to work on, and have them
check their work in Mathematica. I know this is all anecdotal, but I do
find a lot of value in this, and I think my students do too.
--
Helen Read
University of Vermont
Andrzej Kozlowski
make it possible to understand and learn better those areas of
mathematics which are fully accessible to a student with only a pen and
paper. In fact I can see a few reasons why the opposite might be the
case. In many situations I can see clear advantages in performing
algebraic manipulations "by hand" or even "in the head", which is, in my
opinion, the only way to develop intuition. The same applies to
visualisation - while being able to look at complicated graphics can
often be a big advantage, I always insist on students developing the
ability to quickly sketch simple graphs by hand on the basis of
qualitative analysis of analytic or algebraic data. This is again
essential for developing intuition and I am not convinced that doing all
this by means of a computer will provide equivalent benefits.
However, I have no doubt that learning to use a CAS should be an
essential element of learning calculus and the reason for that is not
that a CAS enables one to understand better those aspects that have
traditionally not required it but because it opens up completely new
areas for exploration and well as greatly increasing the efficiency of
many traditional computational tasks. In other words, I see using a CAS
as much an integral part of modern calculus as knowing the basic
techniques of differentiation and integration etc. In the case of the
great majority of people who are learning calculus today one can say
that if they ever find themselves using calculus outside their calculus
class they will be using some kind of CAS to do so. The most efficient
way to learn how to use a CAS in calculus is to combine learning to use
the CAS with learning calculus itself. I don't know if students who use
a CAS in their calculus courses are better at the kind of things that
can be done without a CAS than students who learned calculus the
traditional way, but I am sure the former (potentially) could do many
things that the latter couldn't and that it is these kind of things that
most often come up "in the real world".
Andrzej Kozlowski
> getting through the course. So, why should they do better?
>
> Also, Mathematica off the shelf is not a great educational tool. It
does too
> much at a high level with commands like Solve, Integrate or Limit.
That's
> all fine, but students need something I call "hierarchical depth", the
> ability to do mathematics at different levels and see how things work.
It is
> somewhat ironic that as a computer algebra system, Mathematica (and I
> suspect most systems) are poor at providing the kind of algebraic
> manipulations that students need to work with. They are hierarchically
thin.
> This all could be provided, but it takes more work.
>
>
> David Park
> djm...@comcast.net
> http://home.comcast.net/~djmpark/
>
>
>
Richard Fateman
> Try contacting the folks at the Calculus & Mathematica project at
> University of Illinois and Ohio State. They should be able to point you
> to, or provide, the objective studies they did on performance in their
> Calculus & Mathematica program vs. standard courses.
>
> As I recall, one of their methods was to compare grades of the two
> groups in subsequent courses.
>
> Unless I recall incorrectly, I posted about that very project to this
> group before. Whether or not I did, please do try to follow up on my
> suggestion above.
>
[[executive summary: Very thin evidence of sometimes positive results]]
In
http://library.wolfram.com/infocenter/Articles/3227/
a thesis at Southern Illinois Univ,
entitled
The Effects of a Calculus Course Based on the Computer Algebra System
Mathematica on Subsequent Calculus Dependent Undergraduate Course
Performance
A key result is
"The two groups were compared by means of final grades in a physics
course that had calculus or concurrent enrollment as a prerequisite. It
found a nonsignificant difference in the mean grades of the two groups
in this case. "
(there were differences when students learned calculus first and then
mathematica. I think there is a problem with self-selection here, but
hard to say. I could not get the thesis to read.)
Then there is this, from the atelier of Jerry Uhl ..
http://www-cm.math.uiuc.edu/studies
which suggests that Mathematica helps, but looking at more info at
https://cm.math.uiuc.edu/?q=node/30
suggests it helps not much.
See the sales-brochure-style info at
which suggests that people who takes these courses go to excellent
graduates schools like Cal-Berkeley and get jobs at great places like
Wolfram Research.(pardon the snark).
It also suggests that significant differences between the
calculus+mathematica (C+M) vs
calculus-mathematica courses may also include substantial extra staff
and hand-holding associated with C+M.
