Mathematica and Education
fizzy
Mathematica in the classroom, etc... I just took a course in Advanced
Electromagnetic Engineering and I'm happy to say that I did not perform
a single pencil and paper operation. I did all the homework and exams
using Mathematica. Also, in hindsight, without Mathematica I would
never take the course nor can I conceive of how I could without it
although I realize there was a time where people did work without
Mathematica. The amount of homework was horrendous and how you could
do it with pencil and paper operations is beyond me. In fact, I've
gotten so used to Mathematica that if I were told I can no longer use
it in my work, etc. , I would give Science up. That is how attached
I've become to Mathematica and how much more enjoyment I've gotten out
of Science problems using it.
Also, when I have to use other languages I feel like a Programmer and I
dont relish that at all. With Mathematica I actually feel that I'm
doing some thinking and analyzing instead of just writing lines of code.
Jerry Blimbaum
Paul Abbott
wrote:
> Recently, there were several discussions regarding the use of
> Mathematica in the classroom, etc... I just took a course in Advanced
> Electromagnetic Engineering and I'm happy to say that I did not perform
> a single pencil and paper operation. I did all the homework and exams
> using Mathematica.
I assume, then, that the exams were "take-home"? Can you provide more
information about this course? I have given an undergraduate
Electromagnetism course to second-year students. I permit them
(encourage them) to do their assignments using Mathematica -- but the
final exam is a standard 90 minute written paper.
> Also, in hindsight, without Mathematica I would
> never take the course nor can I conceive of how I could without it
> although I realize there was a time where people did work without
> Mathematica. The amount of homework was horrendous and how you could
> do it with pencil and paper operations is beyond me.
As long as the "by hand" operations are understood, using Mathematica to
perform the calculations is fine.
Another barrier to its wider use is that the instructor has to be
prepared to receive Notebooks as solutions to assignments and exams.
Clearly, yours must have been ok about this.
> In fact, I've
> gotten so used to Mathematica that if I were told I can no longer use
> it in my work, etc. , I would give Science up. That is how attached
> I've become to Mathematica and how much more enjoyment I've gotten out
> of Science problems using it.
I more or less feel the same way.
> Also, when I have to use other languages I feel like a Programmer and I
> dont relish that at all. With Mathematica I actually feel that I'm
> doing some thinking and analyzing instead of just writing lines of code.
Though I solve a lot of problems using Mathematica, I would not consider
myself as a programmer either..
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
King, Peter R
(predominantly) non-mathematical branch of science (earth sciences) I am
very concerned about this approach. Sure Mathematica is a wonderful
tool. As a professional researcher I use it all the time for doing
tedious calculations to save time, or to check claculations where I may
have got things wrong and so on and so on. If I didn't think Mathematica
was useful I wouldn't have it and wouldn't subscribe to this list.
But it is still a tool. IT can't know what calculations to do, what
approximations to make and sometimes when there are mathematical choices
to be made. For example there are times when Mathematica's choice of
branch cut doesn't correspond to the one I want to make. Not a problem I
can tell it what I really want. There are times when its choice of
simplification doesn't suite my purpose. Again not a problem I can tell
it what to do or simply carry on by hand if that's easier. But how do I
know when the defaults don't suite my purpose, because I have spent many
years doing things by hand and gaining that experience to know what I
want. I am not convinced that if I had done all my mathematics within
Mathematica I would have gained the same experience. But I am open to
discussion on this if anyone wants to put the counter case. However, I
would need very strong convincing that it is good for students never to
have to do old fashioned calculations on paper. In the same way I think
it is important for children to learn multiplication rather than rely on
a calculator or to learn to write rather than use a word processor.
In particular for practicing engineers they may be out in the field,
away from a computer and be required to do a back of the envelope
calculation by hand. If you have never done it before you will be stuck
and I don't think you could consider yourself a "real" engineer.
So yes Mathematica is great. Yes students should be taught to use it and
use it properly. But please make sure you could have done your homework
by hand (it is often not as bad as you might think!). Perhaps I am a
dinosaur but I have been in meetings which required moderately difficult
numerical calculations which I could do by hand whereas other (younger)
people present were stuck without calculators.
I was once told a quote and I can't remember who it was from "A fool
with a tool is still a fool"
(Incidentally please don't take this personally. I don't know you and so
I have no reason to doubt that you are a perfectly good scientist I am
simply commenting on a current trend for people to run to software
rather than doing it by hand - which in some cases is actually easier).
Peter King
> -----Original Message-----
> From: fizzy [mailto:fizz...@knology.net]
> Subject: Mathematica and Education
>
>
> Recently, there were several discussions regarding the use of
> Mathematica in the classroom, etc... I just took a course
> in Advanced
> Electromagnetic Engineering and I'm happy to say that I did
> not perform
> a single pencil and paper operation. I did all the
> homework and exams
> using Mathematica. Also, in hindsight, without
> Mathematica I would
> never take the course nor can I conceive of how I could without it
> although I realize there was a time where people did work without
> Mathematica. The amount of homework was horrendous and
> how you could
> do it with pencil and paper operations is beyond me. In fact, I've
> gotten so used to Mathematica that if I were told I can no
> longer use
> it in my work, etc. , I would give Science up. That is how attached
> I've become to Mathematica and how much more enjoyment I've
> gotten out
> of Science problems using it.
>
> Also, when I have to use other languages I feel like a
> Programmer and I
> dont relish that at all. With Mathematica I actually feel that I'm
> doing some thinking and analyzing instead of just writing
> lines of code.
>
>
> Jerry Blimbaum
>
>
>
Dave (from the UK)
> Recently, there were several discussions regarding the use of
> Mathematica in the classroom, etc... I just took a course in Advanced
> Electromagnetic Engineering
I did something pretty similar 'Computer modeling of Fields' as part of
an MSc Microwaves and Optoelectronics - I later did a PhD, although not
on this subject.
I am also the author of an open-source program for computing impedance
of transmission lines of arbitrary cross section.
> and I'm happy to say that I did not perform
> a single pencil and paper operation.
Personally I would be sorry to say that. And if you have any sense, I
would not admit it at a job interview.
> I did all the homework and exams
> using Mathematica.
Were you allowed Mathematica in an exam? At UCL they are very
restrictive on the sort of calculators allowed, so something like
Mathematica would not be allowed.
> Also, in hindsight, without Mathematica I would
> never take the course nor can I conceive of how I could without it
> although I realize there was a time where people did work without
> Mathematica. The amount of homework was horrendous and how you could
> do it with pencil and paper operations is beyond me.
I can't help feeling that doing some by pencil and paper is better for
learning.
> In fact, I've
> gotten so used to Mathematica that if I were told I can no longer use
> it in my work, etc. , I would give Science up.
You might well find like I did that I worked for a commercial company
and got presented with something far less capable. I forget what I had
to use at Marconi, but it was next to useless. I was quite glad when I
later worked in the uni again and had Mathematica.
> That is how attached
> I've become to Mathematica and how much more enjoyment I've gotten out
> of Science problems using it.
I think science problems are interesting, not just because Mathematica
exists. If the problem was not interesting, solving it with Mathematica
would not give me any thrills.
> Also, when I have to use other languages I feel like a Programmer and I
> dont relish that at all. With Mathematica I actually feel that I'm
> doing some thinking and analyzing instead of just writing lines of code.
To me, if you use a lower level language you have to think far more.
