NEO-PIAGETIAN THEORIES OF COGNITIVE DEVELOPMENT. Andreas Demetriou. University of Cyprus

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1 Published under the title Neo-Piagetische Ansatze in W. Schneider & F. Wilkening (Eds.), (2006). Theorien,modelle, und methoden der Endwicklungpsychologie. Volume of Enzyklopadie der Psychologie (pp ). Gotingen: Hogrefe-Verlag. NEO-PIAGETIAN THEORIES OF COGNITIVE DEVELOPMENT Andreas Demetriou University of Cyprus Correspondence: Andreas Demetriou, Department of Educational Sciences, University of Cyprus, P. O. Box 537, 1678 Nicosia, Cyprus. Phone: ; Fax: ;

2 Abstract This chapter summarizes five neo-piagetian theories of cognitive development. Namely, the theories of Juan Pascual-Leone, Robbie Case, Graeme s. Halford, Kurt W. Fischer, and my theory. The first three of these theories emphasize the role of working memory as a factor of transition across the main developmental stages of thought from birth to adolescence. Fischer s theory emphasizes the social and the learning dimensions of cognitive development. Finally, my theory in addition to these factors, integrates considerations about special processes in different domains of thought and also the role of self-awareness and self-regulation in development and individual differences. The theories are then compared and contrastively evaluated. The chapter is addressed to advance undergraduate and graduate students and researchers in developmental cognitive science.

3 NEO-PIAGETIAN THEORIES OF COGNITIVE DEVELOPMENT The Current Status of Piaget s Theory Piaget s theory attracted massive interest in the 60s and the 70s. As a result of this interest, there is nowadays a solid body of empirical evidence concerning most aspects of Piaget s theory (see Brainerd, 1978). On the basis of this evidence, a number of conclusions can be drawn about the validity of the various aspects of this theory and its permanent contributions to our understanding of the organization and development of human mind. The studies conducted to evaluate Piaget s theory can be classified into two broad categories. First, many studies tested if the phenomena Piaget maintained that he discovered were indeed real and reliably present. This research can be summarized quite easily in a single statement: Whenever a replication study remained close to the original Piagetian conditions, the same phenomena were observed in more or less the same way at more or less the same age. In the sensorimotor stage, children grasp the various aspects of object permanence as specified in the original Piagetian studies. In the next phase, children do fail the class inclusion, the transitivity, and the conservation tasks until about the age of seven years; children do become able to grasp these concepts during the school years. However they do not take an if-then stance to problems and they cannot reason propositionally, design experiments by systematically controlling variables, and understand proportionality before adolescence, even when they are highly intelligent, according to performance on tests of intelligence. Thus, it is safe to conclude that Piaget was right in both, the phenomena he discovered and their approximate age of occurrence. But is the theory right? That is, is understanding at the successive stages really due to the logical processes proposed by Piaget? Is thought organized in the structures of the whole Piaget associated with each of his main developmental stages? Is development caused by the reasons invoked by Piaget? The question about processes

4 asks if performance on Piaget s tasks really reflects the processes Piaget thought that they reflect. For instance, does really a child s failure to solve the transitivity task indicate that this child cannot reason logically? One of the first and most influential studies in undermining Piaget s position that his tasks tap logical reasoning processes was Bryant and Trabasso s (1971) study on transitivity. In this task children are shown the A and B and the B and C terms together and they are then asked to infer the relation that exists between the A and C terms. For Piaget, failure to make the correct inference was taken to imply lack of the logical structure which can produce the inference. There was an important question though that Piaget failed to consider. What would happen if children do not remember the A and B and the B and C comparisons--or if they do not represent them properly and accurately? Obviously, they would fail the task. In such a case, however, their failure could not be taken as an indication that they lack logical reasoning, because it might well have been caused by their lack of memory or representational efficiency. This is precisely what Bryant and Trabasso have shown. That is, preschool children either forget the comparisons or they misremember them. For example, instead of representing the premises in their mind in comparative terms (e.g., the red stick is longer than the green stick ) they represent it in absolute terms (e.g., the red stick is long ). Even more, Bryant and Trabasso have shown that preschool children can infer the transitive relations if they are systematically trained to correctly represent and remember the premises. Hundreds of studies such as this demonstrated that extra-logical factors such as memory, language and communication, familiarity with tasks and testing procedures, and interest, affect the performance and the developmental standing of individuals at any age. Another stream of studies focused on the organization of the processes involved in the understanding of different domains rather than on their age of first attainment. These studies aimed to test if Piaget is right in assuming that there is an overarching structure of the whole at each major stage of development which underlies understanding over different domains. Thus, in these studies, children were

