A Dynamic Systems Model of Cognitive and
Language Growth |
University of Groningen, Groningen, The
Netherlands
Correspondence may be addressed to: Paul
van Geert, Department
of Psychology, University of Groningen, Grote Kruisstraat 2/1, Groningen
5712 TS The Netherlands.
The problem of quantitative increase of a capacity,
skill, or knowledge base, a major concern for learning theories, is still
largely unsolved in current structural theories of cognitive and language
development. The theories address the problem of the emergence of new
cognitive capabilities in terms of adding or deleting rules (or whatever
the structural components are in the theory at issue) to or from a
knowledge or procedural base. For instance, in a system of production
rules that describes how the child solves the balance scale task (e. g. ,
Siegler,
1983), a specific decision rule is either part of the rule system or
not. In a transformational generative model of syntactic development, the
child's grammar either contains a rule to change the subject-verb position
in questions or not, but it does not contain rules such as "Change the
subject-verb position arbitrarily in 27% of the cases. " Put differently,
the distinctions between developmentally different states of a cognitive
structure occur in the form of discrete steps (see van
Geert, 1987a, 1988c,
for a more general account). However, the performance concerned does not
change in the form of a sudden discrete leap corresponding to a state
shift, but rather follows a gradual, sometimes irregular, increase
(discussed later). Classical structural theories of cognitive development
have accounted for such phenomena by introducing notions such as
décalage or resistance of contents to assimilation by a specific
operational structure (Smedslund,
1977) and distinctions such as between competence and
performance.
In modern structural theories of cognitive
development, information-processing models for example, the quantitative
increase of cognitive capacities is of more central concern. For instance,
some theories rely on the growth of working memory (e. g. , Case,
Marini, McKeough, Dennis, & Goldberg, 1986; Pascual-Leone,
1970) to explain structural changes, whereas others view specific
forms of increase, namely S-shaped curves, as the quantitative analogon of
a structural change (e. g. , Fischer,
1980; Fischer
& Canfield, 1986; Fischer
& Pipp, 1984). The fact remains, however, that the quantitative
aspect of cognitive development—referred to as cognitive growth
in this article—is not the primary concern of structural models. They
explain growth phenomena on the basis of transition mechanisms that are
often peripheral to the structural core of the models. The problem of how
to reconcile nongradual structural changes with gradual change in
performance does not stand on its own. It is related to why in individuals
the levels of development of various skills that allegedly refer to
identical underlying competencies or structural bases are far less
coherent than should be inferred from the underlying structural model (e.
g. , the discussion on the Piagetian structure d'ensemble
concept, Piaget,
1972; Flavell,
1982). Although there is no general solution to the problem of
quantitative increase in structural models, connectionist models of
cognitive development seem to offer a way out of the impasse. The
connections in an association network of cells change gradually, as does
the performance based on the network state (e. g. , Rumelhart
& McClelland, 1987). A difficulty remains, however, in that one
needs an explicit model of a network, one for learning the past tense of
verbs for example, to generate a quantitative learning curve. In this
article, I try to demonstrate that there is a general model of
quantitative increase or decrease in cognitive development, namely a
dynamic systems model of logistic growth. This model is intended to
apply to all theories that subscribe to the idea that cognitive growth
occurs under the constraint of limited resources.
I define cognitive growth as an
autocatalytic quantitative increase in a growth variable following the
emergence of a specific structural possibility in the cognitive system.
Examples include the growth of vocabulary, the growth of subject-verb
inversion in interrogative sentences, and the growth of the correct use of
a strategy in solving fractions problems. There are three properties of
cognitive growth. (a) The increase must be autocatalytic, that is, given
no extrinsic impediments; growth is something that runs by itself. Any
increase that amounts to mere addition from an outside source is not
genuine growth. (b) It must be quantitative; growth is a property of a
variable, the value of which can be expressed in terms of real whole or
fractional numbers, such as the number of words or a percentage of correct
applications. Growth could be used to describe qualitative changes such as
structural development, but that is not the meaning addressed in this
article. Of course, quantitative growth could result in qualitative
changes in a system, but again this is not the major concern here. (c)
Growth must follow a structural possibility of the cognitive system. That
is, observable behavior, such as the use of words or grammatically correct
interrogatives, should be considered to refer,^{1}
for instance, to the growth of vocabulary or to a subject-verb inversion
rule only within the framework of a structural theory that considers words
or interrogatives as the expression of a vocabulary or of a grammatical
inversion rule. Although this may seem rather trivial, it implies that a
growth model is always subsumed under a specific structural model, that
is, a model providing specific cognitive interpretations for observable
data (see van
Geert, 1987a, 1990,
for further discussion).
I define a growth relation in set-theoretic terms
as a relation with its domain a structural property and its range a field
of applications. The growth relation is defined over the cardinality of
this field of applications, that is, as a quantitative property. For
example, the inversion rule is a structural property in some generative
grammar model of language acquisition. Its range of application is those
sentences to which the rule is actually applied. The cardinality of the
range could be defined as the relative number of inversions in
questions—that is, the number of inversions divided by the total number of
questions during a time interval—or as the percentage of correct uses
relative to all sentences in which, according to a correct grammar, the
rule should be applied (e. g. , Labov
& Labov, 1978). The quantitative property of this range is called
growth level, or L. For instance, the number of words a
ctually understood by a child is the child's growth level of the
structural property passive vocabulary, the percentage of correct
questions with inversion is the growth level of the structural property
inversion rule, and so on.
Thus, a growth relation G can be written as
follows:
where a growth relation G is a relation
that maps a structural property S, defined over a time interval
t, onto an ordered sequence of growth levels L_{t1},
L_{t2}, and so on; the growth levels correspond to the
successive quantitative properties of the range of application of the
structural property S. G defines a quantitative
relationship between successive growth levels L, such as
L_{t1}/L_{t2}, which takes the form of a
regular increase or decrease, in other words, of a rate of change. Hence,
growth rate relation R may be defined in set-theoretic terms as a
relation that maps a growth level L onto another growth level,
given a specific time interval between these two levels:
Given the quantitative nature of L, the
relation R actually corresponds with a ratio r, for
where r is the growth rate.
It follows from Equation
2b that there can be no initial growth level with a value of 0 (if
L_{0} = 0, multiplication by any r only
leads to 0; i. e. , the growth level remains 0). Because the initial state
of growth is in principle the lowest possible growth level, and because
this lowest possible level cannot be 0, it should be some arbitrarily
small number (e. g. , one word, or one correct application). This
arbitrarily small number is the minimal structural growth level
of a cognitive property or element. For instance, if making a subject-verb
inversion in a question has become a structural possibility in a child's
grammar, one may expect the child to use a minimal (and probably small)
number of nonimitative questions using inversion. This is the rule's
minimal structural growth level for this particular child. The
growth-onset time is the moment at which a structurally minimal
expression of a cognitive element emerges. For instance, the growth-onset
time of a child's lexicon is theoretically the age at which the child has
acquired his or her first real word. However, such m inimal extension is
not only hard to ascertain empirically, it is also likely that the minimal
set is actually a few items (see the section on germinal phenomena).
Given a specific cognitive theory, a Piagetian or
an information-processing model for example, the structural elements
discerned in this theory, such as skills, concepts, and rules, may be
described metaphorically as cognitive species in a mental ecology: Each
"species" occurs with a specific population (growth level) and relates to
other "species," that is, to other structural elements (cf. Boulding,
1978). For instance, fast growth of the species words in a
child probably will affect, positively or negatively, the growth in the
species grammatical knowledge. This is so because one growth
process may "feed" upon the other. For instance, the onset of grammatical
growth may depend on the acquisition of some threshold number of words, in
that skills necessary to learn new words contribute to the learning of
grammatical rules, and rapid increases in grammatical knowledge consume
part of the time and effort that might be used in building up the initial
vocabulary (Dromi,
1986). I therefore compare the cognitive system of a developing
individual to an evolving ecological system, which is not an ecosystem of
animals and plants, but an ecosystem of cognitive "species" that take the
form of rules, concepts, skills, and so forth. ^{2}
The ecological metaphor is specified in five heuristic principles.
• Given a specific structural model, the human
cognitive system can be described as an ecosystem consisting of species
(i. e. , structural elements such as vocabulary, grammatical rules,
problem-solving skills, and concepts) that entertain growth relationships
with specific fields of application.
• The elements engage in various types of
functional relationships among one another, which are supportive (the
growth in one supports the growth in another), competitive (the growth in
one relates to the decline in another), or virtually neutral.
• The elements show strongly dissimilar growth
rates and growth-onset times.
• The components compete for limited
spatiotemporal, informational, energetic, and material resources.
• (a) There exist more cognitive species (skills,
knowledge items, rules, etc. ) that can in principle be appropriated than
actually will be appropriated by any particular person. (b) In principle,
any cognitive species may occur with any possible growth level. (c) The
set of cognitive species and respective growth levels characteristic of a
person's cognitive system is the dynamic product of cognitive growth under
limited resources.
The previous heuristic principles are reminiscent
of those holding for biological ecological systems in general, and
evolutionary systems in particular. For instance, the fifth principle is
reminiscent of a principle in Darwinian theory, namely that the number of
offspring exceeds the number of organisms that an environment is able to
support long enough for the organism to reproduce (Gould,
1977). From this it follows that the adaptation of organisms to their
environment increases over generations. Likewise, a heuristic claim is
that learning under competition for limited resources favors species that
can be learned more easily than others. Because learnability is dependent
on the set of supporting cognitive resources that together form an
individual's cognitive system, more easily learnable cognitive species
(rules, skills, concepts, etc. , that are more easily learnable in the
individual's current cognitive system) tend over time to become more
frequently represented in such a system than do less learnable ones (e. g.
, see Newport,
1982, for an application to language; van
Geert, 1985).
In the framework of this ecological metaphor, I may
speak about a cognitive grower and its environment. A
cognitive grower can be any of the species in the mental ecology, or any
structural element or component of a cognitive system to which the growth
relation applies (thus, by grower I do not mean an individual
child, but rather the child's vocabulary or the child's use of the
inversion rule in questions). Trivially, a grower is a cognitive species
that grows. The environment is the totality of supporting or competing
resources upon which the grower feeds. Thus, as far as the nature of the
cause, the magnitude, and time of the effect are concerned, I make no a
priori distinction between subject-dependent resources (e. g. , a child's
mastery in dividing numbers will support the learning of rules for solving
fractions) and external resources, such as amount of available models,
tutorial support, and so on (see also Fogel
& Thelen, 1987; Thelen,
1989).
There exist many kinds of different resources that
contribute to cognitive growth. I have just claimed that in principle no a
priori distinction can be made among resources of different types (e. g. ,
biological vs. environmental) in regard to their potential effect on
growth (e. g. , it is not so that in general, biological resources are
more important than environmental ones, or the other way around).
Decisions about the relative importance of resources depend on specific
contexts and circumstances of growth. Nevertheless, for the sake of
conveniently arranging an overview of the different resources, it could be
handy to make a distinction between two dimensions. The first concerns the
origin of resources and distinguishes between internal (in the subject)
and external (outside the subject) ones; the second dimension deals with
the nature of the resources, namely spatiotemporal, informational,
energetic-motivational, and material resources. It is probably so that
only in extreme cases will it be possible to distinguish the exact
contribution of each of these types of resources. After all, the
distinctions have no intrinsic theoretical meaning but primarily amount to
a matter of "bookkeeping. " The following overview is based on the two
descriptive dimensions mentioned.
The concept of internal spatial resources
refers to the limited amount of information one can deal with
simultaneously (Kahneman,
1973; Miller,
1956) or to the limited range of one's working memory (Baddeley,
1976). The size of this mental capacity is believed to increase with
age, either in the form of a literal increase or in the form of increasing
efficiency of information processing (Case
et al. , 1986; Globerson,
1983; Pascual-Leone,
1970). However, the specific nature of the increase is still a much
discussed issue (Case,
1984; Siegler,
1983). Internal temporal resources refer to the time on task
that one is able or willing to invest in a specific cognitive activity,
relative to the number of different cognitive activities carried out over
a specific period. Internal informational resources consist of
the knowledge and skills already present in the subject, which act as the
internal learning or acquisition context for new skills and knowledge and
which may either facilitate or impede the acquisition of specific new
knowledge or skills. Internal motivational/energetic resources
consist of the amount of energy, arousal, effort, activation, and so on
invested in specific acquisition activities (e. g. , Sanders,
1983). The energetic investment during specific information-processing
activities may constitute a distinguishing property of normal and clinical
groups in development (e. g. , van
der Meere, 1988). If energetic investment is defined as a
content-specific variable, it can be called motivation (e. g. ,
see Leontew's
1973 theory in which motivation plays a major developmental role).
Internal material resources amount to the bodily outfit of a
developing subject, for example, the availability of correctly working
sensory and nervous systems.
External spatiotemporal resources are the
spatial and temporal degrees of freedom given to developing or learning
subjects by their controlling environment. Caretakers and educators
explicitly restrict the free-moving space and time of children, with the
often implicit intention to structure this limited space in an optimally
profitable way for the child. This principle is also inspired by the
educator's need for a resource economy in the environment. Valsiner
(1987) described this principle as the "zone of free movement. "
External informational resources primarily amount to the number,
availability, and form of the items that could be assimilated by the
developing and learning subject (e. g. , the lexicon presented by the
speaking environment, the specific ways in which the teaching environment
makes information available to the learner). The third form of external
resources is energetic/motivational resources. These are
task-specific payoffs, such as the sort of reinforcement provided by the
environment after performance of specific activities of the learner.
External material resources are things like food and shelter,
objects such as books and writing paper, and so on.
The availability, nature, and relationships of all
these resources differ greatly among individuals and groups and also
within individuals (e. g. , temporal variations in the information given
to a child or in the nature and amount of the energetic resources
invested). However variable the resources may be, they are always limited,
and as discussed later, this limited availability is one of the major
formative forces in cognitive development.
At first sight it is very difficult, if not
impossible, to characterize the amount of growth support that the
cognitive environment provides to a specific grower in detail (e. g. ,
vocabulary in a specific child) because this support characteristically
amounts to various resources, many of which probably considerably vary
over time. There is, however, a simple way to specify growth support. It
is based on the following considerations. In principle, a grower might
attain any possible maximal level if all available resources were invested
in the grower concerned. However, there is competition for time, effort,
information, and so on from other growers, and this limits any particular
growth process. In fact, if too many resources were allocated to one
specific grower, the whole cognitive system would become unstable and
finally collapse. This is so because any individual grower depends
strongly on the support of other growers and thus has to leave sufficient
resources for the other growers to develop. Therefore, the highest
possible level a grower may attain is automatically limited by the
constraint of the long-term stability of the overall system.
In ecology, this limitation is associated with the
concept of carrying capacity K of the system (De
Sapio, 1976; Hof
bauer & Sigmund, 1988). It expresses the long-term sum of
resources supporting a specific grower over time, and it is specified in
the form of the maximal stable growth level of a particular grower in
this specific cognitive environment (i. e. , stable under the
condition that the present structure and amount of resources do not
drastically change). This definition has important theoretical
consequences. Because it is rather unusual to think in these terms in
psychology, I return to a (pseudo-)biological example. Suppose that there
is a hermetically closed cage that is populated with a couple of flies
that are free of diseases. Suppose also that fixed amounts of food, water,
oxygen, and so on are added per unit of time and that fixed amounts of
waste products, carbon dioxide, and so on are withdrawn. What is added and
withdrawn from the cage constitutes a multidimensional resource structure
(in that each component, such as water, food, and waste, is independently
variable). If the number of flies in the cage is the variable of focus,
this multidimensional resource structure may be transformed into a
one-dimensional measure, namely the maximal stable number of flies that
cage can contain, given all the resources invested per unit of time. Thus,
the multidimensional resource structure has not only been translated in a
one-dimensional variable, it also has been translated in a variable that
is qualitatively identical to the focus variable, namely a number of
flies. Clearly, a change in the focus variable (e. g. , number of very big
flies) coincides with a change in the carrying capacity (the sum of
resources must then be expressed in terms of the number of very big flies
the cage may support). Another advantage of the concept of carrying
capacity is that local variations in the amount of resources supplied do
not necessarily affect the stability of the carrying capacity level
itself. This is so for at least two reasons. First, if K is the
expression of a multitude of resources, relative scarcity in one may be
compensated within certain limits by relative abundance in another.
Second, the variation (if not catastrophic) is a resource factor in
itself: It is likely, for instance, that an irregular food supply with the
same average as a regular one in another cage would have a slight negative
effect on the number of flies the cage can sustain.
In cognitive development, one could say that the
carrying capacity of a cognitive environment with regard to a cognitive
grower such as vocabulary is a one-dimensional function of all the
informational and tutorial support actually given to word learning, the
time and energy spent in word learning, and the material support, such as
books and toys. The one-dimensionality stems from the fact that given the
investment of all these resources, there is a maximal level of vocabulary
growth that can in principle be attained. This maximum level (e. g. ,
50,000 words during the life span in a literary culture or 350 words
during the 1-word period for a bright child) is a quantitative measure of
the sum of resources contributing to word learning during a given term. In
summary, carrying capacity is a function that one-dimensionally expresses
the sum of resources over time in terms of a maximal stable level a grower
may attain given these resources.
