universals and specificities in the structure and of quantitative-relational thought

8/12/2019 Universals and Specificities in the Structure and of Quantitative-Relational Thought

http://slidepdf.com/reader/full/universals-and-specificities-in-the-structure-and-of-quantitative-relational 1/37

INTERNATIONAL JOURNAL OF BEHAVIORAL DEVELOPMENT, 1996, 19 (2), 255–29

Universals and Specicities in the Structure and

Development of Quantitative-relational Thought:

A Cross-cultural Study in Greece and India

Andreas Demetriou

Aristotelian University of Thessaloniki, GreeceAvinash Pachaury

Regional College of Education, Bhopal, India

Yiota Metallidou, and Smaragda Kazi Aristotelian University of Thessaloniki, Greece

This study investigates the structure and development of quantitative thoughtin Greece and India. A total of 297 Indian subjects and 269 Greek subjects,aged from 10 to 16 years, were examined by a battery addressing their ability toexecute arithmetic operations, a battery addressing their proportionalreasoning, and a battery addressing algebraic reasoning. The items in eachbattery addressed four developmental levels. Conrmatory factor analysisshowed, as predicted, that the same model is able to account for theperformance of both cultures. Some differences were observed in the relative

strength of the various abilities. However, the developmental inter-patterni ngof abilities was generally the same in the two cultures. There were nodifferences in capacity-depe ndent sequences but there were some differencesin strategy-depen dent sequences. These ndings are discussed in the context of our theory of cognitive development.

This paper presents a study designed to test our theory on the effect oculture on the structure, developmental inter-patterning, and pace o

cognitive development. We view our theory as a meta-Piagetian theory ocognitive development. That is, it is a theory that was developed with the ai



256 DEMETRIOU ET AL.

spheres of thought with which the theory is concerned. This is the so-callequantitative-relational specialised structural system. However, this study

part of a larger programme of cross-cultural studies that aims to test theffects of culture on all the constructs and dimensions of the developinmind, as specied by the theory. Thus, in this introduction, the premises othe theory about the general organisation and dynamics of mind will bpresented rst. A brief discussion of the relations of our theory with othesimilar theories will be attempted. Then discussion will focus on thquantitative-relational specialised structural system. The assumptions of ththeory in regard to cultural inuences and the ensuing predictions will b

presented next. This extended introduction is necessary, in order to enablthe reader to place the present study in the broader context of our researcprogramme and the programme itself in the broader context in which it wadeveloped.

The Architecture of Developing Intellect

According to our theory, the human mind develops across three fronts. Th

rst involves a set of Specialised Structural Systems (SSSs) that enable thperson to represent, mentally manipulate, and understand specic domainof reality and knowledge. Five SSSs were identied: (1) the qualitativeanalytic; (2) the quantitative-relational; (3) the causal-experimental; (4) thspatial-imaginal; and (5) the verbal-propositional.

Each of the SSSs is dened as a multistructural entity which is composeof several distinct but closely related knowledge acquisition, representation

and processing schemes, skills, or operations. These components arthought to be integrated into an SSS under the guidance of a set of principlewhich govern the functioning and organisation of our cognitive apparatuNamely, the principles of: (1) domain specicity; (2) proceduracomputational specicity; (3) symbolic bias; (4) subjective equivalencedistinctness of cognitive components; and (5) developmental variation. Thas, it is assumed that if several component abilities are concerned with th

same reality domain, bear on the same procedural-computation

properties, tend to be represented through the same symbol system, and arfelt or cognised by the thinker himself as being similar, then they will b



QUANTITATIVE-RELATIONAL THOUGHT 257

whenever the present moment’s decisions need to be mindfully informed bthe decisions made in the past. This map reects the objective architecture o

mind (Demetriou & Efklides, 1989; Demetriou et al., 1993b, study 3).The third front refers to a general processing system that constrains thcomplexity and quality of the information structures the intellect carepresent and process at a given moment in its development. In other wordthis system is regarded as the dynamic eld where information is representeand processed for the time needed by the thinker to make sense onformation and attain the moment’s problem-solving goals. According t

the theory, the processing system is a three-dimensional construct. nvolves speed (the maximum speed at which a given mental act can b

efciently executed), control (the maximum efciency at which a decisiocan be made about the right mental act to be executed according to thmoment’s requirements, as indicated, for instance, by response times tstimuli involving conicting information), and storage (the maximumnumber of information units and mental acts the mind can efcientlactivate simultaneously) (Demetriou et al, 1993b, studies 4 and 5).

Development and Developmental Dynamics

The dimensions of development. According to our theory, developmentakes place along two main dimensions. The rst of those is a quantitativdimension. That is, the person, at least until about 20 years of age, becomencreasingly able to efciently deal with information structures or problem

which involve an ever-increasing number of information units, dimensionor processing steps. As a result, tasks requiring the execution of moroperations are generally solved later than tasks requiring the execution ofewer operations across all specialised structural systems. Variation alonthe quantitative dimension of development is ascribed to changes in one omore of the three dimensions of the processing system.

However, it is not uncommon to observe that tasks which, according to

standard system of analysis, involve the same number of information units owhich require the same number of operations are spaced many years apart i




nvolved or the solution required. That is, an intuitor is a conceptual blocwhich involves a coherent representation of a given salient relation in th

environment. As such, an intuitor is applied on a problem once somminimum conditions are identied in the problem representation. Oncapplied, the intuitor organises the input in an intuitor-specic manner and eads to an intuitor-specic solution (Demetriou et al, 1993b). The notion ontuitor as dened here needs to be differentiated from the notion o

pragmatic reasoning schemas proposed by Cheng and Holyoak (1985; sealso Halford, 1993). Intuitors function as prototypes of frequently occurrinand easily representable relations and they may be very concrete, local, anspecic. Pragmatic reasoning schemas, although they are derived fromeveryday experience like intuitors, are considered to be abstract knowledgstructures applied across several domains. An example is the schema opermission, which is the equivalent of logical implication.

