Learning Curves and Free Energy considerations on them. LO30786

From: AM de Lange (amdelange@postino.up.ac.za)
Date: 11/17/03

Dear Organlearners,

Greetings to all of you.

It is almost two years ago that i wrote the essay

  "Learning Curves or Performance Curves. LO27588"
   < http://www.learning-org.com/01.11/0105.html >

Some fellow learner here in Pretoria reminded me that i still have to
write on the free energy of learning curves.

In that essay I mentioned that already in 1926 Snoddy began to speak
of learning curves and their "power law". Since them many learning
curves had been drawn based on empirical data. Thus their empirical
nature cannot be doubted. Furthermore, every performance artist has
profound experience and valuable tacit knowledge on these learning
curves. Should such an artist be able to analyse these learning
curves, he/she would admit that they indeed articulate his/her tacit

The importance of the essay "Learning Curves or Performance Curves.
LO27588" is that in it I provided a theoretical foundation for these
empirical learning curves. It is based on LEP (Law of Entropy
Production) which is better known as the Second Law of Thermodynamics.
(Perhaps the mathematics in it, although unavoidable, made it opaque
for many a fellow learner.) I made use of the fact that entropy is
created by entropic force-flux pairs. When the entropic force
dominates, the learning curve stays steep whereas when the entropic
flux begins to dominate, the curve begins to flatten. This is depicted
in figure 1.


Look at the first (left) graph of figure 1. It shows three learning
curves for a typical topic. In the steep learning curve (the top
curve) with focussing power b=0.1 the entropic force dominates. In the
flat learning curve (the bottom curve) with focussing power b=0.9, the
entropic flux eventually dominates after an initial steep increase.
Also shown at the top of the left corner is how the time needed
increases (row of vertical bars) for equal steps in progress

We might imagine that the spontaneous progress in performance would
increase up to 1.0 (100%). But the increase in time taken for equal
steps in progress causes an equilibrium before the progress of 1.0
(100%) is reached. Although further spontaneous progress may be
wanted, the equilibrium prevents it. The performance curve (indicated
by a broken curve) after this equilibrium is only possible in a
non-spontaneous manner. It means that the performer has to be forced
by an external agent to accomplish it.

Why is an equilibrium reached before a 1.0 (100%) progress? It is all
as a result of the free energy (two words, one concept) needed to
sustain the progress. The free energy graph is always somewhat of an
inverse to the graph of the entropy production. This is depicted by
the second (middle) graph of figure 1. The topic has a number of
properties with a free energy for each. Some of these properties are
used to define the topic. Adding all the free energies of the topic's
defining properties give the total free energy for mastering the
objective for the topic. This is the beginning point (black square) of
the graph at the LH (Left Hand) vertical axis.

But as the progress in mastering the topic with each try increases,
each of the topic's defining properties become more involved (less
indifferent). Thus the free energy of each defining property decreases
and hence also the total free energy of them for the topic.
Consequently the free energy graph gets lower to a minimum after which
it gets higher again. But at its minimum value (65% progress)
equilibrium is reached so that no further progress is possible. This
65% gives an indication of the mark which the learner will get in a
test (examination) of the performance.

This graph corresponds closely to the free energy graph of a chemical
reaction depicted on the RH (Right Hand) side. A reaction has been
selected of which the conversion also ends at a 65% progress. On the
LH (Left Hand) vertical axis the total free energy of all the
reactants is depicted. During the progress of the reaction, the
remaining reactants become less and so also their total free energy.
Thus the graph gets lower to a minimum value after which it gets
higher again. This increase is caused by the increasing amounts of the
products and thus their total free energy.

On its RH vertical axis the total free energy of all the products is
depicted. Please note that the total free energy of the products on
the vertical RH axis is lower than the total free energy of the
reactants on the vertical LH axis. This is necessary to have a
spontaneous reaction. However, since the minimum free energy value
depicts an equilibrium, no further progress than 0.65 (65%) is
possible. In other words, up to a 65% the reaction is spontaneous
after which it becomes non-spontaneous. The graph can very much be
figured by taking a string between the two hands and let it hung
loosely. The LH indicates the free energy of the reactants and the RH
indicates the free energy of the products.

Let us now explore how the graph of the total free energy of a chemical
reaction can be modified.


On the first, outer, left hand graph the free energy for a typical
reaction is depicted.

Assume that more of one of the reactants is added to the reaction.
Since it also has free energy, the total free energy of the reactants
are increased. Look at the second, inner, left hand graph. It is like
lifting the LH which holds the string. Note how the minimum point
(equilibrium) of the string shifts to the right, stopping at a 85%
progress (conversion).

But when some more of one of the products is added to the reaction,
the opposite happens. Look at the third, inner, right hand graph. It
is like lifting the RH which holds the string. Note how the minimum
point (equilibrium) of the string shifts to the left, stopping at a
25% progress. However, when removing some of one or several of the
products of the reaction, the converse happens. Look at the fourth,
outer, right hand graph. It is like lowering the RH which holds the
string. Note how the minimum point (equilibrium) of the string again
shifts to the right, stopping at a 80% progress. In other words,
adding a reactant or removing a product have the same effect --
shifting the equilibrium to the right.