See also the discussions
http://www.iiuedu.eu/press/journals/sds/sds1_july_2008/11_SECC_07.pdf
The principal thesis by Alwasaie at UI in 2000 which Uhl claims contains
the positive results appears to be unpublished and not available on the
internet. Unusual in this day and age to be so hard to find.
A paper that can be read, thanks to Google Books is here
(skip the first page indexed -- it is blank or missing).
This seems to reflect slightly positively on C+M, but this does not take
into account the strong possibility that students who enroll in the C+M
sections vs. conventional sections are better to start with, or again)
that they may respond to many of the non-mathematica aspects, such as
working in groups, using an open laboratory model rather than lectures, etc.
None of this mostly unpublished material is, I think, terribly
convincing. Over the past 50 years or so there have been many
experiments in improving calculus, and probably any of them can be shown
to have some positive effect -- if for no other reason than the persons
studied are aware they are the subjects of a study and want to respond
to the extra attention.
This does not necessarily affect the views of advocates of CAS as part
of a technical education who are seeking some justification for the
expenditure of time and money in the undergraduate curriculum for
introducing this technology. And I am not personally saying it is a bad
thing. Just not proven. And I can personally attest to the possibility
that people will find a CAS quite difficult and puzzling -- an
additional burden to the course that was already difficult for them.
My own personal view is that students in technical areas should be made
aware of CAS as part of their education the way they (used to be?)
taught drafting, technical drawing, use of a slide rule... However,
engineering curricula are all filled up, so much so that a 5-year rather
than 4-year set of courses is advocated in many places. So I am told
there is "no room".
Murray Eisenberg
calculations, students can focus on modeling and the resulting and
relevant mathematical concepts and methods -- not the details of
carrying out long chains of algorithmic, algebraic steps.
My 45 years of teaching make perfectly clear that, for most students in
calculus, e.g., they are so involved in trying to get the symbolic
manipulations right, they have little or any idea of why they're doing
them. They totally miss the forest for the trees.
The other side of this situation, I regret to say from my experience, is
that the lazier or intellectually weaker students are often incapable of
rising above merely carrying out mechanically the symbolic manipulations
-- many of which they get wrong anyway -- to have much of an
understanding of the higher-level concepts involved.
On 3/7/2010 4:06 AM, Andrzej Kozlowski wrote:
> I have never seen or heard any convincing reason why using a CAS should
> make it possible to understand and learn better those areas of
> mathematics which are fully accessible to a student with only a pen and
> paper. In fact I can see a few reasons why the opposite might be the
> case. In many situations I can see clear advantages in performing
> algebraic manipulations "by hand" or even "in the head", which is, in my
> opinion, the only way to develop intuition. The same applies to
> visualisation - while being able to look at complicated graphics can
> often be a big advantage, I always insist on students developing the
> ability to quickly sketch simple graphs by hand on the basis of
> qualitative analysis of analytic or algebraic data. This is again
> essential for developing intuition and I am not convinced that doing all
> this by means of a computer will provide equivalent benefits....
Andrzej Kozlowski
symbolic computations" can well qualify as something that isn't "fully
accessible" to many students without a computer. Also, I don't think you
would dispute that all students should be able to do "short symbolic
computations" by hand. If they don't, then, in my opinion, they will
never fully understand what it is that Mathematica is doing for them.
Ultimately it is a question of finding the right balance. There are lots
of things in mathematics that one needs to do just once completely by
oneself in order to develop an intuitive understanding of what is
involved. Once this understanding has been developed, there is no need
to perform ever again these often tedious computations and manipulations
by hand. It seems to me that showing students that all such things can
be done by a CAS before they have understood the basic concepts can
sometimes be seriously harmful (of course mostly to the "the lazier or
intellectually weaker" ones - the others would probably not be satisfied
with mere "button pushing"). On the other hand, there are enormously
many fascinating things in calculus and other areas of mathematics that
can't be done by an average student without a CAS either at all or
within a reasonably time period. I think all computer aided calculus
courses should include such examples (and even perhaps concentrate on
them) because it is only such examples that can really convince both
students and sceptical academics that CAS can be seriously useful in
mathematics.
