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Re: Re: Re: learning calculus through mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg108058] Re: [mg108015] Re: [mg107971] Re: learning calculus through mathematica
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 7 Mar 2010 04:06:17 -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>

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.

However, I have no doubt that learning to use a CAS should be an 
essential element of learning calculus and the reason for that is not 
that a CAS enables one to understand better those aspects that have 
traditionally not required it but because it opens up completely new 
areas for exploration and well as greatly increasing the efficiency of 
many traditional computational tasks. In other words, I see using a CAS 
as much an integral part of modern calculus as knowing the basic 
techniques of differentiation and integration etc. In the case of the 
great majority of people who are learning calculus today one can say 
that if they ever find themselves using calculus outside their calculus 
class they will be using some kind of CAS to do so. The most efficient 
way to learn how to use a CAS in calculus is to combine learning to use 
the CAS with learning calculus itself. I don't know if students who use 
a CAS in their calculus courses are better at the kind of things that 
can be done without a CAS than students who learned calculus the 
traditional way, but I am sure the former (potentially) could do many 
things that the latter couldn't and that it is these kind of things that 
most often come up "in the real world".

Andrzej Kozlowski


On 5 Mar 2010, at 10:33, David Park wrote:

> Certainly not every student should be learning Mathematica at the 
earliest
> possible age. But maybe those who are seriously interested in a 
technical
> career and are motivated should. Maybe it wouldn't be a part of 
regular
> secondary school education, but be done on their own, or in math 
clubs, or
> via mentoring over the Internet.
>
> Maybe it's true that CAS have not made a significant positive impact 
in
> technical education. Does that mean people should give up? Maybe we 
haven't
> properly learned how to use them yet. When new technologies come in 
they are
> often used to just make the old approaches more efficient. Usually 
what is
> needed is entirely new approaches. Instead of mass lectures and mass 
exams,
> maybe there should be more self study, more mentoring and more 
mathematical
> essay writing. As things stand now I have the sneaky suspicion that 
students
> just don't know Mathematica well enough and it is another obstacle to
> getting through the course. So, why should they do better?
>
> Also, Mathematica off the shelf is not a great educational tool. It 
does too
> much at a high level with commands like Solve, Integrate or Limit. 
That's
> all fine, but students need something I call "hierarchical depth", the
> ability to do mathematics at different levels and see how things work. 
It is
> somewhat ironic that as a computer algebra system, Mathematica (and I
> suspect most systems) are poor at providing the kind of algebraic
> manipulations that students need to work with. They are hierarchically 
thin.
> This all could be provided, but it takes more work.
>
>
> David Park
> djmpark at comcast.net
> http://home.comcast.net/~djmpark/ 
>
>
>
> From: Richard Fateman [mailto:fateman at cs.berkeley.edu]
>
> There is a substantial list of links to calculus resources at
> http://www.calculus.org/  This includes complete on-line courses.
>
> I have searched in vain for objective evidence that students who learn
> calculus with a computer algebra system at hand learn it better than
> students without such a tool. (e.g. higher exam grades.)
>
>  This is disappointing to people who would like every student to learn

> how to use a CAS at the earliest opportunity.
>
>   Historically, the big success for calc students was using computers

> to plot functions. Handy to understand slopes and areas. Very easy to 
use.
> Not so prone to arithmetic mistakes, though with problems of their 
own.
>
>
>
>
>
>
>
>
>
>

• References:
  • Re: Re: learning calculus through mathematica
    • From: "David Park" <djmpark@comcast.net>