In an agreement with the picture of development sketched above, wfound that the effective functioning of all SSSs at the beginning of developmental cycle is based on SSS-specic intuitors, which help to rais

the functioning of the SSS to the level of the new relations that need to battended to and processed. Gradually, however, the intuitors ardifferentiated into their component elements and relations. These are rxed as identiable and manipulable mental entities through a process osymbolic encapsulation, which pairs newly generated ideas with specisymbols (Demetriou, 1993). Once mentally individualised and symbolicallstabilised, the components of an intuitor can then be freely combined an

recombined according to the specic demands of the problem. Thus, thsecond dimension of development is what we have called representationa

uidisation or what Karmiloff-Smith (1992) has called representationa

redescription . That is, the movement from intuitors to differentiated integrcomponents that, instead of simply running off as undifferentiated wholecan be exibly used in support of each other as the current task goal requireVariation along this dimension is thought to result from strategic changes i

the person’s approach to knowledge acquisition, knowledge handling, anproblem-solving. As analysed here, the symbolic encapsulation an




processing is followed after a few months by changes in the control oprocessing. These are then followed by changes in working memory, whic

are then followed by changes in the SSS under consideration (Demetriou eal., 1993b, study 5). Alternatively, these latter changes frequently come rand enable the person to handle his/her processing resources morefciently, or to directly affect other associated components within the samSSS (see Demetriou, Efklides, Papadaki, Papantoniou, & Economo1993a). Thus, our theory ascribes development to the multisystemic and thmultistructural nature of mind.

Relations with Other Theories

We have discussed the relations of our theory with other theories elsewher(Demetriou, 1993; Demetriou & Efklides, 1988; Demetriou et al., 1993aDemetriou et al., 1993b; Demetriou, Efklides, & Platsidou, 1993c). Thuthese relations will be discussed here only in brief with a focus on thosaspects of the theory closely related to cross-cultural comparisons in regar

to cognitive development.The theory is close to modern neo-Piagetian theories, especially those o

Case (1992), Halford (1993), and Pascual-Leone (Pascual-Leone &Goodman, 1979). It shares with them the assumption, among others, thachange in processing resources is one of the causal factors in thdevelopment of conceptual structures and problem-solving skills anstrategies. However, there are considerable differences between these othe

theories and ours in the denition of processing resources.First, all of these theories assume that a general characterisation of th

person’s competence associated with successive general stages is possiblThus, they all propose very elaborate stage systems that are indented asubstitutes for the Piagetian system. Terminology may vary from theory ttheory but in all theories, development is expressed as a function of thnumber of units or the dimensions that can be simultaneously handled by th

thinker’s processing capacity. As a result, these theories dene intra- onter-individual differences in cognitive development in reference t




cognitive development. However, none of these neo-Piagetian theories hasystematically dealt with the domain issue. In fact, ours is the only cognitiv

developmental theory that has attempted to map the domains of mindspecify their composition, and decipher the principles of their organisationIt is only Case (1992) who has recently recognised that different domains ocentral conceptual structures may exist. The domains that he proposecoincide, by and large, with our domains. Finally, none of the neo-Piagetiatheories mentioned earlier has explicitly recognised the operation of thhypercognitive system as an integral part of the human mind. IndeedPascual-Leone’s ego executives are control constructs but they are vervaguely described in regard to their nature, functioning, and interrelationwith the other levels and dimensions of the mind.

Sternberg’s (1985) triarchic theory deserves special mention. At a generevel, we share with Sternberg the conviction that any theory of intelligenc

needs to involve a strong experiential and a strong contextual component, t is going to be able to explain how intelligence generates intelligen

behaviour in real environments and how it is shaped by these environment

However, in our theory, the experiential-contextual provisions arntegrated with the structural provisions in the assumption (see the two r

principles above) that different domains of thought (i.e. our SSSs) reedifferent domains of relations in the environment. In Sternberg’s triarchthere is no room for an architectural subtheory, as he relegated the problemof thought domains to low-level task models. Admittedly, Sternberganalysis of intelligence at two hierarchical levels, the level o

metacomponents, which are responsible for the monitoring, planning, anregulation of intelligent behaviour, and a lower level which involves thperformance and knowledge acquisition, are reminiscent of our analysis omind into a hypercognitive system and the SSSs. However, there arextensive differences between Sternberg’s metacomponents and ouhypercognitive system, which are not discussed here because of spacconsiderations. Finally, Sternberg has not systematically involved a capacit

construct in his theory and as such development has not been the primarfocus of his research and theorising. As a result, Sternberg’s theory canno




The Structure and Development of the

Quantitative-relational SSS

Structure. Apart from its other properties, reality has clear quantitativproperties which are independent of knowing systems, whether animate oarticial. Reality elements, be they atoms, molecules, stones, humans, owhatever, tend to aggregate or separate so that they increase, decrease or geredistributed in space or time. The quantitative-relational SSS is responsibfor the representation and processing of these aspects of reality. Thus, thSSS is relational in nature because any quantity Q consists of othe

quantities qi . . . qn and it exists as a quantity Q only in relation to othequantities Q 6 n. As an operating system, the quantitative-relational SSnvolves abilities which enable the thinker to (re)construct the quantitativ

relations between reality elements varying along one or more dimensions, awell as the relations between the dimensions themselves. As representational system, this SSS is biased to a symbolic medium whicenables the thinker to focus on quantitative properties and relations an

disregard other properties and relations which are irrelevant to quantitativprocessing. This seems to be the reason for the abstract and arbitrarcharacter of mathematical symbolism. Thus, this SSS involves severcomponent abilities (Demetriou, Platsidou, Efklides, Metallidou, & Shaye1991).

Abilities of quantitative specication and representation. These abilitieenable the person to specify the basic quantitative relations referred t

earlier (i.e. increase, decrease, and redistribution) and encode what specied in terms of a symbolic system that is able to preserve accuratelthese relations. Evidently, the cornerstone of these abilities is the group othe four basic arithmetic operations. These come out of the subject“quantifying” actions on reality (e.g. bringing in and removing, sharingcounting, measuring, etc.) When inter-co-ordinated, these abilities makpossible the construction of the more advanced abilities to be describe

ater. Abilities of dimensional-directional construction. These abilities enablh d h l l i i d b h




directional co-ordination are those that make possible complemathematical thinking, such as proportional and probabilistic reasoning.

Every specic mathematical ability can be reduced to the general abilitiepreviously described or to some combination of these abilities. Solvinsimple problems, which require the use of the four basic arithmetoperations, is based on the rst set of abilities specied earlieUnderstanding ratio depends on the ability to grasp the relation betweetwo levels of a single dimension. Understanding proportional relationdepends on the ability to understand the relation between two ratioTherefore, understanding ratio depends on dimensional constructioabilities and understanding proportionality depends on dimensional coordination abilities.

Algebra is one of the most symbolic expressions of the human mind. Assymbolic system it indicates, probably better than any other symbolsystem, the tendency of the mind to modularise knowledge and processinaccording to the specicities of a domain of reality and its ensuing specialisecomputational demands. That is, it retains from the environment only thos

properties of elements and their relations that are specic to the quantitativaspects of the world (i.e. increase, decrease, etc.). It is self-evident to say thasymbols in algebra constitute the most abstract denition of quantitativentities (i.e. numbers, dimensions, and intra- or inter-dimensional relationone may conceive of. Therefore, from the developmental point of viewalgebra may be seen as the vehicle of representational redescription. That it is used to raise quantitative thought from a level of operation direct

related to raw or rst-order quantitative reality to the second or even higheorder. The means to this end is the nature of symbols themselves; beinpuried from concrete referents, algebraic symbols enable the thinker tunderstand that quantitative dimensions and operations can stand on theown and become objects of thought activity independently of their materireferents.

