Discussion:  All Topics 
Topic:  Intigration 
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Subject:  RE: Intigration 
Author:  Mathman 
Date:  May 4 2006 
>I was
> thinking about more mundane things like calculators or computers,
> not nuclear devices.. :)
Should they be available at all times?
> Ideally, yes.
Sigh. I knew this might be a can of worms, and do apologise if it becomes too
extensive. However, I have heard and seen all the arguments. I can only rely
upon my own experience two ways; as a teacher and as an observer of student
behaviour. I'd then agree in part that the tools are an advantage if necessary,
but they are not necessary in early grades, and can in fact be a
disadvantage.
I have claimed [and still do] to be able to teach, and do it well, the
principles of all of junior and high school mathematics without either computer
or calculator. Consider: it was so done for centuries, and produced those minds
that invented the stuff in the first place. Consider also the boy who, after a
great struggle, agreed that in order to change from mm to cm he would have to
divide by ten, and REFUSED to let go of his calculator to do so.
So, I'd rather suggest without being argumentative, *ideally*, no. That does
not deny reason for their use, just the overuse, and lack of other skills
because of that. I've shown the advantages of "traditional" methods time and
time again, and one more will not make the difference. But when students [and
some teachers] have said at times, "How do you do that so easily?" I can only
reply that I'd show them how if they'd let me. :)
I'd recommend the use in as many ways as possible in high school, but there are
other, more fundamental skills to master in the early years, and grabbing a
calculator at every opportunity has never really been one of them.
While here, for those interested in such stuff, I always liked to form my own
problems now and then, and loved geometry in particular. Here's an example [and
as far from the calculato as you can get, I suppose.] I'd include more, but
thatowuld require too much description, where a diagram would be better. This
is strictly for enjoyment, and to hopefully distract from the topic at hand, yet
add to it in a sense:
It is simple to draw tangents [without software], and I do realise there are two
considerations, to two circles if one is quite smaller than the other. Draw a
radius in one, and a parallel radius in the other. Extend the line joining the
centers. Form a line from the ends of the radii on the two circles to meet that
extended line at a point, P. Form the semicircle through P and the far
intersection of the extended center line with the further cicle [well, either
circle would do. Where that semicircle meets the original circle in an
intermediate point is thepoint of tangency. Join that to P. [See, a diagram
would be much easier, and a plug for high tech.]
It is also even simpler to draw a tangent if the circles are exactly the same
size. The problem then is how to draw the tangent if the circles are *almost*
the same size. The above method will not work since the point P would be
inordinately difficult to locate when far from the circles, and not being
exactly alike, the second is not feasibl either.
The point of all of this is that "mathematics", or "mathematical thinking" is a
long way from number crunching. Graphing techniques are also an advantage if
done first by hand, and the computer/graphing calculator not being such a
dependency. Certainly, when systems of graphs, or very complex graphs need to
be studied, or when simpler graphs have already been studied, they are a decided
advantage. But they are not such a similar advantage to the very young. I have
done a solution to the above problem, and it involves a fair bit of mathematics
for the young to absorb, with a lot of learning in the process. There may be
software to do it simply [a CAD program perhaps?], but that does not teach the
same princuiples in the same manner.
That's about my 2 cents worth. Time now to really retire, I suppose, and watch
the progress ....and we do progress, to be sure.
David.
 
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