Andrzej Kozlowski
Canopus56
> My own personal view is that students in technical areas should be
> made aware of CAS as part of their education the way they (used to
> be?) taught drafting, technical drawing, use of a slide rule...
> However, engineering curricula are all filled up, so much so that a
> 5-year rather than 4-year set of courses is advocated in many
> places. So I am told there is "no room".
Perhaps recent graduates could weigh in on this. Is industry expecting competence in CAS (e.g. Mathematica, etc.) as an assumed productivity tool skill? If so, in which fields? Just wondering. - Kurt
----- Original Message ----
From: Richard Fateman <fat...@cs.berkeley.edu>
Sent: Sun, March 7, 2010 2:01:58 AM
Subject: Re: learning calculus through mathematica
Murray Eisenberg wrote:
> Try contacting the folks at the Calculus & Mathematica project at
> University of Illinois and Ohio State. They should be able to point you
> to, or provide, the objective studies they did on performance in their
> Calculus & Mathematica program vs. standard courses.
>
> As I recall, one of their methods was to compare grades of the two
> groups in subsequent courses.
>
> Unless I recall incorrectly, I posted about that very project to this
> group before.� Whether or not I did, please do try to follow up on my
> suggestion above.
>
� I recall this too. So I did look up online ... Here's what I found.
David Park
calculation, yet they might be good at mathematical thinking if they can get
over the calculation hump. The Rainman was good at calculation but no good
at mathematics.
As an analogy I was never good at cursive writing and early on took to
printing. I won't say how much time I've spent trying to master "&"! I've
read exhortations by teachers claiming students must absolutely master
cursive writing. So if they don't master it, does that mean they will never
likely be good at expressing ideas in writing, say on a computer?
Mathematica can open up mathematics to a wider range of students. But not
out-of-the-box. It is too hierarchically thin, and too difficult to
manipulate expressions to common forms. Approaches may need to be altered.
There needs to be more of an axiomatic approach and a distinction between
what is mathematics and what is "plug and chug". Almost any subject will
have its axiom set and these should be implemented, either in the form of
rules or definitions that carry out some transformation. But the axioms
should not be automatically applied. For some subjects the higher level
Mathematica operations may need to be bypassed. The student would then have
to recognize when various axioms are needed and apply them. Perhaps the
student could bring up a window that listed all the axioms. If a student
were solving problems by this method wouldn't that be considered as doing
mathematics? Also, wouldn't this keep the student concentrated on the
foundations of the subject and give the right kind of practice?
> this by means of a computer will provide equivalent benefits....
eric g
the professor refused to take any homework not done by hand and without
the help of Mathematica or similar softwares... I think certain things
you should do it by hand especially if you are learning. For example,
the typical calculus I problem is to graph a function (finding its
roots, maxima/minima, and limits) in doing it by hand you learn and
practice it, but once you get it and pass the class I just use Plot, an
even if you want to check you are right before turning the homework use
mathematica to check your results.
For a teacher is really nice to have all the resources possible to have
good power-points presentations and to explain things, here I think
mathematica plays an important role...
I learnt mathematics in a third world country, and even I didn't have
money for having a scientific calculator, I learnt many things using
classic calculus books...later in the states where I did my graduate
studies I heavily used mathematica for doing homework and displaying
results... In one occasion part of a solid-state problem was needed to
integrate cosine fuctions times polinomials, it was tedious to do it by
hand (integrating by part or wath-ever other technique I learnt in my
undergraduate)... I know a student who spent a week doing the integrals
by hand.. Another occasion, in a quantum mechanics problem it was a
bonus problem which implied to solve an eigensystem of a 7x7 matrix, I
was the only one who solve it because I programmed with Mathematica....
some morons told me sarcastically, " you solve it because you used
Mathematica" and that was not fair, to setup the indexes and the code
for building the matrix was not trivial to do (but easy for one who
knows to program, the morons didn't know and want to learn to program),
and even the code was for any nxn-matrix....