External resource factors are specific in that they
are in general more directly controllable than internal factors. An
increase in any of the external factors that contribute to a specific
carrying capacity (amount of food in the fly example; amount of parental
help given in the example of word learning) would most likely lead to an
increase in the carrying capacity, that is, in the highest possible stable
level the grower at issue could achieve. Also likely, however, is that the
effect of such increases would not pass beyond a ceiling level-that is, a
level beyond which increase in external resource factors would no longer
positively affect the carrying capacity. This upper limit forms the
expression of the growth limitation inherent in the internal resource
factors. For instance, young children will not learn abstract words—which
they do not understand—no matter how much such words are used and
explained by the environment. This upper limit plays an important role in
Fischer's (1983a;
Fischer
& Silvern, 1985) skill theory. Most such upper limit K
levels will change as a consequence of overall developmental changes in
the cognitive environment. For instance, the child's growing understanding
of multiple relations will make the learning of abstract words possible
(Fischer,
1980) and will lead to a significant rise in the carrying capacity for
vocabulary. In optimal circumstances this rise is partly caused by the
increase in the internal resource factor (cognitive understanding) and
partly by the increase in specific help that the environment presents
following the change in the child. I discuss this mechanism further in the
sections on bootstrap growth.
In summary, the carrying capacity is a
one-dimensional variable closely linked to a specific one-dimensional
growth variable, namely the growth level of a specific grower (e. g. ,
words). It expresses the multidimensional structure of available resources
in terms of the maximal stable level the grower at issue could achieve in
the presence of these resources. Thus, it expresses resources in terms of
the same dimension as the variable that is focused on, namely the level of
a specific grower. Increase in external resources will in general lead to
an upper limit in the carrying capacity, which is characteristic of
intrinsic (but changeable) limitations in the internal resource factors.
If K is the carrying capacity of a grower
and L is its current growth level, the grower has to grow by
K minus L items before it reaches its ceiling (that is,
its growth limit arising from the limited resources in this particular
cognitive environment). Because K is a measure of the resources
available in a cognitive environment to a specific cognitive grower,
(K-L) is a measure of the amount of resources that can still be
used to promote further growth. Thus, the function (K - L) may be
called the unutilized capacity for growth, denoted by U.
As shown in a later section, it is a major component in logistic growth
models.
The form of the growth relation associated with a
growth process affected by a limited carrying capacity is as follows:
Given R, there is a ratio number
r, such that r =
Lt/Lt+i, and this ratio
depends on the unutilized capacity for growth, (K -
L_{t}) or U_{t}.
If a teacher tells an adult nonnative student of
English that sentences beginning with Wh require subject-verb
inversion, the effect on the number of correct interrogative sentences
uttered by the speaker is likely to increase immediately (it is assumed
that the student did not know the rule). In this case, there is a direct
(i. e. , undelayed) effect of an increase in informational resources on
the growth level of the inversion rule. Compare this situation with one in
which a class gets a better English teacher than the pupils had
previously. The introduction of the better teacher amounts to a sudden
increase in informational resources and tutorial support. However, it will
take some time before the effect of the better teaching is actually
observable in the pupils' performance. Even in the example of the
inversion rule, it is likely that a considerable amount of time will be
needed before the student actually uses the rule consistently. That is, it
takes a specific amount of time for the cognitive system to move from a
state producing a performance p to a state producing a
performance (p + ?p). In this respect, the cognitive
system is not different from other complex systems in nature. They are all
dissipative systems (Brent,
1978; Stewart,
1989), and changing them means that a certain amount of inertia,
friction, and resistance toward moving to a higher level of order must be
overcome. This consumes time and energy.
The time lag between a growth state, that is, a
specific growth level and its corresponding unutilized capacity for
growth, and its effect on a later growth state is called feedback
delay, denoted by f. The form of the growth relation of cognitive
growth processes with feedback delay is as follows:
Note that the only difference from Equation
3 is that the index for the L to the right of the arrow has
changed into t + f.
Feedback delay, as a content-specific expression of
the inertia of the cognitive system, is not the same as learning time,
although learning time contributes to feedback delay. For instance, at
later stages of vocabulary learning, more words are learned during an
equal time interval, and this is probably at least partly due to a
considerable decrease in average learning time per word. Learning time is
the average time needed for a word to move from an unlearned to a learned
state (Greeno,
1974). Feedback delay, on the other hand, is the time lag between
states—for instance, a present vocabulary level and a level that is
r·(K - L) percent higher (discussed later).
This time is not necessarily affected by changes in learning time.
Feedback delay is a measure of the inertia,
friction, or resistance that the cognitive system must overcome to move
from its present state to a more developed state. Although actual feedback
delay depends on a myriad of factors, feedback delay as such is probably a
constant property (i. e. , constant given overall constancy of the
cognitive system). This assumption is based on the general observation
that complex systems tend to reduce the degrees of freedom of each of
their components considerably, in that variability in many dimensions
reduces to variability in a single dimension (Haken,
1987; Stewart,
1989; Thelen,
1989). Feedback delay may of course change, but such change will occur
as a consequence of overall developmental changes in the cognitive system.
That feedback delay can indeed be considered a constant should of course
be demonstrated empirically. In this article, I show that a mathematical
model using this constancy assumption yields good mathematical
descriptions of regular and irregular empirical cognitive growth curves.
Although feedback delay is different in different growers and at different
times, the model presented herein is based on the simplifying assumption
that feedback delays for all growers involved in a specific interaction
are equal.
In discussing the concept of growth rate, I have
shown that an initial growth level must be a positive real number, however
small. The concept of "minimal structural growth level" has been
introduced to account for that fact. If the assumption is rejected that
everything that can grow in cognition is innately present in some minimal,
germinal form, then one must explain how the step from a nil state (growth
level is zero) to a germinal state (growth level is an arbitrarily small
positive number) can be made. This step cannot itself be a growth process.
There are three logically discernible possibilities. First, the germinal
state is innately given. Second, the germinal state has been inseminated
from outside the developing subject; that is, it has been taught or
imitated. Third, the germinal state has been constructed by the developing
individual. One may question whether these logical possibilities also
constitute psychologically relevant distinctions. With regard to the first
possibility, presence in a germinal state actually refers to the innate
nature of the concepts and strategies in question. Basic concepts in
particular have an important germinal component. Examples include the
notions of object (Spelke,
1985), humanity (Sylvester-Bradley,
1985), causality (Leslie,
1982), and number (Antell
& Keating, 1983). This component is not the result of intellectual
construction or teaching, nor is it qualitatively similar to the final
state (van
Geert, 1988a). For instance, the germinal
state of the concept of human being probably amounts to a
specific tendency on the part of the baby to pay attention to and interact
with events that are in general typical of, but not exclusive to, animate
objects (Sylvester-Bradley,
1985). The initial state of the concept of causality is probably a
modular type of perception rule operating on mechanical causality events
(Leslie,
1986). The actual onset of growth of these innate germinal states is
probably timed by the growth of conditional or control variables.
The second possibility for making the step from a
nil to a germinal state is by assimilating an externally presented model,
specifically through imitation and demonstration or teaching. This process
refers to the main source of intellectual growth as far as the
transmission and appropriation of culture by every new generation are
concerned. In teaching, the germinal form of a new grower is inseminated
from outside, and its growth is carefully supported and controlled.
However, imitation is a process that leads only to a germinal state of
what has to be appropriated by the subject; that is, it is the starting
point of a growth process. In this respect, imitation is similar to the
effect of allegedly innate skills: What is innate is a specific starting
point and possibility for learning and construction.
The third way in which a new grower can be
initiated amounts to its autonomous construction by the subject himself or
herself. That is, because there is neither an example that can be imitated
nor any innate inclination, the subject discovers a new cognitive
possibility. This is what probably occurs in creativity.
However, the three logical possibilities
discerned—innately present, imitated, and self-constructed—refer only to
potential germinal states, that is, to different types of starting points
or initial states of cognitive growers. They do not make a difference with
regard to the nature of the cognitive growth process itself, which always
amounts to a process of construction. It is never so that a skill, form of
knowledge, or whatever is innately given, or innately given in its
complete form. Such form is always the result of a process of construction
by the subject, regardless of the exact nature of the initial state or of
the resources supplied.
The construction of new germinal forms in cognition
is a major problem of development, originally discussed in Plato's
Menon, which is concerned with the emergence of new forms out of
old forms. This problem is still not satisfactorily tackled by existing
theories of cognition (Thelen,
1989). The process of constructing new cognitive forms is probably
similar to that in biology. Given a specific cognitive (or biological)
structure, there exists a limited domain of degrees of freedom for
constructing new forms (Ho
& Saunders, 1984; Saunders,
1984). The construction of new forms is an intrinsic possibility of a
cognitive system, in that its reproduction over time or its maintenance is
vulnerable to random perturbation (mutation) and to imported models
(imitation; Fogel
& Thelen, 1987; Siegler,
1984). In some cases, these unintended mutations of some existing
cognitive capacity are selected and supported by the external environment.
A good example is the early growth of words, based on meanings given by
the adults to protomeaningful acoustic productions in a baby (e. g. , see
Jakobson,
1960, on the growth of mommy-daddy words). In general,
however, newly emerging forms have to compete with those that already
exist, and although in the long run the new forms will turn out to be more
powerful than existing ones (e. g. , operational as opposed to
preoperational thinking), they are definitely much less powerful at the
time they emerge in a germinal form. In evolutionary biology, a comparable
problem occurs in explaining the emergence of new species, namely the
problem of cladogenesis (Gottlieb,
1984). It is often solved by using the concept of allopatric
growth or allopatric speciation (Mayr,
1976; Simpson,
1983). Allopatric speciation is rapid evolutionary change in a
geographically separated (i. e. , frontier) part of the original species
population. Because the separated part occupies its own small habitat,
relatively isolated from the mainland, it can change under relatively safe
circumstances, with little or no competition from the main species. Later,
the altered species form, if better adapted to circumstances that might
have changed in the meantime, may take over the habitat of the original
main population. Applied to cognitive development, allopatric growth means
that a new capacity, rule, and so on may be constructed by random
variation, selection, imitation, and so on. This may occur in a relatively
isolated and uncompetitive subfield of the field of application of an
already established capacity, rule, or whatever. Allopatric cognitive
growth is a natural phenomenon, because almost all fields of application
of a rule or production system break down into subfields.
These subfields are characterized by differences in
cognitive complexity, difficulty, specific domain of application, and so
on. A particularly clear example is offered in Klausmeier
and Allen's (1978) longitudinal study of concept development during
the school years. The authors distinguished four conceptual rule systems
that form a developmental sequence, namely concrete, identity,
classificatory, and formal levels. They observed that conceptual
development is not equal for all concepts at all levels. For instance,
there is a natural décalage between object, geometric, and
abstract concepts and between concepts within each domain as far as speed
and ease of development are concerned. A very difficult task would be to
construct a new concept rule system (e. g. , a classificatory level) for
the whole domain of concepts at once, but this task would be relatively
easy for limited domains, such as a particular concept from a particular
class (e. g. , the concept of cutting tool from the object
class). By the time a conceptual strategy in a more difficult concept
domain has reached the limit set by its carrying capacity, a more advanced
strategy is likely to be ready in a relatively isolated, easier subdomain
(e. g. , a specific concept). The latter may then be adapted to the
requirements of the more complex conceptual domain with relative ease.
Thus, instead of being some sort of nuisance, as for instance in Piaget's
model, décalages are the key to cognitive development, in that
they create opportunities for allopatric growth of new cognitive forms.
Décalages may occur because fields of application of cognitive
rules, concepts, and so on have a laminar structure; that is, they tend to
fall apart into (weakly) independent subfields. Although the problem of
the creation of new cognitive forms is indeed a central problem that
requires much further scrutiny, in this article I confine myself to
discussing growth following the installation of a germinal growth form.
I start from the simplest possible assumption,
namely that, for all practical purposes, during a given interval
t, a specific cognitive growth process is not intrinsically
restricted and that no feedback delay occurs. Given the growth rate
relations (Equation 2),
where ? is the symbol for difference. Because
there is no feedback delay, Equation
5 may be differentiated:
where e is the base of the natural logarithm,
which is the classical formula for exponential growth. To test whether an
empirical growth curve is actually modeled according to this equation, its
growth rate may be computed with a formula inferred from Equation
6, namely
The empirical growth rate results from taking the
difference between the natural logarithm of two consecutive growth levels
and dividing this number by the number of units of time;
L_{1} and L_{2} represent the size of a
growth variable at two different moments. Possible examples include the
lexicon or the number of problems solved in a standard test. ^{3}
Let me first try to apply this concept of growth
rate to an empirical example. Dromi
(1986) studied the growth of the lexicon in a girl, Keren, between the
ages of 10 and 17 months.
Keren's cumulative lexicon during the one-word
stage compared with an exponential growth curve, where growth rate
rStage and Structure: Reopening the Debate, Norwood, NJ:
Ablex. Copyright 1986 by Ablex. Adapted by permission. )
Figure 1 shows the cumulative lexicon (see also Gillis,
1984, and K.
Nelson, 1985, for comparable data). ^{4}n
= 1 language studies, the measurement error may be considered to be rather
low.
Growth rate may be computed by taking two points on
the growth curve, for example, at t_{27} (where the
number of words is 227) and t_{1} (where the number of
words is 1). Computed growth rate is about 0. 2 per week (i. e. , about 0.
88 per month, which is very high). If the exponential and the empirical
growth curves are compared, one sees that both run closely together until
about Week 27. At this point, the empirical curve levels off, but the
exponential curve continues to increase. It is interesting that the sudden
drop relative to the exponential curve coincides with the onset of
syntactic development, as shown by the child's use of multiword sentences
(Dromi,
1986). My first tentative conclusion from this example on vocabulary
growth is that initial growth rate—in this case, vocabulary growth during
the 1-word stage—should be very high, or at least much higher than later
growth rate. If it were low, it would take too much time to build up a
critical mass in the domain at issue. Thus, initial growth rate should be
quite high. The second conclusion is that high initial growth rate should
drop very quickly after some critical mass has been achieved, simply
because a continuing high growth rate would lead to exhaustion of all
available resources, for example, learning time. For instance, if the
initial growth rate of0. 88 per month in early lexical development were to
continue, by the 16th month of learning the child would have to assimilate
17 items every minute, nights included, to keep up with the growth rate.
This is physically and psychologically impossible. Consequently, a model
that does not explicitly take resource limitations into account seems
untenable.
In the previous example, vocabulary growth slowed
down at the end of the one-word stage for the child concerned. Why or when
exactly this deceleration occurred does not matter, but that it occurs is
necessary, because otherwise the exponential growth would rise too
rapidly. The deceleration could be the effect of an underlying growth
program's putting the brakes on Keren's word learning and accelerating her
syntax learning. However, slight delays in the onset of the preprogrammed
braking action would provide an exponential grower that is only slightly
in advance of others at the beginning of the growth process, with the
opportunity to rise to an extremely high level. The price of this
exponential eruption would be the consumption of almost all the available
resources, and this would seriously jeopardize the growth of potential
supportive skills and knowledge. However, the model includes the
assumption that the collection of supportive and competitive relations
among cognitive growers in a single subject sets intrinsic and specific
growth limits for any individual grower in the form of a specific carrying
capacity K. For instance, there is an intrinsic limit to the
number of words that are accessible to the child at any given moment (MacNamara,
1982; K.
Nelson, 1985). This accessibility is based on various factors, such as
the average word exposition time per day, the frequency and number of
words used by the more competent speakers in the child's environment, and,
in particular, the cognitive accessibility of words. Imagine a child
growing up in a family of dog breeders with tax problems: Although both
words occur very frequently, the young child will easily learn the word
dog but probably not taxes. ^{5}
Let K be the (temporary) carrying capacity
resulting from the limited-resource factors mentioned earlier, and let
L be the number of words already assimilated in the vocabulary.
At any moment, the unutilized opportunity for vocabulary growth is
(K - L). From the definition of growth under restricted
carrying capacity (see Equation
3), one may infer that growth is a function of the number of items
already acquired relative to the maximum of learnable items; that is,
for
Carrying capacity and the unutilized opportunity
for growth inferred from it may take different psychologically operational
forms. For instance, if a minimal number of encounters with an item is
required in order for it to be assimilated, then the more items one knows
the lower the probability that an encountered item will be unknown, and
therefore the lower the probability that a contribution to item growth
will be made. Another possibility is that easily learned items are learned
first (i. e. , they are the first to move from an unlearned to a learned
state), such that learning rate becomes slower as the number of unlearned
items grows smaller. The more difficult an item, the longer it will remain
in the unlearned item set. Finally, U may consist of tutorial
assistance that decreases as the level of mastery of the tutee increases.
Any combination of such factors is also possible.
