1.2 A Formal Approach to the Concept of Development
Basically, the concept of development is concerned with transitions between states, with particular emphasis on the transition from an initial to a final state. Development can therefore be defined as a set of temporally ordered states and transitions. In everyday usage, for instance, development is viewed as what happens between birth and adulthood. From the state âneonateâ to the state âadultâ there are a number of temporally ordered intermediate states, described as toddler, school-child and adolescent. In Piagetâs model, cognitive development begins with the sensori-motor and ends with the formal operational stage, via the pre-operational and concrete operational stages. Brunerâs criterion for distinguishing the states of cognitive development is the mode of representation applied by the subject. Representation starts with the enactive mode (representation of knowledge in the form of action-patterns), then proceeds to the iconic mode (representation-rules adopted from perception) and ends with the mastery of the symbolic mode of representation by the age of seven.
In principle, developmental states do not need to have the stage-like character of the foregoing examples. In a continuous developmental process the states are defined simply as points on the developmental continuum. Furthermore, the temporal span of a developmental process does not necessarily cover large parts of the life. It is also possible to study the development of a particular skill, e.g. the ability to use the notion of volume (GagnĂŠ, 1968), or a particular set of skills, e.g. the development of concepts (Klausmeier, 1976).
In Arièsâ study of the historical development of childhood and family, the first chapter is devoted to the medieval conception of the ages of life (Ariès, 1962). The author refers to a popular thirteenth-century text that describes the ages or periods of life as follows. The first age is childhood, from birth until seven years. It is the age in which the teeth are planted, a fact which explains why the child cannot talk well. The second age is âpueritiaâ and lasts till fourteen. The third age is adolescence. The medieval scholars disagree on the age at which it ends, either twenty-one or thirty-five. âThis age is called adolescence because the person is big enough to beget childrenâŚ. In this age, the limbs are soft and able to grow and receive strengthâŚ.â (Ariès, 1962, page 21). The fourth age is youth, lasting until forty-five. The fifth stage is called senectitude, the sixth is old age, followed by the age called âseniesâ. Human life counts seven ages because of its correspondence with the then known seven planets. Ariès states that the medieval conception of development must be seen within the framework of medieval thinking with correspondencies (e.g. between life and the planets). The ages of life are a system for classifying people. There is no notion of causal or conditional antecedent-consequent relationships between stages of development.
From the foregoing examples we may infer some basic properties of the concept of development.
First, every account of development uses a particular kind of âlanguageâ or vocabulary to specify and describe its developmental states. In the colloquial model we use the concepts of baby, toddler, adolescent and so forth. Piaget speaks about sensori-motor, pre-operational and operational stages. Obviously, the language of development contains more than names of states: it must also be capable of defining the nature and content of these states, i.e. criteria which determine their properties and mutual differences. In a developmental model, two states are particularly important, namely the initial state and the final state.
Second, every account of development implicitly or explicitly limits and specifies the nature of the transitions that may take place. In some cases, the logical structure of the developmental stages prescribes one transitional step at a time and in a fixed order (e.g. the common sense and Piagetian model). Some models may also accept regression (partial reversibility of the transition, such as in the Freudian model or in some Piagetian approaches to the effects of ageing on cognition). Other models may be relatively free: they may allow the occurrence of more than one stage at a time. They may also permit a certain degree of freedom in the order of attainment of the stages (such as in GagnĂŠâs partly branched model for the development of conservation skills). Within each theory, however, the developmental system must proceed from the initial to the final state within the limitations set by the transition rules.
The transition rules do not only regulate the order of occurrence of developmental states, they also describe the nature of the parameters that determine why and when state-transitions take place. This particular aspect of the transition rules might be called the stability rules. Stability rules may be distinguished on the basis of two relatively independent criteria: they define states as either internally stable or unstable and as either open or closed, that is, susceptible or not to changes due to external factors. Maturational theories, for instance, are characterized by largely closed and internally unstable states. Learning theories of development are characterized by states that are internally unstable and open. According to Piaget, the instability of the states is explained as a search for equilibrium (see, for instance Piaget, 1959). When a particular equilibrium is reached, the system is ready for a confrontation with reality that annihilates the existent equilibrium and sets into motion a process towards a new equilibrium. The process stops when a final equilibrium, formal operational thinking, has been reached.