That enforces understanding in *some* ways.
In some cases, using a high level language you can think more about the
problem than the details - and in many cases the details are not
important. Do I really case how to calculate a log - probably not. I'd
use Mathematica, a calculator or years ago a table of logs.
But I think in many cases the use of such a high level language can
allow you to get answers without understanding.
I like Mathematica and introduced colleagues to it at uni. I arranged
for us to buy copies for a Sun and later to get a departmental license.
I was involved to a certain extent on discussions on a campus wide
license.
> Jerry Blimbaum
I can't help feeling your views are rather extream, and one I doubt even
Wolfram Research as a company would share.
--
Dave K
Minefield Consultant and Solitaire Expert (MCSE).
Please note my email address changes periodically to avoid spam.
It is always of the form: month-year@domain. Hitting reply will work
for a couple of months only. Later set it manually.
Katerina Kaouri
only use pen+paper for your exams?
Katerina.
fizzy wrote:
>Recently, there were several discussions regarding the use of
>Mathematica in the classroom, etc... I just took a course in Advanced
>Electromagnetic Engineering and I'm happy to say that I did not perform
>a single pencil and paper operation. I did all the homework and exams
>using Mathematica. Also, in hindsight, without Mathematica I would
>never take the course nor can I conceive of how I could without it
>although I realize there was a time where people did work without
>Mathematica. The amount of homework was horrendous and how you could
>do it with pencil and paper operations is beyond me. In fact, I've
>gotten so used to Mathematica that if I were told I can no longer use
>it in my work, etc. , I would give Science up. That is how attached
>I've become to Mathematica and how much more enjoyment I've gotten out
>of Science problems using it.
>
>Also, when I have to use other languages I feel like a Programmer and I
>dont relish that at all. With Mathematica I actually feel that I'm
>doing some thinking and analyzing instead of just writing lines of code.
>
>
>Jerry Blimbaum
>
>
>
>
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KvS
I just want to give a general response since I'm of course not familiar
with your exact situiation. Having said this, according to me
"I just took a course in Advanced
Electromagnetic Engineering and I'm happy to say that I did not perform
a single pencil and paper operation. I did all the homework and
exams using Mathematica."
is not necessarily a good thing. In the Netherlands there's been a
reform of education with as one of the results the introduction of a
powerful "graphical calculator", that means the possibility to plot, do
some statistics, numerically calculate integrals etc. It turns out that
a lot of people lose touch with the maths behind the scene and the
calculator starts acting as a black box so that one only needs to know
the order of buttons to press to get a result.
Consequently people make typical mistakes, e.g. not being able to
recognize when an obvious numerical error is made in a plot of a
function and not at all knowing what to do if a small modification of a
known problem is posed for which the calculator doesn't work anymore.
My point is just this: Mathematica is fantastic, but only useful in
teaching if the action behind the scenes is also being taught.
Best, Kees
fizzy
Mathematica.....although, I had the feeling that I was the only student in
the class doing that....I took the course as a distance learning student so
I was not in the classroom....and, of course, at the end of it I have a
very large Notebook with all my work.....which is readable, etc. whereas
probably most pencil and paper operations would be potentially confusing,
etc. or hard to read.....
Jerry
----- Original Message -----
From: "Katerina Kaouri" <katerin...@maths.nottingham.ac.uk>
Subject: Re: Mathematica and Education
> yes, Mathematica is great for what you describe but presumably you can
> only use pen+paper for your exams?
> Katerina.
>
> fizzy wrote:
>
>>Recently, there were several discussions regarding the use of Mathematica
>>in the classroom, etc... I just took a course in Advanced
>>Electromagnetic Engineering and I'm happy to say that I did not perform a
>>single pencil and paper operation. I did all the homework and exams
fizzy
really feel that the Wolfram people owe a strong debt of gratitude to the
mathgroup people.....I would have given up on Mathematica on my initial
encounter because I just couldnt work thru the nuances or non-documented
methods which you need to actually do a lot of the work.....I always wonder
how many others feel the same.....
Long Live Mathematica and may it give Endless Pleasure and Assistance to all
its Users....and, of course, continue to help all those like myself who have
difficulty with math....
Jerry Blimbaum
----- Original Message -----
From: "David Park" <dj...@earthlink.net>
Subject: Re: Mathematica and Education
> Jerry,
>
> That's it! That's it! Music to my ears.
>
> David Park
> dj...@earthlink.net
> http://home.earthlink.net/~djmp/
Murray Eisenberg
King, Peter R wrote:
>
> .... But how do I
> know when the defaults don't suite my purpose, because I have spent many
> years doing things by hand and gaining that experience to know what I
> want. I am not convinced that if I had done all my mathematics within
> Mathematica I would have gained the same experience.
That's a very legitimate concern. to my mind it's the principal excuse
for still doing complicated paper-and-pencil calculations when learning.
(Simple paper-and-pencil calculations may be readily justified as
needed to understand what's happening.)
> In particular for practicing engineers they may be out in the field,
> away from a computer and be required to do a back of the envelope
> calculation by hand. If you have never done it before you will be stuck
> and I don't think you could consider yourself a "real" engineer.
But that seems to me to be essentially a "red herring". It's the old
"What will you do if you're on a desert island and don't have access to
a table of integrals?" question. Surely many "in the field" engineers
now carry their laptops or tablet computers with them whenever they're
on the job. And we may be only a short time away from the day that
Mathematica will be available on a calculator/PDA-sized device that fits
into a shirt pocket.
--
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
Paul Abbott
"King, Peter R" <peter...@imperial.ac.uk> wrote:
> I should like to say that as an educator of science students in a
> (predominantly) non-mathematical branch of science (earth sciences) I am
> very concerned about this approach.
However, is it reasonable to expect your students to gain the level of
mathematical expertise that you have (unless they are going to become
professional researchers also)? If not, what should they be taught?
> Sure Mathematica is a wonderful
> tool. As a professional researcher I use it all the time for doing
> tedious calculations to save time, or to check claculations where I may
> have got things wrong and so on and so on. If I didn't think Mathematica
> was useful I wouldn't have it and wouldn't subscribe to this list.
>
> But it is still a tool. IT can't know what calculations to do, what
> approximations to make and sometimes when there are mathematical choices
> to be made. For example there are times when Mathematica's choice of
> branch cut doesn't correspond to the one I want to make. Not a problem I
> can tell it what I really want. There are times when its choice of
> simplification doesn't suite my purpose. Again not a problem I can tell
> it what to do or simply carry on by hand if that's easier. But how do I
> know when the defaults don't suite my purpose, because I have spent many
> years doing things by hand and gaining that experience to know what I
> want.
A question remains though: how much of your accumulated expertise _can_
be automated. In some fields, e.g., summation, computer algebra can
automate nearly all operations of interest (see the books
generatingfunctionology <http://www.math.upenn.edu/~wilf/DownldGF.html>
and "A=B" <http://www.cis.upenn.edu/~wilf/AeqB.html>).
> that I am not convinced that if I had done all my mathematics within
> Mathematica I would have gained the same experience. But I am open to
> discussion on this if anyone wants to put the counter case. However, I
> would need very strong convincing that it is good for students never to
> have to do old fashioned calculations on paper. In the same way I think
> it is important for children to learn multiplication rather than rely on
> a calculator or to learn to write rather than use a word processor.