In line with classical approaches to logistic
growth in biological sciences, I shall (provisionally) assume that there
is no feedback delay, that ?t approaches 0. Thus, Equation
8a may be differentiated to find the classical logistic growth
function (DeSapio,
1976):
for
meaning that the growth level at time t
is a function of a fraction, where the numerator is the carrying capacity
and the denominator contains the product of a constant c (Equation
10) with the exponential function of the negative product of the
carrying capacity K, the growth rate r, and time
t). If K is set arbitrarily to 1 and L is
expressed as a fraction of 1, logistic growth rate in a given empirical
growth process can be computed according to the formula
This equation is applied to the data from Dromi's
(1986) study. Provided the drop in lexical growth rate at the end of
the one-word stage indeed refers to an upper limit of accessible words,
this limit may be estimated at 350 and then set arbitrarily to 1. Then the
number of words at Week 32 (n = 335) is expressed as a fraction
of 1. Equation
11 yields a weekly growth rate of about 0. 28, which is very high.
Empirical curve of Keren's vocabulary growth
compared with logistic curves, where K
Figure 2 shows that the logistic curve, although
roughly similar to the empirical curve in its general S shape, differs
considerably from the latter in terms of its slope. This lack of empirical
fit implies that the chosen growth model—logistic with feedback delay
equal to 0—is inadequate for the present data. However, consideration must
be given to whether there is a form of cognitive growth that can be
modeled after the present logistic/no-delay model.
Fischer and Pipp (1984;
see also Fischer
& Canfield, 1986) claimed that cognitive growth phenomena take
place in the form of S-shaped growth spurts (e. g. , see the growth in
correct application of the concept of sweetness in Strauss
& Stavy, 1982). They further explained that such spurts are seldom
found because standard testing conditions provide a distorted image of
real cognitive growth. To reveal such growth curves, testing practices are
needed in which the subjects are given feedback to their answers and
additional support (see
The growth of arithmetic problem solving under
practice-and-support and spontaneous conditions compared with theoretical
curves (difference and differential form). (Adapted from "Processes of
Cognitive Development: Optimal Level and Skill Acquisition," p. 57, by
Mechanisms of Cognitive Development, New York: W. H. Freeman.
Copyright 1984 by W. H. Freeman. Adapted by permission. )
Figure 3). If the growth rate is computed according
to Equation
11, there is a yearly growth rate of 1. 26. The logistic curve based
on this growth rate approximates the empirical curve only roughly (Figure
3). In a later section of this article, I present a dynamic systems
application of the logistic curve and show how a much better fit with
irregular curves may be achieved.
In the previous section the assumption was tested
that growth is a nondelayed function of L and U (i. e. ,
of a present growth level and the resulting unutilized capacity for
growth, K - L). There was only a rough qualitative fit
to the data, in that both the empirical and the theoretical curve had an S
shape. Before studying the effect of feedback delay on growth, I test the
assumption that growth is only a function of U (i. e. ,
K - L) and not of U and L. The
equation for the curve is as follows:
If there is no feedback delay, Equation
12 may be differentiated to yield
and r can be computed as follows:
An example of this type of growth is offered by Klausmeier
and Allen's (1978) longitudinal study of concept development during
the school years. Four levels of processing conceptual information were
distinguished. Children learn to solve increasingly complex problems at
each level for each specific concept (e. g. , the classificatory level of
the concept noun). Most of the growth curves reported in
Klausmeier and Allen's study take the form depicted in
Growth of noun-concept problem solving at the
classificatory level compared with curve for restricted growth, where
rCognitive Development of Children and Youth: A Longitudinal
Study, p. 12, by
Figure 4. These growth curves are typical of
restricted growth, that is, growth that is solely determined by the
unutilized opportunity for growth U, which is the difference
between the growth level already acquired and the maximal growth level
K. For the empirical growth curve shown in Figure
4, Equation
14 yields an average growth rate of 0. 28 (for K = 1). The
resulting computed growth curve follows the empirical one very closely.
Growth equations (12
and 13)
imply that the conceptual growth measured by Klausmeier and Allen is
entirely determined by negative evidence, that is, by the nature and
amount of the problems the child is not yet able to solve. This follows
from the fact that in these equations growth is not determined by
L—that is, the knowledge the child already has—but by K
- L—that is, the knowledge the child has not yet achieved. There
is no way for the child to know or be confronted with this unknown
knowledge other than in the form of the errors the child makes, the
problems he or she is confronted with that he or she is not yet able to
solve, the corrections made by a tutor, and so on. The form of the
Klausmeier and Allen curves is very similar to that of the classical
learning curves (Hilgard
& Bower, 1966). It is not too difficult to fit these learning
curves by applying equations that define learning as the negative growth
of what the learner does not yet know (see van
Geert, in press). The major problem with Equation
13, however, is that it does not model S-shaped curves (only the upper
part of the S shape) that are characteristic of many forms of cognitive
growth. Thus, it appears that the assumption underlying the logistic
equation (Equation
9) is basically correct and that Equation
13 models only a subset of growth forms. I next test the assumption
that a better fit can be achieved by introducing feedback delay.
The delayed-feedback hypothesis can be covered in a
simple mathematical model, namely one that does not use differential
operators, as with the growth equations discussed earlier, but instead
uses difference operators (Burghes
& Wood, 1985). The assumptions are a growth level
Lt (e. g. , the number of words a child knows at his or
her second birthday) and a carrying capacity K (the maximal
numbers of words the child could acquire and maintain, given the present
conditions of internal and external cognitive support). The state of the
growth level may be inferred at a later moment, where later means
after feedback delay time has elapsed, according to the growth form (Equation
3) as follows:
Also, the rate r' is a function of the
unutilized opportunity for growth. However, if growth rate is to be
treated as a dimensionless variable, the value of which is not dependent
on the absolute magnitude of the variables on which it operates, then
r' should be assumed to be a function of the relative unused
opportunity for growth and a constant intrinsic growth rate r (Hofbauer
& Sigmund, 1988):
Combining Equations
15 and 16
yields the difference equation for logistic growth, that is, the
equation for logistic growth with feedback delay, namely
This iterative equation specifies any point on a
growth curve as a function of a point that occurs a temporal interval f
earlier. Given an initial growth level, it is easy to generate a growth
curve as a set of points at a mutual time distance of f(in another
section, I discuss how to infer intermediary points).
An interesting notational variant of Equation
17a is
for
(in Equation
17b, a is used as a braking parameter).
Equation
17a has a stable solution, in that L no longer changes. To
find this stable solution, the variable part of Equation
17a is set to 0; that is,
It follows that Equation
17a has a stable solution when
The stable solution is an important property of
the logistic growth curve, as shown later. Given an empirical curve, one
could infer the growth rate r by taking two points at a distance
f from one another according to the following equation:
Because f is unknown, any two consecutive points
on an empirical curve can be used to compute a corresponding r.
It can be shown, however, that the overall fit of the theoretical growth
curve increases as the estimated time interval approaches the real
feedback delay (see
The length of feedback delay defines a unique
curve: Differences in growth rate r cannot compensate for
different feedback delay (f = 1 vs. 4 and 8).
Figure 5).
Actually, time does not appear as a real variable
in Equation
17a (in contrast to time in the ordinary logistic growth Equation
9). Time is nothing but an index variable. The unit of time is the
feedback delay. An empirical interpretation of the real length of f may
follow from curve fitting: Given an empirical curve of which the total
growth time t is known, feedback delay equals t divided
by the number of iterations used to reconstruct the empirical curve
mathematically.
The previous iterative growth equation has proved
its validity in a wide range of applications, such as meteorology,
population dynamics, economics, and fluid dynamics (Abraham,
1987; Abraham
& Shaw, 1987; Garfinkel,
1987; Gleick,
1987; Hofbauer
& Sigmund, 1988). Peitgen
and Richter (1986) termed these dynamics Verhulst dynamics,
named after a Belgian 19th-century mathematician and population
researcher. These dynamics have a number of interesting and unexpected
properties, despite the very simple character of the basic equation (Schuster,
1988).
I next explore whether this simple model is indeed
capable of giving a simple mathematical explanation for some of the growth
forms found empirically. I have shown that the ordinary, differential
logistic equation sharply overestimated the first half of the growth curve
in Dromi's
(1986) study and underestimated the second half. With different
feedback delays, for instance 1 and 2 weeks, Equation
20 can be used to find the corresponding rs. The best overall
approximation seems to be one with a feedback delay of 2 weeks and
r = 0. 71 (see
Empirical curve of vocabulary growth (Stage and
Structure: Reopening the Debate, Norwood, NJ: Ablex. Copyright 1986
by Ablex. Adapted by permission. )
Figure 6). The major argument for taking a feedback
delay of 2 weeks is that it provides a better fitting curve than a
feedback delay of 1 week. A feedback delay of 2 weeks is not necessarily
the most optimal solution (although better than 1 week), but it is easy to
test, because the data are based on weekly measurements. An additional
empirical argument is that 2 weeks is the average time for a word to stay
in an underextended state (Dromi,
1986). A still better fit can be obtained by considering the empirical
curve as a two-step process. The first step seems to be an initial growth
period stabilizing at about 25 words (Week 12). Then a secondary growth
period follows, starting at the 25-word level and stabilizing at about 350
words. A good-fitting curve for the second substage has f = 1 week and
r = 0. 35 (see
Mathematical curves of vocabulary growth in second
substage of one-word stage: Differential logistic curve (f = 0) compared
with difference form (f = 1 week, r
Figure 7). The corresponding undelayed feedback
curve either strongly overestimates the initial growth speed or strongly
underestimates the growth speed toward the steady state (Figure
7). The two-substage hypothesis is supported by the fact that the
development of word meaning in Keren's vocabulary proceeded in two
different stages, one in which semantic extensions of newly acquired words
were unpredictable and a second in which those extensions were regular and
closely followed the adult meanings (Dromi,
1986). These two types of meaning acquisition were clearly
differentiated only at about Week 19, although Week 12 was considered the
point of separation. This delay might be caused by the fact that the
observable overt expression of a growth phenomenon is delayed relative to
its actual growth onset or by the fact that the two substages of
vocabulary growth overlap. A general explanation for the possible
two-substage character might be found in the potential differences in
cognitive requirements and contents of initial words (e. g. , those might
be more directly related to sensorimotor discoveries; Brown,
1973; Gopnik,
1984). A related explanation refers to the child's discovery of the
"naming insight" (K.
Nelson, 1985). Such assumptions of course remain to be tested
empirically. In the next part of this article, however, I discuss a
dynamics model that generates stepwise growth, comparable with the
stepwise growth in Dromi's data, without referring to ad hoc structural
explanations. Finally, please note that the growth rates and feedback
delays found in this example are typical of Dromi's subject and do not
constitute general or universal parameters of vocabulary growth. Later, I
discuss other data (Corrigan,
1983) with different lexical growth curves. It is likely that these
parameters vary among subjects and that upper and lower boundaries of
these parameters define "normality. "
An interesting feature of the iterative logistic
growth equation is that it is capable of describing nonlinear growth
phenomena and even near-chaotic growth (Peitgen
& Richter, 1986; Schuster,
1988). For r < 1, the curve has the characteristic S shape
or sigmoidal form seen in the vocabulary data. This may be called
asymptotic growth. For 1 < r < 2, the curve shoots
above the stable solution level, which is the carrying capacity (see Equations
18 and 19),
then drops and moves on in a decreasing vibration, aiming at the level of
the carrying capacity (
A smooth and linear increase in growth rate
corresponds to abrupt shifts in the form of logistic growth curves
(vibration toward stable level K for
rrr
Figure 8, bottom curve). This may be called
approximate growth. The carrying capacity, which is the stable
solution of the logistic growth equation, is a so-called point attractor
for all growth processes where r < 2. For 2 < r
< 2. 57, the growth process has 2n attractors, and growth
takes the form of a periodic oscillation (Figure
8, middle curve), which may be called oscillatory growth.
Above 2. 57, the process loses its periodicity and moves into chaos, or
near-chaotic growth (Figure
8, top curve). The previous processes provide good illustrations of
the principle of phase shifts following boundary transitions of a
gradually and monotonically changing control parameter (Fogel
& Thelen, 1987). Linear increases in the growth parameter lead
growth processes over several sharply distinguished types (asymptotic,
approximative, etc. ).
In the previous section, I discussed cognitive
growth as a form of increase, expressed by the growth rate r.
Obviously, however, there is not only increase, in the form of acquiring
or learning words, skills, and so on, but also forgetting, the loss of
proficiency, and so on. Thus, a forgetting factor needs to be included in
the logistic growth equation. Let m' be a rate of forgetting,
loss of competence, and so on. In accordance with the basic growth form
and Equation
15,
One may assume that m' is the complement
of r'. Thus, whereas r' is dependent on the unutilized
opportunity for growth U, m' is related to U's
complement. Because U = (K - L)/K, its
complement is L/K:
Substituting Equations
16 and 21
in Equation
15' yields the difference equation for logistic growth and
forgetting:
This equation has a stable solution when the sum
of the variable parts is 0; that is,
From Equation
18' it follows that 17a'
has a stable solution when
This means that if a learning-and-forgetting
curve grows toward a stable level, it grows toward a fraction
r/(r + m) of its carrying capacity. It may now
be questioned whether there exists a growth rate r'', producing a
growth process that is identical to a growth process produced by a growth
rate r and a forgetting rate m, with a stable solution
that is described by Equation
22. That is, what is the value of r? if
The answer from solving the above equation is
that
Put differently, the logistic curve for growth
and forgetting is identical to the curve for growth only, provided that
the carrying capacity of the latter is set to
This means that the effect of forgetting or loss
of proficiency merely consists of lowering the carrying capacity, in
comparison with the case in which no or less forgetting occurs, whereas
the growth rate r stays the same. That is, in Equation
17a, the rate of growth is the net result of actual learning and
forgetting processes, and the carrying capacity K is a function
of pure learning and pure forgetting; that is, K is lower the
higher the rate of forgetting. If forgetting were not a resource-dependent
function—that is, if Equation
21 would not hold—there would be an entirely different growth curve.
However, if learning, the sheer cognitive growth increase, is a
resource-dependent function, it would be very surprising if its
complement, forgetting, would be resource independent.
The difference equation for logistic growth reduces
a growth curve to a set of discrete points, at a distance of (n ·
f) from the initial state. The inference of intermediary growth points is
not trivial. For instance, if the logistic curve were to describe the
growth of a fly population, the members of which die at the end of the
season after having laid their eggs, there would be no intermediate
points. If the equation describes vocabulary growth, it actually specifies
a mathematical relationship between any pair of growth levels that are
separated by a time interval f. That is, in this case, there are
intermediary states, because it is unlikely that all words simply pop up
together an interval f later. Rather, they will probably be released in an
exponential way: The more easily learnable words emerge soon after the
process of learning has started, and the closer the end of the f interval,
the more words will emerge per unit time. Thus, given an initial state, a
feedback delay time, and a growth rate, one may compute
L_{t+f} and infer any number n of intermediary
states by solving Equation
24a for r':
It is easier to solve Equation
24b, however, which is based on Equation
7:
This yields r_{e}, which is an
exponential growth rate, for n, the number of intermediary steps
wanted. Any intermediary step between L_{0} and
L_{f} may be taken as a point for applying the logistic
growth equation. Thus, the growth curve may be filled with intermediary
points to any desired level of detail.
Exponential release of items probably occurs only
at growth intervals that are very close to the minimal structural growth
level, that is, close to the realinitial state of growth. A general
algorithm to make growth interpolations for irregular growth far from the
initial state is the fractal interpolation method (Grasman,
1990). It is based on the concept of self-similarity. If there is an
irregular growth process, with quasi-random ups and downs, the sequence of
intermediary points between any two given measurement points can be
assumed to follow a pattern that is geometrically similar to the pattern
of the overall process. I use this property in a reconstruction of
intermediary points in the growth of lexical knowledge.
In real life, cognitive growth and learning are
probably subject to all sorts of random fluctuations, for instance
fatigue, fluctuations in the condition of health, random fluctuations in
the quality and quantity of the information available to a learner, and so
forth. ^{6}
The logistic growth model is quite robust in the face of these random
fluctuations, and this is so for two reasons. The first reason concerns
the nature of the variables involved. The way in which growth rate and
carrying capacity are defined implicitly takes the normal, random
fluctuations of life into account. Carrying capacity is defined as the
maximal stable growth level attainable, given all the resources and
limiting factors of a cognitive environment. This implies that normal,
random fluctuations, insofar as they interfere with cognitive growth, have
been taken care of in the form of a resource property contributing to
setting a specific limitation on the height of the carrying capacity
(comparable to the way in which forgetting is accounted for by the height
of K). Because the carrying capacity and the growth rate form a
sort of weighted sum of a great variety of variables, very significant
random fluctuations are needed to cause small to moderate changes in
either carrying capacity or growth rate.
The second reason that the logistic growth equation
(the difference form) is rather robust in the face of random perturbation
lies in the form of the equation itself. For instance, if at some (or even
each) step of the computation of a growth sequence, a random number of
about 10% is added to or subtracted from the growth rate and carrying
capacity, the resulting growth curve is still very similar to one where no
such significant random numbers have been added. Note, however, that if
L has approached K very closely, the variance of
L is greater or smaller than the random variance of K,
dependent on the mean value of r (high average rs
producing random variance in L that is much higher than the
random variance in K) and the random variance of r.
