One-to-many-mapping in Learning LO27365

From: AM de Lange (
Date: 10/10/01

Replying to LO27300 --

Dear Organlearners,

Greetings to all of you.

Mark Feenstra <> wrote in the
Subject: Valentinus the learner? LO27300

>>Thank you for your reply. I'm often in awe of
>>the ground you encompass in your responses,
>>and appreciate your efforts to reach out of your
>>frame of reference to meet in the space in

I have replied to it with:

>Thank you for your reply. The reason why I cover so
>much ground is not to impress anyone. (I am aware
>of the intimidation effect of complexity.) The reason is
>rather in my own metamorphosis (digestive and
>emergent development) since childhood. I will explain
>it in a different topic under the name
>"One-to-many-mapping in Learning".

Because of the immense diversity in nature and culture of South Africa my
people (the Afrikaners) had to follow a holistic outlook to adapt
successfully in our complex country. This was reflected in our educational
system even long before apartheid. Apartheid came into effect in 1948. I
went to primary school in 1950. In those days each teacher in each grade
got a class assigned to him/her for at least a year. That teacher had to
teach everything in the syllabus -- languages, arithmetic, mathematics,
history, geography, sociology, science, biology, art and religion. The
teacher was like a substituting parent "("loco parentis") to the pupils in
a class.

What a transformation was it not when we got to the high school. Each
teacher instructed only one subject and pupils had to move from classroom
to classroom for different subjects. I was fairly intelligent and could
cope with so many different teachers. But many of my fellow pupils who I
knew almost as well as my brothers and sister, could not adapt to it. I
still remember the faces of many of them (Flippie, Alida, Koos, Marie,
Tjaart, Elsa, etc.). I helped several of them once or twice a week at
school or at home to catch up. But how could a teenager with a teenager's
insight help where even philosophers fail? I smelled no rat.

At university I took each year the maximum rather than the minimum of
different courses required for a degree in science. (I had to work my butt
off ;-) In the masters degree I wanted to do the same, but the chemistry
department would not allow me taking more than one degree while the
physics department knew it and said they had no pains should I take
chemistry in addition to physics. I still smelled no rat.

Then came four years of a research career into soil science. I was in my
element because a soil is a mechanical, chemical, geological and even
biological system. My fellow agricultural researchers (each engrossed into
one discipline) frowned on me when I contacted them for advice outside
chemistry and physics. I still smelled no rat.

As a result of realising my calling I became a teacher. I began to teach
at a high school. I was committed to find out how learners learn
creatively. I was one of only three teachers who had to take care of all
the science (chemistry and physics) at that school. Like each of the other
two teachers, I was given classes in all five grades. My load was roughly
250 pupils a day. Yet I tried to point out every error which each pupil
made in science, whether it be a scientific error or any other error in
the language, mathematics or whatever they used in their science. It is
then when I began to smell a rat.

Learners in the lowest form (std 6) were grateful to me for indicating all
their errors, often asking me how they could improve, even when it was
 not science. I was for them like the teachers they had in primary school
-- one person who took care of their whole intellectual development. But
the higher the grade, the more agitated a learner would become when I
pointed out to them any other errors than a scientific one.

Even worse were my fellow teachers in these other subjects. "How dare you
teach them my subject rather than only your subject science", were their
usual response. When I pointed out to them what I taught were not part of
their curriculum, but actually in that "undefined region" necessary to
make a firm connection between science and their subjects, they would
respond with "Exactly, you waste their time by teaching them things which
are neither prescribed in your subject nor in our subjects."

I was having my first solid experiences in what I later called
"transdisciplinary thinking". I was caring for the wholeness of the
pupils' minds and thus needed transdisciplinary thinking. My fellow
teachers saw nothing wrong with specialisation and were thus unaware of
transdisciplinary thinking. I began to study all the subjects which the
pupils had to study from their level to an advanced level. On the one hand
I wanted to feel the confusion which they felt having to cope with so many
subjects presented unrelated to each other. On the other hand I wanted to
find solid contacts between science and each of these subjects so as to
increase their wholeness of mind.

Unknowingly I began to do exactly what Goethe did. He was a gifted word
artist. To improve on his gift, he sat on many chairs of academy, walked
in its many moccasins and wore its many hats so as to improve his gift as
a word artist. How often he was figured as a jack of all trades and a
master of none. But by becoming a jack of all trades he became one of the
greatest masters of the word art. Through this he also became aware of
what he called "Steigerung", the fractal development of any complex
system. I did the same to improve on my teaching and understanding of
creative learning. I became aware of what I called the "moncat" pattern,
the fractal development of any complex system. I never knew about this
"one-to-many-mapping" of Goethe's own learning and "Steigerung" as its
outcome until many years later.

