Linear thinking LO22836

AM de Lange (
Fri, 8 Oct 1999 17:43:49 +0200

Replying to LO22810 --

Dear Organlearners,

Winfried Dressler <> writes:

>I can make a lot of sense of your contribution, but I cannot
>see, how and why the essence of that sense is captured in
>the change of the formula
>y=mx+c to
>So may be I didn't get THE sense?

Greetings Winfried,

You are a careful reader. That comment was meant to you, Leo
and others who once upon a time had been drilled to say, when
wakened in the deadly hous of early morning, saying unconsciously
"y is equal to m times x plus c"
when asked
"what is the formula of a straight line".
It was meant to those who know the formulae of straight lines,
flat planes, parabolas and parabolic surfaces. It was meant for
those who know that
d/dx (e^x) = e^x
but never realised that e^x refers to changes which dance
for ever all the way in the deepest of their changes.

It was the best which I could do to caution you that we will go into the
very art of mathematics.

What is the art of mathemetics? To create form out of content, EVEN making
content of form when no more form is possible.

Making form out of content is for most people struggling with mathematics
to emerge into order at the edge of chaos. (Here in South Africa people
now have taken over your term "people at grass roots level"). But for the
mathemtican who have mastered the art of mathematics, it is but mere
digestion close to equilibrium. Mathematicians are animals like giraffes
who do not eat grass, but gracefully the highest leaves among trees. But
we all, our mathematical faculty developed or not, have to eat. So let us

The fun of mathematics is to let this crystallised form collapse so as to
use it as content for a higher order of form -- to let order become chaos
for new order. (Mathematicans are a bunch of Trotskyists.)

So how can I encourage you to let the order in
y = mx + c
collapse so as to seek for new form. ASCII code prevents
me to make a vulgar appeal on your creativity. Thus I had
to do it with such finesse with ASCII code that it works.

It seems to have worked because you write "I can make a lot of sense of
your contribution". Perhaps you want to know how it works because you
claim "I cannot see, how and why the essence of that sense".

See the "why and how" when I do it for first year students face to face as

I will first announce
"Today we will discuss the death of linear thinking".
Then I will say that
"You all know the equation for a straight line"
and write on the screen (blackboard or overhead projector)
y = mx + c.
Next I will ask a question "What is linear thinking?"

The students know how to articulate mathematically
a straight line. It helped to increase their tacit knowledge
on linearity in general. But they never had been asked to
articulate linearity without using mathematics. So they
have no experience in doing it. Hence I cannot expect
an articulated answer from most of them, if not all. Thus
I will have to give them experience. Consequently I write
the following on the screen:
(When you see the word "man" or "woman" replace it in
your mind with the simple icons like those used to signal
toilets for males or females. If you do not have enough
mental discipline to do it, please do it immediately on
a piece of paper and then look at that.)
woman = slant x man + cutoff

Then I wait because three things have to happen
(1) They have to make a creative collapse by making
form content
(2) They have to let the barest of new form emerge
from this content
(3) They have to recognise the new form in my articulation
Each process takes time. I look at their faces, trying to
show not emotion what so ever. But obviosuly, my eyes
try to trace their mental processes by means of their body

Suddenly someone will begin to react. A smile or a hand before the mouth,
eyebrows rising or cheecks blushing. More will begin to react, but each
individually, totally ignorant of what his fellow students do. O, is this
trance during three such steps not a lovely thing to observe.

Then, suddenly the LO steps in. Someone laughing or another one banging
with the fist. The LO sustains pandemonium. Some become bewildered --
what is happening? They ask their fellow learners. Some show indignance.
Finally the atmosphere becomes ordered again.

Will this pandemonium ever manifest itself on a internet email list?

Finally I articulate their experience:
"Linear thinking is not to see form in content"
The rest of the lecture I help them to articulate it

Winfired or any other fellow learner, if it does still does work for you,
then it reminds me of the two very old men sitting on a bench, talking
about what age has been doing to them. A pretty young lady walks by. The
one oldster says to the other oldster: "I remember that there is something
which I do not remember."

No creative collapse of an equilibrium to proceed to the edge of chaos is
ever linear. No bifurcation is ever linear.

Linear thinking denies creative collapses and creative bifurcations.

Linear thinking is necessary in creativty during the digestor phase. But
linear thinking can also become the death of creativity.

Linear thinking denies inclusive thinking.

Linear thinking is negative thinking. Linear thinking is judgemental
thinking since it denies changes in changes.

A LO needs to know how to deal with linear thinking. Do not throw also
the baby out with the bath water.

Best wishes


At de Lange <> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa

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