Let us now return to the free energy graph of mastering a topic such
as in the middle graph of figure 1. This graph is reproduced in the
upper, outer, left hand graph of figure 3.


Let us now think of a complex topic which consists of four topics A,
B, C and D. An example which many of you fellow learners experienced
in this list, is irreversible self-organisation. Topic A would then be
the definition of entropy. Topic B would then be the production of
entropy by force-flux pairs. Topic C would then be the derivation of
the concept free energy. Topic D would then be the optimisation of
free energy in living systems by following as organising path the
punctuated equilibrium.

Look at the upper, outer, left hand graph of figure 3. In the
objective of topic A some properties are taken (as reactants) from the
world of experiences of a learner and combined to define topic A. (For
example, should the topic be entropy, the properties temperature and
heat will be used to define it. Most people will have sufficient
experiences of temperature and heat to define entropy.) Within topic A
these experienced properties result in new properties of which the
learner has little, if any, experiences. Thus the learner still has to
seek experiences in these new properties. Let us call them
unexperienced properties.

Next to this graph is the upper, inner, left hand graph. In this graph
some of these unexperienced properties are combined together with
others into topic B. They result within topic B into even more
unexperienced properties. In other words, topic B is experentially
alien to the learner from its beginning (the formulated objective) to
its end (the mastery of the objective). The same for topics C and D.

However, most importantly, topic B is presented as if it has little to
do with topic A. Furthermore, since topic B is not backed up by
experiences, it has to rely on the learner's mastery of topic A. But
since the learner is neither aware of this connection nor concerned
about it, he/she has to begin with much less mastery and thus free
energy of the outcome properties of topic A serving as input to the
properties of topic B. Consequently, the free energy of the LH side of
topic B lowers, shifting its equilibrium to the left (40% progress).

What will happen when the learner is aware and concerned that topic A
and topic B are tightly linked? Compare the upper graph to the outer
left with the lower graph also to the outer left. The upper graph
shows a progress of 65% whereas in the lower graph the progress
increased to 83%. Why? In the lower case the unexperienced properties
of topic A have been used in topic B. It is like removing the products
of a chemical reaction A by using them as reactants in a reaction B,
thus shifting the equilibrium of A to the right. Note how the free
energy of the mastery of topic A is lower in the lower graph than the
upper graph, resulting in this shift of the equilibrium to the right.

Now compare the upper graph to the inner left with the lower graph
also to the inner left. In the upper graph the progress stops at 40%
whereas in the lower graph it stops at 70%. Why? In the loose
(unlinked) case the unexperienced properties produced in topic A are
not added to the defining properties of topic B. Hence the objective's
free energy of topic B (see left axis) is lower in the upper graph
than the lower graph. It is like removing the reactants of a chemical
reaction B by not supplying them as the products of reaction A, thus
shifting the equilibrium of B to the left.

Should we compare all four upper graphs with their counter lower
graphs, the decrease in progress is profound. In topic D, not linked
to topics C, B and A, the upper graph shows a progress of only 10% at
D. But when topics D, C, B and A have been linked all together to form
one complex topic, the lower graph shows a progress of still 52% at D.
In other words, seeking for the wholeness among a number of related
topics like A, B, C and D results in a far better progress for each of
them than being ignorant to wholeness or unconcerned about it.

I have observed this trend thousands of times when marking the tests of
chemistry students. In these tests i always linked a number of topics
together in one problem with subproblems for each topic. The solution to
subproblem A is needed as data to solve subproblem B, etc. I used to
call the complex problem a "moncat" problem. Here the "mon" and the
"cat" are derived from wholeness and sureness which are two of the 7Es
(seven essentialities of creativity. In those days i described them as
   sureness ("identity-categoricity")
   wholeness ("monadicity-associativity")
But presently i prefer the description
   sureness ("identity-context")
   wholeness ("unity-associativity")

The progress of students who had an affinity for sureness and
wholeness was much better than for students who did not care for them.
In fact, their progress was so extraordinary that they left the others
behind to form a "bell curve" of their own in front of the "bell
curve" for the other students. Perhaps most striking was their
enthusiasm for chemistry and its complex topics. This enthusiasm is a
sign of their free energy after they have completed a complex topic.
This is shown by the endpoints of each of the last three lower graphs
which are higher than the endpoints of their corresponding upper

When did i became aware of this trend in performance curves? As a
student (1964) in chemistry 2 i was intrigued, but also horrified, by
a certain complex problem in physical chemistry. It was the only one
of its kind which i had encountered at university. In 1968, after
having finished a MSc degree, i began to study this problem. It
intrigued me more and more. In 1972, as a high school teacher, i began
to design such complex problems myself. It is then when i became aware
of this trend. However, i did not understand it until much later near
the end of the eighties. I first had to learn a lot of irreversible
self-organisation, especially in learning.

As an application, think of giving a presentation of the five
disciplines of a LO (Learning Organisation). When it is done such that
wholeness is neglected and hence that they are considered as
independent, the progress in the fifth disicpline presented will be
profoundly less than in the first discipline presented. I would not be
surprised should the whole audience be sleeping at the end of the
presentation. But when they are presented as linked together to form
the complex topic "managing the LO", the audience ought to be
attentive up to the very end.

With care and best wishes


At de Lange <amdelange@postino.up.ac.za> 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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