Commentary on: Piaget s stages: the unfinished symphony of cognitive development by D.H. Feldman

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1 New Ideas in Psychology 22 (2004) Commentary on: Piaget s stages: the unfinished symphony of cognitive development by D.H. Feldman Peter C.M. Molenaar, Han L.J. van der Maas Department of Psychology, University of Amsterdam, 1018 WB Amsterdam, Netherlands Available online 25 January 2005 Dr. Feldman s paper constitutes a worthy continuation of the select pedigree of thorough neo-piagetian theory constructions. It presents a general theory of stagewise cognitive development, in the spirit of Piaget s naturalized epistemology. Theory construction at the general level at which Piaget operated has, for good reasons, an essential place in cognitive developmental psychology. In particular theories of stagewise cognitive development have wide-ranging theoretical, methodological and practical implications. Any theory of stagewise development involves the specification of stages, periods of relatively stable performance of the developing system. Each stage is characterized as a re-identifiable qualitative whole, having a unique identity that can be named (by means of attaching labels). In this sense a stage is akin to a latent class in psychometrics (cf. van der Maas, 1998). In addition the theory should specify the sequence or sequences of allowed stage transitions. The duration of a stage transition is always negligible relative to the duration of stages (sudden transition). Hence stagewise development as described up to this level consists of a sequence or sequences of punctuated latent class transitions. In this sense stagewise development is akin to the hidden Markov process model in psychometrics (cf. Visser, Raijmakers, & Molenaar, 2000). Corresponding author. address: p.c.m.molenaar@uva.nl (P.C.M. Molenaar) X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi: /j.newideapsych

2 250 P.C.M. Molenaar, H.L.J. van der Maas / New Ideas in Psychology 22 (2004) Often this interpretation of stagewise development in terms of hidden or latent Markov models has been taken to be definitive (cf. Brainerd, 1978). But it does not capture the real gist of stagewise development. Our hidden Markov model representation of stagewise development obtained above belongs to the class of generalized linear models (GLMs) that can very well-describe sudden changes in the dynamics of an ongoing process by representing these as the result of arbitrary continuous parameter variation (for instance, by varying the momentary value of parameters according to a step function), but it fails to explain the sudden changes between stages (Jansen & van der Maas, 2001). The essential ingredient of stagewise development that sets it apart from all other developmental process models is self-organization. The stages and their transitions are not only described, but their emergence and identities are explained from a unified dynamical perspective. The developmental process, once defined and started up, spontaneously organizes its dynamics into a sequence of sudden transitions between long stretches of structurally stable performance (stages). Within each stage, the structure of attractors (including equilibria) stays qualitatively the same (hence the number and types of attractors stay invariant within each stage and can be regarded as the formal label of each stage). A stage transition involves the sudden re-organization of the qualitative structure of attractors characterizing the previous stage into a qualitatively different structure (different number and/or types of attractors) characterizing the following stage. The occurrence of a stage transition is not due to external influences but emerges as the result of the dynamical properties of the ongoing developmental process itself. To elaborate somewhat further the essential differences between stagewise development due to self-organization on the one hand and GLM process models on the other hand, consider how the transition from some stage A to stage B is represented under both approaches. According to the theory of stagewise development, the transition occurs because the momentary dynamical structure A of the ongoing developmental process becomes unstable due to small continuous changes of process parameters. This lack of stability, resulting in a sudden transition to stage B, as well as the qualitative organization of the emergent stage B, are consequences of the nonlinear dynamical laws underlying the developmental process (van der Maas & Molenaar, 1992). In contrast, the transition from stage (latent class) A to stage (latent class) B is represented in a GLM by arbitrary large, ad hoc parameter changes (cf. Raijmakers, 2004). Such parameter changes could be interpreted as the effect of new resources coming available due to, e.g., genetical influences. But the important point is that a GLM-based analysis takes the stages and the transitions between them as given, and consequently such an analysis cannot explain the emergence and timing of qualitatively new stages. This distinction between descriptive and explanatory theoretical models is equivalent to the same distinction made by Brainerd (1978) in his criticism of stagewise development. The irony is, however, that it is Brainerd s latent Markov model that is descriptive, whereas the model of stagewise development due to self-organization is explanatory (cf. Molenaar, 1987).