Note also that robustness is just one side of the
coin. The effect of random perturbations is state dependent. For instance,
in some well-determined regions of growth processes, even very small
random perturbations have important longterm effects, for instance because
a random perturbation pushes the system over a threshold value or because
in some regions small random perturbations are of about the samemagnitude
as the structural changes themselves. In general, however, it may be
stated that ordinary simple logistic growth processes are quite
insensitive to normal random perturbations or moderate random variations
in the carrying capacity and growth rate.
The subsumption of cognitive growth under
ecological principles, particularly that of the struggle for limited
resources, implies that growth processes have a price. That is, the growth
of vocabulary or the increase in mathematical skill requires time and
energy. Provided the "price" of a growth process could be estimated, its
efficiency might also be calculated, and by so doing the relative
efficiencies of different parameter values (e. g. , different growth
rates) could be compared. It is intuitively clear, for instance, that a
delayed growth process with a growth rate higher than 2. 57 is
considerably less efficient than one with a rate of 1. The first uses time
and energy to run through a chaotic oscillation and builds up high growth
levels that are torn down immediately afterward. The growth process with
r = 1, on the other hand, evolves toward a steady-state level of
1 and stays there. One may assume that a final growth level that is about
the height of the carrying capacity is most optimal from a cognitive
economy point of view: It is stable and provides a reliable basis for the
growth of additional skills, rule systems, concepts, and so on so that no
resources are left unused. A simple operationalization of growth
efficiency, therefore, is the average relative distance between the
carrying capacity and the growth level, measured over a fixed time
interval and starting with a realistically low initial state (e. g. , 1%
of K). Previously, I defined the relative distance between
L and K, that is (K - L)/K,
as the unutilized capacity for growth U. Now I call the average
U over a fixed time interval U_{m}. Because the
central issue is cognitive growth, that is, the processes that lead to the
attainment of a steady state, the short-term efficiency of growth
processes is measured, meaning the efficiency over the time needed to
approximate the steady state. The more efficient a grower, the less time
it needs to approach the steady state, and therefore the smaller the
average relative distance between K and L, or
U_{m}. Because the "cost" of learning a word cannot be
compared, for instance, with that of learning to solve a fraction problem,
I confine the comparison of efficiencies to one grower at a time (i. e. ,
I compare the efficiency of different growth rates in vocabulary, for
instance, but I do not compare vocabulary with learning to solve
fractions).
Given a set of growth processes that are identical
except in one variable, which in general will be growth rate, there is at
least one for which U_{m} is minimal:
U^{min}_{m}. I can arbitrarily set the cost in
terms of time, energy, effort, and so on of
U^{min}_{m} to 1. Thus, the relative growth cost
of a grower A (e. g. , a grower with a growth rate
rA) is
where E_{L} is a measure of the
relative "expenses" made to let L increase.
Provided the carrying capacities for all the
compared growers are similar, the only extra cost factor involved is the
cost of maintaining different growth rates. It is assumed that the cost of
maintaining a high growth rate is higher than that of a low growth rate. A
high growth rate implies a higher speed of acquisition and thus requires
better information handling and structuring. From an ecological point of
view, the maintenance of a more complex structure is more expensive in
terms of resources needed than that of a less complex structure (similar
principles occur in thermodynamics; Atkins,
1984). The cost of maintaining the average growth rate that led to
U^{min}_{m} may be arbitrarily set to 1, and the
cost of all other growth rates may be expressed as follows:
Let wr be a weight factor
attached to Er, which, if EL and
Er are considered to contribute equally to the total
growth costs, is set to 1. To compare growers with different carrying
capacities, one should reckon with the fact that maintaining a specific
carrying capacity level uses resources and thus contributes to the
expenses. Consider the most optimal carrying capacity level to be set
arbitrarily to 1—whatever "most optimal" may mean under specific
circumstances—such that the cost involved in maintaining this carrying
capacity is 1 too. Then attribute a weight wK to it. The
final efficiency equation is
Efficiency graphs for different growth rates, where
rrrrr
Figure 9 represents an efficiency graph for an
ordinary logistic growth curve with delayed feedback, with an initial
growth level of 0. 1 and a time interval of 20 growth steps, which is
sufficiently long for most growth rates to approach the final state level
(if any exists). Given the chaotic nature of growth where r >
2. 57, the right part of the graph is not very interesting. However, as
can be seen from Figure
9, there are two local optima, namely about 1. 5 and 1. 86. These
growth rates correspond to the approximate growth type explained in an
earlier section: It is a very fast growth form characterized by a sequence
of over- and undershootings approaching the K level. It is the
sort of growth that can be expected in fast and early forms of learning,
such as the learning of the meaning of words.
Although the differences between the optima and
their neighboring points are small, such optima may be interesting from a
cultural-evolutionary point of view (see Boyd
& Richerson, 1985; Lumsden
& Wilson, 1981). Assume that there exist two alternative forms of
a skill that have largely similar sorts of functional meaning (e. g. ,
ways of solving social conflicts either by democratic or by autocratic
decision). Suppose further that these alternatives are currently evenly
distributed over the population and that their learning by a new
generation is largely a matter of imitation. If both strategies are
equally difficult (or easy) to master, their distribution over the
population will remain identical over consecutive generations. However, if
one is easier to master—in the sense that its appropriation goes more
efficiently than with the alternative—a slight increase may be expected in
the number of people from the next generation who have adopted the more
efficiently learnable strategy, and this relative increase will be
proportional to its higher efficiency. This means that, all other
conditions being equal, of any two alternative strategies, rules, or
whatever whose germinal state is imitation, the most efficiently learnable
form will become dominant in a population and finally exorcise its less
efficiently learnable competitor. However, the effect of acquisition
efficiency is sensible only under conditions where efficiency is a
decisive property of a growth or learning process, that is, under
conditions of extreme survival pressure. Such conditions have most
probably occurred often during the early ages of the human species. This
could explain why basic universal human properties such as language and
social structure are learned very easily, at an early age, and in a more
or less approximate way. Of course this is mere speculation, but it could
be tested by mathematical theories of cultural evolution, such as Boyd and
Richerson's.
The model of cognitive growth discussed so far may
be extended considerably by applying principles of dynamic systems
modeling. Basically, a dynamic system consists of a state space to which a
set of dynamic rules is assigned. A state space is any
n-dimensional space, the dimensions of which consist of the
various degrees of freedom of a system. For instance, the growth of
vocabulary during the one-word stage can be described in the form of a
state space consisting of four dimensions, namely the number of words
acquired, the growth rate, the height of the carrying capacity, and the
feedback delay. For a real-word learner, the state space will consist of a
fragment of a Euclidean four-dimensional space, with number of words and
carrying capacity between 1 and 500, growth rate between 0. 5 and 2, and a
feedback delay of between 1 and 2 weeks. Each point in this state space
represents a potential developmental state of a word learner. The set of
dynamic rules assigned to the state space describes the evolution over
time of any point in the space. In the example, the dynamic rule consists
of the difference form of the logistic growth equation. Thus, for a point
(K = 1, L = 0. 1, r = 1, and f = 1), the
equation describes a line that consists of an arbitrary number of
consecutive states. This particular starting point can be described as the
initial state of the system.
The line starting from this initial state, which
may be alternatively called the trajectory of the initial state
point, amounts to the logistic growth form discussed in the previous
section. The trajectory for initial states where r < 1 leads
asymptotically toward a point where L = K. This point is
an attractor for all trajectories where r < 2. For 1 <
r < 2, the trajectories move toward the point attractor in a
whirling fashion. For 2 < r < 2. 57, the attractor is no
longer a point but a cycle (i. e. , the periodic oscillation discussed
previously). With linearly increasing r, these cycles get twisted
to form twofold, fourfold, eightfold, and upward loops. For r
> 2. 57, the trajectories form extremely complicated open loops. Points
for which r is 1, 2, and 2. 57 are bifurcation points,
or points where the geometry of the trajectories changes qualitatively (e.
g. , from a spiral into a closed loop). Because a four- and often a
three-dimensional state space is difficult, if not impossible, to
represent on paper, it is necessary to compress the dimensions some way. A
good way to compress the dynamic system of logistic growth is to use two
dimensions, namely L/K (i. e. , growth relative to K)
and a dimension representing absolute speed of growth. The latter can be
expressed as Lt+f -
Lt)/Lt. Because for any given
Lt this speed of growth is a direct function of
r, this second dimension is a way of representing different
values of r in the state space.
Growth curves can be represented in different ways:
Figures at the left represent normal time-axis diagrams; figures at the
right are corresponding state space diagrams. (State space diagrams exist
in two forms: line representations, which show the connections between
data points [connected state space diagrams], and point diagrams, which
represent only the data points [disconnected state space diagrams].
Connected state space diagrams provide clear pictures of approximative and
oscillatory growth [second and third rows]. The disconnected type of
diagram is especially useful for representing chaos and near-chaos
evolutions: Local concentrations of data points can be seen immediately
[bottom row]. )
Figure 10 shows the growth types of Figure
8 in the form of their state space diagrams. Such diagrams can reveal
structure or regularity that would not be visible in the ordinary
time-axis diagrams.
Instead of the geometric concept of a dynamic
system, from now on a more intuitive version is used. By dynamic
system, I intend any structure of n one-dimensional
variables that affect one another over time. The way in which they do so
is expressed in the form of difference equations for logistic growth with
different parameters. Thus, the simplest dynamic growth system is the
K-L-r-f system (or simply K-L-r system, because I treat
f as a constant), where only L develops over time. In the
discussion on the ecological constraints on cognitive growth, I explain
that cognitive growth dimensions may entertain supportive or competitive
relationships. For instance, vocabulary and syntactic knowledge are two
growers that seem to compete for the same resources, whatever those may
be, in the first stages of language development (Dromi,
1986). Each grower may be represented by a K-L-r structure,
and both structures may be related by a mutual competitive relationship.
For instance, one may postulate that each growth level inversely affects
the growth rate of its competitor. I use the principle of competition to
introduce a model of regressive or U-shaped growth. Another principle I
mentioned earlier is that the growth equation may be applied recursively.
For instance, the growth rate of vocabulary may be conceived of as a
variable that is subject to growth in its turn and thus can be
characterized by a growth rate rr and a carrying
capacity Kr, and so in principle ad infinitum. This
principle of recursion, in addition to the principle of supportive growth,
is used to introduce a model of bootstrapping phenomena in cognitive
growth. Finally, dynamic models are constructed where the direction of
growth (i. e. , increase or decrease) is treated as an autocausal
phenomenon; that is, increase causes increase, and decrease causes
decrease. This form of positive feedback effect can be used in models of
stepwise and U-shaped growth. In the remainder of the article, I present
several empirical cognitive growth phenomena and try to construct a
dynamic systems model for each of them. These models will consist of
structures of K, L, and r components and their
respective competitive and supportive relationships. Instead of studying
the geometric properties of the state spaces corresponding to these
structures, I discuss ways of fitting empirical growth curves with
trajectories that naturally occur in these spaces. The study of several
adjacent trajectories in these state spaces may reveal specific models of
families of growth curves of which the empirical growth curves are
members.
To provide a short overview of the types of
bilateral interactions (i. e. , interactions among any two variables in a
system), I return to the five heuristic principles mentioned earlier, and
the second principle in particular. This principle makes a distinction
between competitive, supportive, and neutral interactions between
variables. Thus, provided that the effect of one variable on another may
be one of the aforementioned sorts, there are nine possible bilateral
interactions (
Table 1). The column heads specify the effect of
a variable A on a variable B, whereas the row heads
describe the opposite, that is, from B to A. The most
interesting interaction types are 1, 2, and 4 because the neutral type is
only an interaction in the logical sense of the word (it is a zero-order
interaction). Type 1 is a mutual support interaction: Both growers,
A and B, positively affect each other's growth. This
type of interaction is used in models of bootstrap dynamics. Type 4
represents a mutually competitive interaction: Both growers negatively
affect each other. This type of interaction will occur in competition
among alternative strategies and in cognitive takeover phenomena. Type 2
represents an asymmetric form of interaction: Whereas one variable affects
another positively, the latter affects the former negatively. This type of
interaction has played a prominent role in biological models of
predator-prey relationships and is usually modeled in the form of
so-called Lotka-Volterra equations. Interactions of the Lotka-Volterra
type can be used to model the growth of corrective behavior of parents,
for instance, in response to the growth of unwanted habits of behaviors in
their children (van
Geert, in press).
Under various conditions, the growth in one
dimension may force another dimension to change qualitatively. For
instance, a multicellular organism that exceeds a specific number of cells
is, through evolution, forced to abandon its direct cell-environment
contacts as a major form of energy exchange and to develop an inner
structure (e. g. , a structure with a digestive tract). Cognitive and
linguistic rules and strategies can be considered as
information-management and production systems. Just as with the biological
example, the form of information systems is not independent of
quantitative properties of the information they have to manage.
Manageability is a function of resources; that is, it poses no problems if
one has all the time, space, and energy of the world. If an
information-management system reaches a management limit-for example, in
addressing problems that require more information than the system can cope
with because of memory limitations—then it is likely either to remain
constrained by that limit or to be replaced by another, more powerful
information-management system. If the latter is the case, it is called a
takeover. In the previous examples, the takeover is the effect of
scarcity in internal resources, such as time or memory extension. In other
cases, takeovers may be the result of external resource limitations. For
instance, children often use "wrong" linguistic forms that are based on
their immature grammars. Adults initially accept such forms but become
increasingly intolerant as the child's mastery of the correct form
increases.
Cognitive development is full of takeover
phenomena. Examples are the takeover of one-word by two-word sentence
rules, takeover of concrete operatory logic by formal operatory logic in
specific domains of application, or the takeover of one balance scale rule
by another in the balance scale task (Siegler,
1983). In a significant number of cases, takeover phenomena usually
result in developmental regression (Bever,
1982), U-shaped behavioral growth, or oscillatory growth (Strauss,
1982; Strauss
& Stavy, 1982). Such regression is either a complete abandoning of
the older rule or principle or a temporary decrease of the field of an
initially successful application of a specific production rule, concept,
and so on. Regressions have been found in the fields of early object
cognition, concept development, ratio comparison, early imitation,
language acquisition, face recognition, artistic development, intuitive
thinking, gender identity development, and so on (Bever,
1982; Strauss,
1982). A good example of regression toward zero performance level
caused by a quantitative increase in the information to be managed by a
rule system is provided by the development of conservation in 2- to
5-year-olds (Mehler,
1982). According to Mehler, 2-year-olds use a perceptual memory
strategy and are capable of correctly solving a significant number of
simple conservation problems. As perceptual differentiation increases, the
amount of information encoded within a situation to be remembered
increases correspondingly. At a specific point, the average amount to be
remembered is simply too much to be managed by the perceptual memory
strategy. This is the point at which a new strategy, based on the
inference of rules and regularities, is adopted, while the old strategy
quickly disappears. The new strategy, however, leads to a spectacular
reduction—toward zero—of correct conservation performance. Only at the age
of 5 is the regularities strategy back at the performance level of the
2-year-old using the memory strategy. The regularities approach, however,
is much more powerful and has a much higher performance ceiling.
A comparable example concerns the development of
the concepts of temperature and sweetness studied by Strauss
and Stavy (1982). Children aged 4 use identity justifications and
manage to solve about two thirds of the problems correctly. Then they
start to use a method of comparing quantitative properties, which results
in a considerable drop in the number of correct solutions. By the age of
11, they start to understand the connection between identity and
comparison rules and solve most of the problems correctly. The main
question, then, is why at a specific moment should one strategy
(representation system, etc. ) take over the domain of another that has
proved to be quite successful? Why does the cognitive system not choose to
adapt to the carrying capacity or resource limitations level and stick to
its earlier and successful strategy? One explanation could be based on a
Wernerian orthogenetic principle: The cognitive system would show
an intrinsic tendency toward more complexity and rationality. That is, a
logically or informationally more powerful system will inevitably
interfere, at one moment or another, with a less powerful one. Without
claiming that this solution is necessarily false, I present an alternative
answer that is not based on a postulate of intrinsic tendency toward
higher complexity but simply on a dynamic systems model of competitive
growth.
The dynamic system consists of two growers,
A and B. A is the strategy, rule, or whatever
that, at the end of the day, will be the weaker of the two (e. g. , the
qualitative strategy in the sweetness or temperature problems or the
memory strategy with conservation). This relationship between A
and B can be expressed by stating that the final state of
A is considerably lower than that of B, or that the
carrying capacity of A is much smaller than that of B
(e. g. , at the end of the day, the rule strategy will solve many more
conservation problems correctly than will the memory strategy). In
principle, the competition between the two growers is mutual; both
"suffer" from the success of their competitor. A simple way to introduce
this competition mathematically is by making the growth parameter
r of a grower A increase or decrease by some direct
function of the difference between its growth limit, that is, carrying
capacity KA and the actual growth level of the competing
grower B, namely L_{B} (and similarly for the
growth rate of B; see
A dynamics model of two competing growers: The
simplified model (top) specifies only the direction of the negative effect
of one grower on another as a consequence of a competitive relationship.