The reason why I called it the "moncat" pattern ("mon" for "monadicity"
and "cat for "categoricity") is as follows. I felt that should I describe
it with a word in my own mother tongue Afrikaans, fellow speakers would
find it crazy. So I searched for English words to make it more
respectable. Firstly, I could not find a single English word. Secondly, I
did not know how much complexity is required to understand such a word
should I have found it. The Afrikaans word which I had in mind those days
was "opstapeling". Its morphemic translation in English would be
"upstapeling". It is what a toddler do when putting block upon block to
build a tower.

Four years later I had to teach teachers at the CEFT (College for Further
Educational Training) who wanted to shift from teaching any other subject
to teaching science. Their most important reason for such a shift was that
science teachers were so scarce that they were promoted the fastest in the
hierarchy of a school to retain their services. They bickered when I
pointed out to them all their other errors than merely errors in
chemistry. The more I stressed the role of authentic learning in
understanding chemistry, the more they demanded to be told exactly what
chemical information they should memorise so as to pass the examinations.
I convinced only a small minority of them that authentic learning
involving wholeness is far more superior than rote learning focussed on
passing the examination. The rest preferred the system which they were
used to. I clearly smelled the rat.

Afterwards I was employed at my present university to put an end to the
excessive failures of first year chemistry students. By then I was
becoming wise to the cause of their failures. They wanted to sit on one
chair, namely chemistry, expecting to understand it all. But with all the
tricks in my book I bashed their brains with care. I helped them to sit on
several chairs (like biology, geology and physiology) by selecting all
chemistry problems from these subjects. The reason? Trying to learn
chemistry by sitting on the chair of only chemistry often results in a
deformed understanding of chemistry. That is a main reason why so many
first year students failed.

Later some of them as doctorate students in chemistry told me that when
they got into a fix in their research, they would page through their first
year chemistry notes compiled by themselves. When I asked them what they
have found helpful in these notes, they would invariably reply "Nothing,
except that by going through these notes we came curiously under the
impression of what route we have to follow." I think that they were
reminded to transdisciplinary thinking -- to follow a
"one-to-many-mapping" in their research project and hence experience
"Steigerung". This gave them the orientation to know what to do next.

With this bit of history, dear Mark, I hope to have helped you to
transform your awe into understanding. (I am perhaps a masochist for
taking so much intellectual beatings coming from the system ;-) The only
awe which you might still have is for my audacity to keep on questioning
the system despite the stumbling blocks it brought me.

The wear and tear had been tough. But I am like a desert tortoise because
it is the animal most persistent in its mission. Turn it into another
direction and it will turn back to its former direction. Carry it to
another place and it will still direct itself to the place which it
originally wanted to go. But because of its body shape and the complex
terrain it has to pass, its course is staggered ("Steigerung") rather than
straight. (Be careful when you pick it up because its only defence, other
than pulling in its head and legs, is to open up its bowels ;-)

I have told about me as the tortoise. Many of us know the story of the
race between the tortoise and the hare. So what has become of the hare?

The gradual focussing on a discipline of any subject to become an expert
on that discipline and generate as much information as fast as possible on
it is rampant all over the world. It seems to be the one sure truth which
academy believes in. But it is not a truth. It is the very outcome of the
publish or perish syndrome. It sounds so convincingly to state that a
published paper on a specialist topic means that it has past the scrutiny
of expert peers. However, it has passed only one test, namely that it
could be balanced on the point of a needle.

What about the test of wholeness? Now and then someone writes a monograph
on a discipline to cover all the balancing acts on a tip of a needle. It
is assumed to be the test of wholeness. A person who does it (and includes
as many as possible own balancing acts) is soon reckoned to be an expert
on that discipline. This is not for me a pass of the test of wholeness. It
is merely a many-to-one-mapping.

To pass the test of wholeness a one-to-many-mapping is required. For
example, a holistic botanist is someone who have connected botany to many
other academical subjects. That botanist will know a lot of physics,
chemistry, geology, zoology, economy, history, art and even languages. For
example, the names of plants in an indigenous language (like the Xhoi
languages in South Africa) reveals the taxonomic outlook of the indigenous
people to plants. This different outlook is often helpful to solve a
vexing taxonomic problem.

Let us include mathematics in the "one-to-many-mapping" of learning. The
most simple, although very uncommon, symbolic representation of a
"one-to-many-mapping" is by the sign "<". It is called the "order
relation". It means literally "is smaller than". Remember its meaning as
follows. To the left of "<" the gap between the two legs is zero. To the
right of "<" the gap between the two legs is large. Thus if we write "a<b"
we mean "a is smaller than b". For example, "2<3" uses the sign "<"
correctly. But "2<2" or "2<1" uses the sign "<" incorrectly. Why? The left
(towards the smaller gap) is not smaller than to the right (towards the
larger gap).

