MIND, CULTURE, AND ACTIVITY, 4(1), 34-41
Copyright© 1997, Regents ofthe University of Califomia on behalf of the Laboratory of Comparative Human Cognition
Chaos, Competition, and the Necessity to Create
David K. Dirlam
King College
In a study of the frequeticy of usage of developing strategies in 5-year-old to 18-year-old children's
drawings and in 1968-1992 developmental research strategies, Dirlam (1996) showed that iti both
cases, usage declined exponentially as successor strategies emerged. This "general gamma" model
revealed that macrodevelopment involves (a) inevitable declines in primitive strategy usage, (b)
simultaneous use of multiple strategies, and (c) cultural mediation of strategy change. Two basic
questions remained. Why does macrodevelopment involve decline, and what happens after the most
advanced strategy declines? A new chaos-theory model, rooted in population growth models, assumed
that the rate of strategy decline was related to the summed competition from growing, successor
strategies. It fit the data as well as the general gamma and revealed that if no successor to an advanced
strategy occurs, its usage usually becomes chaotic. This implies that developing systems of strategies
cannot maintain stability without creatively introducing new strategies.
Dirlam (1996) found that frequencies of use over time of multidimensional strategies of both
children's drawings and developmental research obeyed the general gamma probability law;
that is, each strategy declines following a negative exponential curve as it is replaced by its
successor. This implied that both were mediated by the same process. The developmental
research strategies, most of which were defined from Danziger (1990), were found in a random
selection of 599 articles from Child Development and Developmental Psychology, 1968 to
1992, whereas the drawing strategies, defined from Lowenfeld (1957) and Piaget and Inhelder
(1967), were found in drawings obtained from 1,222 5-year-old to 18-year-old children in
northeastern New York. The general gamma fits were precise enough to allow accurate
predictions—for example, that the latest developmental research strategy (analytical, contextualized research) would reach peak usage between the years 2004 and 2010 and decline
thereafter. This strategy uses both difference and correlational statistics, multiple social
contexts, and well-described environmental settings. A change in any one of the three
dimensions ofthe strategy would compose a new strategy.
Three processes of strategy development were distinguished by the manner in which the
strategies were preserved: evolutionary strategies by genetic transmission, developmental strategies by automatization, and historical strategies by cultural transmission. Because developmental
research methods could not possibly arise from a genetic basis or from a solitary individual's
interaction with the environment, the common process was considered to be cultural. Thus, a
particular drawing strategy used by a child was considered to be a cultural artifact, as in Cole's
Requests for reprints should be sent to David K. Dirlam, Department of Psychology, King College, Bristol, TN
37620-2269. E-mail: dkdirlam@king.bristol.tn
CHAOS, COMPETITION, AND NECESSITY TO CREATE
35
(1995) conception, which is mediated by a cultural agent, such as an admired peer. This is much
in the same way as the mediation of English children's games studied by the Opies (1969) or of
the literary genres discussed by Cooper (in press).
Two basic questions arosefromthe general gamma model. Why does macrodevelopment involve
decline? What happens after the most advanced strategy declines? Progress on these questions arose
from a suggestion by Danziger (personal commtmication, July 1996) stemming from the issue of
historical prediction. He suggested that I take another look at chaos-theory models for the historical
data. Two previous attempts (in 1988 and 1992) failed to tum up a suitable model. This time,
however, instead of focusing on the equations, themselves, I created spreadsheet examples. In his
excellent introduction to the use of chaos models in developmental theory. Van Geert (1994)
defended spreadsheets for being as practical for building developmental models as concrete is for
building warehouses. I agree, and I add that they are peculiarly well-adapted to the task.
The use of spreadsheets requires that models be developed in a manner remarkably similar to
the developmental process itself. First, a formula is entered into a particular cell. This produces an
outcome that is displayed in that cell. Then, the formula is copied to other cells, where it is
automatically adjusted so that each new cell operates on the results from the prior cell. Textbooks
on chaos theory often give the impression of acomplex branch of differential equations with strange
visual realizations. A spreadsheet, on the other hand, reveals simple, iterated algebraic formulae.
