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The Dynamic Systems Approach inthe Study of L1 and L2 Acquisition:An IntroductionPAUL VAN GEERTUniversity of GroningenThe Heymans InstituteGrote Kruisstraat 2/19712 TS GroningenThe NetherlandsEmail: [email protected]

The basic properties and concepts of dynamic systems theory are introduced by means ofan imaginary, literary example, namely, Alice (from Wonderland) walking to the Queen inThrough the Looking Glass (Carroll, 1871). The discussion encompasses notions such as timeevolution, evolution term, attractor, self-organization, and timescales. These abstract notionsare then applied to the study of first language (L1) and second language (L2) acquisition. Thesteps that should be taken toward the construction of dynamic systems models of L2 learningare described, including a discussion of the difficulties and the theoretical opportunities thataccompany this model building. Finally, a concrete example of model building and of the studyof variability as an indicator of developmental transitions in L1 acquisition is given.

ABOUT 20 YEARS AGO, THE TERM DYNAMICsystems began to appear in the titles of articlesin developmental psychology. Since then, the ap-pearance of the phrase “dynamic systems” hasbeen steadily but rather slowly increasing in thefield, in terms of the number of publications andareas of application. Nevertheless, the dissemina-tion of information regarding dynamic systemshas not been very widespread. In my view, thisis disappointing because I believe knowledge ofdynamic systems can make a significant contri-bution to understanding development. The pro-cess of development cannot be truly understoodif we confine ourselves to looking at it in the usualway, which is through investigating associationsbetween variables across populations (for a discus-sion of this viewpoint, see van Geert, 1998a; vanGeert & Steenbeek, 2005). Part of the explanationof why dynamic systems has not really taken off liesin the confusion that exists in developmental psy-chology about what dynamic systems are and whatdynamic systems theory means.

The Modern Language Journal, 92, ii, (2008)0026-7902/08/179–199 $1.50/0C©2008 The Modern Language Journal

I will first present a description of what I thinkdynamic systems means and show how it differsfrom the descriptive, explanatory, and method-ological practices that prevail in the scientific dis-ciplines that address psychological development,learning, and acquisition, including language ac-quisition. I will then explain how dynamic systemsapproaches the processes of change in general. Iwill then proceed with a discussion of complex sys-tems and present some issues in the acquisition offirst (L1) and second language (L2) that the the-ory of complex dynamic systems might fruitfullyaddress.

THROUGH THE LOOKING GLASS: BASICPROPERTIES OF DYNAMIC SYSTEMS

In Lewis Carroll’s (1871) Through the LookingGlass, the following intriguing dialogue appears:

‘Well, in our country,’ said Alice, still panting a little,‘you’d generally get to somewhere else—if you runvery fast for a long time, as we’ve been doing.’ ‘A slowsort of country!’ said the Queen. ‘Now, here, you see,it takes all the running you can do, to keep in thesame place. If you want to get somewhere else, youmust run at least twice as fast as that!’ (p. 25, 2005ed.)

180 The Modern Language Journal 92 (2008)

To provide some context, Alice and the Queenmove about in the Queen’s “most curious coun-try,” with “ . . . a number of tiny little brooks run-ning straight across it from side to side, and theground between . . . divided up into squares by anumber of little green hedges . . . .” (Carroll, p. 22,2005 ed.) The country resembles a giant chess-board, formed by the squares of land divided bythe brooks. The story, which is a sequel to Alicein Wonderland , is about Alice, a girl who passesthrough a mirror to find herself in a countrywhere everything seems backward. Now let us tryto look at this little scene from the viewpoint ofdynamic systems.

The State and State Space of a System

To begin with, we must identify the main vari-able of interest here, which is the “place” or “po-sition” in the Looking Glass world that is occu-pied by either Alice or the Queen. The placesor positions form a system. A system can be de-fined as any collection of identifiable elements—abstract or concrete—that are somehow relatedto one another in a way that is relevant to thedynamics we wish to describe. Trivially, the possi-ble places or positions in the chessboard worldare related through distances and specified byat least two measurement dimensions, longitudeand latitude (or the geographical x and y axes).At any point in time, the presence of Alice, or ofthe Queen for that matter, will define a particularplace, which is the position she occupies at thattime, for instance, under the tree or on a par-ticular square surrounded by green hedges. Thestate of system, then, is defined by Alice’s position(or the Queen’s, or both, depending on what onewishes the system to contain) on a particular dateat a particular time, for instance, September 22,2006, at 12:09:01 p.m. The chessboard landscapeis the state space , in other words, the space of pos-sible states or positions of Alice. Alice’s—or theQueen’s—position can be specified by two num-bers, one for the longitude, one for the latitude.That is, the state space is a two-dimensional mani-fold (a two-fold, as Alice would probably call it).

The Time Evolution of a System andthe Evolution Rule

What matters to Alice in the first place is howto get from one place to another. That is, if Aliceis in one place, what does she do to get to anotherplace, and how does she get there? Alice providesthe answer: run. To run means to take steps, oneat a time and in succession, in a particular direc-

tion and with a particular speed (in Alice’s “slow”country, walking is a form of running slowly, andstanding still is a form of running VERY slowly).If you run in Alice’s country, your position on themap will change over time. The change of posi-tions over time is called the time evolution of Alice’sposition (including standing still, if you run VERYslowly) and the taking of one particular step at atime is the evolution rule or evolution term. That is,if you apply the evolution rule—taking a step—to any particular place in the chessboard land, itwill bring you to another place in the chessboardland (and since this is a part of the Looking Glassworld, some steps will make you stay in the sameplace). To be of interest, the evolution rule mustbe applied iteratively. For instance, as Alice entersinto the Looking Glass world (the first position orfirst state of the system), she must try to get to theQueen, where she will have the discussion that weread in the first quote. Alice must take one stepto get to her next position, then from that posi-tion take another step to get to the next position,and so forth. Why describe Alice’s path from theentrance to the Queen in the form of such smallsteps? The reason is obvious: The way Alice getsto her destination is by taking one step at a time.If you want to understand Alice’s path, you mustunderstand it as an iterative sequence of steps.

Constraints and Parameters

To understand why she makes the detour, wemust know that in order to take a step you musthave a solid, supportive ground, and water doesnot have those qualities. (If Alice were a bird, theevolution rule of her position would follow verydifferent constraints and thus lead to very differ-ent time evolutions, such as crossing the brookwithout a bridge.) The necessity of solid groundis an example of a constraint that governs the ap-plication of the evolution rule. Many other suchconstraints can be specified; a particularly inter-esting one could be that it requires less effort totake a step down a slope than up a slope. If thechessboard landscape is very hilly, we should infact expand our manifold to three dimensions,including height, in order to be able to explainwhy Alice prefers a particular path (for instance,avoiding steep slopes). But if the slopes are weak,we can just as well stick to the two original di-mensions, since adding height will not add to ourunderstanding of Alice’s path.

The evolution rule—the taking of a step—requires three parameters for its specification. Oneis the length of the step, another is the frequency(how many steps per time unit), and the third is

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the direction of the step. Some parameters can beconstants, such as the length of the step, which,for the sake of simplicity, is assumed to dependsolely on the length of Alice’s legs. Other parame-ters are variable, in that they can be made to vary,for example, by taking more steps at a time (run-ning). These are called control parameters becausethey control, that is, determine, the form of thepath that Alice will take. For instance, if each step(of similar length, as we agreed on) changes direc-tion by 90 degrees in comparison to the precedingstep, Alice will return to her point of departure af-ter four steps.

In the chessboard landscape, the direction ofAlice’s steps is determined by the position of theQueen because that is where Alice wants to goto. Now we are getting closer to complex systems,as the direction of Alice’s steps seems to be de-termined by other elements in the system, whichis the position of the Queen relative to Alice’sown position, and Alice’s wish or intention to ap-proach the Queen, which we have so far taken forgranted.

Complex systems are systems with many com-ponents that interact, meaning that they co-determine each other’s time evolution. Addingthe Queen makes the Alice “system” just suffi-ciently more complex to understand the notion ofa control system. As Alice wants to get to the Queen,the direction of her steps is determined by Alice’sperception of where she is relative to the Queen.A good rule of thumb (for the real world) is totry to take steps in such a direction that the viewof your target position remains as close as possi-ble to the middle of your visual field as you lookin the direction of your next step. Interestingly,in the looking-glass world, Alice must walk in theopposite direction to arrive at the Queen.

