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Abstract Whereas Rudolphs (2006a) article provides adiscussion of mathematical models of time, Yamado and Kato

(2006a) present a particular image of timecircular timeas akey feature of an entirely different model of temporality, namely

people life-span narratives. In the present article, I attempt toapply some aspects of Rudolphs models to actual developmentalresearch, namely math learning by children in special education,

social development, attachment and the timing of puberty. Thedynamics of these processes require not only an understanding of

how they unfold in timeand which model of time to choose inorder to adequately describe thembut also an understanding of

the participants narratives, in an attempt to understand theiractions as intentional, with concerns or goals that determine their

sequencing and timing. I conclude by saying that, metaphorical ornot, the models and narratives presented in the key articles by

Rudolph and Yamada and Kato have contributed to at least myown understanding of developmental dynamics by providing

particular challenges for reformulating existing issues.

Key Words attachment, dynamic systems, narrative, nonlinearity,relationships, time

Paul van GeertUniversity of Groningen, The Netherlands

Time, Models and Narratives:Towards Understanding the

Dynamics of Life

Every now and then an image pops up that has the power to changethe perspective on many other images. Lee Rudolphs (2006a) thefullness of time is such an image (for me, at least). It is exactly the sortof image you would expect to have that property: made by a differentsort of artist (a mathematician), partly understandable, partly not, andpartly because of this ambiguity also very intriguing. The second

article in the Thematic Issue on time (Yamada & Kato, 2006a) had, Imust confess, initially a less James-Bondian effect on me. I was onlymildly shaken and only a little stirred. A salient point in this article wasthe mixture of the familiar use of university students as representativesof the human species with the observation that such students are

Culture & PsychologyCopyright 2006 SAGE Publications(London, Thousand Oaks, CA and New Delhi) http://cap.sagepub.com

Vol. 12(4): 511531 [DOI: 10.1177/1354067X06069952]

Continuing Commentary

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deeply embedded in their own culturemodern Japanand bringalong with them a number of narrative elements that fit in less wellwith the secular view on the structure of time and the world that weoften think should be the standard in scientific inquiry.

The difficult challenge is: how to identify the mathematicalapproachwhich really sounds like a disturbing call from far awaywith the narrative approach, the study of which is framed in thefamiliar format of psychological research but nevertheless refers to aworld of complex and interesting meanings.

My predecessors-in-commentary have pointed to various problem-atic issues in the Rudolph and Yamado and Kato articles, which I hope

to address and maybe also to bring closer to some sort of solution. Animportant point regards the metaphorical aspect of the modelspresented in the keynote articles (in the Thorngate, Dzhafarov andMller and Giesbrecht commentaries in particularall 2006). BothThorngate (2006) and Dzhafarov (2006) explicitly ask how develop-mental psychologists will actually use these notions of time in theirdaily research business.

We Simply Have No Time . . .

It is remarkable that the science of psychology, which from a rathernave but therefore also very fundamental standpoint should be seenas the science of individuals changing and evolving across time, hastransformed itself into a science where basically a-temporal relation-ships between variables distributed across groups now abound.However, the dangers of this hybrid breed lurk around every corner.Since the majority of professionals in psychologylet us take my ownfield, developmental or educational psychology, as an exampleareconfronted with what happens in individuals, they automatically tendto translate the a-temporal group-based statements into statementsabout events.

Heres an example that just occurred to me the other day. A studenthad done an analysis of videotaped math lessons in a class for childrenfrom a school for special education and had looked at the successionof on-task and off-task happeningswaiting for help and getting

help for individual children. According to the literature she hadcollected, there exists an inverse association between off-taskbehavior and mathematical competence. That is, if one measurementis made of the off-task behavior and math skill over a large group ofstudents, higher scores on math skill tend to associate more clearlywith lower off-task scores for examples of such analyses, see (Flood,

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Wilder, Flood, & Masuda, 2002; Gest & Gest, 2005). However, as simplevisual inspection of the off-task versus math skill graph readilyshows, it is not difficult to find pupils with quite good math skills andrelatively much off-task behavior. That is what is usually found if astatistically significant correlation is mentioned.

The studentand many settled professionals in psychology, for thatmatterwould, under normal circumstances, tend to interpret thisnegative-association claim in terms of a temporal event statement ofthe kind this pupil shows much off-task behavior; his math skills arenot going to develop as wanted. This is the sort of tacit, automaticinterpretation that we usually make. If explicitly asked, anyone who

had reasonable degrees on his or her statistics and methodologycourse will say a correlation is not a cause and will say there are excep-tions to the rule. But that neither solves the problem nor reduces theautomaticity of the interpretation. The point remains that there is noautomatic or obvious transformation from the association-type ofclaim specified over a group to an event-type of statement we need ifwe are confronted with real individuals in real contexts. (SeeMolenaar, 2004, for an explicit statistical discussion of the distinctionbetween group-based and individual-based relationships betweenvariables or properties.)