Third, the models of development claim to have empirical applicability. Consequently, they require an appropriate empirical operationalization of the concepts they employ. The theory must permit its user to decide what kind of state or transition can be ascribed to an individual at a certain point of time. In Piagetâs theory, for instance, the inability to solve a conservation problem is an empirical criterion for belonging to the pre-operational stage, whereas, in Brunerâs theory, it is a criterion for the state of iconic representations. The empirical criteria must be decisive and determinable. Decisiveness means that one and only one developmental state answers the criterion (e.g. the appearance of a specific type of behaviour which is found at one developmental stage but not at another). Being determinable implies that it must be possible to decide in an unambiguous way whether or not a given empirical state of affairs is an instance of the criterion. The empirical operationalization of the theoretical concepts and hypotheses is determined by a set of rules that regulate the relationship between the theory and the world of observation, experiment, application and so forth. We shall call these rules the empirical mapping rules. These rules do not only regulate the mapping of theoretical expressions on empirical facts (operationalization) but also the translation of observable events in terms of the theory (diagnosis).
It is clear that the properties we have distinguished constitute a formal description of a developmental theory. In practice, a theory consists of a model which is a particular exemplification of the formal components. The theory describes a model of development which allows the theoretical psychologist to infer the âgrammarâ (i.e. the state language, the transition rules and the mapping rules) that underlies it. Further, theories are not only exemplifications, but also specifications. That is, a theory might reflect only one of the many possible models that might be based on its underlying grammar (the theory might present itself in the form of a particular empirical mapping, for instance - see Figure 1.1).
The first and second property of a developmental model make it comparable with a finite state automaton. The concept of finite state automaton stems from systems-theory. It has proved its merits in cybernetic approaches to behaviour (Ashby, 1952) as well as in transformational linguistics (Chomsky, 1957). A two-dimensional matrix consisting of two co-ordinates containing the first ten natural numbers can be used to demonstrate such an automaton. Every point in the matrix is characterized by an ordered set of two natural numbers. These points constitute the possible states. As transitional rules, i.e. rules which regulate the transition from one point to another, we might take the following examples: first, a transition is only allowed from one point to an adjacent point; second, transitions must be horizontal or vertical; third, transitions take place at arbitrary moments.
Figure 1.1 The formal components of a developmental theory
In Figure 1.2, two developmental lines have been drawn, connecting the initial point (1, 1) and the final point (5, 5). Both lines represent a set of developmental steps which are permitted by the transitional rules and each set represents a possible model of development within the formal framework.
It is very important to state that the set of possible developmental lines may be determined deductively. The set of possible lines is obviously determined by the properties of the framework, such as the two-dimensional matrix, on the one hand, and the transitional rules, on the other. The set of lines, then, can be deduced by rigorously applying the rules to the framework.
I shall give a very brief overview of two theories, Piagetâs stage theory and Klausmeierâs theory of conceptual development, and try to show how the succession of developmental states in both theories reflects a deductive structure. Klausmeierâs theory will be discussed in the light of the relationship between empirical investigations and theoretical statements, which will be dealt with in the next section.
Before I proceed to a very sketchy analysis of Piagetâs and Klausmeierâs theories I must caution against two possible misunderstandings. First, the analysis is based on a very rudimentary image of the theories and I do not pretend that they can be reduced to this elementary form. Second, the assumption that the kernel of these theories is of a deductive nature is not intended to devalue them (see section 1.3 for further discussion, see also Smedslund, 1978).
Figure 1.2 An example of a finite state automaton. A state is characterized by two integers. The arrows indicate two possible paths between an initial state (1, 1) and a final state (5, 5). The transition rules are explained in the text
In Piagetâs stage theory, the final state of cognitive development is called the stage of formal operational thinking (see Brainerd, 1978, for an excellent overview). The basic property of this stage is that thinking is an internal process, consisting of operations that are carried out upon formal, abstract contents. Operations are mental actions. They can be compared with physical actions but they differ from the latter in a number of points. The basic difference is that operations are a sort of abstract actions. Unlike concrete actions, they can go back to their starting-point, i.e. they are reversible. In the formal operational stage, the subject is able to think about the operations themselves, for instance, and he or she no longer depends on the spatiotemporal limitations of the concrete objects and events.