Agreed. As I've previously posted on this newsgroup, I think that Bruno
Buchberger has got it right. See
http://www.risc.uni-linz.ac.at/people/buchberg/white_box.html
However, I think that the potential for discovery using computer
mathematics is underestimated -- and is only taught in very few
University courses.
David Park
I find your remarks very interesting and I think you state the principal
reasons for NOT making the maximum use of Mathematica in education. It
certainly helps to get the objections and perceived limitations on the
table. However, I would like to try, to the best of my ability, to make the
counter arguments.
If I may summarize the reasons you, and others, have put forward.
1) Mathematica allows a student to get an answer without truly understanding
the underlying theory and reasons. Pencil and paper forces the student to
understand things more deeply and provides additional experience.
2) We have to preserve the old skills. In emergencies we may be forced to
fall back on them, such as in the field, in exams without computers and
after the next nuclear war. Good penmanship and mental arithmetic will save
us.
3) Mathematica will automatically make choices for us that we do not
understand. I would like to state this in a more general sense. Students
haven't mastered Mathematica well enough to use it as a reliable tool.
I have often argued here that students should be taught to think of
Mathematica as 'pencil and paper'. They should use it just like they would
use pencil and paper. Theodore Gray has provided us with the wonderful
notebook interface. You can have titles, sections, text cells, equations and
diagrams. It's the style of textbooks, reports and research papers. It goes
back at least to Euclid. So, I don't understand specifically what advantage
real pencil and paper have over a Mathematica notebook, except perhaps that
it is far easier to get away with writing nonsense.
In fact, let's look at the advantages that a Mathematica notebook has over
real pencil and paper.
1) Neatness. And a student can correct and rewrite more easily.
2) An active document. The definitions students write can actively be used
in further derivations. In fact, the student is forced to make these
definitions and assumptions explicit.
3) Permanent record. Not only a permanent record but also a repository of
resources that the student may have developed.
4) Proofing. With a Mathematica notebook you can actually evaluate things
and verify that they work. One can't get away with sloppiness.
5) MORE and DEEPER experience. With a Mathematica notebook a student can
actually do many more, and more difficult, exercises and examples. Many
times, while working through textbooks, I have seen cases where the author
either skipped the demonstration or simplified the case for no other reason
than the difficulty of hand calculations.
6) A literate style. Conventional exercises and tests are usually skimpy
throw away documents. Mathematica notebooks provide a perfect opportunity
for 'essay' style work and develop the skills for technical communication.
Of course, we have to have teachers and students who know how to take
advantage of these features.
As for preserving old skills, I'm not too sympathetic. Are students to be
taught how to sharpen spears (no advanced bow and arrow technology allowed!)
track animals and identify eatable grubs and berries, just in case we get
thrown back into a hunter-gatherer society? It wasn't that many generations
ago when almost all women knew how to weave or operate a spinning wheel.
Should these skills be preserved? Like it or not, we are dependent on
civilization and modern technology. Rather than teaching 'survival skills'
we should make sure that civilization is preserved and advanced. That's the
best chance. If worse comes to worst, some people will learn the
multiplication tables fast enough (and also how to sharpen spears).
The problem of using Mathematica intelligently, and not blindly, is serious.
Most students are not well enough prepared with Mathematica to use it to
anywhere near its capability. Mathematica is not wide spread enough and
students do not learn it early enough. Any student interested in a technical
career could do nothing better than start learning it in high school.
Furthermore, Mathematica is not optimized for students and researchers. When
it comes to ease of use there are many gaps. I believe that Mathematica can
truly effect a revolution in technical education. But it is not as simple as
just installing it on a departmental server. A lot of preparation is needed.
Additional packages geared to student use are needed. Educators have to
learn how to take advantage of the resource. (For example how to shift from
quick calculations to essay type questions.)
From: King, Peter R [mailto:peter...@imperial.ac.uk]
I should like to say that as an educator of science students in a
(predominantly) non-mathematical branch of science (earth sciences) I am
very concerned about this approach. Sure Mathematica is a wonderful
tool. As a professional researcher I use it all the time for doing
tedious calculations to save time, or to check claculations where I may
have got things wrong and so on and so on. If I didn't think Mathematica
was useful I wouldn't have it and wouldn't subscribe to this list.
But it is still a tool. IT can't know what calculations to do, what
approximations to make and sometimes when there are mathematical choices
to be made. For example there are times when Mathematica's choice of
branch cut doesn't correspond to the one I want to make. Not a problem I
can tell it what I really want. There are times when its choice of
simplification doesn't suite my purpose. Again not a problem I can tell
it what to do or simply carry on by hand if that's easier. But how do I
know when the defaults don't suite my purpose, because I have spent many
years doing things by hand and gaining that experience to know what I
want. I am not convinced that if I had done all my mathematics within
Mathematica I would have gained the same experience. But I am open to
discussion on this if anyone wants to put the counter case. However, I
would need very strong convincing that it is good for students never to
have to do old fashioned calculations on paper. In the same way I think
it is important for children to learn multiplication rather than rely on
a calculator or to learn to write rather than use a word processor.
In particular for practicing engineers they may be out in the field,
away from a computer and be required to do a back of the envelope
calculation by hand. If you have never done it before you will be stuck
and I don't think you could consider yourself a "real" engineer.
So yes Mathematica is great. Yes students should be taught to use it and
Matt
should never do anything in Mathematica unless I have first
demonstrated I can do it via hand calculation on paper" until I read
this 'dialogue' on Theodore Gray's website:
http://www.theodoregray.com/BrainRot/index.html . It hits upon many of
the points presented in this thread in a very compelling way.
Matt
G. Raymond Brown
Following this thread for some time, and as a physics and chemistry
educator of undergraduate students, I am firmly in the camp whose
spokepersons are David Park and Paul Abbott. For years I have supported
the courses I teach with Mathematica notebooks and XML files derived from
my Mathematica notebooks posted on course management systems. I have
also strongly encouraged my students to buy and use Mathematica (my
institutions still, despite my urging, do not have site licenses for the
software, and students mostly consider the software too expensive to
obtain for themselves). In general I find that Mathematica provides an
excellent tool ("pencil and paper" as described by David) for
mathematical discourse between instructor and student. It is a joy to
grade and return student papers electronically, and much easier and less
time-consuming (for me, at least) than killing trees with paper
submissions and the attendant physical organization and handling of paper
submissions.
It _is_ necessary that students understand the mathematics underlying the
Mathematica processes. Paul's reference to Buchbach's contributions are
right on the mark here. Students _can_ abuse Mathematica in the same way
that they abuse simple calculators, reproducing the ancient "garbage in
garbage out" result. Use of Mathematica simply does not substitute for
learning the mathematical underpinnings of the subject. What it _does_
substitute for is the tedium of manual manipulation of mathematical
formulae, greatly expanding the scope of problems I can reasonably assign
to students.
Mathematica brings a host of benefits to any party, but IMHO its greatest
benefit to education lies in its enabling of asynchronous mathematical
discourse between students and instructor. Mathematica simply has no
match in this regard. In this role, it greatly facilitates the learning
of mathematics applied to solution of real-world problems.
-GRB-
G. Raymond Brown, Ph.D.
Scientific Program Coordinator
Division of Science and Mathematics
Morehouse College
Helen Read
> Peter,
>
> I find your remarks very interesting and I think you state the principal
> reasons for NOT making the maximum use of Mathematica in education. It
> certainly helps to get the objections and perceived limitations on the
> table. However, I would like to try, to the best of my ability, to make the
> counter arguments.
