is great, ancient and sustaining wisdom in the exponential
symbol. Its Latin origins are exponere – to expound, to make
explain, to make clear, to make manifest, to elucidate, to clear
of obscurity, to unfold or to the illustrate the meaning of. The exponential
symbol, especially when combined with the function symbol,
provides us with profound insight into the nature of change and how the
universal potential is actualised.
(The function symbol is employed here to convey the idea that two quantities have a relationship and changes in one affect changes in the other. They have a functioning relationship. For instance, the size of the human population varies with time. If we know the relationship, then our knowledge of one quantity enables us to know the what the other quantity is.)
functions can unfold the story of our universe for us,
provide great meaning and enable us to predict the consequences
of our actions. Change in the universe does not happen uniformly and
occurs in differing patterns of waves, each with its unique rhythms. The
only thing that is constant is change. Exponential functions is a use of
symbols that can enable us to elucidate our place in the universe and
explain how many changes occur.
greatest shortcoming of the human race is our inability to understand
exponential functions"- Albert
ten year old child can understand the essence of exponential functions
(see sample learning exercises below) and it is probable that this
shortcoming is a failure of our spirit stemming from our capacity to
deny stewardship/change. We simply do not want to know the implications
of our actions and this denial is reflected in our institutions.
self-deceit is how we enable a handful of merchant bankers to generate
trillions of dollars of credit with little or no relationship to our
assets. This self-deceit is why the human population has grown over
three fold in a century while in the same time we have destroyed over
half of the easily extracted mineral oil on which most of our systems
are now based. This self-deceit is how we justify bequeathing debt with
onerous interest on our children.
They should know how easy this calculator is. See how it works to answer the following questions:
change can occur at an increasing or a decreasing rate. In this example
the rate of change doubles with each generation.
this to a constant or linear rate of change:
This grand denial of
stewardship/change is manifest in our school curricula, which associates
the exponential symbol with mathematical symbols. Thus most
people graduate associating the symbol with mind-numbing impenetrable
complexity, the domain of a few myopic geeks. They do not associate the
symbol with art, language, history, geopolitics, biology, economics and
all the variant change of the universe(s).
When the underlying patterns of change are revealed to us we are better able to lead lives that are in harmony with them. We are better connected to the universe. We can be more at one with the greater the ebb and flow - whether it be in understanding how microbe or human populations expand and shrink or how interest rates accumulate and usury drives people into debt and misery or how solar, weather, ocean and other systems form and dissipate or how the cumulative impacts of our abuse of our carbon potential with our discovery, extraction and destruction of resources such as mineral oil and gas affect our children...
Conserve the exponential
symbol by employing it throughout the range of our discourse so our
children embrace their roles as stewards within change in all its
variety. Thus great and
sustainable ideas can spread exponentially throughout our communities.
Examples of exponential change
– the 29th day and then there will be just one day to save
amount of wheat is approximately 80 times what would be produced in one
harvest, at modern yields, if all of Earth's arable land could be
devoted to wheat. The total of grains is approximately 0.0031% of the
number of atoms in 12 grams of carbon-12 and probably more than 200,000
times the estimated number of neuronal connections in the human brain
total number of grains is probably more than all the grains that humans
have ever cultivated in the history : 18,446,744,073,709,551,615 grains
Teachers know best and can create many games based on this exponential function . An example might be to have a child print a message of hope 32 times on a page. They copy and paste the original thought once to give two copies. They copy these two copies to give four and so on.
then cuts the page in two and gives each half (16 messages) to two
children. They in turn cut their half page in two and give each quarter
page (8 messages) to two more children. Those four children cut their
quarter page in two and pass on 4 messages to two other children. At
this point fifteen of a class of 32 has read the message and yet each
student has only had to communicate it to two others. Played once a week
students will begin to sense the power of exponential functions
Links to Dr Albert Bartlett
Video (8 parts)
Page last updated: July 2010