(In the present model, A competes with and thus negatively
affects B and vice versa. The full model of interactions [bottom]
specifies the way in which the competitive interaction has been shaped:
The growth rate of A is a function of the difference between the
carrying capacity of A and the growth level of B [and
vice versa for B]. Thus, the bigger Lr
Figure 11). The expression (K_{A}
- L_{B}) indicates how much of the potential application field of
a grower A is already accounted for by a competitive grower and,
if L_{B} > K_{A}, the powerfulness of
the B form compared with the A form. It thus provides a
good characterization of the relative success of A versus
B (and vice versa). Because the growth parameter r is
also supposed to be constrained by some intrinsic resource level
(otherwise, it would quickly mount toward a rate that far exceeds the
physical possibilities of its carrier system), the logistic growth
equation can be applied to the growth parameter itself. For instance,
specifies that the value of r of the
grower A at time t + f is a logistic function of several
variables, namely (a) a previous value of r of A at time
t, (b) a carrying capacity K for r_{A}
at t, (c) a growth rate that amounts to the difference between
the carrying capacity for A, K_{A}, and the
growth level of the grower B, L_{B} at time
t, and (d) a competition factor c (the bigger the value
of c, the stronger the effect of the competitor).
The system consisting of two competitive growers
A and B is characterized by four equations, namely
provided that
where f is the symbol for the logistic
function; the subscript i refers to the initial state values.
This set of equations describes a system consisting of a more primitive
strategy A, which is rather well developed at the start of the
observations (hence, the initial state values of L_{A}
and r_{A} are significantly higher than those of
L_{B} and r_{B}, for example, the
identity strategy in Strauss's sweetness experiment). Its growth initially
profits from the lack of competition from a more powerful strategy, which
is then still in a sort of germinal state (e. g. , the quantitative rule
in the sweetness experiment).
Assume that unrealistic values of the competition
factors—that is, those that lead to highly instable and chaotic behavior
of the dynamics—have been ruled out in the course of the phylogeny of the
human cognitive system. Given realistic competition factors, the behavior
of the dynamics modeled after Equations
28b, 28c,
28d,
28e
and 28f
is as follows. If the strong grower B (i. e. , the grower that
will achieve a much higher final level than A but with an initial
state level that is significantly lower than that of A) has a
reasonably high initial growth rate, both growers develop as if their
growth was completely independent of the growth of their actual
competitor. That is, in this particular region of the state space of
competitive growth, there is no regression (fallback or U-shaped growth).
For a significantly lower initial growth rate of B, grower
A sets into a chaotic oscillation as B approaches and
passes A's carrying capacity level. As B approaches its
own carrying capacity level, the amplitude of A's oscillation
decreases, and finally A settles down into its final state value
(see
A cognitively weaker strategy A becomes
temporarily unstable as a stronger strategy B takes over its
domain of application, and A stabilizes again as B
stabilizes.
Figure 12; see the Appendix
for details). Probably there are various ways in which this sort of growth
may be expressed in behavior. For instance, the oscillation of A
amounts to quick alternations in the use of the
A-versus-B strategy in problem solving. This is the sort
of phenomenon that often occurs when a more mature cognitive strategy goes
through some sort of breakthrough state (Bijstra,
van Geert, &
Jackson, 1989; Thelen,
1989). If one takes the average growth level in such a dynamics (e. g.
, over three consecutive states), one finds a temporary regression.
However, the following sections discuss much more direct forms of the
regression phenomenon.
U-shaped growth. Cognitive strategies, rules, and
knowledge are reality-driven acquisitions. If a cognitive strategy
continuously leads to errors, it is likely to be abandoned; that is, it
will show negative growth or extinction. Extinction or negative growth can
be easily modeled in the logistic growth equation by inverting the sign of
the growth rate from positive to negative. If such inversion were to
happen whenever a strategy encounters an instance of counterevidence,
cognitive growth would be very chaotic, if not simply nonexistent.
However, according to cognitive developmental studies, as well as from
studies in the history of science, children and scientists alike tend to
stick to their models, theories, world views, or strategies, even in the
face of considerable counterevidence (Elbers,
1986, 1988).
That is, cognitive development is rather conservative. However, once a
paradigm shift has started, it leads rather rapidly to considerable
changes. A good way to build this behavior into the growth models is the
following positive feedback rule: When increasing, continue
increasing; when decreasing, continue decreasing. Thus, as long as a
strategy or rule grows toward some carrying capacity level, it will
continue doing so; that is, it will not invert the sign of its growth
rate. However, if the growth rate is higher than 1, the grower will
overshoot its carrying capacity and inevitably be thrown back under the
carrying capacity level. This fallback then switches the sign of the
growth rate (in accordance with the simple conservative principle just
mentioned), and an extinction process sets in. This process will continue
until the grower passes some minimal structural growth level. This level
is the minimum level of a strategy, rule, principle, and so on, given the
specific structural possibilities of the cognitive system at issue. For
instance, an operationally thinking child will not completely abandon
qualitative strategies in solving sweetness or temperature problems. To
the cognitive system, such strategies are well within the structural
possibilities and are thus likely to be generated anew even when, for some
reason or another, they might have disappeared temporarily. Thus, as soon
as a strategy in the process of extinction falls under its structural
minimum, it automatically rises again, which switches the sign of the
growth rate, and its growth starts anew (provided, of course, that neither
r nor K has meanwhile been reduced to about zero level).
This mechanism, which is based on a positive feedback effect of the
direction of growth on the sign of the growth rate, provides a model for
the so-called cognitive pendulum phenomenon (K.
E. Nelson & Nelson, 1978) and also for regression in cognitive
growth. The model accounts for the fact that strategies, rules, models,
and so on are changed only if their application effectively exceeds the
carrying capacity of the environment, that is, the level above which the
strategies and other things always and automatically lead to errors or to
anything else that is unpleasant to the system, such as a taxing of
resources.
One may object that the conservative rule just
mentioned makes growth extremely vulnerable to random perturbations. This
is true only if the random perturbations are bigger than the increases or
decreases due to either growth or extinction. If that is the case, the
grower as such is intrinsically unstable anyhow, in that the growth level
depends more on random factors than on cognitive acquisition or learning
capacities. Nevertheless, small random perturbations have very interesting
and far-reaching effects on growth, as I show later. First, however, I
specify the dynamic rules for a system of competing growers to which the
aforementioned positive feedback has been added and then demonstrate the
effect of small random perturbations, which have very interesting
long-term consequences on growth.
The equation for the growth rates in positive
feedback systems is actually a modification of Equation
28a, namely
for
which implies that vt is - 1 if
the last growth level is lower than its immediate predecessor and + 1
otherwise. The complete set of dynamics rules is similar to Equations
28b, 28c,
28d,
28e
and 28f
except for the equation on which the growth rates are based (Equation 29).
I demonstrate the behavior of the resulting dynamic system for initial
state values L_{A} = 0. 5, r_{A} = 0. 8,
L_{B} = 0. 1, and a minimal structural growth level,
which is almost 0. For varying initial state values of
r_{B}, there is the following behavior. For the initial
growth rate r_{B} > 0. 33, A and B
simply grow seemingly independently of one another; that is, no regression
occurs. For r_{B} with a value of about 0. 24, A
shows inverted U-shaped growth, which, depending on the exact value of
r_{B}, may amount either to complete or to partial
extinction (see
The growth patterns of competing growers, one of
which is subject to positive feedback, are different for different initial
growth rates of the stronger grower B; for rA
occurs (grower A is indicated by a line formed with either
squares or diamonds).
Figure 13; see the Appendix
for details). For r_{B} with a value of about 0. 1, there
is a prototypical U-shaped growth, as described by Bever
(1982), Mehler
(1982), and Strauss and Stavy (1982;
see Figure
13). For an r_{B} of about 0. 07, grower A
displays M-shaped growth, and for increasingly lower
initial growth rates for B, the growth of A turns into a
chaotic oscillation, exhibiting W-, UM-, or other-shaped growth (Figure
13).
If a small random perturbation p is added
to each successive growth state of A and B (for
instance, between -0. 005 and 0. 005), the growth patterns are no longer
determined by the initial growth rate values. Overall, the same sort of
growth patterns are found as with the deterministic system, but their
connection with the initial values is only probabilistic. The fate of the
competitive system is now strongly determined by those small random
perturbations that occur in the vicinity of the carrying capacity level
(e. g. , 1) on the one hand and the minimal structural level (e. g. , a
very small positive number) on the other hand. This property is
reminiscent of the so-called butterfly effect in some dynamic systems (the
name refers to the fact that, in principle, the weather for the next day
could be determined by a butterfly fluttering its wings at the proper time
and place; Gleick,
1987; Schuster,
1988). The form of the growth curve A is explained by two
factors. First, it depends on the course of the growth rate itself, the
absolute magnitude of which is approximately equal to the first derivative
of the growth level of B (see Equation
28b) and thus assumes a bell shape. Second, it depends on the local
positive or negative value of the growth rate, and this depends partly on
the magnitude of the random perturbation in the vicinity of the carrying
capacity and structural minimum levels (see
The growth rate of the weaker grower A
changes over time. (While the absolute height of rr
Figure 14).
The model of U-shaped growth presented thus far is
only the most elementary model. First, it does not specify the way in
which performance patterns are mapped upon the growth of the competitive
cognitive strategies in the model. One mapping might consist of the
subject following the strategy providing the highest success rate (e. g. ,
as in U-shaped face recognition, Carey
& Diamond, 1977). This results in a set of potential performance
patterns that vary from growth patterns containing one plateau (of
increasing length) to growth with real fallbacks, all dependent on the
speed with which the weaker strategy goes up and down and the stronger
strategy grows (see
Potential performance curves based on competing
growers, one of which is subject to positive feedback (curves follow the
highest performance level).
Figure 15; see the Appendix
for detail. Another pattern might consist of a random oscillation between
both available strategies as the older starts to decline (e. g. , in
conservation intermediates). Finally, if a strategy drops nearly to zero,
this might force the child to apply a new strategy that is not yet
sufficiently developed to deal successfully with the problems he or she
now has to face, which inevitably leads to high error levels (Bever,
1982; Strauss
& Stavy, 1982). Still another possibility is that the actual
performance is itself a grower, the growth rate of which is a function of
either the growth level or the growth speed of the strategies A
and B. Which of these possibilities actually models empirical
performance curves should be decided through a thorough empirical
investigation of the growth process in question.
Second, the resource level or carrying capacity for
the A grower has been held constant in the examples. It is
likely, however, that this level also increases with age, for instance
because of increases in supporting cognitive strategies or because of a
bootstrapping effect (discussed later). Correspondingly, the structural
minimum level of a grower might increase as an effect of general cognitive
growth. This implies that the regressions will become shallower the later
they occur in the course of the growth process.
Third, if individuals vary significantly as to
relevant initial state properties, it is very likely that group data will
actually conceal the U-shaped nature of the underlying developmental
dynamics. An empirical description of a U-shaped growth process, and of
any process of cognitive growth for that matter, should amount to a
description of a state space, where the trajectories correspond to
individual growth curves. The principle of positive feedback effects in
cognitive growers will turn out to be a useful building block of more
complex cognitive dynamics, for instance in dynamics explaining stepwise
growth or irregular growth toward an optimum (discussed later).
The previous sections focus on the problem of
takeover of an inferior by a superior strategy (regardless of how this
superiority had been defined). However, there are many instances of
cognitive growth where, in principle, no such value distinction exists
among alternatives. An example is cognitive "styles" in strategies, such
as globalistic versus analytic problem approaches, a more democratic
versus a more authoritarian way of solving social decision problems in a
peer group, and synthetic versus analytic perception of class
distinctions. A criterion for distinguishing such alternatives from
developmentally unequal strategies or skills is that the former will show
a distribution over a population that is relatively independent of
developmental levels (interindividual distribution) or that a single
individual will use different alternatives in different contexts
(intraindividual variation). A basic idea behind the growth of
alternatives in a subject or in specific problem contexts is that the
final dominance of one strategy over the other is the result of a
competitive growth process among the alternatives. Such competition can be
achieved by making the growth rate of each alternative linearly dependent
on the growth level of its competitor. Thus, for two alternative
strategies A and B, the growth equations are as follows:
(the factors c and c' are
negative numbers expressing the magnitude of the negative effect of
B on A and vice versa).
Despite the simplicity of the equations, a system
of two competing alternative strategies shows a variety of qualitatively
distinct outcomes (see
Threshold phenomena in competing growers: For a
sufficiently large growth rate (rrA, which
suffers slightly more from B than vice versa grows toward a
maximum, then disappears. B grows toward a level equal to
A's maximum, then starts to increase as A decreases [top
right]. If A compensates its competitive disadvantage with a
slightly higher initial [init] growth level, A and B
grow toward a stable ratio [bottom left]. A very slight increase in
A's initial state advantage changes the pattern into the opposite
of the second case: B disappears, and A grows toward an
upper limit. )
Figure 16). First, for growth rates exceeding a
specific threshold level (about 0. 1), the alternative strategies
A and B grow toward a stable level, which, if the growth
rates of A and B are not too different, is approximately
equal to
(if K, r, and c are
exactly similar for A and B, there is a single steady
state that asymptotically approaches the value expressed in Equation
30c). In other words, if alternative strategies grow sufficiently
quickly, they will evolve toward a steady-state ratio that approximates
This ratio corresponds with the probability that
a person will use either alternative A or B in a
specific problem situation.
Second, if the growth rates stay beneath a
threshold value (e. g. , if r = 0. 05), a completely different
type of behavior is found. Provided at least one of the parameters at the
initial state is slightly unequal to its counterpart with the competing
strategy, the most advantageous of the alternatives will grow toward its
maximum, whereas its competitor, after an initial stage of increase, will
drop back and evolve toward zero level (see Figure
16). The steady-state outcome (either A wins and B
disappears or vice versa) depends on sharp threshold values that interfere
in complex ways. For instance, a disadvantageous competition ratio
(A suffers more from B than vice versa) can be
compensated by a slightly higher initial state level, or a slightly higher
initial state carrying capacity, or a slightly higher growth rate. The
qualitative patterns that appear are the following: If one alternative is
significantly more advantaged than the other, its growth resembles a
normal logistic S shape; if both alternatives are less strongly different
in regard to the major parameters, a stepwise growth pattern is observed
in the winning alternative, and an inverted U shape is observed in the
losing alternative. The winner qualitatively evolves from a zero stage
(strategy absent) by means of an intermediary stage, where both
alternatives have about equal growth levels toward a final stage
characterized by complete absence of the competitor. This qualitative
pattern is often observed in several forms of cognitive growth. In several
cases, which depend on the exact ratios among the parameters, temporary
regression may be observed in the finally winning strategy. The
qualitative patterns that result from the present dynamic interaction
between alternative strategies are preserved if parameters, and
particularly the competition factor, vary randomly over time (e. g. ,
under a ±10% variation of the default values). This qualitative
conservatism of the dynamics is important in light of the fact that the
parameters are unlikely to remain mathematically constant in real cases.
The question of what factors determine the value of the parameters cannot
be answered without taking the context of the growth process into account.
For instance, a holistic strategy will suffer less from an analytic
alternative if the child in question is led to use this sort of strategy
in other problem domains, whereas there may be more difficulties in
competing with the analytic alternative if the latter is favored by the
environment. Finally, the occurrence of threshold values and the
compensatory interactions among the parameters are reminiscent of
phenomena that may be observed in education in general and compensatory
education in particular: the observation that success in changing
unfavorable learning conditions depends on crossing threshold levels, the
existence of which is often difficult to predict, and the fact that
conditions of unfavorable learning can be changed via different ways and
under specific conditions.
According to most theories, cognitive development
amounts to a bootstrapping process: Cognitive growth releases the
resources upon which further growth is based. For instance, in Piaget's
theory, the child's activity is based on his or her current level of
cognitive development. These activities bring about experiences that
affect the underlying cognitive structures and lead these structures
toward increasing complexity and equilibration (Piaget,
1975; van
Geert, 1987a). In transactional models (Sameroff,
1975), the nature and level of the child's current development, for
example, of temperament, is considered to release supportive activities in
the caretaker and to change the norms, expectations, and supportive
activities of the caretaker with regard to the child. In language
development, the child's caretakers adapt the syntactic complexity of the
language addressed to the child to the child's assumed level of
understanding, and they speak so-called Motherese (Snow
& Ferguson, 1977).
Bootstrapping dynamics are easy to construct. To
the elementary K-L-r dynamics is simply added a supportive
relationship from L to K (see
A model of a simple bootstrap dynamics. (The
simplified model [top] represents a grower A positively affecting
itself [circular arrow]. The full model [bottom] specifies this bootstrap
effect: Arrows represent effects as described in the logistic growth
equation. The arrow from L to K implies that L
contributes to the growth of its own carrying capacity K. K is
further affected by its proper carrying capacity
KKK. )
Figure 17). That is, the initial carrying capacity
for a low growth level, of some syntactic rule for example, is low also
(and it is made so low because of some tutorial or pedagogical adaptation,
e. g. , because the parents temporarily tolerate grammatical errors and
are not inclined to actively correct the child). The carrying capacity
grows as an effect of increases in the growth level it supports. One way
of structuring this relationship between L and K is to
make the growth rate of K a function of the growth level L.