[David's excellent counter arguments snipped.]
> As for preserving old skills, I'm not too sympathetic. Are students to be
> taught how to sharpen spears (no advanced bow and arrow technology allowed!)
> track animals and identify eatable grubs and berries, just in case we get
> thrown back into a hunter-gatherer society? It wasn't that many generations
> ago when almost all women knew how to weave or operate a spinning wheel.
> Should these skills be preserved?
Colleagues only a little older than I am used slide rules in school. I
never learned to use one; I had an early generation scientific
calculator. I don't believe this has harmed me in any way. I did learn
to interpolate off of trig tables, and probably my teachers were arguing
at the time over whether they should still be teaching that. My brother,
two years younger, never saw trig tables, and I don't think it hurt him
any. Technology advances, and we should make full use of it.
> The problem of using Mathematica intelligently, and not blindly, is serious.
> Most students are not well enough prepared with Mathematica to use it to
> anywhere near its capability. Mathematica is not wide spread enough and
> students do not learn it early enough. Any student interested in a technical
> career could do nothing better than start learning it in high school.
> Furthermore, Mathematica is not optimized for students and researchers. When
> it comes to ease of use there are many gaps. I believe that Mathematica can
> truly effect a revolution in technical education. But it is not as simple as
> just installing it on a departmental server. A lot of preparation is needed.
> Additional packages geared to student use are needed. Educators have to
> learn how to take advantage of the resource.
At my institution, we have a university wide site license allowing us to
install Mathematica on all of our computers, not just those owned by the
university, but also laptops and desktops owned by faculty, staff, and
students. Mathematica is available to everyone literally 24/7.
I have been teaching calculus with Mathematica for 10 years, for the
last 6 of those years in a classroom equipped with 31 networked PCs (one
for each student, plus one for the instructor), and a printer. We now
have two such rooms. The instructor's machine is connected to the
overhead projector, and we have software allowing easy communication
between the student PCs and the instructor. I can, for example,
broadcast my screen or any of the students' onto the projector or onto
everyone's monitor. Unlike the computer labs on campus, these rooms are
designed for teaching, with clear lines of sight from every student to
the teacher and whiteboard, enough space for the instructor to walk
around and interact with the students, etc.
My students use Mathematica routinely. There are weekly assignments
(notebooks that I post online) which they submit to me via e-mail. We
use Mathematica for examples and exploration activities in class; on
homework assignments that they do for practice and do not hand in; and
on quizzes and tests. I prepare many examples for the students to do in
class with Mathematica, and we fire up Mathematica any time we feel the
urge -- e.g. if something comes up in the middle of a lecture that we
want to see in Mathematica. I save these examples from class and post
them on the web, so that students who didn't finish the examples in
class, or couldn't get something to work, or missed class entirely, can
download the solutions later. I also make up lots of worksheets for the
students to do for homework with full use of Mathematica, as I have yet
to find a textbook with the sorts of problems that I would like.
I'm haven't thrown out the baby with the bathwater, however. We still
do, for example, the techniques of integration chapter, trigonometric
substitution and all. But it is clear to me that the students benefit in
all kinds of ways from using Mathematica, and I hope that *some* of the
pencil-and-paper topics will eventually be toned down. (Right now there
are constraints placed on us by the client departments.)
I give two-part tests. Part I is done with pencil and paper, no
computers, no calculators. This part of the test consists of traditional
skill questions that could have been on a calculus test 30 years ago --
e.g., using the chain rule, integrating by parts. I still teach these
topics and expect my students to do them by hand (a) because it teaches
some good mental skills, and (b) so that they have some understanding of
what is going on when they use Mathematica to do these things for them.
When a student finishes Part I of the test, s/he turns it in and picks
up Part II, which is done with the use of Mathematica. The students show
all of their work for Part II in a Mathematica notebook, making a title
with their name, section and subsection labels for the various
exercises, text cells to insert their own comments and answers to
questions, etc. They print their work and staple it to their test paper
when they hand it in.
Here are just a few examples of the sorts of things that go on the
Mathematica portion of the test.
* Calculus I: Here's some complicated function. Find the linear and
quadratic approximations at some given x0. Plot f(x) with the linear
approximation and with the quadratic approximation on an interval that
gives a good view. On approximately what interval does the linear
approximation give a "good" approximation of f(x)? Ditto for the
quadratic approximation.
* Calculus II: similar to the above, but with say, a degree 5 Taylor
polynomial.
* Plot a curve given parametrically. Determine when (what t) and where
(x and y coordinates) the curve intersects itself. Make a new plot that
shows only the "loop" of the curve (in between where it crosses itself).
Find the length of the loop. Find the surface area of the resulting
surface if the loop is revolved about (a) the x-axis; (b) the y-axis;
and (c) various other axes. (Most students will include a
SurfaceOfRevolution plot for (a) and (b) even if I don't ask for it. I
always try to cook the examples so that the surface plots look cool.)
* Here's a series. Verify with a table / plot that the terms are
positive, decreasing, and converge to 0. Make a table and plot of the
first (say) 100 partial sums. [At this point they can *see* that the
series is convergent.] Use the integral test to confirm that the series
converges. Estimate the sum of the series numerically by finding (a) the
partial sum from n=1 to (say) 500, and (b) using integrals to find upper
and lower bounds on the error if the partial sum is used as an
approximation of the sum of the series.
All of my students, even the weakest among them, are quite comfortable
using Mathematica by the end of the semester. On quizzes/tests, they are
permitted to use the Mathematica Help Browser and to raise their hands
and ask me for help if they are having Mathematica issues; they need
less and less assistance by the end of the semester.
The students almost always do better on Part II (the Mathematica
portion) of the test than Part I, despite the fact that the problems on
Part II are usually what I consider more difficult. Often they can catch
mistakes when working on Part II, things they would have done
incorrectly on a pencil and paper test without realizing it was wrong.
For example, they will set up an arclength integral incorrectly, and get
an answer from NIntegrate (of course with Mathematica, we are not
limited to the small number of examples whose arclength can be
calculated exactly)--and they can see the answer is off by an order of
magnitude when they estimate the length of the curve that they plotted.
Or, they'll find the equation of a tangent line and plot it with f(x),
only to find that they messed something up -- the line is not tangent!
On a pencil and paper test, they'd have done these things wrong, not
known it was wrong, and moved on to the next question. With the use of
Mathematica they can *see* something is wrong, and can very often fix
it. They are much better at checking whether an answer is reasonable
with all the graphical / numerical feedback that they get from
Mathematica than without it.
--
Helen Read
University of Vermont
ggr...@sarj.ca
I agree largely that Mathematica (or any math software) can be used to
great benefit.
One point that I believe your argument takes too lightly is the
availability of the technology. Your example of spinning wheels
becoming obsolete is not really valid. Manufactured clothing is
available everywhere, and there is sufficient variety that it is
available at prices that are accessible to pretty well everyone.
Mathematica and even computers are not. There is a large cost to buy
and maintain both a computer and a license. You can argue that
computers are starting to become cheap and common enough that we can
drop computers as a significant cost. Fine, but a Mathematica license
is not cheap. Overall value doesn't always win the day when there are
many other financial obligations to consider.
So what is the student to do when they go to work for such a company?