Why can we say that "a<b" symbolises a "one-to-many-mapping? Well,
consider the series
On the left of the "<" is one and only one number, namely 2. But one the
right are many numbers like 3, 4, 5, .... The one number 2 is mapped to
the many numbers like 3, 4, 5, ....

An expression like a=b is a "one-to-one-mapping". An expression like
a=f(b,c,d,...) is a "many-to-one-mapping". It maps the many input variables
"b, c, d, ..." to the one output variable "a". It is better known as a
"f" of many variables into one variable. Its opposite notation would be
(b,c,d,...)=r(a), the (order) relation "r" of one variable into many
The symbolic expressions a=b and perhaps even a=f(b,c,d,...) may be
familiar to some of you. But the symbolic expressions a<b and especially
(b,c,d,...)=r(a) are unfamiliar to most of you. It means that very few
people have expressed their thoughts as a "one-to-many-mapping" to you.

Authentic learning is a "one-to-many-mapping". Rote learning begins as a
"one-to-one-mapping" and may digress into a "many-to-one-mapping". All the
laws of science, except two, are "many-to-one-mappings". The one exception
is LEC (Law of Energy Conservation) which is a "one-to-one-mapping". The
other exception is LEP (Law of Entropy Production) which is a
"one-to-many-mapping". Thus we ought to expect some pattern between
authentic learning and the "one-to-many-mapping" of LEP.

What about paradigms? In the paradigm of simplicity "one-to-one-
-mappings" [a=b] are common while "many-to-one-mappings" [a=f(b,c,d,...)]
are considered as expertise. But in the paradigm of complexity
"one-to-many-mappings" is its central feature. All living organisms are
"one-to-many-mappings". For example, one mother may have many children or
one flower may result in many seeds. Tracing with the eye all the branches
of a tree to its trunk is a "many-to-one- -mapping". But tracing with the
eye from the trunk to all the branches and then the many more twigs on
which all the leaves and flowers are borne is a "one-to-many-mapping".

I wonder of there is even one artist who would paint a tree from its
leaves and flowers towards its trunk as a "many-to-one-mapping". Art is a
"one-to-many-mapping". The progress of science to new frontiers is also a
"one-to-many-mapping" as Faraday and Einstein showed us. Thus authentic
art and authentic science have "Steigerung"="opstapeling" (stackering,
staggering or "upstapeling"). However, "conform to the unquestionable
leader" leads to a "many-to-one- -mapping". When all curves are forced
into one, they become a straight line. Conformation leads to linear

Most interesting is the role of wholeness in conformistic thinking. It
seems as if all the curves are weaved into one linear line of thinking.
But to accomplish that, all staggered curves of thinking have to be
fragmented from their own context into the one and only context. It boils
down to getting wholeness even to the detriment of sureness. Wholeness and
sureness are two of the 7Es (seven essentialities of creativity). When we
seek the increase of one essentiality (like wholeness) to the detriment of
at least one other essentiality, we are in for deep problems.

Artur Silva from Portugal has begun a delightful series of contributions
on the topic LO's and metanoia. He thinks that a paradigm shift is needed
in organisational thinking. I agree. He thinks of the L in LO not only as
Learning, but also as Living. I agree. Have you fellow learners observed
how much "one-to-many-mappings" are there in Artur's thinking? I did and
it makes me happy. I wonder if he already has become aware of

Authentic learning is a "one-to-many-mapping" while rote learning is
primarily a "many-to-one-mapping". Both begins as a "one-to-one-
-mapping". But which one you fellow learners will be following depends on
your choice. A "one-to-many-mapping" is a choice into the unknown of the
future. But a "many-to-one-mapping" is a conformation to the past.

Creation has the outstanding propensity to go for the unknown future
rather than homing in on the known past. It takes from the past that which
is characterful (like true, good, right and pretty) to create its future.
But Creation avoid backtracking it steps or repeating the same thing
twice. Creation strives towards greater future authenticity rather than
acting towards past authenticity. Thus Creation has creativity.

Learning which has no creativity is rote learning. Learning which flows
from creativity is authentic. Such a learning is always authentic, a
"one-to-many-mapping". Ordinary Organisations (OOs) forces upon its
members "many-to-one-mappings". Learning Organisations offer to its
members "one-to-many-mappings". OOs become less complex while LOs become
more complex. Dear fellow learners, after fifty years I am convinced, the
choice is yours. I am also convinced that authentic learning, whether
focussed on Personal Mastery or Team Learning, is a one-to-many-mapping.

It does not matter how confusing the world becomes because of destructive,
tragic and despicable event lime on 11/09/2001. The creating, learning,
believing and the loving of the human spirit is a "one-to-many-mapping".
The Creator created a "one-to-many-mapping" into Creation so that Creation
can know its Creator. Should we not learn this, how much else can we

With care and 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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