5 examined by large numbers of tasks designed to represent different domains, such as space, number, causality etc. The notion of the structure of the whole suggests that development in different domains should proceed in synchrony and thus performance on tasks representing different domains should be on equivalent developmental levels. This expectation was not confirmed. That is, these studies showed systematically that persons usually operate on different levels in different domains. These findings suggested that that there may be other factors governing the organization of cognitive processes which go beyong Piaget s notion of the structure of the whole (Demetriou & Efklides, 1985; Shayer, Demetriou, Prever, 1988). Obviously, we are in need of a theory that would be able to accommodate the apparent inconsistency between evidence which suggests that, on the one hand, the types or kinds of thoughts discovered by Piaget are real and palpable, whereas, on the other hand, these kinds of thought do not seem to be organized or change in exactly the same way as Piaget maintained. At the same time, they seem to co-exist with other phenomena which have passed unnoticed by Piaget. In this chapter we will present a number of alternative theories that aimed to accommodate this state of affairs. These theories have come to be known as the neo-piagetian theories of cognitive development. These theories have been influenced by several of the traditions that are still dominant in psychology. Specifically, several of these theories were strongly influenced by the information processing tradition that was very strong in the seventies, when disenchantment with Piagetian theory started to emerge. The theorists who believed that the integration of concepts and methods from Piagetian theory with concepts and methods from information processing theory would lead to a better theory of cognitive development have concentrated primarily on the relations between the sequence of the major stages of cognitive development described by Piaget and the changes that occur in the information processing capacity of the developing individual. Specifically, they ascribed to processing capacity the status of the workspace of thinking where mental acts or operations can be represented, transformed,

6 and combined for the sake of problem solving and understanding. Thus, they hypothesized that there is a direct causal relation between the development of the short-term store and the sequence of the stages of cognitive development, such that the improvements in the various aspects of processing capacity cause the transition from the one stage of cognitive development to the next. Three such theories were proposed: The theory of Juan Pascul-Leone, the theory of Robbie Case, and the theory of Graeme Halford. These theories will be summarized below. A fourth theory, namely the theory proposed by Kurt W. Fischer was influenced more by learning theory and socio-cultural theories of development rather than by information processing theory. Thus, this theory specified developmental stages in terms of the skills they involve rather than in terms of their information processing capacity. Moreover, in this theory, transition from lower to higher stages was explained in reference to social and cultural influences rather than in terms of changes in underlying capacity potentials. Finally, a fifth theory, namely the theory proposed by myself, was influenced more by the tradition of differential psychology as much as it was influenced by the cognitivist tradition. Thus, in this theory, there is a strong emphasis in the specification of different domains and levels of thinking and also in the specification of the factors that make individuals to differ between each other. Juan Pascual-Leone: The Theory of Constructive Operators Pascual-Leone (1970) was the first to propose a model of cognitive development in which he systematically attempted to integrate the fundamental assumptions of information processing theory with the fundamental assumptions of Piagetian theory. Specifically, he advanced the view that human thought is organized as a two-level system. The one of the levels is considered to involve several silent operators, which define the hardware of thought. The other is the level of subjective

7 operators, which refers to the functional rules governing the operation of thought as well as the content of thought. Silent operators refer to a number of constructs and functions that define the volume of information that the individual can represent and process at a given time and the style and preferred ways of processing. In other words, these constructs define the potentials or the capabilities of the individual in regard to what information can be processed, how much of it can be processed, and how processing will be done. In fact, the term silent operators conveys the assumption that these operators operate under the surface exerting their influences silently on the functioning of the subjective operators of the second level. That is, they raise or lower the activation value of the mental schemes on which they apply. The level of subjective operators contains the mental operations that the thinker can execute and the concepts or knowledge that she has about the world. In short, it involves all mental schemes that the thinker possesses. In other words, this level involves the structures of thought described by Piaget. These involve the so called figurative schemes that describe reality (that is, perceptions and mental images of reality, verbal descriptions of it, etc.) and the operative schemes that transform reality (that is, mental operations such as identity, composition and reversibility). Therefore, in Pascual-Leone s theory, the first of the two levels of the human mental architecture originates from information processing theory and the second originates from Piaget s theory. It will be seen below that Pascual-Leone invoked the first level to explain both the functioning and the development of the second level. Some of the constructs in Pascual-Leone s theory are primarily directed to the explanation of cognitive development as such and some other constructs are directed to the explanation of individual differences in cognitive functioning and development. Thus, this section is organized into a part concerned with cognitive development and a part concerned with individual differences.

