The mathematician Lancelot Hogben wrote the following paragraph as the
beginning of his book on pure mathematics for the general reader.
“For MY one deer you must give me three of your spearheads”. The earliest men and
women like
ourselves lived about twenty-five thousand years ago. They could say all
this with their hands,
simply by pointing one finger at the deer and three at the
spearheads. The primitive way of
counting with one finger for one thing and three
fingers for other things, was the only kind of
arithmetic they knew. For thousands of
years such people thought of any quantity greater than
three as a heap or pile”.
Hogben’s book ‘Man Must Measure’, published in 1955, was the first account of
mathematics to be illustrated with vibrant coloured pictures (Fig 1). In 69 pages it
took an historical view of the development of the major ideas of the subject from
prehistoric times to the invention of the computer. As a knowledge system about
human culture it could equally well have started with the computer and worked
backwards, to show how our command of natural resources has depended on a
succession of ideas that began with the first moon- calendar. The point is that
setting a syllabus is an idiosyncratic affair, just as ‘subjects’ are arbitrary divisions
of knowledge contrived by individuals or committees to facilitate learning for
standardised examinations.
Fig 1 A pictorial illustration of the concept of solar geometry in navigation
Hogben’s target audience was elementary
general students of all ages. In contrast,
another mathematician, Frank Castle produced a book in 1900 for the elementary
special students who required an understanding of mathematics to solve the
problems encountered in their jobs as ‘artisans’. He dispensed with principles and
reduced the subject to a set of highly practical objectives. This objective is clear
from the first paragraph of Castle’s book of 194 pages, which was illustrated with
dull monochrome graphs and tables.
“To perform his work intelligently, an artisan must have a knowledge of
Elementary Mathematics. When he comes to appreciate for himself the
workman generally finds that even the arithmetic he learnt at school has left
him, and that he remembers little
more than four simple rules and the
multiplication table. Teachers soon discover that
though anxious to learn, a
student of this kind does not wish to lose contact with the
practical requirements
of the workshop- he is impatient of ‘pure mathematics’- so the
question arises
how to teach him mathematics enough, by dealing with the
calculations
themselves which he is actually called upon to make at his work”
Here, Castle is making an important statement about the fundamentals of effective
learning, which is bound up with customisation and targeting information
according to the needs of the learner. The ‘impatient’ learner, in turn, selects and
arranges the information according to the personal knowledge system appropriate
to his or her needs.
Despite mathematics being a subject with boundaries that are more clear-cut than
most, these two authors demonstrate that mathematics is actually vaguely defined
as a curriculum. Also, neither Hogben nor Castle was guaranteed a student
reader who would be able or willing to think like they did. In fact the study of
students’ motivation has shown that success in examinations comes from being
able to assemble information in a flexible personal body of knowledge that is as
arbitrary as the textbook author’s particular arrangement. In this context, Hogben’s
book is the more useful in that it could be used as the scaffold to build a history of
civilisation, where each major advance depended upon the application of
mathematics to solve real problems of expanding wealth and knowledge.
Learning by making mindmaps
All of the above is to take a view of learning as a process of making mindmaps,
where a particular piece of knowledge is linked to another according to the path
chosen by the compiler to build an understanding. It is often said that humans are
inherent organizers, a process known as taxophily. From an early age, children
exhibit taxophila in playing sorting and matching games. We cope with our ever-
changing world by comparing new objects or experiences with those with which
we are familiar, identifying patterns and categorizing what is new into our existing
frame of reference. The emphasis on developing comprehensive systems for
organising knowledge can be seen in the writings of the earliest philosophers,
many of whom still continue to influence our view of the world. For example,
Aristotle's effort to categorize knowledge into groups (such as physics, politics, or
psychology) is reflected in our language, our education, and our science. The
original classification scheme of the U.S. Library of Congress, used between
1800 and 1814, was based on the philosophical works of the 17th century British
courtier Sir Francis Bacon. Bacon’s view was that knowledge had to be organised
primarily for the study of nature and of man as a component of nature, so as to
reduce both to controllable and 'usable' entities. Adoption of the classification
systems of Aristotle and Bacon led to the creation of our traditional subject-based
knowledge hierarchies set out in text books, which are all based on the ‘tree of
knowledge’ with a trunk that branches out into concepts with subordinate
elements as branches and twigs.