Iteration in each new cell builds on the outcome in its prior cell, much like scientific projects build
on the outcomes of preceding projects. Besides the naturalness of spreadsheet iterations, graphical
displays can be added to the same sheet as the formula. This makes the computational results
immediate, like having a micro/macroscope that allows one to view a new generation of cells and
the new flower they comprise simultaneously. In addition to intriguing stirprises in the flowery
graphs, I found a model that fit my data as well as the general gamma, but it did not so much replace
that older model as explain it.
EVOLUTIONARY STRATEGIES AND POPULATION GROWTH
In 1798, Malthus (1798/1959) proposed that populations grow exponentially, like compound
interest, the number in each new generation being l-t-r times the number in the last generation. This
relation can be expressed in the simple formula (see Figure 1):
x'=[l+r]x
(1)
In this formula, each new value of jc' depends on the last value of jc. Before the advent of computer
programming, a little algebra would be used to convert this formula into one with a single unknown,
producing the familiar, "exponential" growth equation used for compound interest. Because
spreadsheets make the conversion no longer necessary, we can leave the equation in the simpler
form.
In 1846, P.-F. Verhulst proposed that population growth does not proceed indefinitely, but that
the rate of growth declines as the population approaches a maximum sustainable value. If A: is the
proportion of this maximum, then he suggested that the rate is reduced by the amount of possible
growth remaining. In other words, the rate becomes r(l—x). This produced his now famous
"logistic" model of population growth (see Peitgen & Richter, 1986; the underlined portion is what
Verhulst added to Malthus's equation):
36
DIRLAM
2S0O .
2000 .
/
1S00 ,
/
1000 .
500 .
0.
1
1
1
•"1
xO
1
1
25
20
^
3D
1
35
Generations
FIGURE 1 Malthusian (exponential) growth 20% growth rate from two individuals.
A
:A A f U A h A^iA A A A A
•
10
15
•
20
:
. > :
:
25
'•
'
30
• :
:
;
35
40
Generations
FIGURE 2 Proportion of maximtim population produced by selected rates of Verhulstian growth.
x'=\l+r(l-x)]x
(2)
With common growth rates, the new term r(l-x) causes a decline in the growth as x approaches
1 (the maximum sustainable population; see the line of wide dashes in Figure 2). Notice that with
r < 0, there is exponential decay (the wide dotted line in the figure). This fact combined with the
evolutionary implication of Verhulst's equation suggested its usefulness for explaining why
development might proceed via decay. It also suggested some intuitively satisfying features of
development that are not readily apparent in the general gamma. With changing positive values
of r, X approaches an equilibrium (r < 2, the wide solid line of Figure 2), oscillates (r > 2; the thin
black line), becomes chaotic (r > 2.57; the thin dotted line), and takes on nonsensical negative
values (r > 3; not shown). Exponential decay, sigmoid growth, oscillation, and chaotic fluctuation
are all common developmental pattems.
CHAOS, COMPETITION, AND NECESSITY TO CREATE
37
LOGISTIC MODELS OF COMPETING STRATEGIES
Strategies do not multiply in a vacuum, but compete at every opportunity with other populations,
or "growers" as they are currently referred to in the developmental literature. Van Geert (1991)
introduced the idea of two competing growers, each with different environmental carrying
capacities, k, and affected by both the level of a second grower, y, and its competitive strength, c:
x' = [l-\-r(l - xlk) - cy\x. Clayton and Frey (in press) modified Van Geert's equation to model
competition between two growers that share the same capacity: x' = [l+r(l -(x + y) / k}- cy]x.
