Dynamic Systems Models

Dynamic systems models (Fischer & Bidell, 1998; Fischer & Pare-Blagoev, 2000; van Geert, 1991, 1998, 2000) have offered new ways to analyze developmental data. A dynamic system is a formal system, the state of which depends on its state at a previous point in time. The dynamic system model of van Geert (1998) was designed around principles derived from the work of Piaget and Vygotsky and has a number of interesting properties. It can account for different types of cognitive growth, such as slow linear increase and sudden discontinuities, within the same system. It can also show how a complex, self-regulating system can emerge from the interaction of a few variables. The model was fitted to a number of developmental data sets, and some important developmental phenomena, including conservation acquisition, were simulated. Links have also been made between dynamic systems models and neural net models.

Dynamic systems models have also been linked to other issues. Raijmakers, van Koten, and Molenaar (1996) analyzed McClelland's (1995) neural net model of the balance scale and found no evidence of the flags indicating discontinuities that are found in empirical data. They suggest that back-propagation models simulate the type of stimulus-response associations that are characteristic of animals and young children but do not simulate the rule-governed behavior characteristic of older children and adults. In many respects, this finding is consistent with the analysis of the model presented earlier. On the other hand, backpropagation models incorporate learning functions that have been missing from models of higher cognitive processes. As we have seen, they show how structured representations begin to emerge as a result of learning input-output functions.

Although there are acknowledged difficulties with dynamic systems models (van

Geert, 1998), they provide much more sophisticated implementations of important developmental theories, including that of Piaget and Vygotsky. This does not mean that Piaget and Vygotsky are fully vindicated by dynamic systems models, but concepts such as equilibration and self-regulation, which are at the core of their theories, do seem to have a new lease on life. Most importantly, dynamic systems models have potential to deepen our understanding of cognitive developmental processes. And, as Fischer and Pare-Blagoev (2000) point out, there are tools based on Lotus 123 or Microsoft Excel that make dynamic system modeling more accessible.

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