8 Explaining Cognitive Development The most well known and researched of the various constructs that Pascual- Leone proposed is Mental power or Mp. Mp refers to the maximum number of independent information units or mental schemes that the thinker can hold simultaneously in mind at a given moment in order to envisage their relations or use them in order to solve a problem given to him. Mental power operates in combination with the Interrupt operator or I-operator. This is responsible for the deactivation of schemes that are not relevant to the current goal. Thus, this operator refers to central inhibition processes that enable the individual to keep attention focused on the goal of interest and thus make use of the Mp available as efficiently as possible. The two operators together constitute mental attention, which is purposefully directed by the individual at some schemes or stimuli at the expense of not attending others. Generativity is a basic property of mental attention, because the schemes or concepts that are simultaneously activated by Mp may be amalgamated or integrated into new schemes. Thus, mental attention is the main factor of cognitive development because it operates as a cognitive mixer which generates new mental units through the combination of the units already available. Pascual-Leone argued that the mental power available is divided to support or boost, in his terms, two kinds of mental schemes or representations. That is, (a) the representation of the goal or the objective of a given moment (i.e., what the thinker has to do, such as for instance to add up two numbers or judge if two glasses contain the same amount of liquid) and (b) the mental schemes that must be activated and processed in order to meet the goal (i.e., the numbers that must be added to each other and the operation of addition or the information about the transformations that were effected on the liquid involved in the testing situation). In other words, according to Pascual-Leone, the mental energy available is spent (i) to keep the goal in focus, because otherwise problem-solving would be undirected and, therefore, unsuccessful and also (ii) the information that describes the situation and the operations that need to

9 be applied in order to meet the goal. In formal terms mental power is defined by the equation following: Mp = e + k. In this equation e stands for the energy needed to boost the problem goal and k stands for the number of mental schemes, both figurative and operative, that can be kept active simultaneously. INSERT TABLE 1 Pascual-Leone maintained that the energy needed to boost the goal is established during infancy and it remains constant after the age of about two years. However, the mental power available for boosting problem-solving relevant mental schemes increases systematically with age. Specifically, it is one unit at the age of three years and it increases by one year every second year until the age of 15 years when it reaches its maximum of 7 units of information. Pascual-Leone believes that the increase in mental power is maturationally determined. Pascual-Leone took considerable pains to demonstrate that the values of Mp for each age year are as predicted by the theory. In the sake of this aim Pascual-Leone and his colleagues devised several tests of Mp, which they used to examine children and adolescents of different ages. To qualify as a test of Mp, a task must satisfy certain requirements. Most important among them is that the subjects must be trained in the task situation extensively until to automatize the responses that they have to execute in order to satisfy the task-goal. This requirement is necessary to ensure that the goal or the constant e in the equation above does not take any of the resources available for keeping in mind the mental schemes needed to solve the task. Then the subject is presented with the units of information which she must combine or interrelate according to the goal in order to produce a goal-appropriate response. For

10 example, in the classical task used by Pascual-Leone (1970) children are trained to associate each of a series of actions (e.g., raise hand, clap hands, open mouth, etc.) with a particular stimulus (e.g., circle, triangle, square, etc.). Once trained, the children are presented the stimuli in succession and they are expected to execute the respective actions. Thus, the capacity of their Mp is equal to the number of stimuli that they can hold in storage, which is indicated by the number of correct actions that they can exhibit. In a task devised by Case (1985) children are shown a series of number digits, the one after the other for one second (for example 7, 11, 15, 19, 10). The children are trained to indicate where the last number has to go in the series after this one is covered too. Thus, the child must be able to remember all of the numbers presented if she is to be able to indicate where the last number is to be inserted, because no number is visible when she is asked to specify the location. Many experiments like this have shown the values of Mp are more or less as presumed by the theory. The second claim that Pascual-Leone advanced was that the increase in Mp is the cause of the transition from the one Piagetian stage or substage to the next. Table 1 shows how Pascual-Leone associates the successive values of Mp with the sequence of Piagetian stages and substages. This claim requires one to demonstrate two things. First, that the Piagetian tasks which can be solved at each of these stages do indeed require the presumed Mp associated with this stage. For this to be possible one needs to analyze the various tasks in order to specify the number of mental schemes that they involve during processing. In Pascual-Leone s terms, this is the mental demand of the tasks. The second step is an empirical one. That is, it would have to be demonstrated that the subjects with a given Mp can indeed solve the Piagetian tasks that have the corresponding mental demand, provided of course that they can and they want to use their specified Mp. This is exactly what Pascual-Leone did. He provided extensive analyses of most of the more standard Piagetian tasks, which cover the whole spectrum of development from pre-operational to formal thought (de Ribaupierre & Pascual-