An earlier study by Demetriou et al. (1991) validated the previou

assumptions regarding the structure of the quantitative- relational SSS. Thstudy involved four types of tasks: (1) tasks requiring the ability to perform




(1993) this is a three-dimensional structure. Three-dimensional structurenecessarily involve a binary operation (in this case the missing arithmeti

operation), at least two elements on which the binary operation is appliedand the result of the application of the binary operation on the elementHalford argues that three-dimensional structures impose a load of threelements, because if one is to be able to specify one of the dimensionnvolved (in this case the missing operation) one must be able to represen

simultaneously all three dimensions given.According to the same analysis, the second level requires the compositio

of binary operations, thereby enabling the thinker to understand quaternarrelations. This level imposes a load of at least four elements. This is sbecause three elements are required to dene one binary operation and ithese problems two operations must be considered to establish thconsistency of the relations involved. Evidently, it is only from this leveonwards that arithmetic operations are integrated with each other into unied system that would enable the person to build, rene, and interrelatdimensions. The level three and level four tasks were not used or analysed b

other workers. However, if one extended the analysis applied on the tasks othe two lower levels, one would have to assume that level three items whicnvolve three missing operations impose a load at least as high as tha

associated with level two items (i.e. four) together with an ability tsystematically plan, execute, encode, and integrate successive processinacts. This is necessary because in order to solve the problem with the threand, even more, the four missing operations, the person must test, compar

and integrate alternative hypotheses about the identity of the missinoperations. Therefore, the levels in the arithmetic operations battery werdened primarily in reference to the number of the operations the subjecwould have to execute (certainly the rst two and probably the third levetems) and secondarily, in reference to the strategies required (the last two

The rst, the second, the third, and the fourth level of the proportionreasoning tasks required the subjects to compare fully equivalent ratios (e.

1:1::2:2), partially equivalent ratios (e.g. 4:2::2:1), ratios with ordered paiwith the two corresponding terms multiples of one another (e.g. 2:1::4:3




terms of the mathematical and strategic expertise they require. According tthe analysis advanced previously about the role of intuitors in cognitiv

developm ent, the lower level items could be solved by analogy to an intuitoand a minimum of mathematical processing (the half and the one-thirntuitor in the case of the rst and the second level items, respectively

However, in the third level items only one (the one-third or one-fourtntuitor) and in the fourth level items none of the two pairs was reducibl

to an intuitor. As a result, these items required the exact quanticatioeither of some or of all of the relations involved. Therefore, the leven the proportional reasoning battery were dened primarily in reference t

the strategies required from the subject, rather than by the number of thoperations the subject would have to execute.

At the rst level of algebraic reasoning tasks, the solution to the problemgiven to the subjects could be directly deduced from the elements given odened by operating on the elements given (e.g. a1 55 8, a5 ?). At level the subject had to co-ordinate two symbolic structures in order to be able tspecify the value of the unknown symbol (e.g. u 5 f 1 3; f 5 1; u 5 ?

However, at this level, the reference structure which was to be used for thspecication of the unknown symbol was fully dened and given to thsubject (e.g. f 5 1). Thus, the subject only had to substitute the value of f ithe second structure and carry out the necessary operations in order tspecify the value of the unknown symbol. At level III, again two structurehad to be co-ordinated in order to solve the problem. However, at this levethe reference structure (r 5 s 1 t) could only be dened in relation to th

structure to be decoded (r 1 s 1 t 5 30). Thus, the subject had to introducand test extra assumptions in the problem in order to specify the value of thsymbol r (i.e. if r equals s and t together and all three of them equal 30,would have to be the half of this value). The items at level IV requireunderstanding that one may operate on totally undened structures in ordeto purely specify their general logico-mathematical relations, which stanbeyond any specic numerical values (e.g. when is it true that L 1 M 1

5

L1

P1

N?).In terms of processing load, the rst level algebraic reasoning item




s true when M 5 P, the subject only needs to realise that M and P aralternative symbols that stand for the same number. To be able t

understand this relation one needs to represent only three elements. That ithe two letters and the abstract idea of number. However, one should havpassed through the process of symbolic encapsulation and representationredescription that would raise one’s understanding of number and numberelations from the concrete and well-dened to the abstract and thll-dened in order to be able to build this representation. Thus, startin

from a rather low level, coinciding in complexity with the rst levearithmetic operations items, the four levels of the algebraic reasoning itemrepresent a movement in using symbolisation from well- to ill-dened anfrom reality- to representation-referenced structures.

Conrmatory factor analysis has shown, as expected, that each of the rthree sets of tasks loads on a different factor and that, in turn, these threrst-order factors load on a general second-order factor. The arithmetoperations, the proportional reasoning, and the algebraic reasoning factowere taken to correspond to the abilities of quantitative specication an

representation, to dimensional co-ordination abilities, and to the ability tcarry out quantitative processing on generalised relations as represented bsymbols rather than on overly quantitative signiers, respectively. This latteability is considered to represent the process of symbolic encapsulation anrepresentational redescription explicated above. The second-order factowas taken to indicate the power of the organisational principles which forcabilities similar in regard to domain, computational, symbolic, an

subjective specicity to be inter-co-ordinated into the same SSS: In this casthe quantitative-relational SSS.

Development. The sequencing of the items representing the four difcultevels in each battery came close to what would have to be predicted base

on the process analysis attempted earlier. That is, within a battery, itemmposing a higher load or requiring more complex strategies always scale

higher than simpler items. The relations between batteries also prove

consistent with this analysis, as indicated by the fact that items thought tmpose the same loads proved to be solvable at the same age. Specically,




evels alternated with the last two levels of the algebraic ability. The abilitto apply all of these three abilities in integration in order to solv

mathematical problems which require this integration appears at around 1years for simple problems and it continues after the age of 20 years.This inter-patterning of the various abilities is evidently consistent wit

the notion of synergetic causality explicated earlier. That is, the basquantitative constructions that are built by the application of the basarithmetic operations are decontextualised by being encapsulated isymbolic structures through algebra. This process enables quantitativconstructions to remain functional and develop, by raising them from thplane of rst-order representations, such as numbers, to the plane orepresentations of representations, such as the letters that stand fonumbers. In this way the mind acquires the exibility which is necessary toperate on multidimensional structures, such as the lower-leveproportionality tasks, when the processing system will acquire the potentito deal with these structures. In turn, operating with multidimensionastructures should open the way for the processing of complex and undene

symbolic structures, such as those represented by the third-level algebrareasoning tasks. Finally, this ability enables the mind to represent anprocess the highest level counterintuitive proportional reasoning tasks another complex tasks that require the integrated application of differencomponent abilities (Demetriou et al., 1991, 1993b).