Attractors and Perturbations

Based on this more complex evolution rule—taking one step at a time in a direction controlledin a particular way by the view of the Queen—Alice will arrive at the Queen’s position after sometime, irrespective of where she starts her journey,if there are no obstacles in the landscape. Thispoint, the Queen’s position, is the attractor state ofthe system, given the system’s particular evolutionrule. Since we are in the looking-glass world, wecan take the liberty to do as Lewis Carroll did andchange the rules to serve our explanatory needs.Let us assume that Alice can indeed move towardthe Queen, while trying to keep the Queen’s posi-tion at a 30-degree angle from Alice’s central fieldof vision when looking ahead. With such an evo-

lution rule, Alice can start from any place in thechessboard land and arrive at a place where herpath will automatically begin to circle around theQueen because that circle maximally obeys the30-degree rule. In this strange case, the system’sattractor is a cyclical attractor ; that is, given the par-ticular evolution rule, Alice will end up in a circlearound the Queen, regardless of where she starts.

Note that since we describe the chessboardlandscape in the form of a two-dimensional mani-fold (longitude and latitude), our system of placesdoes not include information about brooks orhedges. It is as if we observe an Alice who isequipped with a global positioning system trans-mitting her coordinates to us every second, thusreducing the actual landscape to something thatappears as a black box to us, the observers. How-ever, if we know the evolution rule that governsAlice’s path and we notice that at some placesshe diverges from the path that the evolution rulepredicts, we can be pretty sure that this is be-cause there are obstacles in the landscape, such ashedges, brooks, bushes, poison ivy, and so forth.(For simplicity, I will continue to assume that it isstill Alice’s intention to get to the Queen.) Assum-ing that we have no direct observational access tothese obstacles, we can infer their existence onthe basis of the divergence between the real pathand the path that should be followed if the evo-lution rule could be applied without disturbance.When Alice gets stuck somewhere, for instance,because she sees her path blocked by brooks andother obstacles, she will most likely tend to moveaway from them in an attempt to find a path thatleads to her goal. Such places are the opposite ofattractors; they are repellors, or places that pushAlice away from them. However, their function asrepellors depends on Alice’s evolution rule, therules that govern her steps.

Another term for an external disturbance of apath or time evolution of a system is perturbation.For instance, we can block a frequented path, butgiven Alice’s evolution rule, it will be fairly simplefor her to bypass the obstacle and reach her goal.We also can build a wall around Alice and observethat she is no longer able to reach the Queen. Theeffect of perturbations on the course of the trajec-tory can thus be very informative of the nature ofthe rules that govern the dynamics.

Self-Organization and Emergence

Technically speaking, the chessboard land-scape is a multidimensional or even infinite-dimensional manifold, consisting of dimensionsdescribing each place in the landscape as being


occupied by pavement, hedges, poison ivy, water,and so forth. We can specify all the places in thelandscape in terms of their likelihood of being vis-ited by Alice. Some places, like where the Queenstands, or some dead-ends in the landscape whereAlice will get stuck (e.g., between two brooks), willstand out as unique features, that is, salient attrac-tors or repellors, in the landscape. Likewise, somepaths will occur more often, for instance, becausethey are easier to walk on. As time goes by andAlice tends to visit the Queen more often, someof the paths will become more and more salient,as Alice’s steps create clearly visible paths throughthe grass. Thus, before Alice ever tries to reachthe Queen, the chessboard landscape contains lit-erally an infinite number of possible paths to theQueen. However, given Alice’s principle of walk-ing, the infinite number of possibilities will soonbe reduced to a small number of actual trajecto-ries or paths. That is, the landscape gets organizedinto a limited number of endpoints and a limitednumber of paths. Thus, after a while, the land-scape becomes more orderly, more structured ororganized, in terms of how it relates to Alice’s walk-ing. This organization, that is, the reduction ofthe number of most likely paths, is a direct conse-quence of the (mostly hidden or unknown) prop-erties of the chessboard landscape and of Alice’sdynamics. For this reason—the organization be-ing a direct consequence of the dynamics itself—we might call it self-organization, which amountsto the spontaneous formation of patterns.

Self-organization is a characteristic property of(complex) dynamic systems. Self-organization hasinteresting properties: It is flexible and it is adap-tive. Thus, if the Queen’s pawns were to changethe position of the brooks and hedges overnight(a nice example of a perturbation), Alice wouldsoon create alternative paths, hence the flexibilityof this self-organization. The number of preferredpaths is likely to be small in any case, but thereare many possibilities for establishing such paths.The adaptive nature of the self-organization im-plies that the paths are optimally adapted to thenature of the evolution rules, Alice’s walking, butthis adaptation arises from the fact that the means(the preferred paths) are to a great extent cre-ated by the function (the walking to the Queen).Self-organization, the spontaneous occurrence ofpatterns due to the dynamics itself, is a particu-lar form of emergence (Crutchfield, 1994; Pessa,2004).

Emergence is the spontaneous occurrence ofsomething new as a result of the dynamics of thesystem. To give a different sort of example, as longas Alice walks below a certain speed, her pattern of

locomotion will be walking. However, as Alice in-creases the speed with which she moves one leg infront of the other, her walking will suddenly shifttoward a new pattern of locomotion, running. Thewalking and running patterns are not built intoAlice’s legs or brain; they emerge under the in-fluence of the control parameter, which is, in thiscase, speed. The speed of putting one leg in frontof another is the control parameter here. A con-tinuous increase of the parameter, such as slowlydriving the speed up, results in a sudden shift toa new pattern of locomotion (running), and thespeed at which this occurs is the critical parametervalue. The pattern of locomotion (either the typ-ical walking or the typical running pattern) alsocan be called a phase , and the sudden shift fromwalking to running or from running back to walk-ing is a phase transition. If we ask the walking Aliceto increase her speed, she will make the phaseshift from walking to running at, say, seven Won-derland miles per hour (the critical value). Nowwe ask Alice to slowly reduce her speed, and weobserve that she will shift from running to walkingat the speed of five Wonderland miles per hour.That is, the critical value for walking to runningis different from running to walking. This asym-metry is called hysteresis. It implies that transitions(switching from one state to another) depend notonly on the controlling variable (such as Alice’swalking speed) but also on the system’s history(whether she goes from running to walking orfrom walking to running).

Nonlinearity and Stability

So far, I have not said anything about a peculiarproperty of the chessboard landscape describedearlier. From the story, it is clear that Alice canreach the Queen from her starting point, but onceshe is near the Queen she undergoes a strangeinfluence of the royal highness on the outcomeof her taking steps: “Now, here, you see, it takesall the running you can do, to keep in the sameplace,” says the Queen, and indeed, whereas a stepfar away from the Queen brings Alice a bit closerto the Queen, a step in the vicinity of the Queenleads to remaining in the same place. We can saythat the effect of taking a step on the crossing ofa certain distance is a nonlinear property of thedistance from the Queen, with the effect decreas-ing to zero once Alice is near the Queen. Sim-ply said (but not entirely faithful to the originalmathematical meaning), nonlinearity means thatthe effect of the step is not proportional to thedistance that needs to be crossed. Proportionalitywould mean that Alice makes greater strides the

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farther away she is from the Queen, thus implyingthat the length of the step becomes zero if thedistance from the Queen is zero. But this is nota linear system: We have seen that as Alice walksaway from the Queen every (same) step brings hercloser to the Queen, and as she is near the Queen,every (same) step leaves her where she is, at thesame place. This way of remaining in the sameplace is a beautiful illustration of dynamic stability,or stability in a dynamic framework. In the linearcase, stability is literally static: If you do not takea step, you do not move. That is, in the linearcase, stability is the absence of dynamics. But inthe nonlinear Wonderland, in the vicinity of theQueen, stability, or remaining in the same place, isthe effect of taking steps (and doing so with greatspeed, for that matter). Hence, a dynamically sta-ble state of the system is defined as a position inspace where a step from a particular place movesyou to that same place. Although this seems like avery weird and unnatural property, worthy of onlya truly Lewis Carollian universe, it is in fact oneof the most central features of the living world aswe know it, where standing still (stability) is justas dynamic as moving. Another example of non-linearity is the hysteresis effect described in thepreceding section: The critical value depends onthe direction from which you reach it.