11 Leone, 1979; Pascual-Leone & Goodman, 1979). Here only the example of conservation is given as an illustration of this approach to task analysis. Specifically, the reader is reminded that the conservation of the four basic ontological/physical attributes of an object, that is its identity, the mass or quantity of substance it involves, its weight, and its volume is acquired at the age of about 5, 7, 9, and 11 years, respectively. According to Pascual-Leone s analysis, this is so because these four kinds of conservation have a mental demand, that is, they require, Mp of 2, 3, 4, and 5 mental schemes, respectively. Specifically, the conservation of identity requires from the child to represent the fact that nothing has been added or taken away and also the operation or inferential scheme which states that when nothing is added or taken away the condition (or quantity) remains the same. If these two schemes can be represented and focused on simultaneously they will be combined to give the conclusion that therefore, it must still be the same. The conservation of quantity requires to keep in Mp the two aforementioned schemes plus a third one, that is, that the two entities (that is, glasses, balls of plasticine, series of tokens etc.) were equal at the beginning. The conservation of weight requires to activate in mental space a fourth scheme, that is that equal quantities also have the same weight. Finally, the conservation of volume requires two other schemes, in addition to the three schemes involved in the conservation of quantity, that is, that equal amounts of plasticine occupy the same space and the space occupied determines the amount of water displaced. Pascual-Leone and his colleagues (de Ribaupierre & Pascual-Leone, 1979) have provided evidence indicating that tasks differing in their mental demands also differ in difficulty and age of attainment. However, the question of whether they really correspond to different Piagetian stages as claimed by Pascual-Leone is still under dispute (Flavell, Miller, & Miller, 2001 Explaining Individual Differences in Cognitive Development Do all individuals always demonstrate in full the capacity that is characteristic of their age? Obviously this is not the case. According to Pascual-Leone, individual

12 differences in cognitive performance are due to the operation of several other silent operators, which determine how individuals will make use of the two operators that define the capacity available, that is the Mp and the I operators. Specifically, the field operator constrains how information is perceived and conceived. That is, the field operator refers to an orienting mechanism, which drives the person to perceive and represent environmental stimuli as integral wholes and also to emit responses that are compatible to the perception or conception of the stimuli, even if these responses are not appropriate from the point of view of the current goal. For example, perceptual illusions are due to the operation of this factor. In the Piagetian literature, the deception caused by some dominant dimensions in the task configuration of the conservation experiments are also due to the operation of this factor. That is, the tendency to think, for instance, that an elongated row of coins contains more coins than a shorter one, even if they have been made equal a few moments ago, is due to the fact that usually longer arrangements contain more elements. This representation imposes a tendency to perceive the present situation in a particular way which is consistent with it, despite the fact that it is contradicted by the experience of constructing the two rows as equal. According to Pascual-Leone, the strength of this factor is not the same across individuals. The dimension of field-dependence can be used to describe differences between individuals in their sensitivity to the effects of the perceptual organization of reality. That is, field-dependent individuals are more sensitive to the effects of the field factor than the field-independent individuals. Thus, field dependent individuals tend not to use their full capacity in situations where the organization of information activates the field factor to impose a particular interpretation of the situation. Many of the Piagetian tasks, such as the conservations, belong to this category because the perceptual organization of information contradicts the interpretation suggested by logical processing. In these situations, field dependent individuals tend to be deceived by appearances and err, despite the fact that their age and mental power justifies one to expect that they would respond correctly. A series of experiments by Pascual-

13 Leone and his colleagues (Pascual-Leone, 1983; Pascual-Leone & Goodman, 1979; de Ribaupierre & Pascual-Leone, 1979) has indeed demonstrated the relationship between the dimension of field dependence and a relatively slower rate of development in regard to many of Piagetian concrete and formal operational tasks. Pascual-Leone describes two operators for learning, namely logical learning and content learning. Content learning refers to the storage of (domain-specific) information. Logical learning refers to the integration of mental schemes into new more inclusive and more flexible ones because of their repeated co-activation in reality or because of their concurrent boosting by mental power. In fact, so defined, logical learning refers to abstraction processes that enable the individual to capitalize on and make use of his experience for subsequent problem solving. According to Pascual-Leone, differential learning experiences would differentiate individuals in regard to problem solving, even if they posses the same mental power. This is due to the fact that because of these differences individuals would bring different strategies and principles to bear on problems. Moreover, Pascual-Leone describes the operation of emotional and personality factors as potential differentiators of cognitive development. These factors, which emanate from emotional and instinctual drives and needs, influence the activation power of schemes boosted by mental power. Thus, they contribute to the formation of a tendency of individuals to act in idiosyncratic ways. However, the effects of these factors have not been investigated systematically either by Pascual-Leone or other investigators vis-à-vis the other factors which have been investigated systematically. For this reason, they will not be discussed further in the present chapter. Conclusion In conclusion, Pascual-Leone s theory and research was highly important for a number of reasons. First, it demonstrated that there may be different levels in the organization of human cognition and that a more basic process-free, context-free, and non-logical level may set the limits for what and how many processes can be executed