Or even if the company does buy a license, it may not provide a home
license for the employee to use. Even if they do provide a home
license, you suddenly run into a whole bunch of other issues, like who
owns the IP of anything you might produce. No such barrier exists
with the pencil and paper model.
I think the crux of the argument is that if you are taught well, it
doesn't matter much how you were taught. There are benefits of all
approaches. The problem lies in the fact that in reality, regardless
of the system, students will not be taught nor learn optimally. With
course loads and all the other pressures of student life, there is no
way that most students can devote enough time to get the most out of
any teaching system. Similarly, with all their other obligations, few
professors have the time to devote to being truly excellent
instructors. So you have to fall to a compromise; knowing that the
average student is only going to absorb a fraction of what you're
teaching in the time that is available, what is important?
From my experience, all technology, no matter how reliable, has some
sort of failure in a teaching environment (how many times have you
seen the projector not work correctly?). Dealing with these problems
has merit, but it robs valuable time from instruction of other
concepts. How many times have you seen a university/college level
student go to their professor to figure out how to sharpen their
pencil?
Do these problems with technology outweigh the potential gain by
systems like Mathematica? I don't know. I do know that the
effectiveness of any teaching system boils down to the quality and
motivation of the instructors and students and sadly, even the
administrators.
On Saturday, March 11, 2006 at 5:15 AM, David Park wrote:
> Peter,
> I find your remarks very interesting and I think you state the principal
> reasons for NOT making the maximum use of Mathematica in education. It
> certainly helps to get the objections and perceived limitations on the
> table. However, I would like to try, to the best of my ability, to make the
> counter arguments.
> If I may summarize the reasons you, and others, have put forward.
> Of course, we have to have teachers and students who know how to take
> advantage of these features.
> As for preserving old skills, I'm not too sympathetic. Are students to be
> taught how to sharpen spears (no advanced bow and arrow technology allowed!)
> track animals and identify eatable grubs and berries, just in case we get
> thrown back into a hunter-gatherer society? It wasn't that many generations
> ago when almost all women knew how to weave or operate a spinning wheel.
> Should these skills be preserved? Like it or not, we are dependent on
> civilization and modern technology. Rather than teaching 'survival skills'
> we should make sure that civilization is preserved and advanced. That's the
> best chance. If worse comes to worst, some people will learn the
> multiplication tables fast enough (and also how to sharpen spears).
> The problem of using Mathematica intelligently, and not blindly, is serious.
> Most students are not well enough prepared with Mathematica to use it to
> anywhere near its capability. Mathematica is not wide spread enough and
> students do not learn it early enough. Any student interested in a technical
> career could do nothing better than start learning it in high school.
> Furthermore, Mathematica is not optimized for students and researchers. When
> it comes to ease of use there are many gaps. I believe that Mathematica can
> truly effect a revolution in technical education. But it is not as simple as
> just installing it on a departmental server. A lot of preparation is needed.
> Additional packages geared to student use are needed. Educators have to
David Park
I don't know enough about the economics of the CAS software business, but
they may be pricing themselves away from their market. I know people who
love Mathematica but can't afford to keep up with the latest versions. I
believe that Mathematica can be revolutionary in technical education but it
is not now achieving that goal and there are serious obstacles in getting
there.
I am not a teacher and can only comment at second hand but I also think the
present educational model is far from the quality that could be achieved for
students interested in technical careers.
n...@12000.org
"Theodore Gray has provided us with the wonderful
notebook interface. You can have titles, sections, text cells,
equations and
diagrams. It's the style of textbooks, reports and research papers."
I currently do my homeworks/reports using Scientific word, and if I
need to post some results from Mathematica (a plot or some
calculation), I cut/paste the result in. I found this a little easier
actually for typesetting a report with lots of Mathematics in it, but
this is only becuase I am more familiar with SW for writing reports
with, with its short cuts etc...
I'd like to spend more time to learn more how to typeset full reports
in Mathematica, I think that would be easier than the way I currently
do it.
How good is Mathematica Latex output? Can one write a PhD thesis all in
Mathematica for example? any examples on the net of this being done to
look at?
Nasser
King, Peter R
I have now had the time to read all the responses to my initial response
and I can't really argue with the main points, in fact I don't think I
ever stated that Mathematica should not be used in the teaching of
mathematics (with the caveat below). Yes it does enable you to do all
the things that you and others have stated and can enormously increase a
student's abilities to do things. This wasn't the thrust. My concern was
about students who claimed never to have used pen and paper and only to
have used Mathematica. I think this is dangerous. Why?
1) suppose there is a bug (shock horror they do exist) or the student
has mistyped things, how do they check the results if they can't do some
kind of manual check themselves? Can the student do a rough estimate of
what they expect the answer to look like? Do they understand the answer
and what it means? Sure they could plot it out (but then why not just
write a program to solve the problem numerically in the first place).
This doesn't mean that students shouldn't use MAthematica but it does
mean they should also be able to do calculations by pen and paper when
they are comfortable with that they can move on and use the tools that
enable them to do "more advanced" and "more interesting" things.
2) Related to this, actually I am very concerned about the current
generation that has been brought up on calculators. it HAS generated
people who cannot do simple calculations without one. When a student
asks me how to divide 1 by 2/3 because he hasn't got a calculator I get
worried. When I see exam scripts where people give the answer E (for
error) when they take the square root of a negative number I get
worried. More importantly students (not all but a significant minority)
don't actually understand what numbers mean. I see lengths quoted to 10
significant figures (implying a measurement accuracy on the sub atomic
scale). This has happened over a period of probably 20 years and
reflects poor education policies towards mathematics and is probably
beyond the scope of this thread (or indeed this list) but it has
happened because people have taken the attitude why bother to learn to
do multiplication when a calculator can do it quicker and more
accurately than you can. I would be worried to go down the same track
with more advanced mathematics. I strongly believe that the basic skills
should be learnt first on pen and paper and then reinforced using tools
like mathematica. I do also believe that Mathematica can be used as part
of the learning and reinforcing of the basic skills - just not as a sole
substitute. This isn't just an issue of preserving old skills. After all
we bother to teach people to read. Why? technology can give us spoken
text. I think there are some skills (and this includes mathematics) that
are so basic that if we cannot perform them we are missing something.
Also often we are forced to operate without the use of these tools. Such
as in the field, in meetings without access to computers, in companies
that can't afford or don't want to pay for software licenses (I spent
many years working for a large multinational that I had to convince very
hard to buy a single licence for MAthematica because they couldn't see
how it would affect their business performance - this is not uncommon).
3) Why Mathematica (this is the caveat I referred to above). Now this is
probably heresy or blasphemy to this list but there are other computer
tools for doing mathematics. All these tools have there pros and cons.
They all have their quirks some of which distract from the underlying
mathematics (some of which may enhance). There is a danger that students
get caught up with the intricacies of how to do a particular operation
in that particular package rather than the underlying mathematics. You
could argue that the mathematics is the basic "truth" and the
implementation package is something different (a bit like Plato's shadow
worlds). However, this is an interesting philosophical question that I
don't really want to go into here (pen and paper, is if you like,
another package and how much is mathematics limited by our ability to
write things down and solve analytically by hand and how much is it
enhanced by using the power of computers, expecially for visualising
complex data or phenomena). I haven't seen this with mathematical
packages but for other commercial software I have seen students held
back by learning the idosincracies of packages and claiming something
can't be done simply because the software can't do it. In other owrds it
can limit the student's abilities to do things because of the
limitations of the package. Again this is not a reason for not using
Mathematica in education but it is a reason not to rely on it solely and
to teach students there are other ways of doing things (including by
hand or with other packages).
