|Re: Calculator restrictions in schools|
Message #3 Posted by Luca de Alfaro on 30 Nov 2001, 4:45 a.m.,
in response to message #2 by Ernie Malaga
I actually have mixed feelings about the use of
graphing calculators in schools. From what I see
at the TI site, it seems much of calculus is reduced
to learning recipes for having calculators do the job...
But there is also another reason why I am not sure I
am in favor of graphing calculators in schools.
Sure, if you give them a function, they draw it for you.
By having to draw them myself, I quickly learned
tricks to decompose functions, and especially, to
analyze them mentally before drawing them. I can
stare at one, and with a few calculations, graph it in my
mind. This is not only fast, but is fundamental to
the ability of _designing_ functions. Suppose you have
to design a function that goes to zero here, has a
"rounded" maximum there, and so forth.
If you have always relied upon a calculator to draw functions for you, it's likely you don't know where to
begin from - trial and error doesn't really work well. If you are used to decomposing and analyzing functions mentally, you can quickly assemble the proper ingredients.
Moreover, how much does a small graph really tell you?
It gives you the outline of the function, but can you see
from the graph which one is flatter at 0, x^2 or |x^3|?
On a related note, I wonder why people like and use so much
these CAS systems. I can understand that people working out the equations of astronomical bodies need them, but this clearly does not explain all the use. What, then?
I work with math every day.
Usually, the difficulties I encounter are never related to pushing symbols around (gathering, factoring, whatever).
The difficulties have to do with understanding whether a transformation is legal (can you exchange that limit with the summation?), or with how to decompose an expression in a fashion that opens the way to new manipulation (write a probability as the sum of probabilities conditioned by certain events, so that something turns out to be independent, and...).
In other words, the symbol-pushing is really not the difficulty at all: understanding what to do with the expressions, or how to transform them, is.
And I can't imagine that entering expressions in the slug-slow HP48 to do CAS is going to be faster than me doing it by hand with pencil and paper.
I also like to have a record of the symbol-pushing, so I like to use pencil and paper anyway.
Granted, I don't often have to do complicated integrals, but even for those, is HP48 CAS justified?
I do like graphing calculators, but I use the graphing part not to graph functions, but to plot data: statistical data, curve-fitting, experiment results, etc etc. To me, this is the real bonus of graphing calculators.