14 and combined, what reality structures can be represented and understood and, ultimately, what kinds of logical structures can be constructed at a more advanced process-depended, context-depended, and logically meaningful level of cognitive organization. Second, it also presented convincing evidence that there is a developmentally meaningful causal connection between the two levels, such that changes in the more basic level open the way for changes in the more advanced level. Finally, it demonstrated that different traditions or epistemological paradigms in psychology can be brought together to highlight the inter-relations between different aspects of complex phenomena which were studied in isolation within each of the paradigms or traditions. From the point of view of the psychology of cognitive development Pascual-Leone s method of analysis was conducive to the analysis of mental structures in ways that are more empirically testable than Piaget s logical analyses. Important and ground braking as it were, Pascual-Leone s model was criticized on a number of grounds. For instance, Trabasso and Foellinger (1978) reported a study which was contacted to test if Pascual-Leone s values for the k parameter of Mp are indeed as reported by Pascual-Leone. In their study, Trabasso and Foellinger applied a number of stricter experimental and statistical controls for measuring the value of k. According to their results, the k value for children of different ages does not always come as predicted by Pascual-Leone s theory. Variations from the value expected for each age may be due to individual differences in the acquisition of the structural capacity that is supposedly characteristic of each age or to differences in information management strategies. Moreover, Trabasso and Foellinger argued that the assumed Mp-stage-association is misleading because children of the same age do not always operate on the same Piagetian stage whereas children of different ages may operate on the same stage. Thus, one cannot easily draw the neat Mp-stages relationships as maintained by Pascual-Leone. Pascual- Leone (1978) attempted to refute these criticisms by invoking epistemological arguments concerning the validation of the theories. It is a matter of fact, however,

15 that these criticisms seem justified, to a certain extend, because they are based on clear empirical evidence that Mp figures, although in the expected direction, do not come fully as predicted by the theory. According to more recent research, working memory does develop systematically with age (Demetriou, Efklides, & Platsidou, 1993; Demetriou, Christou, Spanoudis, & Platsidou, submitted; Kemps et al., 2000; Mora, 2000). However, the values found vary according to the types of stimuli to be stored and recalled and the conditions of testing (Demetriou, Efklides, & Platsidou, 1993; Demetriou, Christou, Spanoudis, & Platsidou, submitted). These results simply suggest that working memory capacity is not the sole factor that causes the development of thought. Moreover, even if it were fully accurate, Pascual-Leone s model is incomplete in its account of both levels of the mental architecture he proposed. Specifically, his account of the processing system was rather global as it specified and empirically highlighted only one of its parameters, namely its capacity. His account of the higherorder level involving the mental schemes was also incomplete in at least two respects. First, it did not deal explicitly with possible changes in the nature of representation that development may bring about. Second, also it did not deal with the possible differences in the organization of different types of information, such as visual and phonological information, different types of mental operations, such as those employed to deal with causal and mathematical relations or different knowledge domains, such as the domain of social knowledge as contrasted to knowledge about the physical world. In the same line, Pascual-Leone s model did not deal with the possible differences in the demands placed on Mp by different types of representation, mental operations, or knowledge domains. Robbie Case s theory to be presented next attempted to do justice to many of these weaknesses.

16 Robbie Case: The Theory of Executive Control Structures and Central Conceptual Structures Robbie Case integrated into his theory some of the most fundamental assumptions of Pascual-Leone s theory. Specifically, the mental architecture in Case s theory is the same as in Pascual-Leone s theory. That is, it involves two levels, one defined in terms of processing capacity and another one defined in terms of the mental structures that the thinker can build at a given age. Moreover, causality in the relations between the two levels of the mental architecture in Case s theory runs in the same direction as in Pascual-Leone s theory. That is, the capacity of the processing system sets the limits for the kind and the complexity of the cognitive structures that can be constructed at a given age. Thus, changes in the processing system are considered to open new possibilities for the restructuring of thought and problem-solving. However, Case introduced a number of innovations in the analysis of the constructs involved at both levels. These innovations expanded the descriptive and explanatory power of the theory considerably, because they seem able to do justice to the effects of important factors that were rather underestimated in Pascual-Leone s theory. Below we will present Case s theory with an emphasis on those aspects of it that go beyond the theories of Piaget and Pascual-Leone. Structures and Stages in Case s Theory The way cognitive structures are analyzed by Case is much closer to mainstream cognitive psychology rather than to Piaget s or Pascual-Leone s theory. That is, Case analyzed mental structures as systems of goal-directed representations and strategies which are assembled in a particular sequence until to arrive at a solution of the problem at hands. In Case s terms, cognitive structures are to be viewed as executive control structures. By definition, an executive control structure is an internal mental blueprint, which represents a subject s habitual way of construing a particular problem situation, together with his or her habitual procedure for dealing