Setting the carrying capacity to 1 had no adverse effect on the model's fit and reduced the number
of pariimeters. Note that the rate, r, is now reduced by x -h y (not just x alone). Also, note that cy
is subtracted from the total amount of increase of each generation. This allows for a reversal of
growth (i.e., decay), when the competition is high. The result with the underlined portions modified
from Verhulst's equation is the following:
x' = [l+r(l - (;c + -v)Vcv1;c
(3)
I generalized the Clayton-Frey model to any system of growing sfrategies in which competition
is directional, with later strategies suppressing earlier ones but not the reverse (j refers to strategies
with later mean times of appearance than the one in question and i refers to all strategies; as before,
the underlined portions were modified from the prior numbered equation):
jc'=[l+Kl-&)-£ca]j«:
(4)
Equation (4) needs to be restricted so thatx' never goes below zero (in spreadsheets, this requires
inserting the function that chooses a maximum value of 0 or the formula outcome). When each JC'
has been calculated, the probabilities can be determined from the use of all strategies in the system
(i.e., the maximum possible usage):
P(x^-xiljc'i
(5)
Finding the least-squares differences between (5) and the raw data resulted in fits to both the
drawing and research strategies data that were as good as the generalized gamma distributions (see
Figures 3 and 4). The complexity ofthe two models is also comparable (cf. Myung & Pitt, 1995):
(a) botli require a rate parameter; (b) adjusting the starting values {xo) for the competing strategies
system is comparable to combining cells to get frequencies greater than 5 for the general gamma;
and (c) selecting competition weights is comparable in complexity, but more flexible than selecting
predecessors. The excellentfits reinforce the discussion in Dirlam (1996) conceming the similarity
of drawing and research sfrategy development. They suggest in addition that the reason why
development proceeds via decay is that later strategies cause the use of earlier strategies to fade.
THE SURVIVAL VALUE OF COMPETITION AND THE PERSONAL
AND SOCIAL NECESSITY OF CREATIVITY
The stability of all but very small competing strategies systems depends on perpetually adding
new strategies. By the time several strategies have been replaced, new strategies need to have such
a.
a
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0.5
4.5
1.5 2.5 3.5
.1
5.5 6.5 7.5 8.5
Years after age 5.5
9.5
- - -1- - -11 no shape control-no organization
- - O - -12 no shape control-baseline
- - A - -13 no shape control-overlap
- - • - - 22 line control-base line
— O — 3 2 curve or soiid-i^aseline
- - A^ - -23 line control-overlap
- - • - - 24 line control-plane or cameo
— A — 3 3 curve or solid-overlap
— H — 3 5 curve or solid-plane
— 0 — 3 4 cunre or sdid-cameo
— A — 4 3 curve and solid-overlap
—X—..Designs
— • — 4 4 curre and solid-cameo
— • — 45 curve and solid-plane
10.5 11.5 12.5 13.5
b.
- - -1- - -11 no shape control-no organization
- - O - -12 no shape control-baseline
- - A - -13 no shape control-overlap
- - • • • - -22 line control-base line
——O
32 curve or solid-baseline
- - * - - 2 3 line control-overlap
- - • - - 24 line control-plane or cameo
— f t — 3 3 curve or solid-overlap
— B — 3 5 curve or solid-plane
•
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)
$
,
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.
-
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.2
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0.5
1.5
— A — 43 curve and solid-overlap
—X—..Designs
— • — 4 4 curve and solid-cameo
— • — 4 5 curve and solid-plane
^ '
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5 11.5 12.5 13.5
Years after age 5.5
C.
Q
1
.5
.4
3,
• * " •
.2
— O ^ — 3 4 curve or solid-cameo
— A — 43 curve and solid-overlap
—X—..Designs
— - • — 44 curve and solid-cameo
• — • — 4 5 curve and solid-plane
.1
*
i
0.5
- - - ( - - - ) 1 no shape control-no organization
- - O - -12 no shape control-baseline
- - A - -13 no shape control-overlap
- - • - - 22 line control-base line
— 0 — 3 2 curve or solid-baseline
- - A - -23 line control-overlap
- - • - - 24 line control-plane or cameo
— 0 — 3 3 curve or solld-overtap
— B — 3 5 curve or solld-plahe
1.5
2.5 3.5
4.5
5.5 6.5 7.5 8.5
Yeans after age 5.5
9.5
10.5 11.5 12.5 13.5
FIGURE 3 (a) Raw probabilities of drawing strategy usage, (b) Probabilities of strategy usage according to the
general gamma, (c) Probability distributions of competing strategies in drawing.