However, the student in question had observed five children in theclass several times in a row, had analyzed their math workbooks, andhad background information from the teachers about each of thestudents. For the student, the off-task behavior (or whatever else) andmath skills are now put in an entirely different relationship, namely therelationship of co-occurring events on different time scalesthe time scaleof a single math class and the time scale of the school year. Of evenbigger importance is the fact that the relationships now hold for indi-vidual pupils as they interact with their teachers and with each otherover the course of the half hour of the math lesson, or the course of theschool year.

It became apparent that the relationships of co-occurring time scalesconstitute meaningful individual temporal patterns. One pupil, forinstance, who is very good at math in that he can solve the workbookassignments much more rapidly and effortlessly than his fellow

pupils, shows a considerable amount of off-task time, simply becausehe does not have to ask for so much help from the teacher as do theother pupils, which leaves him enough time to make his math assign-ments and meanwhile reasonably amuse himself during his self-pacedoff-task time. What we see here is a dynamic pattern that governs thecurrent pupils math learning in the socially situated context of his

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math classes and school. (For a more extensive discussion of thisexample, see Van Geert & Steenbeek, 2006.) The dynamic patternseems to result from a basically dialectical process, a process ofopposing intentions, namely the intention to do ones school assign-ments and the intention to engage in other sorts of pleasurable activi-ties, such as chatting with ones friends or just dozing off in a way thatis not too conspicuous. (These pupils spend quite some time in theschool bus and thus have to get up early in the morning. For a modelof opposite intentions in learning contexts, see Boekaerts & Corno,2005; Boekaerts, de Koning, & Vedder, 2006). The importance of dialec-tic tendencies, that is, tensions between opposing forces such as inten-

tions, is a major feature in Hoods commentary. Hood (2006) explicitlyrefers to approachwithdrawal dynamics, which is exactly whathappens in the tension between the pupils on- and off-task tenden-cies. This on- and off-task tendency produces a sort of fractal patternin time, as Hood calls it. If I understand Hoods notion of the termfractal right, it should refer to a certain nested property of the pupilson- and off-task dynamics. That is, a particular pupil shows episodesof mainly on-task alternated by mainly off-task activity, but each ofthese main episodes is cut up by smaller episodes of the oppositeactivity (e.g. the on-task episode shows various small episodes of off-task action, which, if we could look into the pupils consciousness, arethemselves cut up into smaller, alternating units). These alternationsare related to the issue of fluctuation, which is a major aspect of realaction and development, and which is also strongly emphasized inHoods commentary. It is likely that the individual action patterns ofthe pupils show their own variability pattern, that is, their owncharacteristic fractal dimension. (See, for a discussion on the import-ance of variability in developmental data, Van Dijk & Van Geert, 2006;Van Geert & Van Dijk, 2002.)

Time is a central feature of the dynamic pattern; more precisely, it isa pattern in time. A formal description of such a pattern would, in alllikelihood, be more similar to some sort of formalized narrative orscript than to the sort of mathematical equation that is commonlyidentified with the notion of a model. It is also interesting to see howthis pattern fits in with the narratives of some of the important protag-

onists, namely the pupil himself and his teacher. In standard psycho-logical research, the pupil himself is seldom asked about his narrative,his view on the dynamic pattern that is so characteristic of his doingand leaning math in the school context, or any other aspect of his life,for that matter; though there are exceptions, for instance when studentsare explicitly asked for their motivations, or why they act as they do


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(see, e.g., Boekaerts & Corno, 2005; Boekaerts et al., 2005; Singer, 2005).In this particular case, the information available to the student alsocontained an informal check of the teachers narrative, that is, thecoherent story the teacher has about the pupil in question, the way hedoes his math and the way he can be taught to do so. This narrative isundoubtedly of great importance for any attempt to try to understandthe dynamic pattern of the current students math learning and thecurrent teachers teaching. The teachers narrative is not nave. Theseare highly competent, well-trained teachers whose narratives incorpo-rate a good deal of psychological theory and who themselves are scaf-folded by an extensive framework of professionals and experts,

including the researchers who come to their classes to study learningand teaching processes. From a point of view of cultural or socialchange, these narratives, which form the expression of educationalprofessionalism and skill, are changing all the time, incorporating indifferent ways the narrative of scientific research and its reception inthe broader framework of society.

