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Re: Re: Re: Mathematica and Education

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65721] Re: [mg65075] Re: [mg64957] Re: [mg64934] Mathematica and Education
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sun, 16 Apr 2006 03:48:46 -0400 (EDT)
  • References: <200603141059.FAA24082@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

[This post has been delayed due to email problems - moderator]


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


On 14 Mar 2006, at 11:59, King, Peter R wrote:

> 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
>
>> -----Original Message-----
>> From: David Park [mailto:djmp at earthlink.net]
To: mathgroup at smc.vnet.net
>> Subject: [mg65721] [mg65075] RE: [mg64957] Re: [mg64934] Mathematica and  
>> Education
>>
>>
>> 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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