Finally I would like to say that the response on this list has been
almost overwhelmingly in favour of using Mathematica in education and I
would support that wholeheartedly. But that support is tempered by the
requirement that the students are actually learning how to do the
mathematics properly, when required they can think on their own feet and
not rely any particular package and that they are learning not just how
to use a tool but how to use the underlying subject.
I would also point out that that the support for Mathematica on this
list is not entirely unbiased (it is after all made up of people who are
Mathematica users and experts). If I went to the other packages forums
(which must exist, I have never checked) I expect i would see them
strongly advocate the use of their own particular package and if I were
to go to the general group of educators I expect i would see a very
different response. It is easy to dismiss them as being behind the times
or out of touch, but they do represent a very big experience bank.
Peter King
> use pencil and paper. Theodore Gray has provided us with the wonderful
> notebook interface. You can have titles, sections, text
> cells, equations and
> diagrams. It's the style of textbooks, reports and research
> it comes to ease of use there are many gaps. I believe that
> Mathematica can
> truly effect a revolution in technical education. But it is
> not as simple as
> just installing it on a departmental server. A lot of
> preparation is needed.
> Additional packages geared to student use are needed.
> Educators have to
> learn how to take advantage of the resource. (For example how
> to shift from
> quick calculations to essay type questions.)
>
> David Park
> dj...@earthlink.net
> http://home.earthlink.net/~djmp/
>
>
János
a small experiment.
I have two boys with age 13 and 9. Both are good students and good
piano players. They compete with each other in as many way they can
think of. The older one is doing only those activities which are his
interest. I bought Mathematica Student edition for him a year ago
but he used it only once or twice. Whenever I showed him different
calculations - like Solve and Simplify to check his school homework,
he listened, looked interested, but he never went back to use it on
his own. The 9 year old is an explorer. If I am not finding
something - because my wife did some re-arrangement - I am just
asking him and he knows exactly where things are. I also bought
Mathematica Teachers Edition and yesterday I installed it on the
younger one's computer. I showed him how to calculate 2/3 + 5/8 ,
how to ListPlot a bunch of random integers, and similar 4th grade
things. One hour after I left him alone he was still ListPlotting
interesting set of numbers and used Play to hear them - with older
brother watching over his shoulder :)
So the experiment will be that I will teach the younger one for any
kind of neat tricks from week to week and watch how much will trickle
down to the older one through pride, envy and competition.
I am also looking for good educational notebooks/packages for their
levels in all areas. Any good suggestions ? /Unfortunately Wolfram
Tunes is out because at home I do not have Internet connection./
János
----------------------------------------------
Trying to argue with a politician is like lifting up the head of a
corpse.
(S. Lem: His Master Voice)
David Park
1) A skilled teacher who has given much thought on how to use Mathematica in
an educational setting and has gained a lot of experience.
2) An institution that has given her serious support.
3) All the students have access to Mathematica all of the time.
4) The students learn Mathematica early in their college career so they will
have easy use of it in their more advanced courses.
I hope people flock to the University of Vermont to see how it's done.
I'm sure there are others also and I hope we hear from some of them.
It was especially interesting that students do better on the Mathematica
assisted portion of their tests than the non-Mathematica portion. And it's
because they can actually try things and see whether or not they actually
work.
The response by G. Raymond Brown was also interesting but here there was
much less support from the institution. It must be much more difficult to
'do Mathematica' halfway, or when only some of the students have it.
The Theodore Gray, Jerry Glynn essay that Matt points us to is priceless.
From: Helen Read [mailto:h...@together.net]
[parts snipped.]
At my institution, we have a university wide site license allowing us to
install Mathematica on all of our computers, not just those owned by the
university, but also laptops and desktops owned by faculty, staff, and
students. Mathematica is available to everyone literally 24/7.
I have been teaching calculus with Mathematica for 10 years, for the
last 6 of those years in a classroom equipped with 31 networked PCs (one
for each student, plus one for the instructor), and a printer. We now
have two such rooms. The instructor's machine is connected to the
overhead projector, and we have software allowing easy communication
between the student PCs and the instructor. I can, for example,
broadcast my screen or any of the students' onto the projector or onto
everyone's monitor. Unlike the computer labs on campus, these rooms are
designed for teaching, with clear lines of sight from every student to
the teacher and whiteboard, enough space for the instructor to walk
around and interact with the students, etc.
My students use Mathematica routinely.
I give two-part tests. Part I is done with pencil and paper...
When a student finishes Part I of the test, s/he turns it in and picks
up Part II, which is done with the use of Mathematica.
All of my students, even the weakest among them, are quite comfortable
using Mathematica by the end of the semester.
The students almost always do better on Part II (the Mathematica
Mike
<n...@12000.org> wrote:
See a recent (sometime earlier this year) posting from Paul Abbott. Paul
provided a link to documents (PhD and honours theses) that had been written
entirely in Mathematica.
Mike
ggr...@sarj.ca
<snip>
> I have also strongly encouraged my students to buy and use
> Mathematica (my institutions still, despite my urging, do not have
> site licenses for the software, and students mostly consider the
> software too expensive to obtain for themselves).
I don't understand this excuse from students. The student edition of
Mathematica (at least when I bought my v5 copy) was less than the
price of most of my advanced Physics/Chemistry/Math textbooks. Most
of those had to be bought new (or relatively new) anyway just because
the publisher changed editions fairly frequently.
And unlike some other CAS software, the Student Edition of Mathematica
doesn't (didn't?) have stripped down functionality. All I lost was the
printed copy of the Mathematica Book, which isn't a big problem with
all the online resources.
<snip>
> Mathematica brings a host of benefits to any party, but IMHO its greatest
> benefit to education lies in its enabling of asynchronous mathematical
> discourse between students and instructor.
I think this is a critical point that hasn't been addressed so far.
From my experience, in undergrad, there were many physics courses that
required relatively advanced mathematical concepts at times when the
student wouldn't have gone through the corresponding math course. So
in the pencil and pen model, it seemed like a large fraction of the
time available for the physics course was spent learning the mechanics
of the required math, when it could have been spent learning the
underlying physics concepts. And it's not like the student wouldn't
have seen the mathematical detail, the syllabus required the advanced
math courses that covered these topics, just they were scheduled for
subsequent semesters.
Paul Abbott
In article <dv69t4$oec$1...@smc.vnet.net>, ggr...@sarj.ca wrote:
> I think this is a critical point that hasn't been addressed so far.
>
> From my experience, in undergrad, there were many physics courses that
> required relatively advanced mathematical concepts at times when the
> student wouldn't have gone through the corresponding math course. So
> in the pencil and pen model, it seemed like a large fraction of the
> time available for the physics course was spent learning the mechanics
> of the required math, when it could have been spent learning the
> underlying physics concepts. And it's not like the student wouldn't
> have seen the mathematical detail, the syllabus required the advanced
> math courses that covered these topics, just they were scheduled for
> subsequent semesters.
I agree that this is a very important point. Later this semester I am
teaching an introductory (second year) mathematical methods of physics
course. I have not found the "perfect" text, but my favourite is
"Mathematical Methods of Physics" by Mathews and Walker (1964). However,
this book is intended for senior physics undergrads. Nevertheless,
through judicious use of Mathematica, I think much of this course can be
taught earlier.