However, even in the mid-19th century, subject boundaries, particularly in the
natural sciences were not hard and fast. A key figure in the presentation of what
could be termed a multisubject global knowledge system was Alexander Von
Humboldt. He took the view that to understand nature required the linking of ‘hard
matter and rocks in place’ with beauty and order represented in poetry, painting
and gardening. This was the theme of his two- volume thesis entitled ‘Cosmos’. In
Volume 1 Humboldt surveys the heavens and the earth, and deals with all things
from stars and nebula to the earth as a planetary body, its geography and
meteorology and life forms from plants and animals to the races of man. He ends
the first volume on the threshold of explaining the human mind. In Volume 2 he
takes up nature subjectively as a product of mind. He reaches a two fold
understanding of our place in nature as a transcendental response to being
engaged in the gigantic physical mechanism we call the cosmos.
Humboldt’s empirical holism influenced the American naturalist and proto-
conservationist Thoreau, but Humboldt did not stem the tide of subject
consolidators who created the Victorian examination system. These subject
divisions are now unnecessary and restricted when it comes to organising current
knowledge about world development on a planetary scale. We have to return to
Humboldt’s insistence that, as individuals, our only view of the world is through
the mindmaps we make of it by which ‘….
the external world blends almost
unconsciously to ourselves and feelings’. In a world of the printed-paper, the
making of mindmaps requires numerous cross- references in the form of
footnotes, and Humboldt’s Cosmos has a bewildering array of these on every
page. Nevertheless, tree structures upon which the contents of textbooks are
arranged, have served us well in the paper world and are now being adapted to
the digital revolution through the use of mind mapping toolkits, which are user-
friendly interactive personal learning tools. Digital knowledge trees, particularly
where the text is reinforced by pictures, have the following advantages.
-
In the paper world, a concept can hang from only one branch. In the digital
world, concepts can easily be classified in dozens or even hundreds of
different categories.
-
In the paper world, a teacher uses only one tree. In the digital world, there can
be a different tree for each person who can also participate in the making of it.
-
In the paper world, the person who owns the information generally also owns
and controls the tree that organizes that information. In the digital world, users
can control the organization of information owned by others.
Barriers to total understanding
Sectorialism and specialisation permeate society in general. Decision-makers
tend to base their decisions on short-term benefits, and often ignore long-term
environmental, social and cultural costs. In addition, well established existing
organizations, institutions and structures, as well as scientific disciplines, are
factors that contribute to support such narrow decision- making. Together with
vested interests in politics, industry and the corporate sphere, these institutional
barriers effectively block change towards more long- term, integrated decision-
making, favouring the environment and sustainability. In particular, these barriers
obscure the opportunities for joint work between social and environmental
scientists in fields that go beyond their disciplinary frontiers.
Education is a key
instrument for cultural change, and future decision-and- policymakers must be
provided with integrated, multidisciplinary education, training and research. But
bridges are needed between disciplines at all levels of education to reinvigorate
ingrained working methods and mind sets, to enable future decision-makers,
families and individuals to resolve the complexities of responding to change within
an integrated, long-term planetary perspective.
Ideology of a making new subjects
The long term consequences of political, industrial and biological management and
development of the environment can only be understood within a knowledge
system that integrates, ecology, economics, the social sciences and technology. It
should connect government and business with families and individuals. As a new
subject it should be structured in order to:
-
recognise the multi-disciplinary nature of economic development has a long-
term perspective
-
improve the effective balance between knowledge about conserving and using
resources;
-
emphasise informed public participation in decision-making;
-
promote the equitable sharing of resources and reduce the risk for conflicts;
-
foster respect for cultural, social and biological diversity
These are the five educational imperatives recognised by the
UNESCO-Cousteau Ecotechnie Programme (UCEP) as keystones in the promotion of global
education for environment and sustainable development.
Cultural ecology is an example of a mindmap of world development produced from
the subject of natural economy that was launched to support world development
by the University of Cambridge Local Examinations Syndicate in the 1980s. It
formed part of the Syndicate’s International GCSE until a few years ago when it
was replaced by ‘environmental management’.
This new syllabus is more technical
and has lost the broad humanist sweep of the original, which was designed to put
‘being environmentally friendly’ at the centre of a new curriculum for living in an
overcrowded world.
