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 symbolpushing 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 slugslow 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 symbolpushing, 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, curvefitting, experiment results, etc etc. To me, this is the real bonus of graphing calculators.
Luca
