"Winfried Dressler" <winfried.dressler@voith.de> wrote:
In Linear thinking LO22810
> In order to be significant to write z instead of y (change of the form of
> the ink on the paper), there should be a change. Not only a change but
> also a change of a change. But with respect to change, both formula are
> the same:
>
> (d/dx)y = (d/dx)z = m
>
> (d/dx)(d/dx)y = (d/dx)(d/dx)z = 0
>
> Even without considering any change, both formula are the same:
Winfried's valuable statements might be misinterpreted (by an absent
minded reader for example):
Misinterpretaion:
Two mountains have the same slope;
And the slope is straight;
Hence the two mountains are the same!
Avoiding the misinterpretation: However, please notice that Winfried uses
the expression 'with respect to'. Therefore, the two mountains are the
same 'with respect to' slope and its straightness. Therefore, the two
mountains are truly not the same in all respects!
Both the original exposition of At de Lange and the subsequent commentary
of Winfried seem to point at a discussion on the 'level of sameness'.
The notion of 'level of sameness' appears to be fundamental to any
discussion on learning. Some of the basic mental functions (that operate
in learning) such as comparison, extension, induction, abduction, etc.,
cannot be conceived without using the idea of the 'level of sameness'.
Coming back to 'linear thinking', one might be able to indicate a 'level'
at which it is the same as 'nonlinear thinking'. So, to take an example
from an earlier message of At, a parabola is the 'same as' a st. line at
the level of differentiability (a mathematical notion). Practically
speaking, if you take a small piece from a st. line and a small piece from
a parabola, you might not be able to distinguish between the two. If you
are, then try smaller pieces.
It seems to me that much of the talk about linear, non-linear, local,
global, systemic, logical, ... types of thinking is really polemic in
nature and serves political purposes in professional groups. However, when
carefully used, they might serve an educational purpose in learning groups
(or learning communities). But then, such an educational purpose, a method
for achieving it, and a method for ensuring that it is actually being
achieved, will have to be clarified then.
DP
--"DP DASH" <dpdash@ximb.stpbh.soft.net>
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