Cultural Inuences on Structure and Development

Our theory makes two strong claims in regard to the structural organisatioand growth of the mind. Regarding structure, it is postulated that the vSSSs described by the theory are something like Thurstone’s (1938) primarmental abilities or Kant’s natural categories of reason, because they reecthe structural tuning of the human mind to the structural organisation oexternal reality (Demetriou et al., 1993b). Given the omnipresence of th

domains of reality as specied by the theory (i.e. the domains of categoricaquantitative, causal, and spatial relations and the universality of th




architectural and constructive peculiarities of the material substrate ohuman mind, that is, our brain. Specically, the processing system is

functional expression of the capacity of our brain to encode and procenformation. Therefore, the status of the processing system constrains thupper level of complexity of the structures which can be constructed for thvarious SSSs or the various SSS-specic components at a given agLikewise, the hypercognitive system represents the ability of the humamind to record and know itself, so as to be able to monitor and regulate iown activity. Two clear implications follow from these assumptions. On thone hand, the more the levels of a developmental sequence are dependenon the condition of the processing system, the more this sequence woulhave to be universal. On the other hand, individuals should possess representation of the structure of their own cognitive system that, to a largextent, mirrors its objective structure. That is, this representation shoulnvolve in some form or another the SSSs and the SSS-specic component

previously described.It must be emphasised, however, that the very same reasons that enforc

the assumption of cultural and social universality also allow for diversitySpecically, the mind-reality tuning, which is the foremost organisationassumption of the theory, enables one to assume that different environmenmay facilitate or require the functioning of one mental unit more than thefacilitate or require the functioning of another one. That is, differenenvironments may differentially actualise processing potentialities intSSS-specic schemes, skills, or routines. In this case, it is to be expected tha

the facilitated or required unit—be it a whole SSS or an SSS-specicomponent in each environment—would be better established and wouldevelop at a faster rate than the other unit. Furthermore, a sequence ontuitor-based, rather than complexity-based, levels for a given system ma

be different over different environments, the difference being dependent othe specic nature of the intuitors used in each environment.

Likewise, the notion of synergetic developmental causality leads to th

assumption that structural differentiation should go hand in hand witstructural cohesiveness. Specically, it has been argued that divergin




both automatic execution and hypercognitive system representation anregulation.

In Dasen and de Ribaupierre’s (1988, pp. 290–291) terms, our theoradopts a universalist orientation to the study and interpretation of cognitivdevelopm ent. According to this orientation: “cross-cultural comparisons ardeemed difcult, but possible if handled with care, universal characteristicof psychological functioning are searched for empirically, and are usuallfound at the level of ‘deep’ structures, with cultural variations at thsurface’. The latter may be qualitative and/or quantitative.” In other word

this approach rejects both the extreme absolutist or etic approach whicclaims that no differences between cultures may be found and the extremrelativist or emic approach which claims that cognitive functioning cannot bstudied or understood outside its regular cultural context.

THE STUDY

The present study examines the structure and development of th

quantitative-relational SSS from the age of 10 to 16 years in two cultureGreece and India. The subjects involved were examined in the arithmetoperations, the proportional reasoning, and the algebraic reasoninbatteries previously summarised. Based on the assumptions of our theorythe following predictions regarding structure and development would havto be corroborated by this study.

Structure. The theory postulates that the structure of mind is a stronuniversal. Therefore, we would expect no differences in the structurorganisation of the abilities represented by the three batteries. In terms oconrmatory factor analysis, the same model would have to be able taccount for the performance attained by both groups. This model woulnvolve: (a) three ability-specic factors—namely, the arithmetic operation

factor that would represent the quantitative specication ability; th

proportional reasoning factor that would primarily represent thdimensional co-ordination ability, and the algebraic reasoning factor tha




factors to the variance of the various component-specic scores. Specicallthe less familiar and the less developed a group is in regard to a give

component, the more the tasks representing this component would have tbe related to the general factor, in comparison with their relation to thcomponent-specic factor. This is because, in this group, the operation othis particular component would be weak; as a result, the general abilitiewould have to take control of processing the particular tasks. Secondly, thermay also be differences in the between-ability relations as reected in thcorrelations between the component-specic factors or in the relationbetween the rst-order factors and the second-order factor. That is, the mordevelopmentally advanced a group is, the more these correlations shouldiminish. This would indicate, in agreement with Sternberg’s (1985assumptions about the effects of intellectual development, thadevelopment brings about the automation and thus the modularisation ocognitive components. The relative contribution of the differencomponents of mind to performance, as well as the relations between thescomponents, are regarded by the theory as weak universals. Thu

differences between cultures or social groups may appear in regard to thesaspects of the structure of mind (prediction 2).

Development. According to the previous analysis, the two groups mabe compared in regard to two distinct, although interrelated, aspects odevelopment: (1) the inter-patterning of different levels of ability botwithin and across sequences; and (2) the pace of ascension from the lower t

the higher levels.In regard to the inter-patterning of a set of levels that represent th

development of the same SSS-specic component, the theory predicts thaunder some conditions there should be no differences between two groupwhereas under other conditions group differences may be perfectlacceptable. Specically, the more a sequence of levels is directly dependenon the condition of the processing system, the greater the extent to whic

two different groups must develop through the same sequence. An exampln the present study is the sequence of arithmetic operations developmen




representation and processing of the relations involved may result idifferent sequencing over different population groups (prediction 3).

In regard to relations between sequences, the theory claims that thalternation of developm ental priorities from the one sequence to the other an integral part of development itself. Therefore, it is expected that somkind of developmental intertwining of the sequences representing thdevelopment of the three abilities to be investigated by the present studshould be observed in both groups (prediction 4).