"Standard" texts like "Mathematical Methods in the Physical Sciences" by
Boas (2006) attempt to be encyclopedic. What I like about Mathews and
Walker is that the focus is on physics and the choice of topics and
examples are the ones that I use regularly, and wish that I'd learnt
when I was an undergrad.
Andrzej Kozlowski
I have been using Mathematica as a basic teaching aid for over ten
years. In fact I nowadays use in on all the undergraduate courses I
teach in very different environments (on the one hand I use it for
teaching "Mathematics for Physics" at Tokyo Denki University in
Japan, and on the other hand "(Financial) Derivative Pricing with
Mathematica@ at Warsaw University in Poland.) So clearly I agree with
most things that have been said in favour of using Mathematica (and
also other CAS - just in case RJF is reading this ;-)). I also agree
with all the comments below. However, there is just one note of
caution I would like to add that seems to have been omitted. I think
it is a big mistake to identify all mathematics with what should be
called "computational" or "algorithmic" mathematics. Many people have
written about the relations and the differences between the two.
Particularly interesting are are various essays by Donald Knuth (see
his "Selected Papers on Computer Science") as well as various
writings of Roger Penrose particularly "The Emperor's New Mind" where
he actually tries to describe the difference between algorithmic and
non-algorithmic thinking. In the early days when computer science
was very new many mathematicians disdained this upstart subject,
which they considered as essentially trivial. One does not often meet
such attitudes today. However, there is now the danger than many
institutions are falling into the opposite extreme and reduce all
mathematics basically to the algorithmic approach (the bad ones do
not even do that, instead they reduce learning mathematics to
memorising algorithms and the worst ones simply to learning how to
push buttons or write commands in a CAS). The result is that fewer
people learn to think geometrically although it was actually
geometric thinking and not computation that was behind most of the
great discoveries both in mathematics and mathematical physics . Non
algorithmic mathematics (particularly various kinds of geometry) is
usually much harder to teach than computational one because to an
even larger extent it depends on inspiration and talent. Hence there
is a strong temptation to eliminate such courses from the syllabus
thus causing irreparable damage to the quality of understanding of
mathematics. I talk to people who teach mathematics and universities
in many different countries and everywhere I hear the same story:
while student's expertise in various aspects of computing has been
increasing by leaps and bounds the quality of their mathematical
understanding has been correspondingly declining. So the point I want
to make is basically this: by all means teach students to use and
understand Mathematica for all kinds of tasks in computational and
algorithmic mathematics but do not give them the impression that the
kind of things you can do with Mathematica is all there is to
mathematics. The remark I first heard about 20 years ago: "Oh, you
are a mathematician, I thought that is all done by computers
nowadays" can be heard even from scientifically educated people, and
I would hate to think that Mathematica is making a contribution to
spreading this totally wrong and harmful idea.
Andrzej Kozlowski
>> use pencil and paper. Theodore Gray has provided us with the
>> wonderful
>> notebook interface. You can have titles, sections, text
>> cells, equations and
>> diagrams. It's the style of textbooks, reports and research
>> David Park
>> dj...@earthlink.net
>> http://home.earthlink.net/~djmp/
>>
>>
fizzy
Hello....
I'm that "student" by the way.....and actually I'm 61 years old.....I
decided to go back to school and take a course in Electromagnetics
especially now that I had a tool like Mathematica to work with.......I write
this reponse because I wonder why you wrote "proudly
proclaimed"...?......as if I should be criticized for what I had done.....as
if the enjoyment that I get from using Mathematica, in whatever way I can to
help me improve my work and , actually, to enjoy it all the more, deserves
some kind of criticism, especially this idea of the glory of 'pencil and
paper'....in fact, now I have this 'beautiful' Notebook with all my
calculations, clear as a bell.....which I can use over and over again
whenever I need ....which I can re-read easily.....and soon I will take the
second sequence in the course........I've worked with a theorist at
work...and I've watched him write pages and pages of 'pencil and
paper'....and often to find a mistake on the first page....and then have to
re-do all the other pages...it took me years of coaxing and convincing to
get him to use Mathematica.....and that's because , like you, he believed in
this assinine idea of "a fool with a tool is a fool"....he just didnt get
it.....now he finally uses Mathematica ....and has learned that when he
makes a mistake on the first page, he only has to correct that, re-run his
program and then get 'correct' results....took me 5 years of continual
persuasion to get him to use Mathematica......."No, a fool who wants to be a
fool , remains a fool....a fool who finds a tool and works hard at it and
learns how to use it and learns from a group like Mathgroup how to use it
correctly is no longer a fool or, at least, he's much less of a
fool"....and, as far as Mathematica goes, I find myself a very happy
scientific fool....
I think you really misunderstand the nature of the average scientist and
engineer....he wants to improve his technical skill at all
costs.....Mathematica and Mathgroup have helped me enormously to improve my
technical skills.....and by the way, I always use a calculator to add
numbers and fractions....always....because I hate that idiotic and tedious
procedure....and when I do it by hand, I invariably come up with the wrong
result.....much better to be a fool and use a calculator and get the right
result...
Let me ask this....if using Mathematica is so easy, so straight forward, and
so on, then why is there Mathgroup at all??...
And the answer is really very simple.....it's not....and knowing the math
and how to use it is not straight forward....and the number of interesting
and enlightening comments by the people in Mathgroup who are very astute at
mathematics and who explain the math behind the commands is very
enlightening......
I have learned that those people who know only the Mathematica Command
structure will not really get very far with Mathematica.....and that only
those people with 'in depth' comprehension will ever get the full benefits
of Mathematica....I am not mathematically astute....so I have to have
Mathematica help me....but I always try to learn from it....in any way that
I can....it and Mathgroup have helped me in countless ways to understand
mathematics better.....and since I'm a physicist this learning has been
invaluable to help me understand physics much better...hence, the final
result, that is, much more enjoyment in the work that I do....which is
really the bottom line, for me anyway.....
Jerry Blimbaum
----- Original Message -----
From: "King, Peter R" <peter...@imperial.ac.uk>
Subject: Mathematica and Education
> David,
>
> I think actually we are possibly talking at cross purposes. Although I
> also think this leads to some interesting points. I was responding in
> the first instance to a post from a student who had proudly proclaimed
> that he had done all of his assigments without using pen and paper at
> all and only in Mathematica. My view is that this is dangerous for the
> reasons outlined. And in fact my concern is not for the professional
> technical expert (or the prospective ones). In my experience a good
> student who is capable and motivated to understand the material will do
> so and tools like Mathematica can make this an even more exciting
> experience. For them I would agree 100% with what you have said.
> However, poorer students will simply learn how to use the tool as a
> black box. Superficially they will look as skilled as the others because
> they can produce a glossy presentation and they can end up causing alot
> of trouble (I am absolutely serious about this I have seen it - not with
> Mathematica but with other commercial software, which is where my fool
> with a tool quote came from). for weak students they will think that
> learning how to do something substitutes for learning how to understand
> something. As for education for the masses actually I do think there is
> a point. As well as learning about Shakespeare, Bach, Beethoven,
> Michealangelo, Cezanne and so on every civilised person should know
> Newton's laws (with Einstein's amendments), thermodynamics, trigonmetry,
> algebra, calculus and the principles of scientific study.
