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Re: Re: Re: Re: learning calculus through mathematica

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  • Subject: [mg108083] Re: [mg108058] Re: [mg108015] Re: [mg107971] Re: learning calculus through mathematica
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 8 Mar 2010 06:11:36 -0500 (EST)
  • References: <hmiiop$3v3$1@smc.vnet.net> <hmlf25$jsp$1@smc.vnet.net> <3434740.1267699632618.JavaMail.root@n11> <201003050933.EAA29405@smc.vnet.net> <201003070906.EAA01345@smc.vnet.net> <4B93C325.6040307@math.umass.edu>
I don't see any real contradiction between this and what I wrote. "Long 
symbolic computations" can well qualify as something that isn't "fully 
accessible" to many students without a computer. Also, I don't think you 
would dispute that all students should be able to do "short symbolic 
computations" by hand. If they don't, then, in my opinion, they will 
never fully understand what it is that Mathematica is doing for them.

Ultimately it is a question of finding the right balance. There are lots 
of things in mathematics that one needs to do just once completely by 
oneself in order to develop an intuitive understanding of what is 
involved. Once this understanding has been developed, there is no need 
to perform ever again these often tedious computations and manipulations 
by hand. It seems to me that showing students that all such things can 
be done by a CAS before they have understood the basic concepts can 
sometimes be seriously harmful (of course mostly to the "the lazier or 
intellectually weaker" ones - the others would probably not be satisfied 
with mere "button pushing"). On the other hand, there are enormously 
many fascinating things in calculus and other areas of mathematics that 
can't be done by an average student without a CAS either at all or 
within a reasonably time period. I think all computer aided calculus 
courses should include such examples (and even perhaps concentrate on 
them) because it is only such examples that can really convince both 
students and sceptical academics that CAS can be seriously useful in 
mathematics. 


Andrzej Kozlowski

On 7 Mar 2010, at 16:15, Murray Eisenberg wrote:

> One reason is very simple: by using a CAS to do many long symbolic 
calculations, students can focus on modeling and the resulting and 
relevant mathematical concepts and methods -- not the details of 
carrying out long chains of algorithmic, algebraic steps.
>
> My 45 years of teaching make perfectly clear that, for most students 
in calculus, e.g., they are so involved in trying to get the symbolic 
manipulations right, they have little or any idea of why they're doing 
them.  They totally miss the forest for the trees.
>
> The other side of this situation, I regret to say from my experience, 
is that the lazier or intellectually weaker students are often incapable 
of rising above merely carrying out mechanically the symbolic 
manipulations -- many of which they get wrong anyway -- to have much of 
an understanding of the higher-level concepts involved.
>
> On 3/7/2010 4:06 AM, Andrzej Kozlowski wrote:
>> I have never seen  or heard any convincing reason why using a CAS 
should
>> make it possible to understand and learn better those areas of
>> mathematics which are fully accessible to a student with only a pen 
and
>> paper. In fact I can see a few reasons why the opposite might be the
>> case. In many situations I can see clear advantages in performing
>> algebraic manipulations "by hand" or even "in the head", which is, in 
my
>> opinion, the only way to develop intuition. The same applies to
>> visualisation - while being able to look at complicated graphics can
>> often be a big advantage, I always insist on students developing the
>> ability to quickly sketch simple graphs by hand on the basis of
>> qualitative analysis of analytic or algebraic data. This is again
>> essential for developing intuition and I am not convinced that doing 
all
>> this by means of a computer will provide equivalent benefits.
>
> --
> Murray Eisenberg                     murray at 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
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