The differential effects of age and first grade schooling on the development of infralogical and logico-mathematical concrete operations
Introduction
Piaget and Inhelder (1969) describe middle childhood, specifically 7–12 years of age, as the phase of concrete operations. The essence of the move from the sensorimotor stage to that of concrete operations is a shift from action to thought (Davies, 1999, p. 316; italics in the original). Piaget viewed concrete operations as a major turning point in cognitive development (Piaget & Inhelder, 1969). When children attain this stage, their thought bears a much closer resemblance to that of adults than to the preoperational child: it is flexible, organized and logical (Berk, 1991). The 5–7 shift is also marked by considerable psycho-physiological changes: although brain maturation proceeds steadily throughout childhood, there appears to be a spurt between 6 and 7, which correlates with a number of changes in perceptual and cognitive abilities that appear at about age 7 (Case, 1985; Janowsky & Carper, 1996).
Specification of the causal model underlying the acquisition of concrete operations (e.g., Piaget, 1972) is particularly challenging. A central issue in this respect regards the relative contributions of chronological age (i.e., maturation and the associated accumulation of experience) and schooling to this developmental process in Western cultures, where the acquisition of concrete operations, typically around age 6 (the 5–7 shift), coincides with the beginning of schooling. Unsurprisingly, this issue – a special instance of the broader nature vs. nurture controversy – has been the focus of a longstanding debate in developmental literature (Christian, Bachman, & Morrison, 2001).
According to the Piagetian approach, schooling, like any other experience, is expected to promote the development of concrete operations if it provides children with appropriate operative exercises (Laboratory of Comparative Human Cognition, 1983). A debate has thus focused on the school's ability to provide such experiences. Researchers such as Goodnow and Bethon (1966) and Kiminyo (1977) suggested that since schooling does not provide children with direct experience with the environment it may slow down the acquisition of concrete operations. In contrast, others (e.g., Cole & Bruner, 1971) have suggested that schooling promotes the acquisition of concrete operations by increasing children's analytic attention to perceptual features of operational tasks, by providing them with conceptual schemes (including specialized language that makes distinctions critical to performance) and by suppressing alternative explanations of the transformations inherent in the concrete operational tasks (e.g., “action magic”).
The neo-Piagetian approach (e.g., Case, 1998) is more specific regarding the role of schooling in cognitive development. This approach considers culture, and particularly schooling, as a powerful factor responsible for producing the pattern of conceptual development commonly observed during childhood in modern societies. However, the neo-Piagetian approach does not address the issue of the possible variability among concrete operational tasks with respect to their susceptibility to the effects of cultural factors in general, and of schooling in particular.
In contrast, differential predictions regarding the relative effects of age and schooling on performance in concrete operational tasks may be formulated on the basis of the theoretical approach suggested by a French-Swiss team of researchers (Larivee, Normandeau, & Parent, 2000), which we will refer to as the “francophone” approach. This approach distinguishes among operational tasks in terms of the relevance and efficiency of different information processes to their successful solution. Specifically, hypotheses about different processes at play in the Piagetian tasks rely on two distinctions:
- (a)
Piaget's distinction between logico-mathematical (LM) and infralogical (IL) operations (Piaget & Inhelder, 1947). LM operations deal with relations of similarity and difference between discrete objects (e.g., classification, seriation, number); IL operations refer to the relation between the object itself and its parts, and include notions of space, time and conservation of quantities. The literature (e.g., deRibaupierre & Rieben, 1995; Lautrey, de Ribaupierre, & Rieben, 1985) distinguishes between three main types or domains of IL operational tasks:
- (1)
The physical domain, including conservation of substance, weight and volume (Piaget & Inhelder, 1941) and Islands or construction of volumes (Piaget, Inhelder, & Szeminska, 1948).
- (2)
The spatial domain, including sectioning of volumes and unfolding volumes tasks (Piaget & Inhelder, 1947).
- (3)
Mental imagery, including folding of lines and folds and holes tasks (Piaget & Inhelder, 1966).
- (1)
- (b)
The distinction between a propositional and an analogical mode of representation (e.g., Lautrey et al., 1985). This distinction—like that between intuitive and formal thinking (Globerson, 1989), between automatic and controlled processes (Schneider & Shiffrin, 1977), or between Realization and Formalization (Reuchlin, 1973)—is rooted in William James’ (1890/1950) suggestion that human reasoning involves two distinct processing systems: one that is quick, effortless, associative and intuitive (i.e., System 1) and another that is slow, effortful, analytic and deliberate (i.e., System 2) (e.g., Alter, Oppenheimer, Epley, & Eyre, 2007; Evans, 2003; Kahneman & Frederick, 2002). Although not without controversy (see Kruglanski & Thompson, 1999; Osman, 2004), dual-process theories have been used widely by developmental, cognitive and social psychologists (Alter et al., 2007).