Timescales

Although the issue of timescales does not ex-plicitly feature in The Looking Glass, it is not diffi-cult to imagine a situation in which the distinctionbecomes apparent. Imagine Alice saying that shewalks from Point A to Point B, and some LookingGlass character—for instance, the Looking Glasstortoise—then objects that she does not walk, butthat what she does is raise one foot, bring that footto another place, shift her balance, raise anotherfoot, and bring that foot to yet another place,and that all this raising and placing is all there is.Alice and the tortoise are clearly referring to dif-ferent timescales. Alice’s is the long-term timescaleof locomotion, going from place A to B, and theother is the short-term timescale of taking a stepand iteratively combining these steps into the actof locomotion. Thus, the dynamic constituents ofthe long term are single steps, and the dynamicconstituents of single steps are movements of thelimbs and body. To a certain extent, it makes nodifference to the long-term timescale how, exactly,Alice takes the steps. If Alice had one leg tied, shecould hop from A to B, and if she had hurt herfoot and were walking with crutches, she wouldsort of limp from A to B. (Note that these patterns,

hopping for instance, are typical examples of self-organizing patterns, given particular constraints,such as the tied leg.) In each case, her path wouldnot lead across brooks or ponds or over walls(which would be the case if the dynamics of herlocomotion constituted a form of flying, for in-stance). However, the short-term dynamics of herlocomotion will affect the long-term dynamics insubtle but interesting ways. With the crutches ortied leg, for instance, Alice would automaticallytend to avoid rough terrain that she could easilynavigate if both legs were unhampered. Thus, theform of the long-term path will reflect the con-straints and possibilities of the short-term dynam-ics of her walking. In fact, if we for some reason areunable to directly observe the short-term dynam-ics, some of the observable properties of the long-term dynamics (the sort of path Alice takes) cansuggest the nature of the underlying short-termdynamics.

Summary: Through the LookingGlass to Development

Of course, the Looking Glass world is entirelyfictional and, at times, physically impossible. Itsproperties nevertheless illustrate some of the ba-sic principles of dynamic systems. I have used thisfictional world to demonstrate that dynamic sys-tems is not a specific theory but that it is a gen-eral view on change, change in complex systems,in particular, or, systems consisting of many in-teracting components, the properties of whichcan change over the course of time. The devel-oping person, embedded in his or her environ-mental niche, is an example of such a complexsystem, which, for simplicity’s sake, I will call thedevelopmental system (Ford & Lerner, 1992). Theproperties that we saw in the Looking Glass worldalso will very likely apply to the developmentalsystem: It will follow a particular time evolutiondepending on its evolution rules; its time evo-lution will depend on particular constraints andparameters; it will move toward particular attrac-tor states and be affected by perturbations; it willbe characterized by self-organization and emer-gence of a characteristic structure; its change willdepend, in a nonlinear fashion, on particular driv-ing forces; its stability, in the form of its attractors,will be as dynamic as its actual change; and finally,it will be characterized by different, interactingtimescales.

In the next major section, I will discuss somepossible applications of dynamic systems to thestudy of L2 acquisition. Where examples are not


available, I will resort to illustrations from theL1.

DYNAMIC SYSTEMS AND L2

Resolving Potential Confusion

Dynamic systems already has a long history (see,e.g., van Geert, 2003, for an overview), and thereis comparatively little disagreement in mathemat-ics, physics, or biology as to what “dynamic sys-tems theory” means. However, the situation isdifferent for the developmental sciences. If onesearches the developmental literature on dynamicsystems—which is not extensive, but steadily in-creasing in terms of publications and fields ofapplication—one is likely to first come across pub-lications that define dynamic systems theory as atheory of embodied and embedded action (espe-cially in the publications of Thelen and Smith;see, e.g., Smith, Thelen, Titzer, & McLine, 1999;Thelen & Smith, 1994). In essence, cognition,thinking, and action are explained as dynamicpatterns unfolding from the continuous, “here-and-now” interaction between the person andthe immediate environment. A particularly cleardescription, in the context of cognition and in-telligence, comes from Smith (2005): “The em-bodiment hypothesis is the idea that intelligenceemerges in the interaction of an organism with anenvironment and as a result of sensory-motor ac-tivity. The continual coupling of cognition to theworld through the body both adapts cognition tothe idiosyncrasies of the here and now, makes itrelevant, and provides the mechanism for devel-opmental change” (p. 205). The dynamic systemat issue is the continuous coupling between theorganism and its environment, showing a timeevolution that takes the form of intelligent action.

A second line of thought emphasizes the ma-jor qualitative properties of complex dynamic sys-tems. The developmental system is viewed as aself-organizing system, showing attractor states,nonlinearity in its behavior, emergence, and soforth. The inspiration for this view comes from thestudy of dynamic systems in other fields, which hasdemonstrated that such systems show the prop-erties in question. An eloquent defender of thisview is Marc Lewis, who has primarily focused onsocial interaction, emotions, and personality, and,recently, has shifted his attention to brain devel-opment (Lewis, 1995; Lewis, 2000; Lewis, 2005;Lewis, Lamey, & Douglas, 1999).

A third approach is the one that I have de-fended for about 15 years now, which is that dy-namic systems is basically a very general approach

to describing and explaining change, focusing onthe time evolution of some phenomenon of in-terest, including the principles or “rules” that de-scribe this time evolution (for general overviews,see van Geert, 1994, 2003; van Geert & Steenbeek,2005).1

Fortunately, the differences in viewpointsamong the embedded–embodied, the qualitative,and the general approach, if I may call them bythese names, are smaller than they appear to be atfirst sight. (Although some points of disagreementremain, especially with regard to the way in whichcognition, or language for that matter, should bestudied in a dynamic systems framework.) If weapply the general approach to systems that arecomplex and sufficiently permanent—such as hu-man beings, societies, and also more transientphenomena such as an action or a conversation—we soon will find out that they display a host ofinteresting qualitative properties, namely, emer-gence, self-organization, attractor states, nonlin-earity, and so forth. Thus, the application ofthe general approach to systems that are of in-terest, say, to students of L2 acquisition, leadsto the properties focused on by the qualitativeapproach.

The relationship with the embedded–embodied approach is a little more complicatedand requires some understanding of the conceptof dynamics on various timescales, as introducedin the preceding section. When a person sayssomething in response to what another person hasjust said before, an observer can confine oneself todescribing the utterance as a sentence, compris-ing a particular syntactic structure and categories.There is no doubt that this is an adequate (thoughcertainly not exhaustive) description of what theobserved speaker does at the time of uttering.The question is, of course: How does this sentenceemerge, in the real context and timeframe of itsoccurrence? This is the sort of question that arosewith Alice claiming to be walking (a shorthand de-scription of a macroscopic pattern, so to say) andthe imaginary tortoise claiming Alice was not walk-ing but in fact moving one leg (or part of a leg)at a time. The tortoise was thus referring to themicroscopic, short-term appearance of the macro-scopic property called “walking.” The short-termdynamics of language use and understanding isan example of the short-term dynamics of humanaction and communication, and its understand-ing requires an understanding of intentionality(goal-directedness, the interest-driven nature ofaction), and embodiment and embeddedness inthe context (see, e.g., Clark, 1997, 2006; Clark &Chalmers, 1998; Harris, 2004; Love, 2004).

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At present, we are only beginning to under-stand some of the short-term dynamics of howactions, including language, emerge through theconfluence of internal abilities and external, con-textual affordances. However, this relative lack ofunderstanding at the “microscopic” level shouldnot prevent us from trying to understand the dy-namics of a particular phenomenon, such as L2learning, at the macroscopic level of longer time-frames (e.g., months or years during which theL2 is learned). However, the short-term dynamicsthat explains the actual production and under-standing of sentences on the spot is character-ized by a number of interesting properties, one ofwhich is termed “fluctuation and variability” (e.g.,Thelen & Smith, 1994). These properties mustbe taken into account by any model focusing onlonger term timescales and not be disregarded as,for instance, measurement error. I will revisit thisissue later.

In summary, the first approach to dynamic sys-tems discussed in this section refers to a particu-lar, in general, short-term, timescale of phenom-ena that also can be dealt with on the long-termtimescale of macroscopic change. However, thebasic principles of dynamic systems theory in thegeneral sense must apply to all timescales. (This isthe reason connectionist network models, whichoperate on the timescale of input–output rela-tions in the brain, are entirely compatible withdynamic systems principles and are thus exam-ples of dynamic systems [Pessa, 2004; Thelen &Bates, 2003].)

