A dynamic systems model of cognitive and language growth.

“Van Geert, P. (2009) Complex dynamic systems of development.”

“To combine these two developmental curves, Robinson and Mervis use a PRECURSOR MODEL as proposed by Van Geert (1995). In a precursor model, there are two variables, a predecessor and a successor.”

“In the model of van Geert (1991, 1994) the carrying capacity (Ki) of one process is made dependent on the growth (x) of other processes (so-called bootstrap dynamics).”

“van Geert, P. (1991). A dynamic systems model of cognitive and language”

“This is the type of pattern frequently seen in the acquisition of language skills as well as other cognitive growth processes (VanGeert, 1991).”

“That is, instead of the normal Verhulst equation, X^{ = r Xn (1 - Xn), the equation X^{ = r X, (1 - X\) + N must be used, van Geert (1991) already gave an interpretation for the additional variable N.”

“As the first few words are acquired, these exert greater leverage, allowing the rate of acquisition to increase (cf. Elman et al., 1996; van Geert, 1991).”

“They found that the precursor mechanism proposed by van Geert (1991) fit their data.”

“…when such methods address intersectionality (Fischer & Kennedy, 1997; Hartelman, van der Maas, & Molenaar, 1998; Singer & Willett, 2003; van Geert, 1991, 2003) and do not generalize findings to all human beings or all members of a group (M. Cole, 1996, Fischer & Bidell, 2006; Ghavami,…”

“Logistic growth equations can take several different forms, and van Geert (1994) recommends the following as the best starting point for modeling hierarchical growth for an action, thought, or emotion, designated as Grower B:”

“, & van Geert, P. (1998). Jumping into syntax:”

“Even Elman et al. (1996), who argued that the vocabulary spurt may not reflect any underlying cognitive change, assumed that there is an observable spurt in the rate of word learning (see also Bates & Carnevale, 1993; van Geert, 1991).”

“…in developmental systems, it is the hallmarkexplicit purpose of studying the structural profile of developmental transitions (e.g., Fo- principle in DS approaches to development and provides the foundation for a group ofgel & Thelen, 1987; Lewis, 2000; Thelen & Ulrich, 1991; van Geert, 1991, 1994).”

“Their model, based on van Geert (1991), says that a certain level of vocabulary is a precursor for the emergence of plural forms and that the rapid growth of plurals has a temporary slowing-down effect on the growth of the vocabulary.”

“This period of rapid growth is referred to as the 'vocabulary spurt' or 'naming explosion' by developmental psycholinguists (Dromi, 1987; McCarthy, 1946; Nelson, 1973; Goldfield & Reznick, 1990; Van Geert, 1991; Bates & Carnevale, 1993; Elman et al., 1996).”

“Transitions in motor, cognitive, and linguistic development have been modeled by the application of DS concepts that are particularly concerned with the structural profile of change (Case et al., 1996; van der Maas & Molenaar, 1992; van Geert, 1991, 1994).”

“Codependency between two mutually reinforcing processes has been studied in formal theories of growth and change (van Geert, 1991).”

“This dynamic growth model has been thoroughly explained elsewhere ( Van Geert, 1991,”

“…(cf. Port & van Gelder, 1995), developmental psychology (e.g., Fischer & Bidell, 1997; Levine & Fitzgerald, 1992; Thelen& Smith, 1994; van Geert, 1991), organizational behavior (Axelrod & Cohen, 2000; Guastello, 1995), and political sociology (e.g., Axelrod, 1984; Nowak & Vallacher,…”

“Rather, " the more we have studied learning abilities, the more impressed we have become with their specificity " (Trivers, 1985, p. 102).”

“…and Dynamic Systems Theory (de Bot, Lowie, & Verspoor, 2007; Ellis, 2007, 2008a; Port & Van Gelder, 1995; Spivey, 2006; Thelen & Smith, 1994; van Geert, 1991) as they apply to usagebased theories of language (Barlow & Kemmer, 2000; Bod, Hay, & Jannedy, 2003; Bybee, 2005; Bybee & Hopper,…”

“…have been articulated in areas as diverse as visual perception Brunswik, 1956), reading (Seidenberg and McClelland, 1989), social psychology (Kelley, cognitive development (Anderson, 1981), infant attachment (van Geert, 1991), motor (Carlson et al ., 1989), and auditory perception (Massaro, 1987).”

“The first type of models have been introduced by Van Geert (van Geert, 1991, 1994a, b, 1998a, b, 2003) The second type of model is typical of the work of Thelen, Smith, Spencer and others (Smith & Thelen, 1993; Spencer & Schöner, 2003; Thelen & Smith, 1994).”

“…systems are recursive in nature, involving an iterative process whereby the product (system state t + 1) of the equation’s initial run feeds back to the equation as a new initial system state to produce in turn a later system state, t + 2, and so on [van Geert, 1991; van Geert & Steenbeck, 2005].”

“…of many dynamic systems approaches is to formally implement developmental processes to shed light on how behavior changes over time (Spencer, Blumberg, McMurray, Robinson, Samuelson, & Tomblin, 2009; van der Maas & Dolan, 2006; van der Maas & Molenaar, 1992; van Geert, 1991, 1998; Warren, 2006).”

