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Re: Re: Re: learning calculus through mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg108152] Re: [mg108015] Re: [mg107971] Re: learning calculus through mathematica
• From: eric g <eric.phys at gmail.com>
• Date: Tue, 9 Mar 2010 06:24:08 -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 took two graduate level applied mathematics class and in one of them 
the professor refused to take any homework not done by hand and without 
the help of Mathematica or similar softwares... I think certain things 
you should do it by hand especially if you are learning. For example, 
the typical calculus I problem is to graph a function (finding its 
roots, maxima/minima, and limits) in doing it by hand you learn and 
practice it, but once you get it and pass the class I just use Plot, an 
even if you want to check you are right before turning the homework use 
mathematica to check your results.

For a teacher is really nice to have all the resources possible to have 
good power-points presentations and to explain things, here I think 
mathematica plays an important role...

I learnt mathematics in a third world country, and even I didn't have 
money for having a scientific calculator, I learnt many things using 
classic calculus books...later in the states where  I did my graduate 
studies I heavily used mathematica for doing homework and displaying 
results... In one occasion part of a solid-state problem was needed to 
integrate cosine fuctions times polinomials, it was tedious to do it by 
hand (integrating by part or wath-ever other technique I learnt in my 
undergraduate)... I know a student who spent a week doing the integrals 
by hand..  Another occasion, in a quantum mechanics problem it was a 
bonus problem which implied to solve an eigensystem of a 7x7 matrix, I 
was the only one who solve it because I programmed with Mathematica.... 
some morons told me sarcastically, " you solve it because you used 
Mathematica" and that was not fair, to setup the indexes and the code 
for building the matrix was not trivial to do (but easy for one who 
knows to program, the morons didn't know and want to learn to program), 
and even the code was for any nxn-matrix....

so many times I have got the put down "you solve because you used 
mathematica"
... when the true is "you solve it because you know to program, and you 
use mathematica rather than C because you are not a masochist"

my advise to learning calculus is try to know as much a possible in 
whats going on, if you are using mathematica to solve an integral you 
should know what method could be used to solve it, use mathematica for 
checking results/answers, use mathematica to gain insight when you are 
stuck in something complex. if learning physics, try to know how to 
derive the formulas rather than use them, learn/understand as many 
derivations as possible, that will train your basic algebra skills as 
well as gaining full understanding of the details.

try to proof or understand proofs of theorems, that is the pinacle of 
the human brain... and I think mathematica can't help in that.

bests,

eric






On 03/05/2010 01:33 AM, 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.
>
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• References:
  • Re: Re: learning calculus through mathematica
    • From: "David Park" <djmpark@comcast.net>