Steps in the Construction of a Dynamic System for L2

A State Space Description of Developmental Trajec-tories. A characteristic feature of a dynamic systemsapproach is that it is geometric in nature. In theLooking Glass example, the system described Al-ice’s changing position in a world that, basically,was described by means of two coordinates, lat-itude and longitude. Other dimensions can beadded to the system as needed to specify anyadditional variable by which the system is char-acterized. We thus arrive at a mathematical ob-ject that consists of an abstract space of n di-mensions (with n = any arbitrary number). Thestate of the system at any point in time is repre-sented by the system’s coordinates on all its di-mensions as a point in the n-dimensional space,or state space . The bad thing about this is thatit is a pretty abstract way of describing concretethings; the good thing is that we do not need toknow or measure each of the dimensions to beable to use this geometric concept of the system.

This notion of state space will allow us to describeand understand certain general properties of thesystem.

Imagine a state space containing all the di-mensions (variables, properties, etc.) a researcherneeds to more or less exhaustively specify the stateof any conceivable L1 and L2 user and/or learner,and let us, for simplicity, confine ourselves to L1speakers who wish to learn an L2. Somewhere inthis state space there is a region where we canfind the L1 speakers who have not yet begun tolearn the L2 in question, and somewhere we canlocate a region where we will find those L1 speak-ers who, according to whatever criteria we wishto use, have mastered the L2. In order to studycomplex dynamics—of which L2 learning is mostlikely an example—it is strategically wise to con-ceptually reduce the complex system to a singledimension or very simple state space, in whichthe qualitative properties of the system dynamicscan be observed. In the given L1–L2 state space,we easily can imagine a single dimension that de-scribes the distance between the “starting point”region and the “end point” region. This simpli-fication does not entail an ontological claim, inthe sense that it does not imply that “in reality”L2 learning is a one-dimensional process, whichclearly would be absurd. Instead, the simplifica-tion is a mathematical projection (in the sense ofprojective geometry), projecting a complex pat-tern onto a highly reduced state space, which,if done properly, opens the complex pattern forstudy and description.

Empirically, this one-dimensional distance canbe viewed as “L2 proficiency,” which, for now,remains a somewhat vague term. The importantthing is that the steps an individual takes on hisor her way to L2 acquisition can be rank-orderedon this dimension (ties not excluded). What a dy-namic systems–oriented researcher would wish toknow, in this case, is the following: Let �d/�tbe the distance toward L2 covered over sometime interval �t (a day, a week, a month, etc.).How does �d/�t relate to time, that is, the timepassing from some arbitrarily determined start-ing point of the person’s L2 learning to some ar-bitrarily determined end point of L2 learning?On a short-term timescale (e.g., days), for in-stance, �d/�t is likely to map out a nonsmooth,“jerky” function, which represents short-term fluc-tuation and is characteristic of the short-term dy-namics of L2 use and learning. On a longer termtimescale (e.g., months), �d/�t is likely to fol-low a path of accelerations and decelerations,which is characteristic of a stage-wise pattern ofacquisition.


I shall discuss an example based on a free (andidealizing) interpretation of a model advanced byPienemann (1998). The model describes the L2acquisition of German word order in the form offive stages that represent a particular grammati-cal structure. Let t0 be the time at which a personstarts to learn German, and tf a later point intime at which that person uses correct word or-der. The stages imply that the frequencies of useof each of the word order types change. In thebeginning, a person will almost uniquely use thestage 0 structure, and as time passes, the propor-tional frequency of more advanced structures willincrease. Finally, the proportion of correct wordorder structures will approach 1 (i.e., 100%). It islikely that the proportions of the structure typeswax and wane, until only the correct structures re-main. I have simplified and idealized this situationby assuming that the structures form overlappingwaves of usage frequencies (see Siegler, 2005, for ageneral overlapping waves model), as representedin Figures 1a–1d.

FIGURES 1a–1dImaginary Successions of Frequency of Use of Five Developmentally Ordered Types of Word OrderStructures (Left) and the Corresponding One-Dimensional Developmental Distance Representation (Right)

The state space describing the dynamics of Ger-man word order acquisition thus requires five di-mensions, each representing the frequency of useof a particular word order structure. The develop-mental distance vector runs from 100% use of themost primitive structure to 100% use of the mostcomplex structure. If, for simplicity, we put thestages on equal distances from each other in termsof increasing complexity, we can weigh the stagesquantitatively by 0, 1, 2, 3, and 4, respectively, 4 in-dicating the most complex or correct structure. Atany point between t0 and tf , the learner’s positionon the distance dimension is the weighted sumof (0 ∗ p0+1 ∗ p1+2 ∗ p2+3 ∗ p3+4 ∗ p4)/4, for p0

to p4 equals the proportions of usage of eachof the structures at a particular point in time.When we apply this equation to the overlap-ping waves from Figure 1a, we obtain a stepwisegrowth function for the one-dimensional statespace of word order acquisition (Figure 1b). Notethat if the frequencies overlap differently, as inFigure 1c, the resulting one-dimensional curve

Paul van Geert 187

approaches a continuous S-shaped curve (seeFigure 1d).

What this example illustrates is that it is pos-sible to transform a qualitative model of struc-tural stages into a geometric model of quanti-tative change over time. As with any model, thequantitative model is as good as the assumptionson which it has been based, and in the currentexample these assumptions are highly simplifiedand idealized. The assumptions have neverthelesshelped us link qualitative and quantitative repre-sentations of the same phenomenon.

The choice of the state space dimension(s) re-flects the organization of the system under ob-servation. A complex system (i.e., the sort of sys-tem we are interested in) shows a high degreeof organization—a high amount of coordinationof its many components. In the case of complex,developing, and psychological systems, such as anL2 learner, this organization or coordination is ex-pressed in the form of a few dominant, or salient,variables. An example of such a salient variable isthe word order from the preceding illustration;another example is the one-dimensional notionof proficiency that characterizes a particular per-son’s mastery of an L2. These salient variables arecalled the order parameters of the phenomenon inquestion, for instance, L2 learning of German orEnglish.

Distinguishing a one-dimensional, general vari-able such as L2 proficiency or mastery is primar-ily a matter of descriptive choice, but this choicemust be descriptively adequate; in other words, itmust reflect something that is real or functional.A complex system such as the L2 learning sys-tem requires various perspectives, various orderparameters and state space descriptions, which,together, capture the L2 learning phenomenonin all its richness.

Order parameters can have a categorical for-mat, such as the word order structures distin-guished by Pienemann (1998), or they can have acontinuous, dimensional format, such as the pro-ficiency or mastery level parameter. Continuousdimensions come close to the notion of a metric,or ruler, along which the states of the system canbe measured in the classical sense of the word(see, e.g., Fischer & Bidell, 2006, and Fischer &Rose, 1999, for an application in developmentalpsychology in general). In linguistics and psychol-ogy, continuous order parameters often are rep-resented in the form of tests providing a contin-uous or quotient score (an intelligence test is aprime example). However, the test and the statespace dimension are not the same thing: They re-fer to each other, but it is not necessary to have

a test (e.g., of L2 proficiency) in order to fruit-fully apply a particular state space dimension ororder parameter to a system (e.g., L2 proficiencyas a convenient descriptive construct). In somecases, the dimension or order parameter can beempirically mapped onto a series of variables forwhich several tests are available, without implyingthat the dynamics must be described by means ofa state space with as many dimensions as thereare tests (or as many dimensions as there are fac-tors in a factor analysis or principal componentanalysis).

The Short-Term Dynamics of Learning and the Roleof Intentionality. In order to obtain a good un-derstanding of the long-term dynamics of devel-opmental and learning processes, it is importantto specify the short-term dynamics of action. InL2 learning, actions involve communicating withother people, such as native speakers, listeningto conversations, studying grammar in the text-book, making assignments, and so forth. Theseactions contribute to learning and development.However, learning and development determinethe nature of the actions carried out (comparelistening to a conversation by a novice and listen-ing to that same conversation by a more advancedL2 learner, in terms of the information they canretrieve, or what they actually learn, from that con-versation).