Let us recapitulate. In the example we saw a student leaving thedomain of statistical associations between properties in a sample orgroup of individuals and entering a new realm, that of dynamicpatterns applying to individuals. First, the dynamic patterns implytime as a space, which is Rudolphs central argument (Rudolph, 2006a).That is, in order to specify the dynamic logic of a pattern involvingsocial interactions between a pupil, other pupils and a teacher andexplaining the relationships between the chosen variablesoff-task,on-task, asking and receiving helpyou need some sort of descriptivespace in which time is a crucial property. (I will avoid saying that it isa variable of such a space, because its relationship to constituting thatspace is considerably more complicated than being just an orthogonaldimension.) That is, you need a space in which those elements can beordered in such a way that they constitute the pattern. I will addressthe issue of order in time in the next section. Second, the dynamicpatterns are special and typically human in that they contain self-images, that is, reflections on their structure and properties by theparticipants of the dynamics. These self-images are dynamic self-images. That is, they are embodied in the form of the narratives of the

main protagonists. Those narratives are changing across various timescalesfrom the short to the long termand are essential functionalaspects of the dynamic pattern in question. I will discuss this aspect inthe third section.


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Ordnung Muss Sein . . .

Order and Sequence in Time and Narratives

A central component of Rudolphs approach to time is the notion ofsequence or order of elements or members of a set. One may followDzhafarovs (2006) critique and question to what extent Rudolphsmodel is indeed a model of time and not of peoples concepts of time.Maybe the question can be resolved by pointing out that Rudolph aimsat describing psychological time, which I understand as a personsexperience of order and duration of events that are meaningful fromthat particular persons point of view.

The basic types of ordering are total or linear order, partial order andcyclical order (see the section on Time as a Kind of Space, Rudolph,2006a, pp. 169171). The Generative Life Cycle Model referred to inYamada and Kato (2006) is an example of how narratives have incor-porated a combination of linear with cyclical ordering.

From the above, it is clear that the elements that are ordered in timeare elements defined by a particular persons acts of attention. I wouldtend to identify the content of these acts of attention with events, thatis, meaningful unities from the point of view of the person, which havea fuzzy status in that their boundaries are not crisply defined (hencethe states of ambivalence that alternate with the states of attention).Events form nested structures (i.e. consist of main events andcomponent events, up to the acts of attention) and thus require a morecomplex time structure than that of a single non-hierarchical sequence.

If both linear and cyclical ordering are possibilities in psychologicaltime, we need a psychological systema particular personwho canboth make distinctions and see similarities. However trivial thisrequirement may seem, it is highly relevant for a psychology spanningthe distance from the psychophysical and biological to the meaning-bearing and cultural aspects. In this regard, it is interesting to see thatone of the commentators, Dzhafarov (2006), has made a major contri-bution to the mathematical modeling of the psychophysics of distinc-tions, and thus to specifying how making perceptual distinctionsdefines psychological time as a sequence of (distinguished) events. AsDiriwchter (2006) makes clear in his commentary, the distinctions that

deveopmental scientists make on an entirelty different level of aggre-gation, namely in their observations of developmental trajectories, arealso based on their situatedness.

A person cannot experience a cyclical pattern if that person is unableto see the similarities among the corresponding points in the cycle, andneither can a person experience a sequence of events if the person


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cannot see differences. Basically, if any day is psychologically speakinglike any other day, there is no time in the linear, progressive sense ofthe word. There is, at most, a short-term cycle, namely the day cycle.Clinical problems consisting of a persons inability to make sense ofand give meaning to life might be closely related to an aberration inthe persons perceiving and experiencing time.

The way people see differences and similarities is deeply affected bythe culture they live in and by their developmental history and future.Culture and biology cooperate in the cyclical organization of processesand events. Cycles occur within and across individuals. From thephysical-biological standpoint, the life cycle, which forms the topic of

Yamada and Katos article, is a cycle that repeats across individuals.However, for religious beliefs that postulate reincarnation, the cyclicalnature of life is also a biological fact for the individual (although theform of reincarnation postulated need not reduce to a nave biolo-gism: see, e.g., the way reincarnation is defined in various forms ofBuddhism). The fact that reincarnation is not a scientific truth(whatever that may be) does not in any way reduce its causal orconditional effect on the life cycles of many people, namely thosepeople for whom the narrative of reincarnation or life after death is anexistential fact and whose lives are, to various extents, determined bythis narrative.

Cycles abound in the cultural organization of our lives. I teach andwill thus be confronted by the cycle of the classes I teach, I haveworkdays and workweeks and years that contain cycles of work andholidays. These are examples of relatively formalized cycles that areincorporated in cultural and societal practices, artifacts, habits, and soforth. Some cycles are more accidental and informal. Examples arefriendships or romantic relationships with other persons. They mighthave a cyclical character, with different frequencies and intensities fordifferent people.