Regarding the rate of development , it would of course be consistent witthe theory to nd that individuals in one group of any population ascend thevels of a developmental hierarchy at a faster rate than individuals i

another group. This would happen if the rst group of subjects is privilegeas compared to the second in regard to whatever factors, endogenous oexogenous, individual, social, or cultural, affect the development of thgiven ability. Unfortunately, no theory is available that would enable one tmake predictions in regard to possible developmental differences betweeGreeks and Indians. However, a global prediction can be made on the basof earlier empirical research. Specically, many studies have shown thawestern children tend to outperform children growing up in nonwestercountries on tasks such as those used in the present study (see Dasen, 197Dasen & de Ribaupierre, 1988; Shayer, Demetriou, & Prevez, 1988). Baseon this evidence, one may predict that Greek children may perform bettethan Indian children on some or all of the abilities investigated herHowever, there is no empirical basis for a more differentiated prediction i

this regard.

Method

Subjects

Greek subjects. A total of 269 Greek subjects were examined. Thes

subjects were a subsample of a sample tested by Demetriou et al. (1991). Thdistribution of these subjects across grades, SES, and sex can be seen i




TABLE 1

Composition and Age of Samples

Indians (High SES) Greeks (High SES) Low SE

Grade Gender Age SD N Age SD N Age

5 F 138.720 2.492 25 126.154 3.532 13 –M 138.640 2.481 25 125.188 3.885 16 –

6 F 140.625 1.929 24 139.071 5.121 14 –M 140.692 3.095 26 134.467 3.335 15 –

7 F 149.625 4.030 24 151.455 3.908 11 152.37

M 149.231 3.338 26 152.167 3.682 18 154.00

8 F 159.143 4.269 21 163.176 4.202 17 166.12M 164.586 3.561 29 162.588 4.874 17 163.85

9 F 165.500 1.987 20 173.706 3.687 17 178.15M 167.533 4.747 30 176.158 5.408 19 180.93

10 F 187.625 7.418 24 183.857 3.485 7 186.66M 188.000 2.558 23 185.778 4.494 9 188.85

Note: Ages are given in months.

Indian subjects. A total of 297 Indian subjects drawn from the same schoogrades as the Greek subjects were examined. The distribution of thessubjects across grades, SES, and sex can be seen in Table 1. In terms of theducation of the parents, these subjects were comparable to the middle-claGreek subjects. That is, at least one of the parents of the Indian subjects ha

some kind of tertiary education (i.e. 14 years of education). Like their Greetest mates, the Indians were of urban residence, as they lived in Bhopal.

Task Batteries

Of the four batteries analysed in the introduction, the rst three only werused here. The fourth which required the integrated application of thabilities tapped by the other batteries was not used in this study because th

tasks involved were verbally very demanding. Table 2 shows all of the itemnvolved in each of the rst three batteries.




TABLE 2

The Items used in Each of the 3 Batteries and the Item Discrimination Level

(I) Arithmetic Operations Level (II) Proportional

Reasoning Level (III) Algebraic Ability Lev

1. (4 6 3) 5 12 (I) (2,2) vs. (3,3) (I) a 1 5 5 8; a 5 ?

2. (3 3 5) a 5 5 10 (I) (1,1) vs. (3,3) (I) a 1 b 5 43; a 1 b 1 2 5

3. (6 o 2) 5 3 (I) (1,1) vs. (2,2) (I) 2a 1 5a 5

4. (8 6 3) 5 5 (I) (1,2) vs. (2,4) (II) e 5 f 1 3; f 5 1; e 5 ? (I

5. (2 o 4) * 2 5 6 (II) (4,2) vs. (2,1) (II) m 5 3n 11; n 5 4; m 5 ? (I

5. (4 6 2) a 2 5 6 (II) (2,4) vs. (3,6) (II) 2a 1 5b 1 a 5 (I

7. (7 o 3) ˆ 5 5 9 (II) (2,1) vs. (4,3) (IIIA) e 1 f 5 8; e 1 f 1 g 5 (II

8. (12 6 3) * 25

8 (II) (2,3) vs. (1,2) (IIIA) 3a2

b1

a5

(II9. (4 $ 2) a 3 5 2 (II) (1,3) vs. (2,5) (IIIA) x 5 y 1 z; x 1 y 1 z 5 30;

x 5 ? (II

10. (3 $ 2 a 4) 6 3 5 7 (III) (6,3) vs. (5,2) (IIIB) Mult. n 1 5 by 4 (IV

11. (2 * 3 a 3) $ 5 5 7 (III) (4,2) vs. (5,3) (IIIB) When is true:

L 1 M 1 N 5 L 1 P 1 N (IV

12. (7 a 5 ˆ 6) o 2 5 6 (III) (2,3) vs. (3,4) (IIIB) When 2n . 2 1 n (IV

13. (3 6 2) o 4 5 (12 $ 1) * 2 (IV) (5,2) vs. (7,3) (IV)

14. (8 a 4) $ 5 5 (4 ˆ 2) * 1 (IV) (3,5) vs. (5,8) (IV)

15. (2 * 4) 6 2 5 (6 o 2) $ 3 (IV) (5,7) vs. (3,5) (IV)

Notes: The roman numerals in parentheses show the level addressed by an item.

For the purposes of the structural models tted to the performance attained on these batteries and report

below, three mean scores were created for each battery. These mean scores were created as follows:

OP1 5 (1 1 2 1 5 1 10 1 13); OP2 5 (3 1 6 1 7 1 11 1 14); OP3 5 (4 1 8 1 9 1 12 1 15).

PR15 (11 41 71 10 1 13); PR25 (21 51 81 11 1 14); PR35 (31 61 91 12 1 15). AL15 (1 1 41 7

10); AL2 5 (2 1 5 1 8 1 11); AL3 5 (3 1 6 1 9 1 12).

The abbreviations OP, AL, and PR represent the arithmetic operations, the proportional reasoning, an

he algebraic reasoning battery, respectively. The numbers in parentheses indicate the item number in table.

The proportional reasoning battery. This battery involved 15 items whicare shown in column II of Table 2. All but one of these items were selectefrom a battery devised by Noelting (1980). In groups of three these item

were addressed to four levels of difculty. The three items of the rst set anthe three of the second set were addressed respectively to the rst (full




were told that the black cups were lled with red paint and the white onewith solvent. The cups under each jar were supposedly going to be emptie

nto the jar, so as to prepare two paint mixtures. The subjects were asked tchoose one of three answers regarding the “redness” of the paint mixtur(mixture A would be more red; mixture B would be more red; both will bequally red) and explain their answer.

The algebraic ability battery. This battery involved 12 items which arshown in column III of Table 2. Most of these items were rst used bKüchemann (1981). In groups of three, these items were addressed to thfour levels of the development of algebraic reasoning explicated in thntroduction.

Scoring

The arithmetic operations battery. Responses on each of the items werevaluated on a pass-fail basis. A subject was considered to have passed atem if he were able to specify all unknown operations in this item.