Cultural ecology starts from the premise that knowledge about Earth’s future is
preferable to blind ignorance, and that there has to be a ‘global syllabus’ with
room in it for teaching that order is better than chaos; creation better than
destruction, and that gentleness is preferable to violence, and forgiveness to
vendetta. This gives a powerful socio-ethical dimension to cultural ecology, which
encompasses what is ethical and sacred about our ecological position on a minor
planet at the edge of a vast empty cosmos. Pessimists would say that ethical
constraints cannot hold against the human race as an evolved group of
disputatious tribal animals hard-wired to conquer the solar system. Nevertheless,
the British teachers who created the mindmap of cultural ecology believed that
they had to try to provide a map for withdrawal from the war against nature.
As a concept, cultural ecology starts with the principle that human cognition differs
from the cognition of all other animals, primarily because it is intrinsically a cultural
phenomenon. In this context, the practical focus of cultural ecology is on education
about the cultural rules of behaviour for sustainable living. Culture is the patterning
of our interactions with the human environment, and it is not susceptible to design
principles appropriate to the creation of bridges and software programmes. But it
is susceptible to the highly personal design principles of a gardener or an
architect.
Engineering models are appropriate to ordered systems in which cause-and-effect
relationships can be discovered and verified, and where those relationships repeat
themselves in a predictable manner. In such systems, efficiency rules – we want
the most efficient system, one in which each component of the organisation is
optimised in order that the system as a whole can be optimised. Examples from
the social sphere are expense rules, compliance procedures, quality standards,
legal structures and so on. The problem is that human behavioural systems, in
respect of their cultural context are not ordered systems and, ironically, the path to
their optimisation is to allow sub-optimal behaviour in their parts. This is because
society is multi- structured in terms of groups and their histories, and full of
individual flair in its creation. Pluralist human systems need to be effective,
whereas machines are efficient; the two are not necessarily the same thing. This
leads to the following two categories of systems, social ones and ideational ones:
-
Culture as a social system is about what people do or make. This is the pattern
of residence and resource exploitation that can be observed directly,
documented and measured in a fairly straightforward manner.
-
Culture as an ideational system comprises shared ideas, systems of concepts
and rules and meanings that underlie and are expressed in the varied ways
that humans chose to live. Culture, so defined, refers to what humans learn,
not what they do and make. Ideation is the way in which humans provide
standards and structures for deciding what is... for deciding what can be... for
deciding how one feels about it... for deciding what to do about it… and for
deciding how to go about doing it.
The emphasis within cultural ecology was to provide an ideational scaffold. The aim
was to present ideas about living a sustainable life, not to provide prescriptions
for living it (i.e. it was to be biased towards the second of the above categories,
not the first).
The big ideas behind sustainability through conserving resources, its educational
pillars as it were, are categorised as being about ‘people and place’, ‘nature’,
‘non-violence’, and ‘environmental management’. These ideas about conserving
resources are placed alongside ideas about exploiting resources connected with
biodiversity, nature’s production and managing natural resources for human
production (Fig 2). Both of these major divisions are placed within the historical
context of the origins of the ideas, with case histories to exemplify how the ideas
are expressed in the real world.
Fig 2 Mindmap of cultural ecology
The other principle behind the creation of the cultural ecology mindmap was to
demonstrate how to build such maps for crossing subject boundaries and
establish relationships to assemble a personal body of knowledge for living
sustainably. Technically, this is a straightforward task of demonstrating mind
mapping with commercial software such as 'Mindmanager' and 'Inspiration'.
In summary, when producing the mindmap the organisers of cultural ecology were
trying to answer the question how should people learn in order to be educated
citizens and to find and do interesting and important work in the 21st century?
This is really about a lifelong learning perspective for action, beginning with the
following proposition:
If the world of working and living relies on collaboration, creativity, definition and
framing of problems and if it requires dealing with uncertainty, change, and
intelligence that is distributed across cultures, disciplines, and tools—then
community education programmes should foster transdisciplinary competencies
that prepare people for having meaningful, productive and sustainable lives in
such a world.
To deliver cultural ecology as an information package for behavioural change in
people living in the 21st century, the following additional questions need
answering. How do people go about knowing what they know? What is the
contribution of the environments in which the knowing is accomplished? What are
the limiting factors to be removed?