Personal Relationships and Attachment: An Example of Complex

Nested Cycles

Let us take the issue of personal relationships as an example of thecomplexity of psychological time, related to the fact that cyclical

processes intertwine or become superposed, thus making the cyclicalcharacter of the constituent processes less obvious or less easy to recog-nize. The example I will focus on is that of attachment, a popular themein developmental and clinical psychology and which can be defined asthe tendency to seek closeness to another person and feel secure whenthat person is present (as the Wikipedia tells us). Thus defined, the


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notion of attachment must by necessity also entail separation, forminga cycle of seeking proximity, then separation (e.g. because one seeksproximity to an other person), then proximity, and so forth (Field, 1996;see also the approachwithdrawal theme discussed in Hoods [2006]commentary). In fact, the whole notion of the strange situation, whichis the default experimental set-up to study attachment in youngchildren, is based on this implicit cyclical nature of attachment. The factthat attachment develops at a particular age in babies is related to thedevelopment of their ability to make distinctions (among people whoare their caregivers or not) and see similarities (recognize the return-ing caregiver as the person to whom they are attached). Attachment is

a process that extends across the whole life cycle (see, e.g., Klohnen,Weller, Luo, & Choe, 2005; Paterson, Field, & Pryor, 1994).

The question is of course how this extension throughout life shouldbe understood from the point of view of time, and psychological timein particular as discussed by Rudolph (2006a). The more or lessaccepted approach is to conceive of some identical kernel or entitysuch as an attachment model or an attachment representationthatemerges in early childhood and is conserved over a lengthy period oftime, or eventually transformed as a consequence of experiences(Shaver, Belsky, & Brennan, 2000; Treboux, Crowell, & Waters, 2004;Waters, Merrick, Treboux, Crowell, & Albersheim, 2000, 2003). From adynamic point of view, and in the perspective of Rudolphs model oftime, what actually occurs is a nested sequence of cyclical patterns, thatis, short-term patterns of proximity-seeking and separation and of theassociated emotional appraisals embedded in patterns of continuouslyextending cycle length. Examples of the latter are patterns of encoun-ters with relevant persons (caregivers, peers, friends, romanticpartners, etc.) and life-cycle related patterns of growth (increasingindependence during childhood, sexual maturation during puberty,growth into adulthood, the family cycle, etc.). The whole structurelooks a bit like Rudolphs Balinese calendar with the structure ofnested cycles of unequal length. The question is: what in this nestedset of cycles is actually conserved? What is the similarity? This is theanalog of the standard question about the stability of the attachmentmodel or attachment representation, asked in terms of the dynamic

patterns that occur across time.My assumption is that what is similar in these cyclical patterns of

proximity-seeking and separation is different all the time. That is,between one such cycle and the next (or some later one) there is likelyto be much similarity, but what is similar for cycle (a) and (b) is not thesame thing as what is similar for cycle (a) and (c), so to speak. The


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similarities and the differences are determined by the dynamics of thepattern we call attachment. That is, from a mathematical or modelingpoint of view, the similarities in attachment patterns across an individ-uals life trajectory are both a consequence and a cause of the dynamicsgoverning the attachment process over the lifetime. The reason whythe same thing can be both a consequence and a cause of itself lies inthe presence of time. Because of the way attachment patterns dynam-ically unfold, certain similarities with previous such situations emergethat become part of the persons history and narrative and thus co-determine the dynamic unfolding of future attachment patterns. Anordered sequence of attachment events exists where each event has

both similarity with and difference from other events, and that arecharacterized by an iterative structure (the outcome of the previousevent plays a role in the unfolding of the next one).

Research on the life-span development of attachment often refers toentities such as attachment models or attachment representations andcaptures these models or representations by means of questionnaires,interviews or observation scoring systems. Questionnaires and inter-views are (relatively) formalized ways for obtaining an image of apersons personal narrative. However, in the questionnaire context, thepersons narrative is taken out of its genuine temporal frame and trans-formed into some a-temporal reflection (eventually with distortionsknown as measurement errors) of the other a-temporal entity, theattachment model or representation. That is, there are basically twodifferent views. In one, the questionnaire is a formalized narrative thatis treated as an image or recording of a particular entity, the attachmentrepresentation or model. In the other, there is a narrative dynamics anda dynamics of attachment events (of proximity and separation) andthese dynamics are coupled. Thus, the narrative itself is an event, orbetter a series of eventsfor instance dialogs among friends orpartners, personal stories or complaintswhere one event, that is, oneparticular story, begets another event, that is, a later or successivepersonal story that reflects on the persons own feelings and opinionsregarding his or her attachments to other people. The coupled natureof the narrative and of the attachment dynamics follows, in one way,from the fact that these narratives themselves play an important role

in the dynamics of attachment across life, depending on their form,frequency, emotional intensity and the way these narratives arereceived by others and thus take part in dialogs. However, those narra-tives have their own rules and conditions, and their dynamics are notnecessarily similar to the dynamics of the events to which they referand relate. For instance, for some persons or for some periods of a


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persons life, the narrative may be far behind or be far ahead of theactual attachment patterns it refers to.