In principle, there are two ways in which the carrying capacity
K can be a function of the growth level L, namely as a
function of relative change in L on the one hand and the absolute
level of L on the other. I shall first discuss a dynamics based
on relative increase.
K's growth might depend on the relative
increase in L, that is, on how much L has increased or
decreased over the past period relative to its absolute magnitude or to
its previous increase. This is the sort of increase that might be expected
in transactional social models of development. For instance, in the
interaction games studied by Bruner(1975;
Bruner
& Sherwood, 1976) and Wertsch
(1979), a subtle interplay occurs between the information and guidance
provided by the mothers and the behavior of their young children. The
activity of the mothers technically amounts to a continuous raising of the
carrying capacity of a growth variable (e. g. , knowledge of language or
of the structure of games) by raising the demands as well as the
complexity of the examples and corrections given. One may assume, however,
that mothers, or educators in general for that matter, will be sensitive
to the relative growth of knowledge in the child, that is, to the speed
with which the child progresses on the aspect of knowledge at issue. More
precisely, the rate of making more complex help and information available
probably will increase if the child proceeds quickly and probably will
decrease if the child no longer shows considerable relative progress (this
sort of feedback principle is nicely illustrated in Bruner,
1975). In addition, one may assume that mothers or caretakers in
general will differ in the level of sensitivity with which they will
regard the child's behavior as a signal of his or her needs and
developmental level (Ainsworth,
Blehar, Waters, & Wall, 1978). This sensitivity can be expressed
in the form of a damping function, or a function that either magnifies or
reduces the effect of a growth level on the growth of the carrying
capacity:^{7}
The dynamic rules for the complete system amount
to the following:
which reads for Equation
32a, the effective carrying capacity K_{ef} is a
logistic growth function of its previous level, the change in the growth
level L, a factor dK damping the effect of
L, and a maximal carrying capacity KK; for Equation
32b, growth level is a logistic growth function of the effective
carrying capacity K_{ef} and the growth rate r;
for Equation
32c, the factor damping the effect of L on K is a
function f' (e. g. , a simple linear function) of the sensitivity
of the system to changes in L, or sL; and for
Equation
32d, the initial level of the effective carrying capacity
Ki is significantly lower than its highest possible
final state level KK.
Thus far, I have assumed that bootstrapping occurs
in an upward direction. However, downward movements also occur in
cognitive growth. For instance, it is likely that the initial growth rate
of a fast grower like vocabulary is much higher than later growth rates,
regardless of the damping effect of the carrying capacity (see the section
on exponential growth without feedback delay). That is, especially with
young children acquiring basic skills and knowledge, one may assume that
the initial phase is characterized by very fast learning, whereas the
learning rate will decrease as a consequence of increasing skill or
knowledge level. Note that in the ordinary logistic model it is the
absolute speed of growth that decreases as L approaches
K, while the growth parameter r remains constant. The
present model postulates that r itself shows a decline.
Therefore, assume that the growth rate, of vocabulary growth for example,
is also subject to negative growth, in that its carrying capacity, that
is, the growth rate's potential final state, lies well below its initial
state level. The bootstrapping principle may now be applied to the growth
of the growth rate by assuming that the initial growth rate grows toward
its much lower final carrying capacity as a result of the increase in the
cognitive growth level (e. g. , vocabulary). Thus, as far as changes in
the growth rate are concerned, the bootstrapping leads downward, instead
of upward, as with the carrying capacity discussed previously. For the
sake of the argument, let me demonstrate the effect of absolute growth
level (e. g. , vocabulary) on the changes in the dependent variable
(growth rate of the vocabulary). This effect amounts to the principle that
the more words the child knows, the lower the rate with which the
vocabulary grows, regardless of the actual carrying capacity for the
vocabulary. That is, a competitive relationship is postulated from a
growth level to its underlying growth rate. Again, one should assign some
sort of damping factor to the variable vocabulary level to account for
different levels of sensitivity within the dynamic system at issue. A
dynamics consisting of a K-L-r structure with the decrease in
r dependent on the increase of L gives rather trivial
results (trivial in the sense that it yields growth curves where the
growth rate decreases faster toward the end than with constant
r). However, the competitive relationship from growth level to
growth rate should be added to the dynamics discussed in the previous
section, a dynamics that contained a supportive relationship from growth
level to carrying capacity (see
A model of a more complex bootstrap dynamics based
on the model from L is a factor reducing, instead of increasing,
the value of r. This bootstrap dynamics starts with a low initial
K and a high initial r and stabilizes at a level where
K is high KKKr is low. )
Figure 18). Thus, to the dynamics rules (Equation
32a, 32b,
32c
and 32d)
are added:
which reads for Equation
32e, the effective growth rate r_{ef} is a logistic
growth function of its previous level, the growth level L, a
factor dr damping the effect of L on
r, and a carrying capacity K_{r}, which amounts
to a stable final level of r; for Equation
32f, the damping factor dr is a function f'
(e. g. , a simple linear function) of the sensitivity of the growth rate
r to the growth level L; and for Equation
32g, the initial level of the effective growth rate is significantly
higher than its final state, which is close to Kr. I now
give two empirical examples of bootstrap dynamics in which both forms of
adaptations, upward adaptation of the carrying capacity and downward
adaptation of the growth rate, are incorporated. In the first example, the
main assumption is that parents adapt their help, correction, and support
in function of the relative increase in a child's competence, that is, in
terms of whether more or less progress has been made than during a
comparable preceding period.
Ainsworth
et al. (1978) have studied different forms of parental responsivity
and their effects on the development of attachment in infants. Parents
have been found to differ in their sensitivity to the infants' behavioral
signals. This sensitivity is expressed in the form of responses toward
these signals, in providing help and support, in giving the child a
specific amount of independence, and so on. Mothers could be distinguished
on the basis of differences in sensitive responsiveness: Some were highly
(i. e. , optimally) sensitive, others were highly insensitive, and a third
group was characterized as inconsistently insensitive. The last group is
characterized by a tendency toward overstimulating their infants (Belsky,
Rovine, & Taylor, 1984), in addition to inconsistency in giving
support (Ainsworth
et al. , 1978). Paternal sensitivity has an effect on the quality of
the child's attachment (Belsky
et al. , 1984) and probably also on later personality and cognitive
development (Campos,
Barrett, Lamb, Goldsmith, & Stenberg, 1983; Van
Ijzendoorn, 1988).
The dynamics modeling the developmental effect of
parental sensitivity has the form depicted in Figure
18 and is modeled according to Equation
32a, 32b,
32c,
32d,
32e,
32f
and 32g.
In this model, parental sensitivity amounts to the damping factor that
mediates the effect of the growth of a developmental variable (e. g. ,
attachment behavior and sociocognitive understanding in the child) on the
growth in the support and help given by the parent. This support and help
should adapt to the increasing competence of the child. That is, it should
take the form of a carrying capacity that increases as the growth level of
the child's competence at issue increases. In the model, this increase in
the carrying capacity is the effect of a growth rate, which is a function
of the relative increase in a growth level of the child's competence and a
damping function representing the parent's sensitivity. I do not specify
the child's growth level L, but one might imagine it representing
the quality of the child's attachment, for instance (assuming the latter
is a continuous one-dimensional variable), a sociocognitive competence,
the growth of which is believed to depend on the quality of the
parent-child interaction, or a personality property such as ego control or
ego resiliency (Block
& Block, 1980). If the dynamics are run with a low-sensitivity
factor, the carrying capacity, which is a function of parental support and
responsiveness, grows slowly. In fact, it cannot keep a sufficient
distance from the growth level of the child's competence L. As
L comes within a critical distance of K, K and
consequently, also L no longer increase; that is, they get
trapped into a point of stability that is well beneath the maximal level
that K and L could actually achieve (see
Parental responsivity and support determines the
carrying capacity for a growing competence in the child; different
sensitivity levels varying from hyper- to insensitive produce different
growth patterns and final states. (The dynamics is described in L
to K has been mediated by a damping factor, which is a linear
function of the parent's sensitivity. )
Figure 19, top right; see the Appendix
for details). With an optimal sensitivity factor, K and
L grow smoothly toward their maximal level (maximal given the
complete system of resources and possibilities in a particular
environment; Figure
19, top left).
If the sensitivity is above optimum, the carrying
capacity grows too fast, which corresponds to overstimulating the child
(providing him or her with much more support and help than actually
needed, given the child's present level of growth). The fate of high
growth rates is that the resulting growth level starts to oscillate in an
almost chaotic way. This also occurs in the present dynamics, and this
corresponds with the observation that the parents' responsiveness is
actually inconsistent (sometimes too high, sometimes too low; see Figure
19, bottom right). With increasing magnification of the sensitivity
factor, the behavior of the carrying capacity (e. g. , of the support and
scaffolding of the parents) becomes chaotic, until it (almost
neurotically) oscillates between its lowest and highest possible values.
This usually results in a cognitive growth process that shows a mild
oscillation around some low final state value. It is interesting to see,
though, that the growth rate of the grower L may sometimes
compensate for the damaging effect of oversensitivity. It may have a sort
of appeasing effect on the neuroticism of the carrying capacity (e. g. ,
the parents' deployment of resources, attention, and help) and lead toward
a final state comparable with that achieved with an optimal sensitivity
level (Figure
19, bottom left). One may therefore conclude that the present dynamics
simulates the theory and empirical findings on parental sensitivity quite
well, as far as the relation between the level of sensitivity, the
resulting parental support, and the resulting growth in the child are
concerned. From the previous simulation, it follows that the effect of
decreasing sensitivity on the stable final state value of the growth
level,^{8}
given all other initial state values are equal, is nonlinear (see
The effect of varying parental sensitivity levels
upon the final state of a growing competence that depends on parental
support, modeling, and so on. (Parental sensitivity is specified in the
form of a damping factor. The figures represent increasingly small
portions of the sensitivity scale, i. e. , 0. 01 < damping factor <
0. 01, 0. 01 < damping factor < 2. 01, 0. 65 < damping factor
< 0. 85, and 0. 704 < damping factor < 0. 714. In the latter
region, the final state effect is strongly nonlinear, showing a chaotic
succession of maximal and near-minimal final states. The chaotic nature of
the dynamics in this region amounts to the fact that very small
differences in the sensitivity parameter lead to fundamentally different
outcomes. )
Figure 20; see the Appendix
for details). Running the dynamics with various sensitivity values
demonstrates this clearly. As can be seen in Figure
20, the separation between optimal and hypersensitivity effects does
not take the form of a simple dip in the final state effects. Rather,
there is a chaotic band in which the final state effect of sensitivity
values, which are very close to one another, is strongly different (i. e.
, either about zero or maximum). This chaotic zone separates two
qualitatively different ways in which the environmental support system
adapts to the needs of the grower (i. e. , either with too high to
hypersensitivity, or with optimal to low sensitivity).
By syntactic growth I mean the increase in
the relative amount of correct use of some specific syntactic rule.
Examples are the growth in the correct use of plurals and present
progressive, studied by Brown
(1973), and the growth of inversion in Wh- questions (e. g. ,
"What mama is doing" vs. "What is mama doing?", Labov
& Labov, 1978). Data on the growth of inversion are shown in
Empirical growth curves of percentage correct use
of inversion in wh- questions. (Adapted from "Learning the Syntax
of Questions," p. 23, by Recent Advances in the Psychology of
Language: Language Development and Mother-Child Interaction, London:
Plenum Press. Copyright 1978 by Plenum Press. Adapted by permission. )
Figure 21. To construct a dynamics model for growth
of inversion, I make three assumptions. First, syntactic growth, such as
inversion, is an example of a bootstrap process. That is, I assume that
the carrying capacity, exemplified for instance by parental effort
invested in providing examples of correct sentences and in correcting
errors and by the child's effort in experimenting with particular
syntactic constructions and paying attention to the latter, is a function
of the growth of competence in using the syntactic rule already attained
by the child. Second, I assume that in syntactic growth it is the absolute
level of competence of the child to which parents are sensitive in regard
to providing help, correction, and support. That is, growth in carrying
capacity is not a function of relative increase in growing competence, but
of the absolute level attained. In the present example, this absolute
level has a psychologically relevant meaning, in that it can be compared
easily with an absolute standard, namely correct grammatical use. Mature
language users have very clear intuitions about the idiomatic
grammaticality of sentences; ungrammaticality in the language of immature
or nonnative speakers is quite salient information. This situation is
different from the previous example (e. g. , play behavior and event
knowledge), where no such absolute standard exists or can be applied
easily to characterize the child's absolute level of competence and where
only relative increase provides useful information. Third, I assume that
the empirical growth curve represents the growth in one underlying
variable, namely the child's competence in using the inversion rule. This
level of competence determines the probability that a correct form will be
used (and inversely, that an error will be made; Spada
& Kluwe, 1980). It is expressed in the amount of errors produced,
relative to the amount of correct sentences.
The dynamics is of the type represented in Figure
18. The dynamics rules consist of rules presented in Equations
32b, 32c,
32d,
32e,
32f
and 32g,
but Equation
32a should be replaced by
which reads that the effective carrying capacity
K_{ef} is a logistic growth function of its previous
level, the growth level L, a damping factor dK,
and a maximal final state level KK. In accordance with
the previous dynamics model, it is assumed that K increases
toward a carrying capacity that yields 100% correct responses and that
r decreases toward some minimum level. Instead of using constant
damping functions, a curvilinear relationship may be introduced. That is,
the effect of L on K and r is more strongly
damped the closer L is to either its initial or its final state.
The latter implies that the sensitivity of the system to the syntactic
growth level is greater at the beginning than at the end (and the other
way round for the former case).
A mathematical simulation of growth of correct
inversion rule in questions, based on a bootstrap dynamics (see
Figure 22 shows a simulation of the growth of
inversion in What sentences (see the Appendix
for details). Basically, it applies the principle of oscillatory or
near-chaotic growth with a carrying capacity that itself grows as a
consequence of the growth in the use of inversion in What
sentences. The high initial growth rate grows toward a much lower final
state, also as a consequence of growth in the inversion rule, thus
accounting for a considerable reduction in the amplitude of the growth
oscillations toward the end state (the 100% correct level). By
intrapolating intermediary growth states^{9}
between the first and second points of the curve, a complete set of
intermediary growth points can be computed (e. g. , weekly data points,
instead of monthly averages presented in the empirical study). Figure
22 shows two different interpolation strategies, one based on random
numbers, another on the self-similarity method described earlier. In this
case, the self-similarity method yields a strongly oscillating pattern,
which is probably not in accordance with the empirical findings. It is
also easy to study the effect of small random perturbations (e. g. ,
ranging between -1% and +1% of the growth rate involved) on the evolution
of the growth curve. It seems that, although the actual form of the curve
may vary rather drastically as a consequence of such perturbations, the
qualitative form of all these curves (irregular oscillations with
diminishing amplitude as L approaches K) is rather
robust. In fact, this finding is in accordance with individual data, which
show rather strong intraindividual differences for different sentence
types that are not strongly different in terms of complexity (Brown,
1973; Labov
& Labov, 1978). Notwithstanding these intraindividual differences,
the qualitative nature of these empirical curves is similar for all the
types of syntactic structures studied.
There appears to be an obvious fallacy in the
previous dynamics model of syntactic growth: Correct use cannot grow
higher than 100%, whereas the simulated growth curve may easily overshoot
the 100% level, provided growth rate is high. This problem could be solved
by interpreting overshoot as error, as an overgeneralization error for
example. More important, however, the model is rather unlikely from a
psychological point of view. One of the major discoveries of cognitive
developmental research is that the child's errors and mistakes are in
general not based on trial and error, or sheer ignorance, but are rather
the expression of underlying rules. This is true for cognitive and
language growth alike. For instance, the use of the noninverse strategy in
questions seems directly based on the subject-verb-object strategy (Quigley
& King, 1980; Slobin
& Bever, 1982), or on a "head-initial" parameter setting (Atkinson,
1986), or whatever rule is considered a major sentence-formation rule
in the early stages of syntactic growth. Inversion, on the other hand, may
be based on imitation of model sentences, an alternative setting of a
syntactic parameter, and so on. Essential to this discussion is the
understanding that these rules are competitive: The noninversion rule is
consistent with major sentence-formation principles existing in the
child's current grammar, whereas the inversion rule is consistent with the
environmental sentence models. Assume that these rules are in fact
separate growers, that their domains of application grow in a logistic
way. Also assume that as the growth level of the correct rule (i. e. ,
inversion rule) increases, the explicit support for this rule also
increases, for instance because the child tends to notice more and more
examples of this inversion rule in the language of the environment or
because the inversion gets established as a structurally coherent rule in
the child's grammar. Put differently, the carrying capacity for the
correct rule increases as a consequence of a boot-strapping process. On
the other hand, it is likely that the support for the wrong rule decreases
as a result of growth in the use of the correct rule. For instance,
parents probably tend to be more tolerant with regard to syntactic errors
when the child starts to use a specific construction, such as a question,
and less tolerant of errors as the child's capacity to use the correct
rule increases; parents probably tend to provide less corrective modeling
when a new sentence form has just emerged; and the child probably pays
much less attention to parental corrections of syntactic constructions
when a construction is new than later on. In fact, a whole range of
factors contribute to the increase in the carrying capacity of the correct
rule and the decrease in the carrying capacity of the wrong rule.