In a recent series of papers (Steenbeek & vanGeert, 2005, 2007, 2008; van Geert & Steenbeek,2005), a dynamic model of action in the context ofdevelopment has been presented. Its major com-ponents are the person’s concerns (goals, inten-tions, etc.), the actions available to realize thoseconcerns, the person’s subjective evaluation of therealization of those concerns through emotionsand emotional expressions, and, finally, a directsocial component that accounts for the “conta-giousness” of the actions of others and their im-pact on the actions of the person through im-itation. A person’s goal relates to—but is notthe same as—an attractor in the state space asdescribed earlier. For instance, a person wishesto learn a foreign language and wants to passan exam with excellent marks. These goals co-determine the person’s actions, but the actionsalso are co-determined by a host of other fac-tors. An important additional factor is the per-son’s preferred balance between the goal at issue(e.g., pass the language exam) and other goals(e.g., the person’s wish to go out or relax, or theperson’s job and family interests). The attractorstate—the person’s having learned the foreignlanguage, the marks obtained on the exam—in


fact self-organizes as a result of the dynamic rela-tionships among all the factors involved. (For anapplication in the context of children’s play, seethe publications mentioned earlier in this para-graph.) An important factor of the short-term dy-namics of actions related to learning and devel-opment is the real-time emotional evaluation of aperson’s actions in light of his goals or concerns(Posner, Russell, & Peterson, 2005). Positive emo-tions (both self-conscious and other-related) areof primary concern (see, e.g., Dewaele, 2005, andImmordino-Yang & Damasio, 2007, in the contextof L2 learning).

Tschacher and Haken (2007) recently havepresented an interesting general explanation ofintentional action, which can help explain theordered and self-organizational nature of actionsand the resulting developmental and learningtrajectories. First, developmental state spacesare characterized by what Tschacher and Hakencall a gradient . On the long-term timescale, thegradient is the difference between lower andhigher levels of development (learning, mastery,etc.). On the short-term timescale, the gradientrepresents the distance between the intendedstate, or goal state, on the one hand and thecurrent realization of that state on the otherhand. Second, in complex state spaces consistingof many descriptive dimensions, such as the L2developmental state space or the state space of L2learning activities, a system (e.g., an L2 learningindividual) theoretically can take any routeleading from a lesser to a more developed state.That is, theoretically, any trajectory in the statespace has the same probability of being taken bythe acting system. In reality, however, we see thatthe number of trajectories is limited (there arelikely to be more than one, but in general notmore than a few; Shore, 1995). The limitation ofthe trajectories to only a few is an example of self-organization. Tschacher and Haken (2007) claimthat complex systems must resolve gradients bymeans of optimally dynamically efficient routes,which are in fact like minimal-energy solutionsto crossing the distance from, in this particularcase, a lower to a higher level of L2 proficiency.Tschacher and Haken’s model thus attempts toexplain why the number of possible developmen-tal trajectories is limited, and why the numberof possible action trajectories leading from agoal to its perceived, satisfactory realization alsois limited. These trajectories are the result ofself-organizing processes, which include but arenot limited to explicit intentions and actionplans. In this sense, both action and long-termlearning and development are examples of“organized self-organization” (van Geert, 1989).

Static Versus Dynamic Accounts of L2. In psychol-ogy, or any other science that deals with humanbehavior, including language, it is customary to in-directly approach the dynamics of a phenomenonby finding out how the distribution of a particularphenomenon in a group (population or sample)is related to distributions of other phenomena inthe group. For instance, one might wish to knowthe association between a student’s oral perfor-mance on the one hand and the student’s taskgoal orientation on the other hand. One can ap-proach this question by taking a large sample ofstudents, measuring their oral performance andgoal orientation, and calculating the correlationor some other indicator of association betweenthe two measures, which, as we will find out, takesa positive value. This association indicator is a so-called static measure (there is absolutely no pe-jorative or evaluative meaning associated to thisterm), meaning that it reflects the position ofstudents in a group relative to each other. (Fora recent example relating to adaptive languagelearning, see Woodrow, 2006. Note that I am usingthis relationship as an example about which I willspeculate freely in order to try to illuminate thedifference between a static and a dynamic model.I am in no way claiming to discuss or criticizeWoodrow’s findings here.) We can assume thatthe positive correlation means that we will tendto find higher or better task orientation with stu-dents who have better oral performance and, bydefinition, also the other way around. Technically,we can specify this relationship as follows:

yi = f (xi )

with y representing oral performance and x taskorientation (or vice versa). The f stands for func-tion and means a positive association here, whichallows us to predict a value for x, given a value of y,within certain limits (the error term, or percent-age of explained variance).

However—and this is a very important point—the fact that an association holds over a popu-lation or sample does not necessarily imply thatsuch an association holds for a time evolution ortrajectory, that is, a dynamic system. Thus, if we fol-low the time evolution of oral performance andthe time evolution of task orientation in a partic-ular subject—which is where the evolution takesplace—we do not necessarily find that higher taskorientation is related to higher performance.

This possible absence of positive relationshiphas nothing to do with eventual random fluctua-tion or error. What I mean to say is that it is verywell thinkable that a static, positive relationshipover a population (including all kinds of errorfluctuation) might correspond with a negative,

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dynamic relation (or anything else) in any or even-tually all of the individual members of the pop-ulation. Think, for instance—and this is purelyspeculative but not impossible in my view—that aperson’s goal orientation, including the strengthof his intention to perform well and invest moreeffort in accomplishing the task, increases as heor she perceives the task as more difficult and de-manding, as in reading and understanding textsof high lexical and syntactic complexity, for exam-ple. It is not unlikely that such demanding tasksrequire the student to invest more effort in as-pects other than oral performance, and thus thathis or her oral performance in such tasks is onaverage lower than in tasks that the student con-siders easy and not requiring a stronger task ori-entation than he or she is used to. Thus, seenin a dynamical perspective against the student’sown standards of performance and perception ofcompetence, the relationship between task orien-tation and oral performance might be a negativeone, whereas that same relationship is positive ifthe topic of analysis is the sample of students. Thecrucial thing here is that a dynamic relationshiprepresents a process mechanism, whereas a staticrelationship over a sample represents an associ-ation between the distributions of two variablesover the sample.

The finding that a static and a dynamic re-lationship are not the same thing is an exam-ple of the so-called ergodicity problem (Hamaker,Dolan, & Molenaar, 2005; Molenaar, 2004; Mole-naar, Huizenga, & Nesselroade, 2003). The prob-lem refers to the possible similarity of a spaceaverage (e.g., an average or association) calcu-lated for an ensemble of participants, and a timeaverage (i.e., an average calculated over a se-quence of steps in an individual). The speculativeexample of the oral performance and task orien-tation relationship is an illustration of a situationin which this similarity does not necessarily apply.Although this example is speculative, there existsresearch in a different field that shows this in-equality empirically. A study by Musher-Eizenman,Nesselroade, and Schmitz (2002) showed that therelation between perceived control and academicperformance found in a cross-sectional or classicallongitudinal design (few repeated measurementsover a comparatively long time) was differentfrom the relation found when relatively short-term within-person change patterns were studied.Thus, the short-term dynamics of perceived con-trol and academic performance in individuals isdifferent from the long-term or cross-sectional re-lationship. If this reasoning is applied to the oralperformance versus task orientation example, it

would hold that whereas in the short run therelationship between the two is negative, in thelong run the relationship becomes positive again,in that a long-term increase in goal orientationin an individual can be associated with a long-term increase in that person’s oral performance.(For an example of a distinction between rela-tionships holding over a sample and holding be-tween time points in a dynamics of language learn-ing and instruction, see van Geert & Steenbeek,2005.)

In order to avoid possible confusion arising outof the ergodicity discussion, I must add that I donot mean to say that development is some sortof internal, solipsist thing. Dynamic rules or rela-tionships pertain to particular dynamics, and inthe case of L2 learning, this is the dynamics ofan individual learning the language in a contin-uous interaction with other persons who use theL1 and L2 or teach the L2. There is no doubtthat L2 learning is a highly social process. In addi-tion, the ergodicity discussion does not imply thatindividual cases, or even exceptions, are more im-portant than what is common among individualsor groups. In a dynamic systems approach, a sam-ple is the collection of dynamic trajectories, someof which are more common than others. A validdynamic model must thus be able to explain thecommon pattern as well as the uncommon pat-terns and exceptions.