This phenomenon might relate to the phenomena of partial orderingor complex ordering that Rudolph refers to in his article. Anotherphenomenon, which is crucial for the issue of attachment as well asdevelopment, and which illustrates the developmental effects of partialorderings of time, is that of social encounters, that is, all forms of socialinteraction that require particular adaptations from the participantsand that can have long-lasting effects. (By encounters I do not refer toanything that is primarily hostile: encounters can be positive, neutralor negative.)

Social Encounters, Partially Ordered Time and Trees

Attachments require encounters. The most basic and far-reachingencounter is that between a newborn and his or her parents. It is anexpected, literally self-made encounter with somebodythe neonatewho is, on the one hand, entirely new, that is, a new person, and, onthe other hand, also familiar, that is, the product of the parents genes,and thus to a considerable extent familiar and recognizable. (For aparticularly illuminating account of the tensions between various timescales in social development, see Cairns, Garipy, & Hood, 1990.) Thebirth of a childmore precisely but less culturally relevant, the concep-tion of a childis a literal branching event in time, the production ofa new life that begins, by a mature life that can procreate. It is also, likeany other major life event, a good example of a partially orderedacyclic graph (see Figure 4 in Rudolph, 2006a, p. 194, right-handimage). More concretely, an event such as the birth of a child (or othermajor life event) does not only branch towards the future, but alsobranches back into the past. It can almost literally change the past, inthat it brings certain past events to the fore that make sense withrespect to the current major life event, or give sense to that event. Aless happy example is that of an adolescent or young adult goingastray, thereby almost literally redefining his and his parents past, inthe sense of bringing events to the fore that foreshadowed theproblem or educational actions of the parents that caused it.

From a viewpoint of ordered events in time, the birth and growth of

a child and its being raised by its parents require the coordination, andthus temporal ordering, of two fundamentally different sequences,namely that of beginning life and that of nurturing life. Attachment, inthe classical sense, provides a good example of the almost paradoxicalordering of events. Under normal circumstances, birth will beaccompanied by the sudden emergence of a strong parental bond or


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attachment (the suddenness will be different for the mother than forthe father, who has not known the physical intimacy with the newbornchild during the months of gestation). However, the childs attachment,in the true sense of the word, will emerge only later, approximatelyaround the age of 6 months. In cultural and economic circumstanceswhere a high level of good-quality care is the norm, we tend to see theprocess of growth and education as a relatively seamless and smoothprocess, that is, as a single sequence of events that after about twodecades leads to a happy and competent young adult who will thenseparate from his or her parents to start the cycle anew. Problematicevents and problematic relationship between parents and children are

seen as exceptional, in that they are not expected as the normal case(a particularly interesting case in this regard is the discussion on thepotential aversive effects of total time spent in day care: see Belsky,2002; NICHD, 2006). Instead, they are seen deviant and serious and aconsiderable amount of social effort is invested in treating them. But inreality, the problematic nature of the relationship may be more naturalthan the unproblematic onenatural in the sense that it is more likelyto occur, given the problematic dynamics of two (or more) very differ-ent time lines or event sequences: namely, on the one hand, the timeline of the growing child, and, on the other, the time line of the adultparent. The problematic nature becomes more apparent when theresources are scarce, when there is poverty or lack of education. Theseconditions put additional pressures on the timing problem, that is, theparent and childs problem of reconciling the tempo and timing ofcrucial events.

Child labor is a characteristic example: the pressures of survivalcompel the parent to force the young child into a role it is not yet readyfor and that will seriously hamper the further sequence of its growthand development, for instance by denying the child the right to school-ing and education. The choice for child laborwhich is hardly a choicefor the parents or the childrenis again an example of a bifurcation,that is, the creation of a particular node on the tree structure of time,as Rudolph (2006a) describes it. An event such as the incorporation ofchildren in the labor force is not a complete bifurcation from the formalpoint of view. The potential event sequence it creates is more like the

a-cyclical graphs, where certain branches might fall back on others.That is, children who, in general, encounter adverse circumstancesearly in life, such as being put to work, but also being abused or other-wise maltreated, can eventually at a later time be given the chances foreducation and growth they had missed at an earlier age. However, thebranching back is less likely and hard, and sometimes impossible.


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Another example of the complexity of timing and temporally orderedevents that is also of considerable importance to the issue of attachmentbeyond childhood is the timing of puberty, especially in girls. Sexualmaturation is an extremely important event. From an evolutionarypoint of view, it must occur at the right time, which means that it mustrun parallel with the timing of additional events required for success-ful reproduction. There is a large body of evidence showing that thetiming itself is a result of events that preceded it, and that it is in itselfa cause of later events, for better or for worse. For instance, pubertaltiming is responsive to ecological conditions earlier in life, which mightbe beneficiary as well as adverse. Such conditions may lead to either an

