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COMPUTER SIMULATIONS 247
computer simulations, that they oversimplify the
real world. Complexity theory argues that, in many
respects, the real world, though highly complex, is
built on comparatively simple rules that give rise
to such complexity (see also Gilbert and Troitzsch
2005: 10).
Simulations have been used in the natural
sciences and economic forecasting for several
decades. For example, Lewin (1993) and Waldrop
(1992), in the study of the rise and fall of
species and their behaviour, indicate how the
consecutive iteration – repeated calculation – of
simple formulae to express the iteration of a limited
number of variables (initial conditions), wherein
the data from one round of calculations are used
in the next round of calculation of the same
formula and so on (i.e. building in continuous
feedback), can give rise to a huge diversity of
outcomes (e.g. of species, of behaviour) such
that it beggars simple prediction or simple causeand-effect
relationships. Waldrop (1992: 241–2)
provides a fascinating example of this in the
early computer simulation program Boids, where
just three initial conditions are built into a
mathematical formula that catches the actuality
of the diverse patterns of flight of a flock of birds.
These are, first, the boids (birds) strive to keep a
minimum distance from other objects (including
other boids); second, the boids strive to keep to
the same speed as other boids; third, each boid
strives to move towards the centre of the flock.
Some of the key features of simulations are:
The computer can model and imitate the
behaviour of systems and their major attributes.
Computer use can help us to understand the
system that is being imitated by testing the
simulation in a range of simulated, imitated
environments (e.g. enabling researchers to see
‘what happens if’ the system is allowed to run
its course or if variables are manipulated, i.e.
to be able to predict).
The mathematical formula models and interprets
– represents and processes – key features
of the reality rather than catching and manipulating
the fine grain of reality.
Mathematical relationships are assumed to be
acting over and over again deterministically
in controlled, bounded and clearly defined
situations, on occasions giving rise to
unanticipated, emergent and unexpected,
wide-ranging outcomes (Tymms 1996: 124).
Feedback and multiple, continuous iteration
are acceptable procedures for understanding
the emergence of phenomena and behaviours.
Complex and wide-ranging phenomena and
behaviours derive from the repeated interplay
of initial conditions/variables.
Deterministic laws (the repeated calculation of
aformula)leadtounpredictableoutcomes.
In the field of education what is being suggested
is that schools and classrooms, while being
complex, non-linear, dynamical systems, can be
understood in terms of the working out of simple
mathematical modelling. This may be at the
level of analogy only (see Morrison 2002a), but,
as Tymms (1996: 130) remarks, if the analogue fits
the reality then researchers have a powerful tool
for understanding such complexity in terms of the
interplay of key variables or initial conditions and a
set of simple rules. Further, if the construct validity
of such initial conditions or key variables can be
demonstrated then researchers have a powerful
means of predicting what might happen over time.
Three immediate applications of simulations
have been in the field of educational change
(Ridgway 1998), school effectiveness (Tymms
1996), and understanding education systems. In
the former, Ridgway (1998) argues that the
complexity of the change process might be best
understood as a complex, emergent system (see
also Fullan 1999).
In the second, Tymms (1996) indicates the
limitations of linear (input and output) or
multilevel modelling to understand or explain
why schools are effective or why there is such
a range of variation between and within schools.
He puts forward the case for using simulations
based on mathematical modelling to account for
such diversity and variation between schools; as
he argues in his provocative statement: ‘the world
is too complicated for words’ (Tymms 1996: 131)
(of course, similarly, for qualitative researchers
Chapter 10