Finding the Evolution Term and Explaining theLong-Term Dynamics. In order to try to obtain abetter understanding of this ergodicity problemand of the principles of dynamics in general, con-sider the simplest possible format of a dynamicsystem, which is:

yt+1 = f (yt ) and xt+1 = g(xt )

The equation says that the next state (at time t+1)of a variable y or x is a function (f or g) of thevariables’ preceding states (at time t). If this is so,then any next state is a function of its precedingstate, and the equation produces a picture of thetime evolution of y and x:

yt → yt +1 → yt+2 → yt+3 → yt+4 → . . . and

xt → xt+1 → xt+2 → xt+3 → xt+4 → . . .

This is a simple application of the iterativeness (orrecursiveness) explained earlier. Thus, the firstquestion to answer is: What is the content of, orthe mechanism behind, the enigmatic f , assumingthat y represents L2 proficiency, the variable qual-ity or level of L2 performance of a person? The


simplest possible answer to this question is that,if L2 performance is part of a learning–teachingprocess, the change of L2 performance likely is tobe some form of accretion, some form of addingsome performance quality to the preceding stateof L2 performance quality. An accretion modelof lexical learning implies that a person learns afixed number of words a day (plus or minus somerandom variation due to purely external causes).From a learning-theoretical2 point of view, the ac-cretion model is very unlikely because it views thelearning, at any point in time, as completely inde-pendent of what the person already knows and ofwhat the person does not yet know.

Studies of learning and development convergeon the finding that the increase of performance,skill, or knowledge over time is proportional tothe performance level already attained, propor-tional to the difference between what is alreadylearned and what has yet to be learned, and pro-portional to the available resources (e.g., the qual-ity of the teaching, the student’s motivation, etc.).This proportionality leads to various models oflong-term change, the most likely or applicableof which is the so-called logistic growth model , themodel of growth under limited resources. Thisdynamic growth model has been thoroughly ex-plained elsewhere (de Bot, Lowie, & Verspoor,2007; van Geert, 1991, 1994, 2003). Under idealconditions, it leads to a smooth, S-shaped patternof increase of performance, which stabilizes at alevel it is able to maintain given the available re-sources. For an L2, this might include such re-sources as the frequency with which high-quality(native) L2 is heard by the L2 learner, the fre-quency of use, and the speaker’s linguistic tal-ent (whatever that may mean). However, as statedbefore, the dynamic growth model must reckonwith the properties of the short-term dynamicsof actual L2 production and L2 reception, whichrequire explanation at a different timescale. Asstated before, a prime example of such a prop-erty is the natural fluctuation of performance overtime.

In the preceding section, I discussed the ex-ample of oral performance and task orientation,which demonstrated the importance of a particu-lar coupling of variables. Thus, task orientation islikely to affect oral performance over time, and wehave seen that this relationship need not be thesame as the relationship found over a sample ofparticipants. The relationship between variablesrelates to a particular property of dynamic sys-tems, namely, the possibility to dynamically couplethe variables in various ways. The formal formatof a coupled dynamic system is as follows:

yt+1 = f (yt , xt ) and xt+1 = g(xt , yt )

meaning that the next step of y depends on thepreceding value of itself and of another variable,namely, x, whereas the next step of x depends onits preceding value and the preceding value of y.The variables y and x can refer to processes withina person and also to processes between persons.For instance, y can represent a level of proficiencyin a particular L2 learner, such as the type of wordorder frequencies in the learner’s L2 production.x, however, can represent corrections or demon-strations made by an L2 speaker or L2 teacher,following the L2 learner’s apparent mistakes. Ifthe learner is sensitive to these corrections andassimilates them, his or her L2 level will changeand induce other types of corrections, if any areneeded, in the L2 teacher.

In the simplest possible sort of model, the re-lationship between one variable and another canbe positive, as is the case if increased task orien-tation results in better oral performance. Or thisrelationship can be negative, as is the case if anincrease in task orientation is due to perceivedtask difficulty eventually resulting in lower per-formance than if the task is easy or less demand-ing. These relationships are unidirectional; thatis, they go from one variable to another but notthe other way around.

More interesting dynamics result from mu-tual relationships between variables. Take, for in-stance, the following speculative example. It islikely that in the early stages of L2 learning, L2 userequires considerable effort, in that it is not yet amore or less automatic task and requires moni-toring, explicit attention, and so forth. More ef-fort will, in principle, result in better performance(all other things, such as prior L1 and L2 knowl-edge, support given, etc., being equal). Good per-formance, if it is recognized as such by a morecompetent L2 conversation partner, will in prin-ciple help to maintain the speaker’s motivationand thus the investment of effort. However, highlevels of effort lead to fatigue, and fatigue is likelyto affect performance negatively. Hence, the rela-tionship between performance and fatigue, via ef-fort, is mutual but asymmetrical. It is positive fromhigh performance (via high effort) to fatigue inthat high effort makes fatigue increase. It is neg-ative from fatigue to performance, in that higherfatigue results in lower performance. This typeof asymmetrical relationship, which can be com-pared to a predator–prey relationship, is likely toproduce oscillating patterns, waves of high perfor-mance and high fatigue. (This is, of course, an ide-alized example). However, the effort component

Paul van Geert 191

depends clearly on the level of mastery of theL2. The more L2 use becomes an automatic taskperformance, the less it will appeal to the effortcomponent, and the less it will be a cause of fa-tigue. But fatigue from a different source (e.g., jetlag at the beginning of a conference) will still af-fect L2 performance, even in an experienced L2speaker. Thus, the nature of the dynamic relation-ships changes over time and is also context andorigin dependent.

To return to the ergodicity problem and the is-sue of static versus dynamic models, assume thata researcher has a sample of participants at differ-ent levels of performance and at different levelsof effort required to achieve that performance.The relationship between performance and fa-tigue over this sample will likely be anomalous;that is, the correlation may be small and not sta-tistically significant. However, there is no directlink between this null relationship over the sampleand the fatigue–performance relationships thatgovern the dynamics of L2 acquisition and per-formance in all the independent individuals com-posing the sample. In individuals, the link is stillpresent and influencing the individual dynamicsto a considerable extent. However, the nature ofthe link itself is subject to developmental changeand thus not uniform over participants.

Short-Term Dynamics and Fluctuation. The short-term dynamics of language refers to the mech-anisms operating on the actual production andunderstanding of language, for instance, in con-versations or oral or written monologues, suchas the homework assignments of an L2 learner.Fluctuation, or intraperson variability, is a char-acteristic feature of the short-term dynamics oflanguage use (de Bot, Lowie, & Verspoor, 2007;Ellis, 1985, 1994; Pienemann, 2007). Fluctuationsoriginate, for instance, from the availability of al-ternative forms or strategies (e.g., various patternsof word order) and from contextual variations(e.g., monologues versus conversations with na-tive or nonnative L2 speakers). Fluctuations thuspartly originate from the long-term dynamics ofL2 learning, which makes certain forms, strate-gies, or rules available or not.

Short-term dynamics also are likely to con-tribute to the long-term dynamics of the L2 learn-ing process. Variability in linguistic productions ofL2 learners among different action contexts pro-vides possibilities for spontaneous explorations ofpossible forms or rules of the L2 that is beinglearned, for instance, because this variability inproduction allows for corrections or rephrasingby the L2 teacher or native speaker. Since the

fluctuation is a product of the underlying short-term dynamics (e.g., the dynamics of sentence for-mation or the dynamics of a conversation), thefluctuation must not be treated as noise, that is,as variability due to some independent, externalsource of disturbance. Rather, the fluctuation, ifproperly described and understood, may containinformation about the underlying short-term dy-namics and its effect on the long-term dynamicsof increasing L2 performance. An example fromL1 learning will be discussed in the next section(Bassano & van Geert, 2007).

Another example, also in early L1 acquisition,comes from a study with Marijn van Dijk on vari-ability peaks in the use of spatial prepositions,marking the transition to rule-governed use ofthose preposition types in the child’s language(van Dijk & van Geert, 2007). Other examplesof this phenomenon are presented in the arti-cle by Verspoor, Lowie, and van Dijk (this issue).In short, the theoretical specification of the evo-lution term explaining the time evolution of L2performance on the long term will have to reckonwith the consequences of the short-term dynam-ics, such as variability. These short-term dynamicsare the subject of study of disciplines such as cog-nitive psychology (Costa, La Heij, & Navarrete,2006) or neurocognition, or dynamic systems asdefined by Thelen and Smith (1994; see also Col-unga & Smith, 2005, and Jones & Smith, 2005,for an application of this theoretical approach toword learning and use).