The proportional reasoning battery. Performance on each of the items waalso evaluated on a pass-fail basis. To pass an item the subject was requireto select the correct choice for this item. It was decided not to basevaluation of performance on this battery on explanations (which werrequired and they are available) in order to free, to the extent this is possiblany possible differences between the two ethnic groups in the attainment othe levels of proportional reasoning from possible differences in their verbfacility to deal with this kind of problem.

The algebraic ability battery. Responses on the algebraic reasoning itemwere also evaluated on a pass-fail basis. A subject was considered to havpassed an item if he/she was able to provide a mathematically acceptablsolution to the problem.

The scores of 0 and 1 were allocated to failure and success on an itemrespectively, in all batteries. It may be noted here that preliminary analysby means of Shayer’s analysis of discrimination levels (Shayer et al., 1988

ndicated that all items in the arithmetic operations and the algebraireasoning battery and all but two items in the proportional reasoning batter




of about 30, 30, and 45 minutes was required for the completion of the threbatteries, respectively. No time limitations were imposed on the subjects.

Results and Discussion

The Horizontal Structure of Abilities

In order to test the predictions concerned with the horizontal structure oabilities, the data were analysed through a sequence of conrmatory factoanalysis models, tted with the latest version of the EQS programm(Bentler, 1993). For technical as well as substantive reasons, three meascores were created for each subject to represent his/her performance oeach of the three test batteries, giving a total of nine scores for each subjecEach of these scores was the mean of the performance attained on eacrespective third of the items involved in each battery. It should be noted thaall developmental levels and content variations were represented in eacmean score of each for the three batteries. The raw scores involved in eacmean score are specied in the notes to Table 2.

Reducing a large number of raw scores to a limited number orepresentative scores is an approach suggested by proponents of structurmodelling (Gustafsson, 1988). This manipulation increases the reliability othe measures fed into the analysis and it therefore facilitates thdentication of latent variables or factors. The researcher needs to b

sensitive to factor-identication problems when conrmatory factor analysand structural modelling are used. These methods pose strong demands o

the reliability of measures analysed. Moreover, three scores were usebecause having only two may cause problems in the identication of factoand having more is either unnecessary or technically cumbersome (seBentler, 1993).

The zero-order correlations on which the various models were tted arshown in Table 3. It can be seen that the scores representing the same abilitwere generally very high for both the Greek and the Indian subjects. Th

correlations between scores representing different batteries werconsiderably lower than the within-battery correlations.




TABLE 3

Correlation Matrix, Means, and (Standard Deviations) across the 2 Ethnic Groups of the

Scores Subjected to Structural Modelling

1 2 3 4 5 6 7 8 9

A. Indian Group (N 5 297)1. AL1 1.0002. AL2 0.752 1.0003. AL3 0.809 0.741 1.0004. OP1 0.523 0.540 0.529 1.0005. OP2 0.652 0.611 0.659 0.669 1.000

6. OP3 0.612 0.581 0.602 0.667 0.669 1.0007. PR1 0.534 0.496 0.531 0.345 0.505 0.398 1.0008. PR2 0.403 0.379 0.452 0.179 0.378 0.267 0.524 1.0009. PE3 0.486 0.419 0.474 0.309 0.453 0.350 0.669 0.475 1.00

Means: 0.492 0.428 0.398 0.578 0.482 0.510 0.535 0.507 0.50SD: 0.311 0.262 0.376 0.249 0.278 0.314 0.274 0.260 0.27

B. Greek Group (N 5 269)1. AL1 1.0002. AL2 0.691 1.0003. AL3 0.704 0.695 1.0004. OP1 0.410 0.528 0.496 1.0005. OP2 0.538 0.632 0.588 0.757 1.0006. OP3 0.455 0.516 0.497 0.698 0.773 1.0007. PR1 0.289 0.338 0.358 0.412 0.414 0.407 1.0008. PR2 0.334 0.342 0.382 0.373 0.392 0.403 0.779 1.0009. PR3 0.340 0.361 0.378 0.309 0.358 0.363 0.696 0.828 1.00

Means: 0.545 0.627 0.492 0.753 0.717 0.751 0.571 0.518 0.51SD: 0.293 0.253 0.326 0.252 0.279 0.301 0.342 0.337 0.33

Note: The abbreviations AL, OP, and PR represent the algebraic reasoning, the arithmetoperations, and the proportional reasoning battery, respectively.

still be signicant over the general factor, or whether any of them woul

become nonsignicant. A nonsignicant ability-specic factor would impthat the component ability represented by this factor does not stand on i




TABLE 4

Results of the Tests of Fit of Nested Factor Models Across the 2 Ethnic Groups

Fit of Models ChangeFactor

Model Included Culture c2

df CFI P c2

df P

1 QR In 259.385 27 0.859 0.001 – – –G 614 318 27 0 647 0 001

In the present case, all nine scores were prescribed to load on the generfactor. The three scores for arithmetic operations, the three for algebrai

reasoning, and the three for proportional reasoning were prescribed to loaon the hypothesised arithmetic operations, algebraic reasoning, anproportional reasoning factor, respectively. The four successive modenvolved the following factors in succession: (1) the general factor; (2) th

general and the arithmetic operations factor; (3) the general, the arithmetoperations, and the algebraic reasoning factor; and (4) the general, tharithmetic operations, the algebraic reasoning, and the proportionreasoning factor. Each of these four models was tested separately on thperformance attained by each ethnic group. The aim was to test the status oeach of the abilities in each of the two groups. Table 4 presents the statisticof the various models.

It can be seen that the introduction of all factors resulted in a highlsignicant improvement of model t in both ethnic groups. It can also bseen in Table 4 that the t of the complete model, which involved all foufactors, was excellent in the case of both the Indian subjects {c2(18)5 24.12

P 5 0.15, comparative t index (CFI) 5 0.996} and the Greek subjecc

2(18) 5 21.431, P 5 0.258, CFI 5 0.998}. Therefore, it can be concludethat the basic structure of the quantitative-relational SSS is the same in thtwo groups. That is, it involves the general and the specic componentassumed by the theory.

This nding justies testing a further stricter assumption. Specically, thathe performance of the two ethnic groups is not only organised by the sam

four underlying constructs identied earlier, but also that the relativcontribution of each of these constructs to the variation of performance o



t h e p

e r f o r m a n c e a t t a i n e d b y t h

e I n d i a n

( l o a d i n g s i n i t a l i c s ) a n d t h e G r e e k s u b j e c t s

( l o a d i n g s i n r o m a n n u m e r a l s ) . T h e

n t t h e

a r i t h m e t i c o p e r a t i o n s , t h e

a l g e b r a i c r e a s o n i n g , a n d

t h e p r o p o r t i o n a l r e a s o n i n

g c o m p o n e n t s , r e s p e c t i v e

l y ; c r , t h e

e p r i n

c i p l e s o r g a n i s i n g t h e c o m

p o n e n t s i n t o a n S S S .