The description of the final state provides a particular point of view from which we may interpret the meaning of the behaviours and abilities of the initial state-child. For Piaget, the neonateâs and infantâs behaviour does not yet reflect the intellectual properties ascribed to the final state. A large number of complex reasons, which we shall not discuss here, can be brought forward to support the assumption that the infantâs intellectual life is completely determined by external actions. âExternalâ does not exclude mental events such as perception and feeling, however, but implies that the entire process of thinking does not take place internally. The infantâs thinking is not internal and not operational but âactionalâ. It is limited, therefore, to the material and spatiotemporal properties of the concrete objects and events.
The order in which internalization, the transition from action to operation and from concrete to abstract contents, is acquired is determined by the meaning of these concepts. Since operations are defined as âabstractâ, mental actions, it is necessary that internalization occurs before the transition from concrete action to operation. Actions as well as operations can be carried out with concrete contents, but abstract contents require operational activity and cannot be dealt with on the action-level. The emergence of abstract, formal contents, therefore, depends on the presence of operations.
In principle, the transition from external to internal, from action to operation and from concrete to abstract content may occur quasi-simultaneously, i.e. they might succeed each other with such short intervals that the successive stages are almost unobservable. For the theory, it does not matter however, how much time it takes for proceeding from the initial to the final state. The mapping of a deductive theoretical kernel upon empirical variables, such as a specific course of time, belongs to the empirical aspect of the theory.
If we summarize the orders in which the various acquisitions must take place we find the following theoretical states of development. If the final state is characterized by thinking as an internal process of operations carried out upon abstract contents, the initial state must be characterized by non-internal (external), non-operational (actional) and non-abstract (concrete) thinking. The initial state has been given the name of sensori-motor stage of development. The first developmental acquisition, which makes all the others possible, consists of the emergence of internal thinking processes. The second developmental state, in which action with concrete contents has become internalized, is called the pre-operational stage. This stage is characterized by a number of particular properties that are due to the fact that thinking is still actional, i.e. that it is not reversible. Interiorization of the thinking process is a necessary condition for the emergence of the next acquisition, namely operational thinking. The actions have become reversible and organized in logical systems. Thinking is still carried out with the kind of content employed in the foregoing state, namely concrete contents. Piaget calls this state the concrete operational stage. The final step in the developmental process concerns the transition from concrete to abstract contents. This transition cannot be made before thinking has become operational. Summarizing, the definitions of the concepts employed in the description of the final state tolerate no other succession of developmental states than the one described in Piagetâs theory.
The fact that the order of the Piagetian stages, namely from sensori-motor to formal operational thinking via pre-operational and concrete operational thinking, represents a deductive necessity does not imply that the actual Piagetian theory is also deductively true. In the actual theory the formal, deductive kernel that we have sketched has been mapped upon a number of empirical variables. Piaget assumes, for instance, that the developmental process lasts roughly until the middle of the second decade of life, that the transition from pre-operational to concrete-operational thinking occurs at the age of seven, and so forth. It is clear that the truth of this empirical mapping can only be decided on empirical grounds. The relationship between deductive and empirical aspects of a developmental theory will be the topic of the next section.
Figure 1.3 A proposed conceptual deep structure of Piagetâs stage theory.
(a) shows the state transition rules consist of simultaneous and successive conditions of occurrence of state-properties
(b) shows the state-transition rules determine the conceptually permitted succession of developmental states; the change of one state-property corresponds with a developmental state-transition; double arrows indicate conceptually incompatible state-properties, the resulting states - indicated by an asterisk - are inexistent
(c) shows the basic state properties can be mapped onto spatial axes: external versus internal on the horizontal axis, operational versus actional on the depth axis, concrete versus formal on the vertical axis