Transitions in Early Language Development: AnExample of Dynamic Systems Model Building

A Hypothetical Model of Three Stages in Early Lan-guage Acquisition. In a study I carried out in col-laboration with Dominique Bassano, an attemptwas made to build a dynamic model of long-termchanges in language production, focusing on sen-tence length as an indicator of underlying prin-ciples of L1 production (for details see Bassano& van Geert, 2007). The model that we wished tocheck goes back to a hypothesis about L1 produc-tion that has been around for almost 30 years now.It assumes that, in the construction of a genuinesyntactic language, children begin with a stagein which one word expresses a complex referen-tial meaning (“holophrase”). The hypothesizedgenerating mechanism of language at this stageis therefore called the “holophrastic generator.”In a second stage, the child is assumed to sim-ply combine these complex holophrases into ut-terances consisting of a few (two or three) suchunits, and thus generate language by means of


combination, hence the stage of the “combina-torial generator.” It is assumed that, while com-bining words, children become sensitive to theway the language spoken by the linguistic envi-ronment solves the combinatorial problem, whichinvolves the use of syntax (order rules, word corre-spondence, etc.). Thus, the third and final stage,according to this hypothesis, is the stage at whichlanguage is supposed to be based on a “syntac-tic generator.” These stages are assumed to occurin the form of overlapping waves. The alternativehypothesis is that language development is a con-tinuous process (whatever the exact nature of theunderlying continuous processes).

Data and Method. The main set of data usedin this study came from the longitudinal cor-pus of one French girl, Pauline, who was studiedfrom ages 1;2 to 3;0. Additional data are fromanother French child, Benjamin, who was stud-ied from age 2;0 to 3;0. (For analyses of the chil-dren’s language development, see Bassano, 1996.)Data were obtained using a free speech samplingmethod. For each child, frequencies of one-word,two-word, three-word utterances, and so on (W1,W2, W3, etc.) were calculated for each monthlysample (120 utterances) and for the subsamples(60 utterances and 30 utterances, respectively).Figure 2 shows the smoothed curves of the rawdata, based on a Loess smoothing procedure (lo-cally weighted, least-squares smoothing), whichprovides a representation of the changes in thesentence frequencies and is able to capture even-tual temporal regressions and accelerations in thegrowth rate.

A Dynamic Model of Stepwise Grammatical Devel-opment. The aim of this model is to explain thequantitative evolution of three types of utterancesor phrases, namely, one-word, two- and three-word, and four-word-and-more phrases, calledW1, W23, and W4+, respectively. These types ofphrases are supposed to be linked with the linguis-tic generators described earlier. Before proceed-ing with the model, two remarks should be made.The first is that the relationship between a partic-ular type of phrase, for instance, W1, and its as-sumed underlying generator, is in itself a transientrelationship in that it changes over developmentaltime. Thus, an early W1 is considered a product ofthe holophrastic generator, but a late W1, occur-ring in the midst of W23 and W4+ phrases, is mostlikely an expression of the assumed syntactic gen-erator. This changing relationship must be takeninto account when interpreting the meaning ofthe linguistic data. Second, the model we wish to

build does not explain the emergence of a newgenerator out of an old one. In order to do so, adifferent type of model would be required, show-ing how a procedure of word combination, forinstance, grows out of a procedure of holophraseuse, based on various kinds of linguistic inputs,due to increasing participation in communicativeevents with other, and in principle, (more) maturespeakers. Moreover, the model—which is a modelof long-term change—does not include the short-term dynamics of phrase production; for instance,it does not provide an explanation of how and whychildren, in their actual speech producing pro-cesses, make a choice between a holophrastic andcombinatorial generator at developmental stageswhen both are available.

The term explain might give rise to confusion.Of course, the dynamic model might be said to“explain” this choice by invoking the notion ofa random selection between two available gen-erators, given the expected frequencies at someparticular moment in time. However, this model-theoretic random selection is not intended as aliteral description of the actual short-term processof phrase production and does not in itself implythat a language-producing child literally makes astochastic response choice. In fact, for reasons ofsimplicity, the long-term model treats the short-term process as if it were a stochastic choice asconvenient for the time being and leaves the de-scription of how the process actually occurs inthe short term of phrase production to a theoryand design that studies this short-term timescaledirectly.

In line with comparable growth models of long-term change, we postulated that the hypothesizedgenerators—which are hierarchically ordered interms of assumed developmental complexity andorder—were characterized by bidirectional asym-metric relationships (see Figure 3).

Relationships run from the less to the moreadvanced generator and from the more to theless advanced.

First, the relationships from the less advancedto the more advanced generator are supportiveand conditional . For instance, the holophrasticgenerator explains the increase in the productionof one-word phrases. It is assumed that a mini-mum level of one-word phrase production is nec-essary to make the transition to a combinatorialgrammar, which generates two- and three-wordphrases. This minimum required level is neededfor the combinatorial generator to emerge, hencethe notion of a conditional relationship fromholophrastic to combinatorial generator.3 Thesupportive relationship between holophrastic and

Paul van Geert 193

FIGURE 2Raw Data (Top) and Smoothed Curves (Bottom) of W1-, W23-, and W4+ Sentences of Pauline Between 14and 36 Months of Age

Note. W1 = one-word utterance; W23 = two- and three-word utterance; W4+ = four-word-and-more phrase.

combinatorial generator implies that there is a lin-ear, positive relationship between the number ofW1 phrases actually produced and the increasein the number of W23 phrases. That is, “sim-pler” formats, such as W1 phrases produced bya holophrastic generator, are assumed to stim-ulate the production of more complex formats,such as W23 phrases produced by a combina-torial generator. I see this as a very simple, ba-sic developmental assumption, which can be ex-plained on the basis of classical developmental

and learning-theoretical rules or principles (seevan Geert, 1998b).

Second, the relationships from the more ad-vanced to the less advanced generators are compet-itive . Such a relationship implies that an increasein the use of the more advanced generator is re-lated to a decrease in the use of the less advancedgenerator, which will contribute to the latter’s de-cline (see MacWhinney, 1998 and van Geert, 1991for a more thorough discussion of the origins ofthe competitive relationship).


FIGURE 3Competitive and Supportive Relationships AmongHypothetical Syntactic Generators in L1 Acquisition

The competitive relationship is, in all likeli-hood, a transitive relationship. That is, it holdsfor all generators that are less complex or less ma-ture. Thus, a competitive relationship from thesyntactic to the combinatorial generator also im-plies a competitive relationship from the syntacticto the still older holophrastic generator.

The next step is to specify a mathematicalmodel that provides a formal representation ofthe set of relationships. For instance, the support-ive relationship entails a positive effect of the pro-ductivity of a more mature generator on a lessmature one. Thus, if Ct represents the frequencyof W23 at time t , and St represents the frequencyof W4+ phrases at time t , the increase in St, whichwe can write as �S/�t, is proportional to Ct. Fur-ther, if the parameter d stands for that proportion,�S/�t depends on d ∗ Ct. In one of the precedingsections, we have seen that the effect of a growthfactor or learning factor on a variable is likely tobe proportional to the current level of that vari-able. That is, the magnitude of the effect of C onS is proportional to the current level of S. If e isa parameter that stands for that particular pro-portion, we can now combine the two effects in asimple equation that specifies the effect of C of S,

as follows:

�S/�t = d ∗ Ct ∗ e ∗ St

If, for simplicity, we set d ∗ e equal to a value c, wefind that:

�S/�t = c ∗ Ct ∗ St

which is the expression for a supportive relation-ship. If c takes a negative value, the equation ex-presses a competitive relationship.