A l l

l o a d i n g s w e r e s t a t i s t i c a l l y s i g n i c a n t , e x c e p t t h e t

w o l o w e r

n t h e

r e s p e c t i v e f a c t o r i n t h e m

o d e l ( A ) r e p r e s e n t i n g t h

e I n d i a n s a m p l e .




each of the observed variables to which it is related would be exactly thsame for the two groups. To test this assumption, a multiple sample analys

was run. In this analysis, the complete model involving the four factorspecied previously was tted to the performance of the two groups undethe assumption that the loading of each variable on each of the factors twhich it is related would be equal in the two groups. The t of this modealthough quite good, was not statistically acceptable {c2(54) 5 106.62P 5 0.001, CFI 5 0.984}. However, the t of this model improvesignicantly, as indicated by the difference between the two models {c2(185 60.826, P 5 0.001}, and became very good {c2(36)5 45.798, P 5 0.127, CF5 0.997}, once the across-groups equality constraints were released. This the model shown in Fig. 1A: According to the gural conventions ostructural modelling, the three arrows going from a factor (i.e. the construcsymbolised by circles) to a task (i.e. the constructs symbolised by squarendicate on which factor each of the tasks was prescribed to load.

An inspection of the loadings suggests some interesting similarities andifferences between the two ethnic groups. Specically, the general factor

very strong in both groups. In fact, in both groups this factor accounts fomore variance of the arithmetic operations and algebraic scores than thcorresponding ability-specic factors. However, the strength of the abilityspecic factors varied noticeably from the one group to the other. Tharithmetic operations factor was moderately strong in both groups; howevethe algebraic factor was very weak in the Indian group, as indicated by thfact that two of the loadings of the algebraic scores on the respective facto

were nonsignicant. In contrast, the corresponding loadings wermoderately high and signicant in the Greek group; likewise, thproportional reasoning factor was much stronger in the Greek groupalthough it was also satisfactorily strong in the Indian group. Thereforealthough all factors were present in both groups, the more specialised factoseem to be more well established among the Greeks. We will return to thnding later, when the discussion section will deal with development.

The presence of a rst-order general factor does not necessarily imply ththe ability-specic factors themselves are inter-co-ordinated into a gener




statistically indiscriminable from the model previously discussed {c2(44)57.996, P 5 0.077, CFI 5 0.996}. Attention is drawn to the fact that th

oadings of the ability-specic factors on the general factor were signicanand very high in both groups. This nding indicates that the quantitativerelational SSS does exert its organising power in both cultures.

The two models just explicated indicate, in agreement with prediction that all constructs assumed by the theory are observed in both culturagroups. However, none of the models tested earlier provide exanformation about the interrelations between the ability-specic factor

This information is provided by the model shown in Fig. 1C {c2(44)5 57.96P 5 0.077, CFI 5 0.996}. In this model, which was also tested on the twgroups at the same time without imposing any equality restriction, all threability-specic factors were taken to be independent factors that werallowed to correlate. All three correlations were statistically signicant iboth ethnic groups. However, the correlations between the factors of thGreek subjects were considerably lower than the corresponding correlationof the Indian subjects. The reader is reminded that the model shown in Fi

1A suggested that the ability-specic factors of the Indian subjects werweaker than the ability-specic factors of the Greek subjects. This ndingwhen combined with the pattern of the correlations between factors shown Fig. 1C, indicates, in agreement with prediction 2, that the stronger th

ability-specic factors become, the more autonomous of each other they geIt will be shown later that the differences between ethnic groups in the ratof development do t squarely with this interpretation.

The Vertical Structure of Abilities

The investigation of the vertical structure of abilities aimed to test how thvarious items scale both within and across the three battery scales for each othe two ethnic groups. Identifying the scaling of the items should provide thbasis for testing the developmental predictions of the theory.

Rasch rating scale analysis (Rasch, 1980; Wright & Masters, 1982) waused for this purpose. That is, the pass-fail scores on the set of 42 items wer




did succeed on the items requiring ability of this level or loweAlternatively, they also suggest that an item occupying a given position o

the scale is solved by subjects found to have “this or more” ability.Inspection of Table 5 suggests that there are interesting similarities andifferences between the two ethnic groups in regard to the inter-patterninof the various abilities. In regard to the scaling of abilities within each of ththree batteries, the performance of the Greek subjects was very close twhat might have been predicted based on the rationale that guided thbatteries’ construction. That is, in the case of the arithmetic operations anthe algebraic reasoning battery, theoretically lower level items always scaleower than items constructed to represent higher level abilities. This wa

generally also true for the proportional reasoning battery; however, in thcase, there were some inversions in the scaling order of some level II, IIIAand IIIB items, indicating that the boundaries between these levels are novery clear.

The scaling of performance attained by the Indian subjects was versimilar. That is, the scaling of the arithmetic operations did conform

perfectly to expectation. It is interesting to note here in regard to tharithmetic operations items that, as expected, in both groups items involvinthe operations of division or multiplication scaled higher than itemnvolving the same number of addition or subtraction operations. Th

algebraic reasoning items also scaled according to expectation, althougthere were some minor deviations: That is, some inversions were observed ithe scaling of level III and level IV items. As in the case of the Greeks, mo

deviations were observed in the scaling of the proportional reasoning itemSpecically, the three level I items did scale lower than all other items. LevII items alternated with level IIIA items, and level IIIB items alternated witevel IV items. Thus, whereas in both groups the easiest items were clear

differentiated from the most difcult items, there was no cleadifferentiation between the items of intermediate difculty.