3 P.C.M. Molenaar, H.L.J. van der Maas / New Ideas in Psychology 22 (2004) We agree with Dr. Feldman s point of view concerning the importance of stagewise development in Piaget s theory of cognitive development. In fact, stagewise development due to self-organization constitutes a distinct theoretical model, the application of which is not limited to cognitive development. For instance, stagewise development plays important roles in advanced mathematical biological modeling, in particular in the field of biological pattern formation (e.g., Murray, 1989). It has important theoretical implications, one of which is that self-organization creates irreducible variation, i.e., logically independent from genetical and environmental influences (Molenaar, Boomsma, & Dolan, 1993). This so-called third source of phenotypic variation has been shown to explain higher proportions of variation than genetical and environmental influences in quantitative genetical studies and consequently its existence puts an upper limit to the fidelity of cloning. In addition, stagewise development due to self-organization has important practical consequences. For instance, the effects of interventions will be different during a phase of high stability in comparison with their effects during a phase of low stability close to a transition. But what about methodology? Even the most solidly rationalized theoretical exercise has to be validated against empirical data, hence the question arises how one can subject stagewise development to empirical testing. Given the complexity of the theoretical model of stagewise development due to self-organization, it will be obvious that subjecting it to appropriate empirical testing is not an easy task. We have derived a principled methodology for empirical testing of the presence of stage transitions (van der Maas & Molenaar, 1992; Hartelman & Molenaar, 1999). In its most basic form this methodology involves the detection of special characteristics associated with genuine stage transitions (so-called catastrophe flags, where catastrophe is the mathematical term for stage transition). These characteristics include the occurrence of a sudden transition, an increased vulnerability to interventions, longer durations before equilibrium is re-established after intervention, etc. This new methodology has been successfully applied in various empirical studies (e.g., Hosenfeld, van der Maas, & van den Boom, 1997; Wimmers, Savelsbergh, van der Kamp, & Hartelman, 1998; Jansen & van der Maas, 2001; van Rijn, Someren, & van der Maas, 2003). The question how our methodology for the detection of stage transitions can be generalized to accommodate stochastic influences has proven to be difficult. The situation has unfortunately been complicated by the introduction of approaches that can be proven to be fundamentally flawed (like the technique proposed by Guastello). Only Cobb s maximum likelihood technique is based on sound statistical argumentation, although its implementation requires special care (e.g., Wagenmakers et al., 2004, in press). An improved computer program for the maximum likelihood fit of the so-called cusp transition model can be obtained from the authors on request. Presently this maximum likelihood approach is generalized to accommodate single-subject time series. We conclude that stagewise development involves a distinct (nonlinear dynamical) paradigm with special theoretical, practical and methodological implications. It is based on recent advances in the mathematical and physical sciences and compatible

4 252 P.C.M. Molenaar, H.L.J. van der Maas / New Ideas in Psychology 22 (2004) with well-known instances of stagewise development like the transition from a liquid to a gaseous phase (stage) in thermodynamics. In this paradigm transitions constitute pivotal events of special interest, because they define the emergence and qualitative structure of stages. Only at a stage transition can the number and types of attractors of a developmental system change. Given that, as we indicated above, the number and types of attractors define the formal nature or character of a stage, and given that during a stage this formal character stays invariant, it follows that stage transitions only can occur at the beginning of a new stage. In contrast, Dr. Feldman locates stage transitions at the midpoint of each stage. The postulation that stage transitions are located at the midpoint of each stage may not be acceptable according to the model of stagewise selforganizing development. If this is indeed the case, then is required to construct a new principled methodology with which stages can be identified and transitions can be detected. To the best of our knowledge such new methodology is lacking. Alternatively, one could try to accommodate Dr. Feldman s use of recursive procedures to the model of stagewise self-organizing development, because such recursions (called automorphisms in mathematics) can be represented as instances of the same systems of differential equations that constitute the natural starting point of our catastrophe model of stage transitions. This brings us to our closing remarks in which we would like to place the model of stagewise self-organizing development in a somewhat wider context. We already indicated its successful applications in mathematics, physics and biology. Hence the model is not specifically tied up with all the details of the Piagetian theory of stagewise development. On the other hand, it is possible to apply the model to investigate potential substage transitions of much more limited nature than the main Piagetian stages, thus accommodating Dr. Feldman s appeal to within-stage transitions. The rationale for such more limited applications is evident from the successful attempts to apply the cusp catastrophe model to Gestalt switches in ambiguous figures (Stewart & Peregoy, 1983). Such Gestalt switches, involving alternating re-centerings of the perceptual field, show many of the catastrophe flags mentioned above without implying large-scale qualitative changes of perceptual information processing. The recent spectacular progress in nonlinear dynamics, especially the discovery of several types of the so-called strange attractors, has shown the existence of various alternative kinds of stage transition (commonly called singularities in the mathematical literature concerned). Like any statistical methodology, the fidelity of the model of stagewise self-organizing development has to be investigated (e.g., in large-scale Monte Carlo studies and mathematical analysis) within the wider context of these alternative types of transition. An initial step in this direction has been made by Molenaar and Newell (2003), but much, much more efforts are required. Suffice it to say that these efforts will be well-spent because the whole paradigm of stagewise self-organizing development is based on solid mathematical grounds.

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