It can be shown that growth and learning pro-cesses are processes of limited increase (see vanGeert, 1991). A trivial example of a limitation isthe amount of time a person can spend producingphrases (which is certainly limited by a person’slife span). A less trivial example, relating to growthof the lexicon, is that such growth is limited by thenumber of words available in the language. Thisreasoning leads to the following model, which isthe coupled logistic growth model. For simplicity,it first will be applied to the increase, over time,of the number of W1 phrases uttered by a child,for instance, on an average day. Given that a childmust build up a lexicon from an arbitrarily smallnumber of first words, and thus from an arbitrar-ily small number of possible one-word phrases (H,for holophrases), the number of one-word sen-tences thus will grow over time. This growth is, aswe have seen, most likely proportional to the levelof H already attained. Thus, the increase of H isexpressed as, �H/�t = b ∗ Ht, with b as a propor-tion that specifies the rate of increase. However,as I have just stated, that increase is limited. Thesimplest way of specifying this limitation is that b,the rate of increase, will decrease as H gets biggerand thus approaches its limitations, and this de-crease is proportional to H (let us say the symbola represents this proportion). This relationshipcan be expressed mathematically in the followingway:

�H/�t = (b − a ∗ Ht ) ∗ Ht

We also have stated that H “suffers” from the in-crease of more complex phrases, based on thecombinatorial generator. Thus, the increase in Hover some time t can be expressed as follows:

�H/�t = (b − a ∗ Ht ) ∗ Ht − g ∗ Ct ∗ Ht

Applying a comparable logic to the growth of W23(the C-generator utterances) and W4+ (the S-generator utterances), we arrive at the followingcoupled dynamic system:

�H/�t = (b − a ∗ Ht ) ∗ Ht − g ∗ Ct ∗ Ht

− h ∗ St ∗ Ht

Paul van Geert 195

�C/�t = (i − j ∗ Ct ) ∗ Ct + k ∗ Ht ∗ Ct

− l ∗ St ∗ Ct

�S/�t = (m − n ∗ St ) ∗ St + s ∗ Ct ∗ St

which describes the time evolution of W1, W23,and W4+ phrases. (Remember that the connec-tion between a type of phrase and a generator,for example, W1 and the H-generator, changesover time, which makes the connection betweenthe model and the empirical data more compli-cated.) This is a dynamic system described by athree-dimensional state space (the H, C, and S di-mensions). For simplicity, it can be represented bythree separate one-dimensional time evolutions(the time evolution of H, C, and S separately).

Mathematically, this model is equivalent to thefollowing notation (for reasons of simplicity andwithout loss of generality, we can confine ourselvesto the H variable):

Ht+1 = Ht + b ∗ (1 − Ht/KH) − g ∗ Ct ∗ Ht

− h ∗ St ∗ Ht

with KH the limit level of H (the highest possi-ble level H can attain under the present limit-ing circumstances). This K parameter has an in-teresting property: Since the limit level, specifiedby K, is a function of all the available resources,whether they be internal or external, K amountsto a single-value representation of all the avail-able resources. The fact that it is at all possible torepresent the very complex structure of resources(memory, effort, help given, motivation, talent,available books, etc.) by a single value is a di-rect consequence of the fact that the resourcesare represented in function of the dynamic effectthey have on the growth and stabilization of thevariable at issue (H, in this case).

Finally, the coupled-dynamics model specifiesonly the supportive and competitive relationships,not the conditional relationships, which havebeen left out from the equations to avoid furthercomplication. They can be added to the equationsin the following way (the example is limited to theC-component):

�C/�t = 0 if Ht < HP

�C/�t = (i − j ∗ Ct ) ∗ Ct + k ∗ Ht ∗ Ct

− l ∗ St ∗ Ct if Ht ≥ HP

with HP the conditional value H must have for Cto grow (e.g., if HP is set to 100, it is assumed that achild must have at least 100 different words beforehe can start making two-word combinations in a

productive way, and thus for the combinatorialgrammar to increase in terms of productivity).

The next question is, of course, what the valueof these a, b parameters should be. They can beestimated by mere guessing and then by tryingout the model with these guessed values to seewhether the model based on these parameter val-ues fits the data. However, they also can be esti-mated by means of parameter optimization pro-grams that calculate the best possible set of pa-rameters, the set for which the model producesthe best possible fit with the data. Since the W23and W1 phrases occurring later in developmen-tal time are “absorbed” by the emerging syntacticgenerator (meaning that a late W1 is producedby the S-generator, and no longer by the origi-nal H-generator, for instance), the data must berescaled in order to account for this absorptionprocess. This rescaling was done by normalizingthe data to values between 0 and 1 and by subtract-ing a baseline value. Figure 4 shows the result ofthis optimization procedure, demonstrating thatthe model provides a good fit with the normalizedand rescaled data.

Evidence of Stage Transitions? The dynamicmodel predicts that the probabilities of W1,W23, and W4+ sentences increase and decreasesmoothly. For instance, every day the expectedfrequency of W1 phrases decreases a little bit,whereas the expected frequency of the W23 grad-ually increases. This means that for any episode oflanguage production long enough to arrive at re-liable frequency counts—say, an episode of a cou-ple of hours or days—the expected frequencieswill change smoothly. Since the observed frequen-cies depend on the expected frequencies, whichare in fact probabilities of production, the ob-served frequencies will, of course, fluctuate overdays or hours. These fluctuations must be withinthe bounds predicted by the probabilities of pro-duction, represented by the smoothed frequen-cies that the dynamic model so nicely simulates.So far, all this is an expression of an orderly world.However, we began this article by visiting Alice’sWonderland, and so we will end. It is time to imag-ine some strange events.

Just assume that around the age of 22 months,Pauline produces only W1 phrases on even cal-endar days and only W23 phrases on uneven cal-endar days. Thus, in a truly Alicean way, it is ei-ther Holophrase or Combinatorics for Pauline(around the age of 22 months, that is; thingsmight change afterward). In other words, the two


FIGURE 4Smoothed Data Curves Compared With Curves Generated by the Dynamic Growth Model

generators are in harsh competition, and thereis an age (i.e., 22 months) where both have thesame chance to win the contest and thus to de-termine the sort of phrases that will be utteredduring a particular day. If this were so, we wouldobserve a maximal fluctuation over days between100% W1 and 100% W23 for some time. Thisday-to-day fluctuation will thus be considerablygreater than the fluctuation we expect to findif the generators wax and wane in the smoothway of the dynamic model, which so nicely fitsthe smoothed data. However, this condition of in-creased fluctuation is less Looking Glass–like thanone might expect. In fact, increased fluctuationwas exactly what we found in Pauline’s and Ben-jamin’s data. We found two periods of fluctuationwhere the variability was significantly greater thanshould be expected on statistical grounds, giventhe observed probabilities of the three phrasetypes (details can be found in Bassano & van

Geert, 2007). What this finding means is thatreality lies somewhere between the continuousdynamic model and the discontinuous world in-spired by the Looking Glass magic. That is, as a newgenerator emerges, there is a short period duringwhich the child feels (moderately) “torn” betweentwo generators, and this is the kind of thing oneexpects to find during true developmental transi-tions (see, e.g., Ruhland & van Geert, 1998 for anapplication of catastrophe theory to language de-velopment). To put it differently, the study of fluc-tuations and intra-individual variability may addto our understanding of the underlying develop-mental processes. It also helps us understand how,in development, continuity and gradualness onthe one hand and discontinuity and transition onthe other hand go together (see van Dijk & vanGeert, 2007). To avoid any misunderstanding, thespecific dynamical phenomena discussed in thepresent chapter pertain to specific phenomena in

Paul van Geert 197

L1 development and do not necessarily general-ize to L2 development. However, the principles ofdynamic systems—types of relationships betweencomponents, occurrence of changing levels ofvariability, and so on—are assumed to apply toany developmental process (see de Bot, Lowie, &Verspoor, 2007, for an application of these princi-ples to L2 learning).

CONCLUSION

There are various ways in which dynamic sys-tems theory can be applied to language learningand acquisition, and there are different timescalesand phenomena to be explained. Researchers canresort either to making mathematical models or todetailed analyses of fluctuations in the data. Thesepossibilities still do not exhaust the richness of thisapproach. It must be emphasized that dynamic sys-tems is not an approach that allows for simple gen-eral answers, and there is so much that has yet tobe explored and tested. It is not an approach thataims to abandon the current, established way ofstudying language growth, learning, and change.However, an understanding of dynamic systems iscrucial if we want to go beyond the static or struc-tural relationships between properties or variablesand wish to understand the mechanism of devel-opment and learning as it applies to individuals.The road toward an understanding of the dynam-ics of L1 and L2 acquisition and learning is steepand long and paved with difficulties, but it is aroad well worth taking.

NOTES

1 This entire overview of names and publications ishighly selective in that it does not do justice to manyothers who have made contributions to the field.

2 The term learning-theoretical refers to the viewpointof classical or general learning theory (see, e.g., Mazur,2005).

3 The minimal level hypothesis is related to the crit-ical mass hypothesis in lexical development (see, e.g.,Marchman & Bates, 1994; Bates & Goodman, 1997,2001).

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