so many times I have got the put down "you solve because you used
mathematica"
... when the true is "you solve it because you know to program, and you
use mathematica rather than C because you are not a masochist"
my advise to learning calculus is try to know as much a possible in
whats going on, if you are using mathematica to solve an integral you
should know what method could be used to solve it, use mathematica for
checking results/answers, use mathematica to gain insight when you are
stuck in something complex. if learning physics, try to know how to
derive the formulas rather than use them, learn/understand as many
derivations as possible, that will train your basic algebra skills as
well as gaining full understanding of the details.
try to proof or understand proofs of theorems, that is the pinacle of
the human brain... and I think mathematica can't help in that.
bests,
eric
On 03/05/2010 01:33 AM, David Park wrote:
> Certainly not every student should be learning Mathematica at the earliest
> possible age. But maybe those who are seriously interested in a technical
> career and are motivated should. Maybe it wouldn't be a part of regular
> secondary school education, but be done on their own, or in math clubs, or
> via mentoring over the Internet.
>
> Maybe it's true that CAS have not made a significant positive impact in
> technical education. Does that mean people should give up? Maybe we haven't
> properly learned how to use them yet. When new technologies come in they are
> often used to just make the old approaches more efficient. Usually what is
> needed is entirely new approaches. Instead of mass lectures and mass exams,
> maybe there should be more self study, more mentoring and more mathematical
> essay writing. As things stand now I have the sneaky suspicion that students
> just don't know Mathematica well enough and it is another obstacle to
> getting through the course. So, why should they do better?
>
> Also, Mathematica off the shelf is not a great educational tool. It does too
> much at a high level with commands like Solve, Integrate or Limit. That's
> all fine, but students need something I call "hierarchical depth", the
> ability to do mathematics at different levels and see how things work. It is
> somewhat ironic that as a computer algebra system, Mathematica (and I
> suspect most systems) are poor at providing the kind of algebraic
> manipulations that students need to work with. They are hierarchically thin.
> This all could be provided, but it takes more work.
>
>
> David Park
> djm...@comcast.net
> http://home.comcast.net/~djmpark/
>
>
>
Murray Eisenberg
I agree that students should learn to do some symbolic calculations by
hand. And they have to do some longer symbolic calculations by hand,
not for the sake of getting an answer that way, but for the experience
of concentrating on a problem that requires many steps.
Three issues here are: (i) which symbolic calculations? (ii)
calculations of what complexity? and (iii) for how long should they have
to do them by hand?
Two examples.
Re (i): Should we be teaching the techniques of indefinite integration
the way it is ordinarily done in calculus courses -- given that symbolic
integrators often use different algorithms from what we do by hand?
Re (ii): Is it worthwhile to take student's time, e.g., to integrate a
rational function whose denominator is the product of, say, the square
of an irreducible quadratic and the cube of a linear factor? Isn't it
enough for students to do MUCH simpler examples with paper and pencil
and then leave these more complicated versions to a machine?
Or, take a typical Calculus I minimization problem about a drowning man
in the lake, with the lifeguard running along the (straight) shore at
one rate and then swimming toward the man at a different rate. If the
rates are cooked up right, then this is quite reasonable to ask a
student to do with paper and pencil.
However, change the problem to a more realistic one of, say, finding the
path a light ray takes in traveling in air and then in water, where the
two rates are 30 cm/ns and 22.5 cm/ns, respectively. The equation for
the critical point of the total time of light travel reduces to a 4th
degree polynomial with some unpleasant coefficients. In principle,
students could apply the quartic formula to find those roots, but is it
in any sense reasonable to ask them to do it? To me it seems like a
perfect opportunity to resort to a CAS to work through the entire thing,
if not just to solve (symbolically or numerically) the quartic equation.
[I first learned this example, and using Mathematica for it, from the
wonderful calculus course and book developed by Frank Wattenberg.]