inhibition or an acceleration of the onset of puberty. The effects of earlieror later puberty, on the other hand, are non-linear, being most notablein the extremes (both positive and negative; for an overview, see Boyce& Ellis, 2005; Ellis, 2004; Ellis, Essex, & Boyce, 2005). The problematic orbeneficiary effects of early or late timing of onset or puberty are not amatter of a linear ordering of events, as if early menarche itself wouldlead to adverse effects on other variables, such as psychosocial adapta-tion or birth-weight of the first offspring. What matters is that suchtiming issues cause a problematic (or beneficiary) coordination, that is,mutual ordering, of different time lines or event sequences. Examplesof such event sequences are timing of romantic dating and first sexualintercourse, the nature of the partners and potential providers, whomay or may not be able to support the offspring, and so forth. From thepoint of view of the participants, that is, the people who are actuallyinvolved in these issues, timing relates to the subjective order of mean-ingful life-events in their own life and in the life of other persons withwhom they are intimately connected (educators, parents, peers,romantic partners, etc.). The complexity resulting from this confluenceof many events with many different, eventually contradictory andemotionally-laden interpretations is a good example of the Ryokoconcept discussed in Yamada and Katos reply to the commentators(Yamada & Kato, 2006b).

The Change of Stories and the Story of Change

Time is crucially related to dynamic systems, that is, systems where oneearlier event generates some later event (for a definition, see Weisstein,1999). As Rudolph (2006a) also emphasizes, a dynamic can refer tochange as a flow, that is, continuous change, or as a mapping of eventsonto one another, that is, as a discrete event pattern. The simplestpossible expression of a dynamic system is x/t =f(x), in the case of


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a flow or xt + 1; =f(xt) in the case of a map (f(x) means is a function ofx, andfis the so-called evolution law or evolution function).

The notion of a dynamic pattern that I introduced in the discussionof learning math is closely related to a central notion of dynamicsystems theory, the attractor. As Rudolph rightly remarks, the notionof attractor requires a form of (approximative) similarity. Thus, ifapplied to psychological time, it requires a cultural and psychologicalsystem of perception and categorization that can see differences andsimilarities. This system is a product of cultural and historical changeand evolution, but also of individual development in a cultural context.The question that I shall briefly discuss in this section deals with the

relationships between temporal and life-span narratives, as illustratedin Yamada and Katos article, and dynamic systems.

If applied to the event sequence model used in the precedingsections, a dynamic systems model would take the form et + 1; =f(et),meaning that a later event at time t + 1 is some function of an earlierevent at time t. (For a discussion in the field of developmental psychol-ogy, see Van Geert & Steenbeek, 2005.) Rudolphs claim that suchevents are in fact acts of attention alternated by states of ambivalencerefers to a particular property of events as they are interpreted byhuman minds. If we take the mind to follow a stream of consciousness,it is obvious to conceive the mind as operating in some form ofcontinuous time. Events, or the acts of attention that mentally specifythem, are discrete. However, there is no intrinsic incompatibilitybetween the two, because the mind seems to work that way in general.For instance, the color circle is clearly a continuous (and cyclical)phenomenon. Yet the way human beings perceive and speak of colorsis discrete, but it is discrete in a special way (Franklin, Pilling, &Davies, 2005; Goldstone, 1995; Saunders & van Brakel, 1997; seeRudolphs reply to the commentators, Rudolph, 2006b). What humanbeings do is to divide the continuous dimension into overlappingranges of discrete colors. The overlap is crucial: this is where the colorcategories are seen as ambiguous (of course, new color terms can beinvented to categorize the ambiguous reasons, but they will in theirturn consist of smaller overlapping ranges, again with ambiguousoverlaps). The perception of phonemes works the same way, namely

by discrete categories with intermediate ambiguous regions. Acts ofattention, according to Rudolph, seem to work in the same way, in theform of clearly discernible acts of attention with intermediary states ofambivalence or ambiguity. However, Mller and Giesbrecht (2006) findthis notion of acts of attention as discerning the primary events ofpsychological time too abstract and too intellectualistic (p. 226). They


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suggest an approach to such elementary events based on Merleau-Ponty, which puts the intentionality of consciousness to the fore. Inten-tionality or goal-directedness implies a reference not only to thepresent, but also to the past and future. This approach fits in with theexample that I gave at the beginning of this article, which referred tochildren in special education doing math. In order to understand thedynamics of their actions, we need to know their interests andconcerns, which are in a fundamental sense dialectic in nature and leadto a characteristic fragmentation or sequencing of their actions (see theexample of the fragmentation of on- and off-task actions during thecourse of a math lesson).