>
> My anology with calculators is to show that reliance on using a tool
> can weaken the more traditional skills. Not in itself a problem. As you
> very eloquently pointed out I cannot sharpen a spear and hunt for my own
> food, it is easier to go down to the supermarket. If shops ceased to
> exist I would be stuck! The point being that there are some
> cirucmstances when we don't have calculators about us or when it is
> easier or even better not to use them (I think it is a great skill that
> is underestimated and lost to be able to estimate answers to problems
> without doing an "exact" calculation). I make this analogy, not because
> I think that Mathematica is a glorified calculator (it is much more than
> that) but because I remember the discussions 20-30 years ago when
> calculators were first introduced. Many people in education argued that
> they would enable students to do more calcuations, more quickly and get
> on to the "more interesting stuff" because they weren't being held back
> by "boring" caclulations. In fact I think for weaker students the
> opposite has happened. They do the caclulations but don't understand
> them - as I said they don't really understand the concept of numbers at
> all. This hasn't happened with more advanced mathematics (yet?) because
> the tools haven't really been available for so long but the same
> arguments are being made. My concern is that the same mistakes will be
> made and we will end up with a less mathematically literate population
> then before. Now this is not inevitable and so this is the real
> challenge. How can we ensure that these tools are used appropriately to
> improve mathematical skills and not used lazily to reduce them. There
> have been some excellent and thought provoking examples of how people on
> this list have done this. My fear as that these few sites of excellence
> will be outnumbered by rather lazy teachers (& students) for whom
> Mathematica will used as a black box. I hope I am proved wrong.
>
> I must also confess that I have another sneaking fear. I first learnt
> progamming by writing FORTRAN IV on punched cards on mainframes. I could
> probably still do it tomorrow if such things existed. However, the
> energy barrier to learning more modern languages has meant that I
> haven't bothered. OK so I'm a dinosaur and therefore out of touch. My
> worry is for people who have only learnt how to use MAthematica, when
> that is superceded (and again I am sure that it will) will they have the
> ability to move on to something else if they haven't learnt the
> underlying mathemtics properly. However, the mathematics is much more
> enduring.
>
> Anyway I don't really want to keep batting this around, I am sure
> everyone knows my point of view by now. I hope the message I have got
> across and that I suspect we would agree on is: anyone claiming to use
> mathematics must really understand the basic maths (however they are
> taught it); there is no doubt that modern technologies have an essential
> and deep rooted part to play in education, research and practical
> application in mathematical science; the challenge is to ensure that
> this is done thoughtfully and carefully to ensure best practice and
> maintainence of fundamental skills (rather than erosion as I feel has
> happened in some sectors with numeracy skills). I would be very pleased
> and interested to hear about how this can be done (and perhaps I can
> sneakily take advantage of some of these ideas in my own teaching).
>
> Best wishes,
>
> Peter
>
>
Bill Rowe
On 3/15/06 at 6:28 AM, peter...@imperial.ac.uk (King, Peter R)
wrote:
>I think actually we are possibly talking at cross purposes.
>Although I also think this leads to some interesting points. I was
>responding in the first instance to a post from a student who had
>proudly proclaimed that he had done all of his assigments without
>using pen and paper at all and only in Mathematica. My view is that
>this is dangerous for the reasons outlined. And in fact my concern
>is not for the professional technical expert (or the prospective
>ones). In my experience a good student who is capable and motivated
>to understand the material will do so and tools like Mathematica
>can make this an even more exciting experience. For them I would
>agree 100% with what you have said. However, poorer students will
>simply learn how to use the tool as a black box.
While I understand the concern you are raising, I don't see that requiring anyone to use paper and pencil rather than Mathematica as being a solution. After all, paper and pencil are just different tools. They don't confer greater understanding of mathematics than any other tool. In fact, one could make an argument the greater experimentation possible with Mathematica makes it the tool that is likely to result in greater understanding. Of course this assumes the student takes advantage of that capability. And likely, students likely to treat Mathematica as a block box will not avail themselves of this capability.
For example, consider the computation of simple statistics such as the mean or median of a data set. The mechanics of this computation is easily learned and easily done either with Mathematica or paper and pencil. But neither Mathematica nor paper and pencil will confer any understanding as to why one or the other (mean or median) should be used and what they signify about the data set.
>Superficially they will look as skilled as the others because they can
>produce a glossy presentation and they can end up causing alot of
>trouble (I am absolutely serious about this I have seen it - not with
>Mathematica but with other commercial software, which is where my fool
>with a tool quote came from).
That isn't an issue with the software. Instead, this issue is a lack of critical thinking about presentations by the audience. With an audience that employs critical thinking and has the background to understand the presentation, a glossy presentation will not hide a lack of skill in the subject on the part of the presentation creator.
--
To reply via email subtract one hundred and four
bsye...@gmail.com
I do not believe that there is a single way of understanding
mathematical/physical/... ideas.
After all, if there was a single way, we wouldn't see any
examples/exercises
in scientific textbooks, isn't it? The required set of theorems and their
proofs would be sufficient in this case.
I fully support Andrej's comments, but I have the feeling that an important
issue is missing in this discussion. Mathematica can be a valuable tool in
this context due to the option to combine text (typesetting), graphics, and
programs (symbolic, algorithmic, numerics, etc.). This makes Mathematica a
wonderful tool for EXPRESSING ideas.
I strongly emphasize this property to our students and encourage them to be
fluent with Mathematica. I still let them be aware of not using it as a
"black box" by giving them problems that require innovative and critical
thinking.
Also there are enough ways to a professor to check the theoretical level and
skills of the students and prevent them from making Mathematica a "black
box" or a "magic calculator". This might require a little more dedication
but it benefit with better students.
One may find solutions to theoretical problems in various theoretical
subjects available today on the Internet. Using this "blindly" is similar to
using Mathematica as a "black box". Although we have no control of using
such "resources" we can control the final level that student need to have in
order to pass the final exams which should have a substantial theoretical
part.
As was described earlier in one of the posts, Mathematica was used as the
"machinery" to study a course in electromagnetic fields. Although I find it
a little bit extreme in this specific case, I find no harm if this is used
to enhance understanding of the theory and basics of the subject.
I remember well while being an undergraduate (many years ago) most of the
students had difficulties in ectromagnetic fields and electromagnetic waves.
The text books at the time were lacking enough enlightening examples
(especially in EM waves). Mathematica would be great for that matter (but
did not exist at the time). I just wonder how Mathematica was allowed to be
used in the final exam. This (at least for me) seems a "line crossing" of
the academic institute..
As for Andrej's remark of better programming skills and less theoretical
skills of current day students, I find it (painfully ) true. But, still, I
will not bet on such students to be leading something in the future of their
professional life. Such students will always be on the (lower) technical
side and the better ones with the better theoretical capabilities and
understanding will determine the path. Being in high-tech employee (for
example) does not guarantee of having strong "theoretical skills".
Summarizing, one need to define what are the building blocks of knowledge
that are needed to be taught to the students and what is the best way of
teaching that. Also, one need to define the "studying habits" of the
students. Mathematica certainly can be a helpful tool for both with a
careful use and control from the teaching staff.
with best regards
yehuda
On 4/16/06, Andrzej Kozlowski <ak...@mimuw.edu.pl> wrote:
>
> [This post has been delayed due to email problems - moderator]
>
>