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MathGroup Archive 2006

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Re: Re: Mathematica and Education

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65014] Re: [mg64957] Re: [mg64934] Mathematica and Education
  • From: "David Park" <djmp at earthlink.net>
  • Date: Sat, 11 Mar 2006 05:15:58 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

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.

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.)

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

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
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



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