The question is: will we ever have a mathematical theory thatexpresses the evolution law of events, or acts of attention, for thatmatter, that is, will we ever have a dynamic systems theory of themental and of the development of the mind? The events that constituteour world are of incredible complexity, and it seems very unlikely thatsuch a theory will ever be achieved. However, to a certain extent, wealready have such a theory. It is a symbolic theory, it is highly algorith-mic (its principles are learnable in finite time) and it is used to predictand rule the stream of eventshuman actions and experiences ingeneralthat it pretends to explain. This theory is nothing else thanwhat is called, in somewhat disparaging terms, folk psychology,nave psychology, theory-of-mind, and so forth. (There is a lot ofdebate about the status and function of folk psychology as a theory ofthe mind: see, e.g., Stich & Ravenscroft, 1994.) It summarizes the mean-ingful events in a (relatively) short list of fundamental descriptiveterms (names of emotions, of psychological states such as desires, etc.),it contains a few powerful evolution terms, such as the notions relatedto intentions and desires, actions to fulfill those intentions, and so forth.It accounts for subtle and less subtle variations among cultures in theway psychological events are interpreted and has a number ofuniversal features that reflect the universal features one would expect,given the fact of biological universality under cultural variations.

From a dynamic point o view, this theory takes the form of anongoing sequence of self-related narratives and self-related dialogs.These narratives and dialogs take the form either of inner speech or of

overt dialog and communication with others. They specify the personsintentions, views of the past and the future, the persons evaluations,positions with respect to others, and so forth. More precisely, they forman ordered sequence in time that expresses a specific dynamics, that is,person-specific way of iterating the narratives and dialogs (in the sensethat any next dialog is to some extent a function of the previous ones).


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It is important to note that these narratives and dialogs follow theirown dynamics. (See Miller, Cho, & Bracey, 2005, for an example in thecontext of working-class children.) They are representations of whathappens to the person, but they are very special representations in thesense that they specify what is important for the person (or even morespecifically, in the sense that they specify what is important for theperson to specify). They are embedded in the dynamics of the personsaction and attention in general and thus interact with other dynamiccomponents or aspects determining the persons life-cycle events.There will be many situations in which the dynamics of action and thedynamics of narratives in the person are incoherent or even collide

(think, for instance, about analyses of behavior showing how behavioris determined by simple imitation of others, whereas the personhimself ascribes the behavior to his own free will; Chartrand & Bargh,1999; Ferguson & Bargh, 2004). However, these collisions, or whateverit is that marks the relationship between the dynamics of the narrativeand the dynamics of anything else that influences human action,sustain the process of action as a whole and explain the unique natureof the dynamics of human beings.

A generalization of this viewpoint can be found in an article byMichail Zak (2000). Zak is a physicist working at the Jet PropulsionLaboratory of the California Institute of Technology in Pasadena. Hisapproach is entirely from the viewpoint of building adequate mathe-matical models of dynamics of living beings. His main point is that inorder to model the dynamics of living beings, that is, the patterns oftheir behavior over time, one cannot confine oneself to what he callsNewtonian properties. Living beings possess a crucial non-Newton-ian (his words) property, which is a self-image, however primitive thatself-image may be. This self-image is in fact a probability space, that is,a space defining the relevant probability dimensions for the organismat issue, for instance the probability of obtaining food in a particularenvironment following a particular action. The probability spacefundamentally relates to the notion of time, since it defines theorganisms future (in the sense of probabilities). Zak argues for a differ-ence between the motor dynamics and the mental dynamics of theorganism, and builds an explanation of intelligent behavior in general

on the basis of the coupling of these dynamics.The dynamics of narratives and dialogs is, in my view, an example

of the mental dynamics (or maybe it is the mental dynamics). As statedbefore, narratives and dialogs need not be true images or reflectionsof the motor dynamics. If that were so, they would not have thefunction they have now and which is crucial for understanding the


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dynamics of the entire system, which is the function of exploring theprobability space that is of concern to the organism in question.

Cutting up the Developmental Pie

A major theme addressed in Rudolphs and Yamada and Katos repliesto the commentators concerns the distinction of states or events indevelopmental time. That is, what is the set of possible, developmen-tally meaningful differences between individuals? When a personincreases in ageconsider any time scale at which this increase iscounted, from milliseconds to decadesthe person changes in all

different sorts of ways. His lips move when he speaks, he eats asandwich, reads a book, goes to college, raises a family . . . But whichof these changes are developmental changes? That is, which of thepossible distinctions one can make among possible states across timeare developmentally relevant distinctions?

In a series of articles that I published some twenty years ago (VanGeert, 1987a, 19877b, 1987c, 1988), I argued that this is exactly whatclassical theories of developmental do: they give you a recipe to cutup the developmental pie (basically by telling you how the pie isbaked, but the metaphor is clearly carrying us away . . .). On the super-ficial side, those theories often provide a model of stages, such as thesensori-motor, the pre-operational and the concrete and formal opera-tional stages of Piaget. On the more subtle side, these stage distinctionsare based on descriptive dimensions that enable one to make develop-mentally relevant distinctions, for instance the dimensions ofexternalinternal, actionaloperational and concreteformal which, inmy view, underlie the relevant distinctions made in Piagets theory. Imyself used simple graph theory to formalize the major properties ofthe ways in which developmental theories (at least a subset of theoriesthat drew my attention at that time) cut up developmental time, thatis, describe developmentally relevant distinctions. If I understand itright, simplicial complexes, which Rudolph amply discusses in hiscommentary, are generalizations of graphs, and thus may be applieddirectly to the sort of analyses of developmental sequences that Ipresented in these eighties papers. It is interesting to note that, once we

have a formal way of dimensionalizing the space of developmentaltime, we can not only distinguish among the states of a system that wethink are developmentally relevant, but also get the dynamics for free.That is, by cutting up the developmental pie into pieces, we implicitlydefine potential developmental pathways.