17 with it. All executive control structures will be presumed to contain at least three components: (1) a representation of the problem situation, that is, a representation of the conditions for which the plan is appropriate, and in which children sometimes find themselves; (2) a representation of their most common objectives in such a situation, that is, the conditions which they desire, and toward whose achievement their plan is directed; and (3) a representation of the strategy they employ, that is, the set of mental steps that they develop for going from the problem situation to the desired situation, in as efficient manner as possible. (Case, 1985, pp ). This conception of comes directly from the classical work of Newell and Simon (1972) on problem solving. According to Case, this analysis of cognitive structures has the advantage that it permits to specify at one and the same time both, the structural or componential composition of what children can represent or do and also the processes which they can plan and execute in order to attain a goal. To exemplify the inter-dependency between the structural and the procedural aspects of mental structures, Case introduced a specific notation. An example of this notation is given here in reference to a rather simple situation. Imagine, for instance, a young infant who wants to grasp an interesting object but in order to be able to satisfy this desire she must first remove another object which stands in the way. PROBLEM SITUATION OBJECTIVES * Interesting object (toy) at X > * Play with object at X * Second object on path to X > * Move second object out of path * Second object within reach at Y * Move hand from Z to Y STRATEGY 1. Move arm in direction Y. 2. Push object at Y with hand,

18 until it moves. 3a. Check availability of object at X. If available, go to 3b; otherwise, return to Step 2. 3b. Move hand toward object at X. 4. When object touches toy at X, pick it up and begin to play with it. It is clear that this structure can be viewed either as a description of the processes that the child needs to go through in order to solve the problem or as a description of the components involved at the various phases of this process. As a process description, the child s engagement with the problem starts with noting the object in her visual field and with the activation of the desire to play with it. The attempt to grasp the object makes the child realize that there is another object which stands in the way. This realization generates a second sub-goal, which is to remove this object in order to attain their main goal. Once this is understood, the problem is psychologically solved because the child knows what steps are required in order to reach the desired final state. Thus, the deployment of strategies evolves in the reverse order from the order of the goals and the relevant representations. That is, the problem starts because a goal is set but the deployment of strategies go from those which are needed to remove the barriers to those needed to realize the ultimate desire. As a structural description, Case s analysis points to two aspects of mental structures. That is, first, it points to the different types of representations, conditions, and processes that they involve. Second, it also points to how complex a mental structure is, as it specifies the number of representations, goals, and strategies that the child must be able to keep in mind and coordinate with each other in order to solve a problem. Stages in Case s Theory

19 Executive control structures are the building blocks of developmental stages. Specifically, Case argued that executive control structures undergo two types of change with the progression of age. That is, structures change in the type of mental units they involve and in the number of units that can be assembled or integrated in a plan of action. Case argued that there are four types of executive control structures: sensorimotor, interrelational, dimensional, and vectorial. Sensorimotor structures evolve in the age phase from 1 to 18 months. The units involved in these structures are perceptions and actions that can be performed on the objects. For instance, the perception of an object may cause a desire (e.g. I want to hold this object in my hand) which in turn activates the actions that can satisfy the desire (e.g. extend hand towards the object and grasp when hand touches on the object). Interrelational structures evolve in the period from 1-1/2 to 5 years. The basic units in these structures are simple relations between actions or representations. For instance, children at this phase have a basic understanding of relational systems such as balance beams. Thus, they can understand that the arm of the beam will tip to the side which has the larger number of weights on it. Dimensional structures evolve in the period from 5 to 11 years. Dimensional structures involve relations between the relations that can be represented at the preceding stage. Thus, they constitute dimensions in the sense that representations or concepts about the same reality are placed along a continuum which provides information about their variations, such as increases or decreases. As a result, at this phase, intra-dimensional relations can be grasped easily and there also is a grasp of global relations between dimensions, such as the relation between weights and their distances from the fulcrum in the balance. Vectorial structures evolve in the period from 11 to 19 years. In this period the relations between dimensions are fully elaborated so that complex relations of covariation, such as those involved in proportional relations, can be grasped. Note that the four types of structures correspond to the stage of sensorimotor, preoperational, concrete operational, and formal operational stage of Piaget s theory, respectively.