Analytical processing is typical of stimuli composed from “separable” dimensions (“separable stimuli”) and holistic processing is typical of “integral” dimensional combinations (“integral stimuli”; Kemler Nelson & Smith, 1989). At issue is whether a multidimensional, multicomponent stimulus is differentiated into its constituents properties or whether it is treated instead as an undifferentiated unitary whole:
For separable stimuli, dimensions are the entities on which processing operates. The effective stimulus is the concatenation of these separate properties. But for integral stimuli (…), the components have no immediate psychological status. Instead, stimuli composed of integral dimensions are organized by their overall similarity relations (which are influenced by, but not processed in terms of the constituent dimensions). Accordingly, the critical way to distinguish analytical and holistic processing is to ask whether processing is structured by component properties (as it is when the properties are differentially attended or weighted) or organized by overall similarity relations, directly apprehended, between the stimuli as wholes (Kemler Nelson & Smith, 1989, 117–118).
The distinction between a propositional and an analogical mode of representation may enable understanding of the different processes at work in the two types of operational tasks:
In a propositional mode of representation, relations between objects and representations are arbitrary. The different units are usually assembled through rules that are extrinsic to the representation (e.g., logical rules). The nature of a propositional mode is analytical or separable … and processing is likely to be sequential. This mode is therefore particularly adapted to solving LM tasks. In contrast, the analogical mode is more global or holistic (integral) and embodies, in a single representation, units of information and their spatio-temporal relations. It maintains a certain isomorphism between the external events and their representation, which makes it a likely candidate for solving IL tasks.1 (deRibaupierre & Rieben, 1995, p. 6; italics in the original)
According to the “francophone” approach, therefore, there is an interaction between type of operational task and processing mode: a propositional (i.e., analytical or formal) mode is more adequate for dealing with LM tasks, whereas an analogical (i.e., intuitive or holistic) mode is more adapted to IL tasks. Furthermore, because, according to this approach, formal schooling in modern societies consistently grants a more important role to analytical or propositional processes (deRibaupierre & Rieben, 1995, p. 7), whereas out-of-school experiences typically involve analogical representations (Larivee et al., 2000), the interaction between type of operational task and processing mode leads to differential predictions regarding the sensitivity of IL and LM tasks to school and out-of-school experiences: successful performance on LM tasks (which rely on the propositional mode) is mainly promoted by schooling, whereas performance on IL tasks (which rely on an analogical mode) develops mainly through everyday experience (see footnote 1).
Section snippets
The empirical evidence
To the best of our knowledge, this hypothesis has not been explicitly examined to date. In fact, the only relevant and valid, however partial and indirect, empirical evidence comes from two studies performed in the last decade, which, relying on an entirely different theoretical approach, examined the effects of age and schooling on the development of quantitative skills between ages 5 and 7. The theoretical approach underlying the first study (Bisanz, Morrison, & Dunn, 1995) attributes
Purpose of the present study
We suggest that additional investigation of the differential effects of age and schooling on the acquisition of LM and IL concrete operational tasks in modern societies is necessary at this point. Such investigation should involve additional IL and LM concrete operational tasks, particularly LM tasks that are not explicitly taught in school, different and possibly improved methodologies, and different populations. Our study is one step in this direction. The main purpose of the study was to
Participants
The sample consisted of all (580) first and second grade students, aged between 5 years 10 months and 7 years 10 months, attending five elementary schools in which Hebrew is the language of instruction, in Jerusalem, Israel. The schools were selected so as to represent all SES levels. For methodological reasons the sample included only the students born between January and October of the appropriate year for their grade.2
Results
Because the study was performed at the beginning of the school year in grades 1 and 2, we refer to first grade students as kindergarten graduates and to second grade students as first grade graduates. That is, the schooling difference between the two groups is 1 year of schooling in the first grade.
Table 4 presents the estimated effects of 1 year of schooling in grade 1 and an average of 1 year of chronological age (in kindergarten and grade 1) on each task. Both effects are expressed in S.D.
Discussion
The results of this study point to a clear interaction between type of operational tasks and the effects of chronological age and schooling on their successful completion: the development of performance on class inclusion and transitivity, and to a lesser extent classification (clearly LM operational tasks) is mainly attributable to schooling, whereas acquisition of conservation (a prototypical IL task) is almost entirely due to age. The consistency of the results across the three LM tasks and
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