There could be an asymmetric competitive
relationship between the growth level of the correct strategy and the
carrying capacity (i. e. , environmental support) for the wrong strategy.
That is, the support for the correct strategy is probably directly
dependent on the growth of that strategy, whereas the support (carrying
capacity) for the wrong strategy decreases as the growth level of the
correct strategy increases. Psychologically, the latter amounts to a
decrease in the environmental tolerance of and support for the wrong
strategy as the mastery of the correct strategy increases, whereas on the
other hand, the support for the correct strategy does not decrease when,
temporarily, the mastery of the wrong strategy increases. A wrong strategy
is probably also characterized by the fact that it is tolerated until it
exceeds some threshold (i. e. , until it exceeds its environmental
tolerance level, which is nothing other than its carrying capacity) and is
then explicitly discouraged and rectified until its use falls below some
minimal threshold (e. g. , until it is no longer noticeable). After that,
the wrong strategy may grow again until it again shoots above its
tolerance, and the cycle may start anew. I have already described a
dynamics for this sort of process, namely the positive feedback cycle in
regressive growth (described in a former section), which is typical of
weak cognitive strategies. The complete dynamic system for the competitive
growth of a correct and a false strategy is represented in
A dynamics for competitive growth in a correct and
a wrong strategy. (The simplified model [top] specifies that both are
bootstrap growers [the circular arrows refer to a positive effect of each
grower on itself]. The correct strategy negatively affects the growth in
the wrong strategy; i. e. , there exists a competitive relationship from
the correct to the wrong strategy. The wrong strategy is subject to
positive feedback [the broken "pf" arrow]. The full dynamics model
[bottom] specifies the nature of the negative effect of correct strategy
on wrong strategy: The growth level of the correct strategy negatively
affects the carrying capacity of [i. e. , the support given to] the wrong
strategy. The positive feedback cycle holds for the carrying capacity of
the wrong strategy. Both strategies grow in accordance with a bootstrap
dynamics as depicted in KK
Figure 23.
The dynamics rules for each grower are similar to
those described under bootstrap dynamics (dependent on absolute growth
levels) and regressive growth.
A mathematical simulation of the growth of the
correct present progressive form; curve shows monthly averages of the
percentage correct per week based on separate growth of correct and wrong
strategy. (Although the curves [top] are slightly out of phase at the
beginning, the theoretical curve follows the oscillations in the empirical
data quite closely. The theoretical curve is based on a simulation of the
growth of the correct and the wrong strategies [bottom]. The top panel was
adapted from A First Language: The Early Stages, p. 256, by
Figure 24 (see the Appendix
for details) shows a mathematical simulation of Brown's
(1973) data on the present progressive in 1 child, which are based on
setting initial state values of all the parameters involved in the
dynamics and letting the system run in accordance with the equations
described in the previous section. The curve shows the monthly average of
the percentage of correct uses of the rule. Advantages of the dynamics are
that it also yields a theoretical reconstruction of weekly averages and
that it shows the growth of the correct and the false strategy separately
(Figure
24, bottom). Such theoretically reconstructed data can then be checked
against the empirical data as a further test of the postulated dynamic
model.
Bootstrap dynamics of interacting and competitive
growers are very rich, in that many different types of building blocks can
be used (e. g. , direct or indirect effects of L on K
and r, effect of either absolute level of L or relative
increase on K and r, and either positive feedback
effects or not). Although each of these dynamics yields growth curves that
are qualitatively very similar, the quantitative nature of these curves
may be characteristic of different dynamics architectures. Much work still
needs to be done to reveal the properties of these dynamics and to see how
far they provide valid models of empirical cognitive growth processes.
In preceding sections, I discussed a particular
form of adaptation of the carrying capacity called bootstrapping:
The tutorial environment adapts its support to a current, low growth level
and raises that support as a consequence of increase in the grower. This
process amounts to a temporary adaptation of the effective carrying
capacity, which then moves toward an intrinsic carrying capacity level
that remained constant during the whole growth process.
I have shown that carrying capacity is a measure of
the overall support a cognitive environment may lend to a specific grower.
This specific amount of support is expressed in the form of a potentially
stable upper limit a grower may attain, that is, the carrying capacity
level. It follows that if major changes in the cognitive system occur, the
inferred carrying capacity, for vocabulary growth for example, will change
accordingly. The problem that is addressed in this section is the
following: Because a grower (e. g. , vocabulary) is an intrinsic part of
the overall cognitive system and because the carrying capacity is
determined by the overall properties of that system, what will be the
contribution of growth in a single variable (e. g. , vocabulary) to
changes in the overall system and thus to changes in its own carrying
capacity? In the context of this question, the bootstrap dynamics
discussed in previous sections is a very particular tutorial adaptation of
the carrying capacity. The carrying capacity as such is not changed, but
it is dissociated into an effective carrying capacity and a sort of
background carrying capacity toward which the effective K evolves
as a consequence of growth in the dependent variable (e. g. , grammatical
rule use). This type of positive adaptation of support can also be
observed in what Fischer and coworkers (e. g. , Fischer
& Canfield, 1986; Fischer
& Pipp, 1984) called "practice and support" conditions of testing
skills. Practice and support are offered in function of the student's
increase in mastery of a skill, and this greatly enhances the speed with
which the skill grows. A related concept is Vygotsky's "zone of proximal
development" (Vygotsky,
1978).
An opposite form of adaptation occurs when, as a
consequence of significantly low performance, for example in mathematics,
a student's curriculum is changed (e. g. , the student moves to another
curriculum where mathematics is no longer an obligatory subject). These
changes are coercive tutorial adaptations of the carrying capacity, and
they are not the major concern of this section. More subtle changes in the
carrying capacity might result from the negative effect that poor
mathematics performance, for instance, might have on the student's
attention to mathematics-related information, to effort spent in doing
exercises, and so on. In this case, a slow downward growth of the carrying
capacity for mathematics occurs as a consequence of low mathematics
performance in the student. However, if a person's cognitive environment
too easily lowered the carrying capacity of relatively slow growers (e. g.
, mathematics knowledge) and similarly raised that of quick ones (e. g. ,
knowledge of pop music), the person concerned would rapidly change into
some sort of "idiot savant" (it may be that that is what is characteristic
of idiot savants, namely that their cognitive system adapts too swiftly to
differences in growth rates of the components). I explore which principles
of K adaptation are more adequate.
A first adaptive principle implies that carrying
capacities should grow slowly toward the growth level of slow growers,
whereas a second holds that they should grow quickly away from the growth
level of fast ones. Although the first principle is probably intuitively
plausible, the second requires some explanation. Fast growers (e. g. ,
r > 1. 5) overshoot their carrying capacity level. The more
they overshoot, the lower they will fall at the next step. It follows,
then, that if K grows not toward but away from L, the
next K level will always anticipate the next L level.
This can be implemented easily by making K grow as a function of
(K - L)/K. ^{10}
A third adaptive principle is that K should grow as a function of
the relative distance between K and L, that is, (K -
L)/K, which is the unutilized capacity for growth, U.
The default principle is that the growth rate of K should be
higher as the distance from K to L is larger. These
K-adaptation principles can be summarized in the following
equations:
meaning that the next state of the carrying
capacity K is an exponential function of the previous state
Kt and a growth rate, which is the product of a damping
factor d and the unutilized capacity for growth
Ut; d is a function of the growth rate
r and yields a large negative number if r is very low, a
number slightly bigger than 1 if r is very high, or a positive
number that is practically zero for all intermediary values of r.
The fact that an exponential instead of a logistic growth form is used for
Kt+f implies that no intrinsic limits were set to the
upper level that the cognitive grower may attain. It is expected that the
upper limit of the grower will result from the way in which the growth
rate r is determined by the parameters in the model.
Adaptation of the carrying capacity K to a
slow grower as a function of negative value of the unutilized opportunity
for growth, -U.
Figures 25 and 26
(see the Appendix
for details) show the effect of K adaptation on slow and fast
growers. The effect on a fast grower is of special interest: K
increases and decreases such that oscillations will be damped, and
K and L move toward a common stability point that is
considerably higher than the original carrying capacity level (Figure
26). The natural interpretation of this effect is that people who are
considerably better in some cognitive ability (mathematics, language, etc.
) will achieve a higher level of mastery and, similarly, that cognitive
environments tend to invest more resources into fast growers than in
others. Using an efficiency equation (Equation
27), I can show that raising K and damping the oscillations
is more efficient than keeping K constant and tolerating
high-amplitude oscillations of the growth level. Such raising and damping
is not necessarily the effect of intentional tutorial activities, but
rather the result of a simple K-adaptation principle.
Figure
25 shows the effect of an adaptation of (-U) to a slow
grower, resulting in a considerable decrease of the final state of K.
The fact, however, that the growth of K toward a slowly
growing L depends on U, that is, on the relative
distance between L and U, could be a disadvantage.
Assume, for instance, that the growth of vocabulary in a child is very
slow. A fast negative adaptation of the resources needed to build up a
vocabulary (i. e. , the vocabulary's carrying capacity) would finally
result in a very poor vocabulary. Because in a complex cultural
environment even a "minimal" vocabulary should be rather extensive, it is
better not to adapt the carrying capacity too soon. One way to accomplish
this is to increase the damping factor d from Equation
34a. Another way is to make K dependent not on (-)U
but on the inverse of (-)U: (-)1/U.
Equation
35a implies that K adapts only very little as long as its
distance from L is still big and adapts faster the closer
K and L approach each other. The functionality of this
form of adaptation lies in the fact that in the middle of the S-curve,
absolute increase is rather considerable, even for low growth rates,
whereas in the vicinity of K, the growth of L
decelerates anyway. Thus, if K would wait to adapt to L
until L has sufficiently closely approached K, implying
that L's absolute growth would have decreased considerably, the
system could have spared the cost of maintaining a high K when
L is approaching only slowly, while still achieving a
sufficiently high final state of L. Thus, U can be
substituted in Equation
34a by 1/U. The effect on growth of adapting K in
accordance with this last equation is very interesting. Instead of
settling down to a steady state, the grower and its carrying capacity
start to meander in a sort of narrowband random walk, rather reminiscent
of stock exchange variations (see
If carrying capacity adapts to growth level
L as an inverse function of the unutilized capacity for growth,
or as a function of (-1/U), L and K follow a
quasi-random meandering course (detail of left graph is shown at bottom of
panel). (Extremely small initial state differences in the damping factor
may cause dramatic differences in the course of the growers only after a
considerably long interval; in some cases, unexpected drops or rises of
about 25% may occur. Lines represent separate growers that differ only in
the damping function. The left differs from the right only in the initial
state condition. )
Figure 27; see the Appendix
for details). Dependent on the height of the damping factor, sudden leaps
and dips may be observed. The whole system is also very sensitive to small
differences in initial state conditions. For all practical purposes, the
growth level and its carrying capacity behave as if they follow a random
evolution, which in general stays within a small margin and now and then
shows unexpected leaps and dips that are rather considerable. However, the
evolution is not the result of random factors but is completely
deterministic. More precisely, a deterministically evolving grower may
provide a source of randomlike perturbations to other growers that depend
on it. Thus, a growth system in which the present adaptation principles
hold produces its own random perturbations. In complex systems, random
perturbations are important in that they may determine the long-term
evolution of the entire system, given that they occur at points where the
system is in relative instability (Prigogine
& Stengers, 1982; see, for instance, the section on pendulumlike
growth processes and the effect of small random factors therein). Finally,
in ongoing research, I am trying to determine the evolution of carrying
capacities in terms of competition and support among growers in more
complex systems (whereas in this article K has been treated as a
single factor). It can be shown that systems of competitive and supportive
effects from a multitude of growers on one another nevertheless result in
one-dimensional K factors; for each separate grower. The
evolution of systems of 10 to 15 interacting growers shows various
classical cognitive growth phenomena, such as evolution toward a stable
end point, transient regressions in some growers, stable regressions (some
growers disappear), stepwise growth, and so on (see van
Geert, in press).
Stages are traditionally accounted for in terms of
structural models. Because an analysis of the way in which structural
theories explain stages would far exceed the scope of this article, the
reader is referred to other publications in which this work is undertaken
(van
Geert, 1986, 1987a,
1987b,
1987c,
1988a,
1988b).
Note that the current concept of stage has shifted away from the
classical Piagetian view of an overall state definition in terms of unique
and characteristic structural features of the entire cognitive system, to
a view that is much more content specific. Stages may occur within, and
not necessarily across, cognitive domains or dimensions, and they may be
characterized by various qualitative as well as quantitative properties
(Levin,
1986). The question I address here is in how far the present growth
model, which is explicitly gradualistic, may account for stage phenomena.
Stepwise shifts in the magnitudes of characteristic
variables are frequently seen as the major indication of stage shifts (Fischer
& Canfield, 1986; see also Fischer,
Pipp, & Bullock, 1984; Globerson,
1986), whereas a stage as such corresponds to a temporarily stable
level (Fischer,
1983a). Stage shifts do not necessarily amount to the construction of
entirely new skills, structures, and so on characteristic of the new
stage. Those skills and structures were often present long before the
onset of the stage shift, but they existed in germinal form (as
décalage phenomenon, or as an innate or at least very early
generic concept, a possibility even Piaget took seriously; Piaget,
1968). The germinal form may last for a rather long time before it
starts to grow. Growth occurs in the form of a spurt followed by a
leveling off toward a steady final state. This is exactly what the
logistic growth model explains, and it does so in purely quantitative and
gradualistic terms. In fact, what makes the difference between an
apparently slow and quasi-linear increase in a variable and an almost
quantum-leap-like emergence of the steady state of a variable is the
height of the growth rate factor rather than some hidden structural
factor, such as a restructuring of an underlying rule system causing the
growth spurt. This underlying restructuring, if any, might be the result
rather than the cause of a growth spurt. That is, a potential
restructuring of the underlying rules (or whatever generative structure is
assumed to cause performance) probably constitutes a response to the
increasing pressure on the cognitive rule system that follows from the
fact that the application of a rule or principle steadily grows and thus
requires a more efficient or more powerful system than the one already
available in order to be able to manage the significant increase in the
domain of application.
A quantum-leap-like shift in a major variable may
amount to the effect of growth in either an underlying resource variable
or a control variable. By resource variable I mean a variable
that significantly contributes to the carrying capacity of a grower. For
instance, the carrying capacity of complex problem-solving strategies is
clearly dependent on the size and efficiency of working memory. In fact,
this is the sort of hypothesis put forward by several neo-Piagetian
researchers (e. g. , Case,
1985; Pascual-Leone,
1970) who related Piagetian stage transitions to increase in working
memory. By substituting different values for L and K in
the difference equation of logistic growth (Equation
17a), one can demonstrate that the effect of stepwise increases in
carrier capacity, for instance as a consequence of working memory growth,
is significant only for growth variables that are close to their carrying
capacity level, or their upper growth limit. This is so because the
absolute speed of growth strongly reduces in the vicinity of K
such that any sudden rise in K will suddenly increase the
distance between K and L, which automatically results in
a higher absolute growth speed expressed in the form of a jump. In this
connection, a short discussion of functional and optimal
levels of skill development, as they are termed by Fischer
and Pipp (1984), may be relevant. The functional level, measured under
conditions of low support, increases only slowly and more or less
linearly. The optimal level, measured under optimal contexts (i. e. ,
optimal practice and support) shows an S-shaped growth that is
significantly higher than the functional level. What is probably witnessed
in this case is the effect of different growth rates in the environmental
help and support as expressed by the carrying capacity. If the growth rate
of K is significantly lower than that of L (the growth
level of some specific skill), L will very closely follow the
slow, gradual increase of K in that L is always
asymptotically close to its upper level as determined by K.
Phenomenally, slow growth—although theoretically S shaped—appears in
the form of a slow linear increase. If the help and support level grows
much faster than the skill level itself—because the environment is
particularly sensitive to the growth of the skill level in the subject,
for example—L, provided its growth rate is high enough, will show
the characteristic S shape of logistic growth and the high upper limit
that is associated with high and adequate help and support. Quantum-like
shifts may be the effect of a rise in a resource, control variable, or
both. I interpret control variable (Fogel
& Thelen, 1987; or order parameter, Haken,
1987) to mean any variable other than the carrying capacity and the
growth rate that determines the growth in a dependent variable. In fact, a
control parameter acts as a timing device for the dependent variable. That
is, the dependent variable cannot start growing until the control variable
starts growing or until the latter has reached some threshold level. A
simple way to model a control system is the following: The growth rate of
a dependent variable D has a very small initial state value and
grows as a function of the growth level of a control variable C.