20 Development within each of these four main stages evolves along a sequence of four levels or substages, namely the level of (1) operational consolidation, (2) unifocal coordination, (3) bifocal coordination, and (4) elaborated coordination. As implied by their names, structures of increasing complexity can be understood or assembled at each of the four levels. Successive stages are not unrelated, however. According to Case, the final level of a given stage is at the same time the first level of the stage following. Thus, development is recycling in Case s theory. That is, when the structures of a given stage reach a given level of complexity (which corresponds to the level of elaborated coordination) a new mental unit is created and the cycle starts up from the beginning. At the level of operational consolidation the basic unit of a given stage gets formulated and established. For instance, at the first level of operational consolidation in the sensorimotor stage, infants strive to become able to control their senses and the movements of the related parts of the body, such as the movements of the head, for the sake of staying into contact with interesting stimulation. Or, alternatively, they strive to control the movements of their hands in order to activate the sucking reflex by putting their hands in their mouth. However, they cannot coordinate such units with each other for the sake of attaining results that require the coordinated application of both of them. In one of the tasks used by Case to study development during the sensorimotor stage, a simple balance beam was devised that can be operated by young infants. This beam involved a flat arm which can go up and down if pushed. Under this end a bell was fixed which produced a sound when hit by the arm. Infants at this level can look at the arm producing the sound and re-fixate their eyes correctly on the arm when this is moved by the experimenter but they cannot strike the arm themselves in order to produce the sound. This coordination is effected at the second level of unifocal coordination. At this level, the infant can strike the arm at one end in order to make a bell, which is under this end to ring. This is so because, according to Case, at this level the child can represent situations involving two integral components, such as the sound and the

21 movement of an adult which caused the sound. This representation leads to the formulation of a two-parts objective (e.g. (1) re-initiate pattern of movement (2) by repeating the same movement as the experimenter), which, in turn, suggests the strategies to be used (e.g. (i) move arm from present position to position X and (ii) strike). At the next level of bifocal coordination the bell is fixed above the end, which is opposite to the one that must be pushed down. Thus, although the movement to be produced is the same, the infant needs to integrate an extra pointer in her representation of the situation and of the objectives. Specifically, it needs to be understood that the sound is not produced by the end pushed down but by the other end when it goes upwards. Thus, the objectives at this level are as follows: (1) ring bell at other end; (2) re-initiate movement at other end; (3) move hand to beam, at x. This adds up to the strategies the extra need of monitoring what is going on at the other end, together with the actions involved in the behavior of the preceding stage. Finally, at the level of elaborated coordination infants can build complex sensorimotor structures which have the characteristics of reversibility in action. That is, in the task addressed to this level the bell is under the other end. Thus, if it is to make it ring, the infant needs to understand that the downward movement of the end that hits the ring is caused by pulling the other end up. Thus, the task at this level involves four objectives: (1) ring bell at other end; (2) make beam go down at other end; (3) pull this side of beam up; (4) move hand to beam at point x. Thus, the task at this level requires from the infant, in addition to the strategies needed at the previous level, to monitor the direction of the arm movement. This structure is already interrelational because it involves relations between different and opposing actions which are integrated to produce a desired result. When attained, they are equivalent with the level of operational consolidation of the interrelational stage. When consolidated, these structures can be coordinated in pairs. Thus, at the stage of operational coordination the child can understand how the movement of the beam is blocked by a block placed under it so that he first removes

22 the block and then pulls the beam up in order to make the bell ring at the other end. At the subsequent level of bifocal coordination the child understands all of the aforementioned relations and in addition understands how to use a peg to move the arm. Finally, at the level of elaborated coordination, children understand how the upward-downward movement of the two sides of the arm can be produced by using a light weight on the one side and a heavy weight on the other side. The attainment of this structure already implies that weight is represented as a dimension. Thus, the interrelational elaborated coordination corresponds to the first level of the dimensional stage, namely dimensional operational consolidation. In a normal balance beam which can balance by various combinations of weights which can be hanged from various distances from the center, children at this level can examine each side in order to specify which one looks heavy and which one looks light and predict which side will go down accordingly. At the next level of operational coordination the child counts the weights on each side of the balance and picks the side with the larger number of weights as the one to go down. At the next level of bifocal coordination, the child takes into account the distances as well. Thus, when the number of weights on the two sides are (about) equal, the child picks the side with weights at greater distance from the center as the side to go down. Finally, at the level of the elaborated coordination, both the weights and the distances on each side are counted and their multiplicative effect is noted before it is predicted which side will go down. Obviously, this structure is already vectorial as the forces on each side are conceived as products of two factors (i.e. weights and distances) which are interrelated. Therefore, this structure is considered equivalent to the first level of the vectorial stage. At the vectorial operational, bifocal, and elaborated coordination adolescents become able to apply increasingly complex mathematical procedures in order to determine the relative torgues exerted on each side of the beam. Vertical and Horizontal Structure