38
- - A - - 1 1 1 simplification
- - O - - 1 2 1 social
- - • * • - - 1 1 2 localized
— A — 2 1 2 anaiyticai iocaiized
--•--122situated
211anaiyUcai
221 anatytical social
222 anaiyticai situated
0.0
12
16
20
Yrs after 1968
- - A - - 1 1 1 simplification
- - O - - 1 2 1 social
--*--112locaii2ed
— A — 212 anaiyticai Iocaiized
- - • - - 1 2 2 situated
— D — 2 1 1 anaiyticai
— A — 221 anaiyticai social
— • — 2 2 2 anaiyticai situated
12
16
years after 1968
-A--111sitnptiflcation
- O - - 1 2 1 sociai
- - * - - 1 1 2 iocaiized
A— 212 anaiyticai iocaiized
-•--122situated
211 anaiyticai
0.1
221 anaiyticai sociai
222 anaiyticai situated
12
16
Yrs after 1968
FIGURE 4 (a) Raw probabilities of research strategies, (b) General gamma probabilities of usage, (c) Competing
strategies system for research.
39
40
DIRLAM
high growth rates to get established that in the absence of competition from later strategies (i.e.,
in the absence of innovations), their continued growth will become chaotic. If the strategies in
question are evolutionary, then the low points in chaotic growth (see the thin dotted line in Figure
2) make a grower particularly vulnerable to extinction. This finding implies that the existence of
an effective competitor increases a species' chance of survival.
In the context of developmental strategies, as any nondrinking alcoholic is aware (see Lave &
Wenger, 1991), the extinction argument does not apply. Developmental strategies may return after
falling to zero frequency for long periods of time. Likewise, apparently abandoned historical
strategies (e.g., genocide or even human sacrifice) may resurface with agonizing frequency. Even
so, Clayton and Frey (in press) noted that despite their high growth rates, chaotic growers do not
compete successfully against slower rate, nonchaotic growers. These conclusions strongly support
the value of innovation to both individual and social stability.
So what happens to the prediction of the demise of the analytical, contextualized approach in
developmental research? The competing strategies model suggests that it is possible for this last
strategy not to decay in the time frame predicted by Dirlam (1996), because no competitor may be
found. But if a successor strategy is not found, then we can expect the last strategy's usage to begin
fluctuating in a chaotic fashion, endangering the whole enterprise. A reasonable and testable
projection is that within a decade, chaos modeling will have sufficient competitive strength to
begin causing the usage of difference combined with correlation statistics to level off. To be a
competing developmental research strategy, and not just a new analytical technique, however, such
modeling will need to be combined with the multiple social contexts and well-described environmental settings that characterize sociocultural research today.
ACKNOWLEDGMENTS
Research funds were provided from the Mellon Foundation and the Appalachian College Association.
Special thanks go to Kurt Danziger for advice and encouragement through several years ofthe
history project, Michael Cole for encouraging this article and its predecessor, Daniel Bowell for
assistance in securing sources, Daniel Fetters for clarifying various forms of Verhulst's equation,
and Keith Clayton and Barbara Frey for the manuscript that included their competing strategies
equation. Step-by-step data analysis details and an Excel example can be obtained by letter or
e-mail (dkdirlam @king.bristol.tn.us).
REFERENCES
Clayton, K., & Frey, B. (in press). Inter- and intra-trial dynamics in memory and choice. In W, Sulis & A. Combs (Eds.),
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Cooper, C. R. (in press). Genre and genres. In C. R. Cooper & L. Odell (Eds.), Evaluating writing: Second edition. Urbana:
National Council of Teachers of Enghsh.
Danziger, K. (1990). Constructing the subject: Historical origins of psychological research. Cambridge, England:
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CHAOS, COMPETITION, AND NECESSITY TO CREATE
41
Dirlam, D, K. (1996). Macrodevelopmental analysis: From open fields to culture via genres of art and developmental
research. Mind, Culture, and Activity, 3, 270-289.
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