In this dynamics, the growth rate of the control variable functions
as a timer for the onset of a very quick growth process in the dependent
variable (see
A Grower G, which seems to appear
suddenly, grows as a consequence of growth in its control variable
C (right). (If the control variable is an autocatalytic grower,
its growth-onset time depends on its initial state level [top right vs.
bottom right]; the autocatalytic timer [at] determines the growth onset of
dependent growers [
Figure 28; see the Appendix
for details). An example of a powerful control parameter for a variety of
cognitive growth processes is given by Mounoud
(1986) and concerns the ability to embed information in new contexts.
Mounoud suggested that the sudden growth of cognitive skills such as
reading, writing, and formal thinking around the age of 6 to 7 is made
possible by the emergence of the general cognitive capacity to extract
information from one context and embed it in another. Another example of a
control parameter in a Piagetian model of cognitive development is the
probability with which a specific sort of cognitive conflict will arise
(e. g. , a conflict between two opposing strategies for solving the same
type of problem). If the growth of some specific form of logical
understanding is indeed dependent on the probability that a specific sort
of cognitive conflict arises and if this probability itself grows
logistically, then the growth of the logical understanding will be timed
by the underlying probability growth and take the form of a
quantum-leap-like increase as shown in Figure
28 (top left). It is also possible to make an autocatalytic version of
the timing device dynamics, that is, a dynamics where the timing of a
sudden quantum leap is a function of the absolute growth of the dependent
variable itself. The growth equation for such a dynamics is as follows:
which means that the growth rate of the growth
rate of L is an exponential function of the absolute increase of
L over the interval (t - f) until t. It can be
shown that in this dynamics, the duration of the initial state period is
approximately an inverse logarithmic function of the magnitude of the
initial state of the growth rate. The long initial state is then followed
by a quantum leap in the growth level, which amounts almost to a sudden
emergence of the variable at issue (see Figure
28, right). Autocatalytic timing functions like the present one might
provide a model for cognitive capacities, the timing of which seems to be
maturationally determined (i. e. , they are apparently not dependent on
experience or learning).
A sequence of so-called conjunctive stages may
result from separate growers with different growth rates. A conjunctive
stage sequence (Van
den Daele, 1968) can be defined as one in which each later stage
encompasses the field of application of a former stage, in addition to new
and more complex applications. In fact, conjunctive stage sequences amount
to takeover phenomena. For instance, in Piaget's model, the formal
operational logic takes over the domain of application of the concrete
operational logic and adds to this the domain of specific formal
operational applications. The sequence of such stages could result from an
initial state in which all systems are present in a germinal state. The
differences in growth rates and resource level account for the stage-like
sequence. For the moment, assume that the growers entertain competitive
relationships, which is quite likely anyhow, and that the positive
feedback principle applies. Under these conditions, a temporary regression
of a former stage may occur by the time it is surpassed by its successor
(
Three conjunctive growers with different growth
rates produce a steplike sequence and temporary regression after takeover
by a cognitively more powerful grower (e. g. , sensorimotor,
preoperational, and operational cognitive strategies).
Figure 29; see the Appendix
for details). Although the general shape of the interaction between the
conjunctive growers is rather robust, the specific form, that is, whether
and when regressions will occur, may rather strongly depend on small
random factors (e. g. , maximally 0. 0001, as in the simulation from Figure
29). Takeovers and conjunctive growth are often, if not always, an
indication of hierarchical relationships among growers. For instance, a
growing skill A takes over the former domain of application of a
skill B because B is a structural component of
A or because A cannot start growing if B
remains beneath a specific threshold level. For a growth model to explain
long-term cognitive growth in various domains it should contain a model of
such conditional relationships among growers. A model of conditional
relationships requires an elaborated structural model of skills, rules, or
knowledge describing their composition in terms of structural
constituents.
Finally, the concept of stage or
substage may be applied to stepwise changes in a single variable
(see Fischer,
1983b, for several examples). For instance, Dromi's
(1986) vocabulary growth curve shows a temporary flattening which
probably marks the transition to another substage during the one-word
period. It is highly probable that the flattening of the learning curve
following the onset of syntax is also only temporary and that it will be
followed by a (probably slight) increase in the growth rate of new words,
ending in a final leveling when the ultimate carrying capacity is
approached. Corrigan's
(1983) data on vocabulary growth in 3 children show a pattern of
successive increases and decreases of the growth rate, resulting in a
humped growth curve (see
A mathematical simulation of empirical data on the
growth of vocabulary and maximal length of utterance. (Horizontally, the
diagrams show the increase in vocabulary [top], the number of new words
acquired each month [middle] and the maximal length of utterance each
month [bottom]. Vertically, the diagrams correspond to data from 3
children [John, Mindy, and Ashley] and simulated data based on the
dynamics from Levels and Transitions in Children's Development,
San Francisco: Jossey-Bass. Copyright 1985 by Jossey-Bass. Adapted by
permission. )
Figure 30). Such stepwise growth forms probably
reflect the effect of oscillating growth in a sort of attentional resource
variable. The time and effort a child may invest in specific learning (e.
g. , learning new words) is limited and could be controlled by some sort
of activation-of-attention function attached to a specific grower (e. g. ,
words or applications of a syntactic rule). This function actually
determines the average amount of time and effort allocated to the learning
process to which it is attached and thus specifies the child's interest in
or motivation to perform some sort of acquisition task. It is also likely
that this resource function is subject to positive feedback growth of the
type discussed earlier—in other words, that it rises as a consequence of
the success (progress and growth) of the dependent grower and that it
tends to fall when some intrinsic resource limit (i. e. , its carrying
capacity) is crossed. This will, of course, negatively affect the growth
rate in the dependent variable. However, this fallback implies that more
time and effort will become available to some other acquisition process.
For instance, one may assume that the child tends
to temporarily invest much time and effort in learning new words and, as
this investment decreases, that the time and effort not spent on word
leaning will be invested in the growth of syntax expressed in the form of
a growing mean length of utterance (MLU; assuming that this is a
reasonable measure of overall early syntactic growth. Individual MLU data
show that the increase is not smooth, but rather irregular; Brown,
1973, Pérez-Pereira
& Castro, 1989).
The dynamics of this process is represented in
A model of a dynamics explaining long-term stepwise
growth. (Two bootstrap growers C and D [circular arrows]
grow antagonistically as a function of a resource B. Whereas
B is positively affected by B, D is positively affected
by the inverse of B; i. e. , KLB-D
arrows]. B is subject to positive feedback [hence, the broken
"pf" arrow] and is positively affected by a grower A. There is a
competitive relationship from C to A; i. e. , C
negatively affects A and feeds upon the growth of a variable
A. )
Figure 31. In the present dynamics, two parameters
are important. The first is the strength of the competition relation. If
it is low, the growers grow smoothly and independently of one another to
their proper maximum. The stronger the competition, the more steps occur
and the more irregular they are. Mathematical implementation of a dynamics
with rather strong competition leads to a picture of stepwise increasing
growth curves, with mutually exclusive plateaus and rises in the growers
that compete for the same attentional resource (see
Growth curves following a stepwise course, based on
the step dynamics from C and D in the dynamics from
B component. The A component from
Figure 32; see the Appendix
for details). A second important parameter is the amount of damping of the
effect of the oscillating resource function. For instance, it is not
necessarily the case that if the child's attention to syntactic aspects of
language is temporarily minimal, syntactic growth simply stops (an
implicit assumption made in the model from Figure
32). Rather, under such circumstances, growth rate decreases but is
not reduced to about zero. This phenomenon may be used to model the Corrigan
(1983) data mentioned earlier. Corrigan compared, among others,
vocabulary growth with growth in maximal length of utterance in 3 children
between 10 and 27 months of age. Her data can be mathematically modeled
quite well by a dynamics of the sort represented in Figure
31, given that the effect of the oscillating resource variable is
sufficiently damped (see Figure
30; see the Appendix
for details). Again, this simulation is based only on a set of initial
state values, which then develop deterministically in accordance with the
equations based on the dynamics from Figure
31.
This discussion of how the growth model may account
for stage phenomena has been restricted to quantitative aspects.
Qualitative shifts, in the sense of the emergence of new forms, constitute
in fact the major challenge to any developmental approach (Thelen,
1989). In principle, the quantitative approach presented here could
contribute to the explanation of qualitative changes, for example by using
the growth mechanism as a major transition factor in synergetics-type
models. In such models, many variables cluster into simple structures that
are characteristic of different developmental stages.
The major idea behind the cognitive and language
growth model discussed in this article is that cognitive growth occurs
under the constraint of limited resources, with either mutual support or
competition for resources among the cognitive growers that constitute a
person's cognitive and language systems. This point of view explicitly
subsumes cognitive growth under the general laws of thermodynamics: The
cognitive system is a system carrying complex information and as such is
far from thermodynamic equilibrium. Increasing its order and information
load consumes time and energy. The cognitive system is also a
self-organizing system, maintaining and increasing its own order, provided
sufficient resources are available. Its form is to a great extent
determined by the fact that these resources are limited.
I have shown that cognitive growth can be modeled
mathematically in the form of a logistic difference equation, which
applies to all—or at least to a very significant majority—of the variables
involved in cognitive growth processes. All such variables interact and
react with one another, thus making even relatively simple growth dynamics
complex, transactional events. The growth curves resulting from such
dynamics were often very difficult, if not impossible, to predict on the
basis of simple linear extrapolation of initial state properties. A
dynamic systems approach like the present one might change the meaning—at
least the connotational meaning—of concepts such as deterministic,
random, predictable, and so on. For instance, in several cases,
growth sequences appear very similar to random sequences, although they
behave completely differently from real random sequences, for instance in
that under specific circumstances they may evolve toward stable points,
which random sequences will never do. Although the equations involved are
very easy to understand and involve no complicated mathematics, they are
very difficult to solve; that is, it is very difficult to give general
answers to questions such as under which conditions specific equations
will lead to stable solutions, to damping of oscillations, and so on.
Thus, further mathematical scrutiny of the equations presented here is
necessary.
I have discussed several dynamics without entering
deeply into the psychological and process interpretations of these
dynamics and their components. The major aim of this article was to show
that dynamics of the sort I have discussed may provide a plausible general
model of several cognitive growth processes. The real work of thoroughly
testing the empirical merits of specific dynamics applied to specific
developmental fields and of finding psychological interpretations that can
stand up to empirical scrutiny is still to be done. Moreover, several
obvious theoretical questions have not been answered. I give two examples.
One concerns the application of mixed interactions, that is, interactions
in which one affects another positively whereas the other affects the
first negatively. An example of mixed interaction is the negative support
dynamics, such as paternal correction following the growth of unwanted
skills or habits. Another question still to be answered concerns the
simulation of interactions in more complicated systems of cognitive
growers, for instance, systems consisting of 10 or more growers which
interact with each other, the effects of random perturbations on the
long-term course of processes, and so on.
From a methodological point of view, applying the
present dynamics model would ideally require longitudinal individual
studies with a sufficiently dense measurement schedule and with maximally
reliable data. This is especially so in cases where growth functions show
strong interindividual variability. Unfortunately, this requirement is
much more than most current studies in cognitive development can offer. If
individual growth functions do not differ too considerably in general
shape and rate, cross-sectional and mixed designs may be used to test
several growth patterns. For instance, in a mixed design with at least two
consecutive measurements for each age group, one may use differences in
the slopes of the curves or the degree of instability of measurements over
time as indicators of an underlying growth form. Cross-sectional group
data should be used selectively. For instance, if group data reveal a
transient regression, then such regression should be characteristic of a
significant portion of the individual growth curves, if such curves would
be available (and provided the groups developed under largely similar
cultural and social-historical cohort conditions). If cross-sectional
group data fail to show regressions, this may be due either to the fact
that individual regressions have compensated one another or to the fact
that no such regressions have occurred. Methods in which multiple tasks
are presented to children and related to specific developmental functions
may also be used to reveal different underlying growth processes (Fischer
et al. , 1984).
Another methodological aspect of the model is that
intraindividual instability of growth data is not considered a weakness or
a sign of unreliable measurement, but rather as an essential
characteristic of cognitive growth. Of course, reliable measurement
remains an essential condition for theory building. On the other hand, one
should not automatically imply stability over time as a criterion of
reliability of measurement, because there are many forms of growth
conceivable in which temporal instability is a major structural
characteristic of growth instead of some sort of aberration.
As far as the underlying assumptions of the
logistic growth model and the dynamic systems approach apply to
microgenetic events, such as the growth of attention or skill during a
single experimental session, the model may also be used to describe
short-term changes in behavior or short-term learning effects. Finally, in
this dynamics model, the data from a group of subjects on a specific form
of growth (e. g. , of inversion rules in Wh-questions) should not
be considered for means and overall data but should be viewed as a
collection of trajectories specifying a state space. The developmental or
growth model should provide a general model of this state space and
explain which individual trajectories are theoretically possible and which
are not. My approach does not start from the idea that in each process of
development some orthogenetic line of development should exist that is
typical of a group of subjects as a whole. Rather, the dynamics model
describes cognitive and language growth as a constrained bundle of
individual growth possibilities or trajectories that proceed as a result
of a specific underlying dynamics.
1
Refer is used in the semantics sense; that is,
observable behavior should refer to theoretical concepts, such as a
grammatical rule. It is not implied that observables should refer to
discursively interpreted mental states or structures, such as "rules in
the head" or whatever; the reference relation does not imply the assignm
ent of a specific ontological status to the theoretical concepts used such
as "rules," "concepts," and so on.
2
The metaphorical term cognitive species is similar
to several terms introduced by scholars who have applied evolutionary
analogies to the problem of the cultural transmission of knowledge and
skills. They have proposed several terms to describe the units of such
transmission; Dawkins (197 6)
used the term meme as the cognitive analogon to gene; Lumsden and Wilson
(1981)
used the term culturgen; see van
Geert (1985)
for an overview.
3
This formula is identical to Haldane's formula for
relative speed of change in evolution (Simpson, 1983).
4
The parameters and equations necessary to
reconstruct the present theoretical curve are described in the text.
Details are described in the Appendix.
In view of the intensive data-gathering procedure normally used in these
5
It is very likely that cognitive and physical
accessibilities are also subject to growth. However, particularly in the
first phase of word acquisition, one may expect that vocabulary growth
rate will be much higher than the growth rate of the accessibility
threshold, and in general, of the carrying capacity for vocabulary.
6
Fatigue and motivational changes may be an effect
of the growth process itself; for example, motivation may decrease as a
function of effort invested in the growth process. In this particular
case, such changes are not considered extrinsic "random" factors.
7
When no subscripts referring to time (e. g. ,
subscripts t, t + f) are shown, it is assumed that the argument left of
the equal sign should take the subscript t + f, whereas the arguments
right of the equal sign take the subscript t.
8
Final state has been operationally defined as the
state after 100 iterative applications of the growth equation, that is,
100 f after the initial state); after this interval, all but the slowest
trajectories have settled into a stable state, or at least into a state
that changes only very minimally; if f = 1 week, 100 f is about 2 years,
which is a sufficiently long developmental interval in view of the fact
that the developmental processes discussed in this section take place
during infancy and toddlerhood.
9
Instead of making an intrapolation on the basis of
an exponential increase between the first and second growth points, it has
been assumed that intermediary growth points follow an oscillating course
geometrically similar to the overall oscillation of the empirical growth
curve.
10
For instance, when L is bigger than K, it will drop
back under the K level, and the higher L is, the deeper it falls. If L
> K, then the growth rate of K will be negative, and thus K will
decrease relative to the distance between K and L. That is, at the next
growth state, K and L will thus be in each others' proximity. However, the
closer L is to K, the closer it will stay to K in the next growth state.
Consequently, K and L evolve toward a stability point, which would not be
the case if K would not adapt (see Figure 26
If carrying capacity K adapts to a rapid
grower as a function of the unutilized opportunity for growth,
+U, chaotic oscillations of the growth level L are
damped, and K and L move toward a common point attractor
at a higher level (top curve); if K does not adapt, L
follows a chaotic course (bottom curve).
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Appendix A Equations, Initial State Values, and
Constants Used in Growth Sequences Depicted in Figures
All growth sequences depicted in the figures are
described, except those for which a description is given in the main text.
The mathematical symbols correspond to those in the equations mentioned.
The variables include real variables, meaning those changing during the
growth process, and constants. Of the real variables, only the initial
state values are given. If the equations used do not correspond to
equations explicitly discussed in the text, they are presented in this
appendix. The difference form of the logistic growth equation is either
presented by its equation number (17a),
or presented in abridged form—that is, with the symbol ƒ followed by its
arguments, which are separated by a comma. The arguments are presented in
a fixed order: L-r-K. For instance, K = ƒK,
d · L, KK)
means that the growth of a carrying capacity
K is a logistic difference function of the level of K
occurring a time interval f ago, a growth rate that is a product of a
damping function d and the growth level of a grower L
(also a time interval f ago), and finally a carrying capacity that
corresponds to the highest possible stable level of K.
Received: March 5, 1989. Revised: December 29,
1989. Accepted: June 27, 1990. |