In his 1992 review of Jean Piaget's theory, Harry Beilin compared its influence on the study of cognitive development to that of Shakespeare on English literature. Any theorist who has studied cognitive development in children from the middle of the twentieth century onward has been a neo-Piagetian in that broad sense. However, the subject of this entry is a smaller group of researchers who have called themselves neo-Piagetians. Because the neo-label directs attention back to a theory's origins, most neo-Piagetian researchers eventually chose other names that focused on their theories' new elements. Thus Kurt Fischer's neo-Piagetian theory became known as skill theory (Fischer & Pipp, 1984; Schwartz & Fischer, 2004) and Robbie Case's evolved into central conceptual structure theory (Griffin, 2004).
Neo-Piagetian theorists kept Piaget's idea that the progress of cognitive development is more like climbing a series of stairs (the stages) than walking smoothly up a ramp. They also agreed with Piaget that biological maturation sets broad upper limits on the kinds of thinking children are capable of doing at particular ages. All of the neo-Piagetians also shared Piaget's conviction that children's thinking reflects their developing internal mental structures (Case & Okamoto, 1996). However, each of the neo-Piagetians combined this general premise with ideas about the influence of experience on development that were more analytically specific and more attuned to cultural and individual differences than anything found in Piaget's theory. For example, they considered how biological maturation of the central nervous system during the first years of life increases the speed with which children process information—an idea introduced in 1970 by Pascual-Leone. The neo-Piagetians linked that maturing processing speed to increases in working memory capacity that, in turn, set upper limits on the complexity of problems a child is likely to be able to solve.
The neo-Piagetians also have drawn on information-processing and linguistic theorists' ideas about the contentdomain specificity of cognition. Piaget's theory evolved to include provisions for unevenness in the sophistication of a child's reasoning across different kinds of problems, but this domain specificity is more fundamental to neo-Piagetian theories. The influence of the information-processing paradigm also is evident in the neo-Piagetians' tendency to analyze Piaget's and other cognitive tasks in ways that highlight why one problem may be more or less difficult than another, even within the same content domain. Australian neo-Piagetian Graeme Halford (Andrews & Halford, 2002) became especially well known for his focus on this kind of task analysis.
The neo-Piagetians also adopted principles from social-cognitive theories such as that of Lev Vygotsky. These principles allowed them to give more extensive consideration than can be found in Piaget's work regarding how culturally determined experiences and minute-by-minute interactions with teachers and peers influence a child's intellectual performance. Although biology sets upper limits on performance, a child's culture and everyday
experience might not provide the information and practice needed for development up to that limit. Neo-Piagetian theories include mechanisms to account for cultural and socio-economic differences in patterns or rates of cognitive development. These theories also are compatible with the idea that individual children differ in their ability to absorb the experiences available in their culture.
With their joint consideration of biology, the precise requirements for doing a task, and the contributions of experience, the neo-Piagetian theorists have elaborated on possible mechanisms of developmental change, the how of development that was only sketched in very general terms by Piaget (Case, 1984; Case, 1996; Griffin, 2004; Schwartz & Fischer, 1994).
The neo-Piagetians' interests in mechanisms of development and in optimizing learning offer educators more explicit guidance than is available in Piaget's theory. A number of neo-Piagetians have proposed theories with relevance for classroom practices. However, Canadian psychologist Robbie Case (1945–2000) probably is the neo-Piagetian whose work has had the most influence on education.
Case might be considered the quintessential neo-Piagetian researcher because he dedicated his own theorizing and research efforts to working explicitly within the historical framework of Piaget's ideas. However, Case tried to remedy some of the deficiencies in Piaget's theory by incorporating ideas and methods from other traditions, especially Vygot-sky's social-constructivist theory, information-processing theories, linguistics, and new findings in developmental neuroscience (Case, 1996).
Like Piaget's own grand theory, Case's neo-Piagetian theory is hard to summarize because it developed—changing substantially across the span of Case's own research career and with work that his colleagues continued after his death. In his early work, Case focused on the broad implications of central processing speed and working memory span (e.g., Case, Kurland, & Goldberg, 1982). These ideas were retained in later versions of the theory, but there was more attention to the nature and development of children's mental representations in each of several cognitive domains—the central conceptual structures. In a 1996 monograph, Case and his colleagues investigated central conceptual structures underlying reasoning about number, space, and narrative (Case, 1996). All of these structures were described as going through stages that were labeled pre-dimensional, unidimensional, bidimensional, and integrated bidimensional and that characterized children's thought at about ages 4, 6, 8, and 10 years. Figures 1 and 2 show the central conceptual structures for number hypothesized to be characteristic of middle-class children in a technological society at about age 4 years (Figure 1) and at about age 6 years (Figure 2). Figure 1 shows that typical 4-year-olds, who are in the pre-dimensional stage, have not yet coordinated two ideas about number—the idea of comparisons between smaller and larger quantities and the idea of counting off a set of objects, saying number names in sequence. In contrast, typical 6-year-olds (Figure 2) have a unidimensional central numerical structure that coordinates several ideas along a number line that they can use to do a variety of arithmetic reasoning tasks.