23 Remember that one of the criticisms leveled against Piaget s theory was related to its failure to explain satisfactorily variations in performance on tasks whose solution was supposedly dependent on the same underlying structure. According to Case and Pascual-Leone, these problems come from the fact that in Piaget s research the operations necessary to solve the tasks and the task-content are frequently confounded. As a result, one cannot be sure if variations in performance are caused by differences in the operations involved or by the content of the tasks. Thus, to test if development at successive levels of representation evolves through the same sequence of levels, as supposed by the model of recycling development proposed by Case, two requirements need to be met. First, one needs to keep the content of the tasks the same across all stages. This is the reason that the same balance beam task was adjusted to be appropriate for testing sensorimotor, preschool, school age children, and adolescents. Second, it needs to be possible to structure the tasks in such a way so as to tap the representational level of each developmental phase. Third, within each level, it must be possible to systematically vary the complexity of the tasks in terms of the goals and strategies required for their solution. It is only under this condition that the hypothesis about the formal equivalence between the sequence of substages involved in each of the main stages can directly be tested. This is the reason that a single task, the balance beam task, was employed to specify the levels of development from birth to maturity. To test that the level sequence holds over different contents one needs to proceed the other way. That is, one needs to construct tasks representing different domains but which are so structured as to scale along the same sequence of levels. In this case, if the level sequence is, as assumed by the theory, a staircase whose levels are defined and constrained by the constructional characteristics of the human mind, then all of the tasks representing a given level would have to be solved or failed across the board and independently of their content. Case presented evidence in favor of this interpretation. That is, Case showed that the sequence of levels described

24 above holds for the development of mathematical thought, spatial thought, social thought, and drawing (Case, 1992). Nature, Functions, and Development of Processing Capacity in Case s Case used the term executive processing space to refer to processing capacity. Case believes that processing capacity involves two integral components, namely operating space and short-term storage space (STSS). The operating space refers primarily to the operations that need to be performed by the subject in order to have the problem goal attained. The STSS refers to the maximum number of mental schemes that the thinker can focus on at a single centration of attention. Specifically, in Case s theory STSS involves pointers or, in more simple language, cues that help remind the person of the products of the operations already executed or of the operations to be executed at the next processing steps. An example is when one has to count how many elements are involved in several groups of objects and at the end recall all values found. In this example the operation of counting as such occupies the operating space component of the total processing space and the values found as a result of counting occupies the STSS. Attention is drawn to Case s view that the contents of the operating space change with development. That is, at each of his four major stages of development the operating space is occupied by sensorimotor, relational, dimensional, and vectorial operations. These are the operations discussed in the previous section. In other words, the development of processing capacity recycles through the same stages already described above in relation to the structure of problem solving skills and processes. It is also notable that Case believes that total processing space does not change with development. At the same time, however, he does not deny that both the operating space and the STSS do change with development. In fact, he proposed that the execution of the operations occupying the operating space becomes more efficient and thus less demanding of processing resources with age. As a result, of the total processing space available, more space is left free to be used for the remembering of

25 mental schemes representing either the results of the application of operations or of more objectives in regard to a situation. In fact, according to Case, this inverse tradeoff between operating space and STSS results in three changes in each main stage which correspond to the three substages or levels already described. That is, the total processing space available at the substages of operational consolidation, unifocal coordination, bifocal coordination, and elaborated coordination is 1, 2, 3, and 4 schemes, respectively Case (1985) presented extensive evidence in support of his position. To show that the development of STSS recycles through the same levels across the four main stages of development he and his colleagues devised STSS tasks appropriate for each the four different stages of development described above. That is, the tasks addressed to each of the four stages required from the child to keep in mind sensorimotor actions, such as seeing or grasping, relational representations, such as words or mental images, dimensional representations, such as numbers, and vectorial representations, such as ratios of numbers. For example, in the test addressed to the dimensional STSS children are shown sets of cards on each of which there are several dots. They are instructed to count the dots on each card as soon as it appears, keep the number found in memory, and recall all numbers when the last card of a set is presented. Children are usually presented sets which involve 2 to 7 cards. Thus, they have to store in memory and recall 2 to 7 digits which stand for the number of dots counted on each card of the successive sets. The test addressed to the vectorial STSS is similar in all respects but one. That is, there are dots of two different colors on each card, for example 6 green dots and 2 red dots, and the child is asked to find out how many green dots there are on the card for each red one and store in memory the result of this operation. Thus, the operation of counting in the first test is dimensional because it involves one kind of elements which can be placed along a single dimension. It is vectorial in the second test because it has to integrate two different dimensions into one.