Re (iii): Once a student has demonstrated basic paper-and-pencil
proficiency in calculus and has graduated to a full-fledged course in
differential equations, is there any wholly defensible point in
expecting hand calculation of integrals such as may arise, e.g., in
applying the method of variation of parameters?
The usual kind of answer I hear to this is something like, "But it
provides reinforcement of what the student learned in calculus by giving
him a chance to practice it again." Which begs the question, of course,
as to whether and why the student should still have to do such "baby"
calculation by hand at this point.
I could rant on further about this, but I'll stop here.
On 3/8/2010 6:11 AM, Andrzej Kozlowski wrote:
> I don't see any real contradiction between this and what I wrote. "Long
> symbolic computations" can well qualify as something that isn't "fully
> accessible" to many students without a computer. Also, I don't think you
> would dispute that all students should be able to do "short symbolic
> computations" by hand. If they don't, then, in my opinion, they will
> never fully understand what it is that Mathematica is doing for them.
>
> Ultimately it is a question of finding the right balance. There are lots
> of things in mathematics that one needs to do just once completely by
> oneself in order to develop an intuitive understanding of what is
> involved. Once this understanding has been developed, there is no need
> to perform ever again these often tedious computations and manipulations
> by hand. It seems to me that showing students that all such things can
> be done by a CAS before they have understood the basic concepts can
> sometimes be seriously harmful (of course mostly to the "the lazier or
> intellectually weaker" ones - the others would probably not be satisfied
> with mere "button pushing"). On the other hand, there are enormously
> many fascinating things in calculus and other areas of mathematics that
> can't be done by an average student without a CAS either at all or
> within a reasonably time period. I think all computer aided calculus
> courses should include such examples (and even perhaps concentrate on
> them) because it is only such examples that can really convince both
> students and sceptical academics that CAS can be seriously useful in
> mathematics.
>
>
> Andrzej Kozlowski
>
> On 7 Mar 2010, at 16:15, Murray Eisenberg wrote:
>
>>> this by means of a computer will provide equivalent benefits.
>>
>> --
>> Murray Eisenberg mur...@math.umass.edu
>> Mathematics& Statistics Dept.
Ingolf Dahl
experience it is not unique. I am working in the field of liquid crystals,
and was investigating a paper, where they measured the optical retardation
of liquid crystal cell, and they could vary both the wavelength of light and
the inclination angle of the illuminating light. Using this data, they
fitted three parameters, characterizing the optical properties of the cell.
This really looked good to us, because we want to measure the same
parameters. However, I used two days with Mathematica to solve the
differential equation they presented, and from that I could (using their
formulas) use NIntegrate to get the retardation as function of all involved
parameters. Then I did some simple plots to check how the retardation varies
when the fitting parameters are varied. Unfortunately I found that the
retardation changed in (almost) the same way, independent of which parameter
I varied... So the problem was as well specified as the equation x + y + z
== 50. It is possible to get a very good fit of the x, y, and z to this
equation, but the value and relevance of one particular solution is null.
Thus it would be possible to specify only one, but not three, parameters
from the experimental data.
So why had not the authors of the paper seen the same? I am 100 % sure that
it is not due to my superior intelligence. Maybe my experience from solving
a similar problem on a Olivetti Programma machine 40 years back helps a bit,
but the thing I think that really matters is that I have appropriate
knowledge about the relevant CAS tool (Mathematica, of course), and I guess
that the same was not true for these authors. It is not appropriate to
reveal their names here.
So the relevant test for the math students is maybe not to test them with
small 2x2 equation systems that can be solved with paper and pen. Give them
instead slightly more involved problems, and a computer with some CAS
systems installed, and measure then the time and the quality of the
solutions. The set of problems you can solve if you have a CAS system to
your disposal is immensely larger than the set of paper-and-pen problems,
especially if you are trained to use the abilities of the CAS.
Ingolf Dahl
> -----Original Message-----
> From: Helen Read [mailto:h...@together.net]
> Sent: den 7 mars 2010 10:04
> To: math...@smc.vnet.net
> Subject: Re: learning calculus through mathematica
>