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In their reply to the commentators, Yamado and Kato (2006b) explic-itly say that they dont think the sort of graph analysis that I presentedin 1988 is a good idea: it is too formalistic and does not sufficientlyaccount for the essential properties of the living developing subject. Imust say that, after about twenty years, I have come to sympathize withtheir viewpoint. Oppositions, ambiguity, fuzziness, paradoxes, they areall essential features of the complex system of development. (See VanDijk & Van Geert, 2005, and Van Geert & Van Dijk, 2003, for discussionsof how fuzziness and ambiguity can be used as methodological andstatistical principles in empirical developmental research.) However, Isee it as a fundamental feature of the complexity of human develop-

ment that it can be represented both by formal mathematical structuresthat emphasize the underlying, eventually rigid logic and by narrativestructures that emphasize the ambiguity and multiplicity of thedevelopmental process. They are both sides of the same phenomenon.

By Way of a Short Recapitulation

When I was a young student at the University of Ghent in Belgium Ihad a professor of logic, Leo Apostel, who had once cooperated withJean Piaget and who had a (modest) cult status because of the original-ity of his use of logic for a better understanding of human cultures andbehavior. I joined an informal work group around Apostel. (Iremember one of the other members was a priest who practicedexorcism according to the Roman Catholic rites as part of psychiatrictreatment.) The aim of the work group was to see if formal logic couldbe used to understand the notion of sacrifice as it appears in many reli-gions all over the world. What these meetings of long ago have taughtme is that mathematics, in this case logic, is a very rich tool that can beapplied in many diverse ways to whatever subject or topic you areinterested in. Just like any tool (e.g. a standard DeWalt percussiondrill), you need to learn how to use it, but you dont really need tounderstand its mechanism (what is under the yellow-and-blackhousing).

Mathematicsfor instance the mathematical model of time presentedby Rudolph (2006a)helps one to better understand and represent

psychological change and development. The application of a mathemat-ical framework is neither a final step nor a step that should only bemade if the available data are hard and thoroughly quantitative. It isnot a final step in the sense that a mathematical view on a particularphenomenonbe it a simple dynamic model, a formalization or the


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application of mathematically founded conceptscan also be thestarting point for further inquiry, or an intermediary point in the senseof a first attempt at putting ones thoughts and findings in a particularformal framework, or indeed a final step, after the data have beencollected and processed. The use of a mathematical framework ormodel to understand the utterly complex phenomenon of the humanlife span and the function of something personal and relatively fuzzyas a narrative or dialog provides a good example of the fact that math-ematics is a tool for understanding reality and is thus applicable tovirtually anything that can be conceptualized, observed or presumed,at any stage of psychological inquiry. The ultimate criterion of such

application of mathematics is not whether it contains some final truth(or some modest version of that) but whether it constitutes a step in along series of steps that lead to better and deeper understanding of thephenomena that form the content of the pheomenena that we are inter-ested in. Some commentators (Dzhafarov and Thorngate) are concernedabout the metaphorical nature of the models provided in the keynotearticles. However, a metaphor can be a tool like any other. It dependson whether it helps progress our understanding or functions as a firststep towards more elaborate and more functional tools. If metaphorscan act as frames that help us order and structure the experiences thatfeature in our research questions and as challenges that stimulate andshape further thinking, they will have served important goals.

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Biography

PAUL VAN GEERT studied Psychology and Educational Sciences at theUniversity of Ghent, Belgium, and was appointed Professor of Developmental

Psychology at the University of Groningen in 1985. His main field ofinvestigation concerns the application of dynamic systems theory to variousaspects of development, including language, cognition and socialdevelopment. He has a strong interest in the theory of developmentalpsychology and in the methodological consequences of applying dynamicsystems thinking to studying development. He is a former fellow of theNetherlands Institute for Advanced Studies in the Humanities and SocialSciences and the Center for Advanced Studies in the Behavioral Sciences,Stanford, California, has held visiting professorships at the universities ofParis and Turin, and is an active member of the Mind-Brain-and-Educationgroup founded at Harvard University. ADDRESS: Paul van Geert, Professor ofDevelopmental Psychology, The Heymans Institute, University of Groningen,Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands.[email: [email protected]; [email protected]]


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