“…of mechanisms into dynamic systems models has intrigued me for a long time, and together with my collaborators I have tried to model several types of such phenomena (see, e.g., Kunnen & Bosma, 2000; Steenbeek & van Geert, 2008a, 2008b; van Geert, 1991, 1994, 1998; van Geert & Steenbeek, 2005).”

“Language development may likewise be characterized by sudden jumps in understanding ( Van Geert, 1991 ), for instance when children start mastering grammar.”

“(p. 347) What is true for cognition has also been shown to be true for language development (de Bot et al., 2007; Larsen Freeman, 1997; van Geert, 1991).”

“This is an example of uncertainty about the exact meaning of observed data described by van Dijk and van Geert (2005), who recommend a particular type of sensitivity analysis in such cases.”

“The choice of the equations was based on the dynamic growth theory and the dynamic growth models developed by Van Geert (1994). The development of living systems in general is well described by means of the so-called logistic growth equation.”

“…the evolution rule can be simplified to a mathematical expression, for instance in the form of a differential or difference equation (e.g. Van Geert, 1991, 1994), but sometimes the dynamics are simply too complex to be represented by such Dynamic Systems Perspective 155Human Development…”

“They found that accelerations in lexical growth (i.e., the vocabulary spurt) were coincident with the onset of word combinations and that lexical growth tended to decelerate with continued growth in word combinations (see also Dromi, 1987; van Geert, 1991 ).”

“Since disengagement is the result of competition between processes which maintain the focus of attention and gaze, and those which interrupt and shift it, changes in its frequency can be expected to be particularly abrupt and rapid ( van Geert, 1991 ).”

“Exemplars are similar if they have many features in common and few distinctive attributes (features belonging to one but not the other); the more similar are two objects on these quantitative grounds, the faster are people at judging them to be similar (Tversky 1977).”

“Such S-shaped learning curves are predicted by the theoretical models of Hull (1943), Klopf (1988), van Geert (1991), and Newell et al. (2001).”

“…more cognitive based approaches (Peters & Menn, 1990; Schlesinger 1988; Slobin, 1973, 1997), dynamic systems theory (Thelen & Smith, 1994; Van Geert, 1991), schema-based approaches in morphology (Dressler, Kilani-Schoch & Klampfer, 2003) and probabilistic approaches to learning grammar…”

“…feedback from the task and from human partners, systemic limits, and iterative change [Branco & Valsiner, 1997; Fischer, 1980a, 1980b; Fischer & Granott, 1995; Fogel, 1993; Fogel & Lyra, 1997; Granott, Fischer, & Parziale, in press; Lewis & Douglas, 1998; Siegler, 1994; van Geert, 1991, 1998].”

“For instance, in parent–offspring settings, parental sensitivity to attachment behavior can mediate growth of a developmental variable (van Geert 1991).”

“This stage-like pattern, involving a so-called phase transition, is distinct from a pattern of acceleration in a continuous growth process (Eckstein, 1999, 2000; Van Geert, 1991, 1998).”

“…as a dynamical system, a proposal championed by Liz Bates, Jeff Elman, and other colleagues (e.g., Bates & Elman, 1993; Elman, 1990, 1995; Elman et al., 1996; Smith & Thelen, 1994; Van Geert, 1991), has had a profound impact on our thinking of the relationship between language and cognition.”

“In an attempt to address the existing methodological issues in identifying the vocabulary spurt, Ganger and Brent (2004; see also van Geert, 1991) employed a mathematical approach by modeling a statistical representation of children's rates of early word learning .”

“Furthermore, in the domain of intelligence Van Der Maas et al. (2006) showed how the “gfactor” in intelligence emerges as a consequence of dynamic interactions between lower level components of intelligence, such as perceptual, memory, decision, and reasoning processes.”

“Rose, 1994; Smith & Thelen, 1993; Thelen & Smith, 1994; van Geert, 1991, 1994,”

“Through mathematical modeling of interacting system elements, van Geert (1991, 1994, 1998) has generated theoretical growth curves that are then compared to longitudinal data, often demonstrating a good fit between ideal and actual profiles (Ruhland & van Geert, 1998).”

“…where flux and individual variation abound, and where cause-effect relationships are non-linear, multivariate and interactive in time (de Bot, Lowie, & Verspoor, 2007; N. C. Ellis, 2007; N. C. Ellis & Larsen Freeman, 2006a, 2006b; Port & Van Gelder, 1995; Spivey, 2006; van Geert, 1991).”

“An example of an application is the study of the use of logistic and other difference equations to model growth (and in particular growth spurts) of various cognitive phenomena, e.g., the growth of a child’s lexicon between 10 and 17 months; cf. (Geert, 1991; 1995).”

“…precursor to the development of grammatical knowledge (Bates & Goodman, 1997; Dixon & Marchman, 2007), although other studies have reported more bidirectional developmental trajectories (Anisfeld, Rosenberg, Hoberman, & Gasparini, 1998; Dionne, Dale, Boivin, & Plomin, 2003; Van Geert, 1991).”

“One already sees indications that developmentalists aremoving in this direction (e.g., Thelen &Smith, 1994; van der Maas and Molenaar, 1992; van Geert, 1991) as are other subdisciplines of psychology (e.g., Fitzgerald & Levine, 1992; Vallacher & Nowak, 1994).”

“Schoner, 1989; van Geert, 1998 ) and indeed could readily do so in ways mathematically consistent with the learning processes assumed within connectionist theories.”