Case and his colleagues (Case, 1996) also described how a sequence of increasingly complete and coherent central conceptual structures might organize children's developing reasoning in other content domains besides number. Although these domain-limited structures were expected to develop more or less in synchrony because they were all affected by common biological limits, the content of each structure also was expected to be influenced by cultural and individual experience, which gave them some independence from one another.
The idea that biological maturation limits the speed with which a child of a given age is likely to be able to process information, or think, is central to Case's interpretation of development. With increasing age, the child's neurons become more extensively coated with a fatty myelin sheath, which speeds transmission of information along the nerve fiber. Another key change with age is that the pattern of connections among the nerve cells in the brain becomes better specialized—unneeded connections drop out and important ones are strengthened with a combination of time and experience. Because older children can think faster than younger children, they can keep more concepts in mind at the same time. Case and his colleagues first demonstrated the link between thinking speed and memory span in a study of counting speed and memory for digits (Case et al., 1982). The principle extends far beyond arithmetic. In any content domain, familiarity and practice can help children process information more quickly and solve more complex problems. However, chronological age and biological maturation set limits on how much can be achieved with practice.
In Case's theory, one of children's developing mental structures is devoted to the representation of space. These spatial structures are reflected in children's drawings, which become increasingly organized along multiple spatial axes as children grown older. A 4-year-old's figure is likely to be floating in space, but a 6-year-old's drawing of a person usually stands on a ground line and may be accompanied by other objects, such as an animal or tree, arrayed along the same horizontal axis. Dealing with the vertical and depth dimensions comes later, and it is not until age 10 years or more that children coordinate all of these dimensions well. Even precociously talented young artists are not likely to represent spatial axes in a way much beyond that typical of their chronological age, although their drawings of each figure may be beautifully detailed and realistic (Case & Okamoto, 1996). Thinking in a coordinated way about multiple dimensions of space also is important in playing games such as checkers, in reading or drawing maps, and in making scientific judgments about relations within a series of objects (Case, 1996).
The ways in which culture might influence the separate development of various central conceptual structures was illustrated dramatically in a study comparing children who had experienced Western technological culture and schooling with those who had not. Case's colleague Fiati found that children who were living in isolated rural villages in the Volta region of central Africa and who had not experienced any Western-style schooling, performed at the same levels as urban children their age on tasks involving the coordination of ideas in a story, a task hypothesized to depend on Case's central conceptual structure for narrative. This showed that there was nothing generally wrong with the children's ability to do complex thinking. However, life in the village had offered very little experience with numerical tasks, and the children's performance on tasks involving Case's central conceptual structure for numbers (see Figure 1) was immature. In less extreme cross-cultural comparisons, Case and his colleagues found that central conceptual structures developed at about the same rate for children in the United States, Canada, China, and Japan. However, socio-economic differences within a culture could be associated with large differences in developmental rate (Case, 1996; Case, Griffin, & Kelly, 2001).
Although some of Case's early theorizing concerned development in infancy and the toddler years (Case, 1985), much of his work was concerned with how children's thinking changes between about age 4 and age 10 or so (Case, 1996). This makes his work especially relevant for educators working with children in preschool and elementary school.
A major implication of Case's work, consistent with the implications of information-processing theories, is that young children should not be expected to think about too many new ideas at once. For example, typical 6-year-olds are much more likely to give the correct answer to the problem “6 + 2 = ?” than they are to the related problem “6 + ? = 8”, which is expressed with a missing addend. To solve a missing addend problem, a child needs to understand several unfamiliar symbols. Case (1978) found that children were more successful with missing addend problems if they were gradually introduced to the non-numerical elements of an equation before doing the arithmetic. Throughout his career, Case was concerned with analyzing exactly what a child needed to know and manipulate in order to solve common school problems and with how age and cultural experience may have left some children unable to meet classroom expectations (Case, Griffin, & Kelly, 2001).
When everyday experience has not helped children develop a particular kind of central conceptual structure, carefully planned instruction has been shown to help children catch up. One of Case's colleagues, Sharon Griffin, developed a compensatory education program for first graders from low-income families that she called Right-start (Griffin, 2004; Griffin, Case, & Siegler, 1994). The goal of this program was to give these children lessons that would fill in gaps in their central conceptual structure for numbers (Figure 2). Children in the program and those in a control group were pre- and posttested using exams containing items influenced by Case's theory, and those who had been in the Rightstart program did better on the posttest.
Case's theory also has contributed to the design of more effective instruction on problems that are likely to be difficult for all children, regardless of their family background, such as understanding rational numbers. Rational numbers are percentages, decimals, and fractions. These topics usually are introduced in the late elementary years and often are not mastered fully even by adults. Moss and Case (1999) found that fourth graders achieved a deeper understanding of rational numbers in a curriculum that focused on step-by-step teaching of a conceptual structure for these numbers. The curriculum started with percentages, using exercises that involved observing and manipulating liquids in beakers and other objects that were partly full. Discussion at this point was all in terms of percentages. Then children were introduced to decimal notation using large number lines on the floor, and links between decimal fractions and percentages were emphasized. Finally, traditional fractional notation was taught and linked with the other two forms of rational number notation. Children who experienced this curriculum had a deeper understanding of rational numbers than children in a control group.
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