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MathGroup Archive 2006

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RE: Re: Re: Mathematica and Education

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65101] RE: [mg65075] Re: [mg64957] Re: [mg64934] Mathematica and Education
  • From: "David Park" <djmp at earthlink.net>
  • Date: Wed, 15 Mar 2006 06:28:13 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Peter,

I don't think that anyone would argue that students and researchers should
never use pencil and paper. What I am arguing is that one can use a
Mathematica notebook just like pencil and paper, and it's advantages (which
I listed in a previous posting) are so great that eventually one will find
oneself doing more and more in the notebook and less and less on the side.
And if one learns how to write, derive and calculate clearly in a
Mathematica notebook, this might actually carry over to pencil and paper
work.

It isn't something I thought up myself but was built into the notebook
concept from the beginning by Theodore Gray. And there are plenty of
examples of good style from Paul Abbott, Stan Wagon and many others. But,
despite its being right there in front of us, many users fail to take
advantage of it. Some users don't use any Text cells or Sections at all.
Some users do most everything in word processing mode, typing expressions in
Text cells, and barely use the kernel. Some users want to design a whole new
GUI without learning how to use the notebook GUI. Some users want to morph
Mathematica into their favorite programming language. The standard
Mathematica usage and notebook interface are very good and if students and
researchers would learn it they would get up to speed much faster.

Let's agree that we are talking about people who are seriously interested in
a technical career. Teaching rudimentary math to the masses is an entirely
different matter. I don't even know what the objectives and issues are for
mass math education. A CAS would almost certainly be too great a hurdle for
most of them so let's leave that out of the discussion. We're talking about
'the happy few'.

Many of your objections relate to calculators and not to Mathematica. I'm
arguing that one should not think of Mathematica as a 'calculator' (although
it clearly can act as one) or as a 'programming language' (although it
clearly is a type of programming language) but as 'pencil and paper' for
writing text, making definitions, doing calculations, derivations and
transformations, and drawing diagrams. The paradigm and mind set with which
one approaches Mathematica can make a hugh difference in getting effective
use from it.

It may not be strictly true that one can do every technical derivation in
Mathematica or that it would be the most efficient way, or lead to the
deepest understanding. But, for all practical purposes, for almost all
students and most researchers, it is the best way. There are far fewer
errors than with hand calculation. There is an easier possibility and time
for alternative approaches. Answers can be more easily checked against
specific cases. Definitions and methods have to be explicitly given and
these are usually the nuts and bolts of deeper understanding.

Learning how to communicate substantive technical ideas clearly is far more
important than being good at mental arithmetic. Mathematica notebooks follow
the standard, time-honored style of technical communication. It is not at
all counter to what students should be using or with standard pencil and
paper work.

It would be nice if there was one CAS that was very good, inexpensive and
everybody used it. That might come about some day when CASs will be a
similar as slide rules were. But in the meantime we have to live with the
world the way it is and let market forces fight it out. It is definitely a
problem.

Students and researchers should be able to think in terms of their subjects
and not have to worry about the 'intracacies' of the CAS they are using. In
this sense I think there are many gaps in Mathematica relating to
convenience of use. These can be filled by additional packages geared to
broad fields. I think good packages will generally follow the Mathematica
paradigm and principally add new convenience commands. (The Combinatorica
package would be a good example.) Also, good packages for students should be
developed with lots of actual usage on existing textbooks. In fact, the best
method of development is to work through a good textbook and try to add
general routines that fill the gaps and remove the 'intricacies' that might
relate only to programming methods.

My interest in Mathematica is in trying to learn some modern math and
physics, specifically general relativity and differential goemetry. I'm
currently trying to write a differential forms package to go with Tensorial.
I'm trying to work through a textbook 'Advanced Calculus: A Differential
Forms Approach' by Harold M. Edwards. (Edwards is a very good mathematician
and teacher, and this is a very good book.) I haven't gotten very far at all
without spinning off a number of routines that make it much easier to work
with the material.

An exercise I was working on yesterday involved deriving, using Cartesian
coordinates!, a 2-form for spherical flow from a source at the origin.
Edwards gives only a plausible derivation where he used 'analogy' and an
'enlightened guess'. With Mathematica I was able to derive an answer
explicitly from the basic principles. I used GramSchmidt to calculate two
orthoginal vectors to the radial vector {x,y,z}. These expressions were
fairly complicated. I could not have done it by hand. I could then write a
general expression for the 2-form that represents the flow, and equations
for the flow through a surface perpendicular to the radial direction and the
other two orthogonal surfaces. Mathematica could easily solve these
equations to get the coefficients in the 2-form. But then I got an extra
Sign[x] factor in my solution that Edwards did not have. (Suprise! Error of
method?) I then realized that GramSchmidt will have a singular line (x == 0)
in the way that I set it up, and the orientation of the vectors might change
when we crossed that line. I wasn't certain of that so I made an animation
as we ran around the equator, and sure enough, there was a discontinuous
flip in one of the vectors as we crossed the x == 0 line. So I could throw
out the Sign[x] factor to keep the orientation the same. Then I realized
that the derivation was no good for x == 0. But I've read about atlases and
charts and realized that I could set up GramSchmidt so that the singularity
was at y == 0, or z == 0, instead - and always get the same final result. So
I learned or reinforced a mathematical method and principle. I wouldn't have
learned any of that if I had tried to 'analogize' it through with pencil and
paper.

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/







From: King, Peter R [mailto:peter.king at imperial.ac.uk]
To: mathgroup at smc.vnet.net

David, (and all the othes who responded),

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



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