In agreement with prediction 3, then, the arithmetic operations sequenc

which was considered to be more capacity-dependent than any othesequence, proved to be the same in both groups. In contrast, th



TABLE 5

Item Calibration of the 42 Items in Each of the 2 Ethnic Groups

Indian Group Greek GroupOP AL PR Logit Error OP AL PR Logit Err

OIV14 4.32 0.34 AIV12 2.99 0.1AIV 12 4.13 0.31 AIV11 2.75 0.1

OIV15 3.08 0.22 AIV10 2.36 0.1OIV13 2.75 0.21 AIII9 1.98 0.1

PRV13 1.95 0.17 AIII8 1.81 0.1AIII7 1.59 0.19 PRV15 1.81 0.1

OIII12 1.42 0.16 PRV14 1.79 0.1AIII9 1.31 0.17 PRIV12 1.68 0.1AIV 11 1.28 0.15 PRIII8 1.61 0.1AIV 10 1.19 0.19 PRV13 1.50 0.1

PRIV12 1.08 0.19 PRIII9 1.34 0.1AII6 1.02 0.18 OIV15 1.30 0.1

PRV15 1.00 0.19 AIII7 1.19 0.1OIII11 0.97 0.17 PRIII7 1.16 0.1

AIII8 0.93 0.18 PRII5 1.10 0.1

PRIV11 0.78 0.17 PRII6 1.06 0.1PRIV14 0.74 0.18 PRIV11 0.99 0.1

AII5 0.62 0.18 PRIV10 0.97 0.1PRIV10 0.56 0.16 AII6 0.90 0.1

OII9 0.34 0.17 OIV14 0.88 0.1OIII10 0.28 0.15 PRII4 0.60 0.1

PRII5 0.28 0.15 OIII12 0.57 0.1PRII6 0.10 0.14 OIV13 0.49 0.1PRIII9 20.27 0.15 OIII11 0.21 0.1

PRIII7 20.37 0.16 OIII10 0.14 0.1PRII4 20.42 0.15 AII5 0.10 0.1PRIII8 20.48 0.16 OII9 20.20 0.1

AII4 20.60 0.15 PRI2 20.43 0.1AI3 20.70 0.15 OII8 20.49 0.1

OII8 20.72 0.16 PRI1 20.72 0.1OII7 20.76 0.17 OII6 20.75 0.1OII6 20.87 0.17 PRI3 20.78 0.1OII5 21.01 0.17 AI3 20.84 0.1

PRI2 21.55 0.16 OII7 20.93 0.1OI3 21.60 0.17 AII4 21.47 0.2

PRI 1 65 0 17 OI 1 51 0 2




there were any systematic differences between the two ethnic groups in thattainment of the quantitative abilities represented by this scale. The r

analysis involved all the Indian subjects, and only the middle-class SEGreek subjects, of each age group; thus, this was a 6 (the six age groups) 3(the two cultures) 3 2 (the genders) analysis of variance. The seconanalysis compared the low SES Greek subjects with their Indian age mateThese two analyses aimed to uncover the effect of culture on the attainmenof quantitative-relational abilities and the possible interactions betweeculture, age, and gender. It was necessary to run two analyses because thIndian group involved only middle-class subjects, and in the group of Greesubjects only the four older age groups involved low SES subjects. Figuresummarises mean logit attainment across age, culture, and SES.

In the rst analysis, which compared the middle-class Greek and Indiasubjects, the main effect of age {F(5,470) 5 113.748, P 5 0.000} and culturF (1,470) 5 148.605, P 5 0.000}, was highly signicant; the main effect o




TABLE 6Percentage Distribution of Levels Across Age, Culture, and Ability according to Analysis

of Discrimination Levels

Arith. Op. Algeb. Reas. Proport. Reas.

Grade Cult. 0 I II III IV 0 I II III IV 0 I II IIIA IIIB I

5 In 50 38 10 2 0 32 68 0 0 0 4 0 14 30 30 2Gr 7 24 21 31 17 0 72 21 7 0 13 0 3 14 21 4

6 In 20 48 20 12 0 28 68 2 2 0 2 6 6 26 24 3Gr 0 14 34 48 3 7 45 28 21 0 10 0 7 34 17 3

gender, however (F (1,470) 5 2.710, P 5 0.100}, was non-signicant. None othe two- or the three-way interactions between these three variables wa

signicant. In the second analysis, which compared the low SES Greesubjects from grade 7 through grade 10 with their Indian age mates, only thmain effect of age {F (3,291) 5 34.354, P 5 0.000} was signicant. The maieffect of culture {F (1,291) 5 0.524, P 5 0.470} and gender {F (1,291) 5 1.33P 5 0.248} was nonsignicant. However, of the various two-way interactiononly the age 3 culture interaction was signicant {F (3,291) 5 11.05P 5 0.000}. Based on these ndings, in the analysis below we will comparthe middle-class subjects across the two cultures.

In order to obtain a more accurate picture of the distribution of leveacross age, culture, and component abilities, Shayer’s analysis odiscrimination levels was employed (see Shayer et al., 1988). Like Rascscaling, this method scales both the subjects in reference to the items and thtems in reference to the subjects. However, unlike Rasch scaling, instead o

simply scaling the items, it draws the cutting lines between differendevelopm ental levels and it ascribes each of the items into the level to whic

t belongs, according to the pattern of successes and failures obtained. Thut can be used to validate directly theory or earlier research regarding th

developmental structure of a series of items, and also to specify thdistribution of levels across the populations of interest. In order to be able tcompare the level attainment of the two ethnic groups across each of th




three batteries, the method was applied separately on the performancattained by each ethnic group on each battery.

Table 6 presents percentage level attainment across age, culture, anbattery. This table clearly suggests that there were no dramatdevelopmental differences between the two groups in regard to any of ththree batteries, although Greeks tended to excel Indians by about one levover all three batteries. Specically, the subjects in each age were spreaover two, and in some cases three, adjacent levels in both ethnic groups. Onof these levels was the same for both of them. However, the other—usuallthe modal—level was for the Indians lower and for the Greeks higher tha

their common level, indicating that developmental ascent over each of ththree scales was faster among the Greeks.

Some differences in level attainment across the three batteries need to bnoted. Regarding arithmetic operations, both groups were found to functioaround level I at grade 5. However, the Greeks clearly attained level IV bgrade 8 whereas the Indians remained on level III even at grade 1Regarding algebraic reasoning, both groups were found to operate modal

on level I at grade 5 and they both clearly reached level IV at grade 1Regarding proportional reasoning, both groups were found to operate othe three higher levels from grade 5 through grade 10. Thus, there was nclear developmental pattern in the development of proportional reasoninIn our view, this is due to the fact that in this case, educationally transmitteskills (e.g. reduction to common denominator skills) intervene witdevelopmental construction and, as such, affect developmental tendencie

The performance of the Greeks in relation to the teaching oproportionality in Greek primary education is in line with thnterpretation. Specically, attention is drawn to the fact that almost half o

the Greek 5th-graders were able—beyond any expectation—to succeed othe level IV proportional items. However, the performance of the Greekdropped systematically at the next two grades. It seems that there is a cleaexplanation for this pattern of performance. Specically, in the Greeeducational system, proportionality is systematically taught throughou

grade 5. Thus, the children at this grade possessed recently acquiredomain-specic skills that seem responsible for their success. Interesting




n each of the three batteries, cannot show whether the differences betweethe two groups originated from differences in the specic abilitie

themselves or from the general processes underlying all three componentThe method of choice for answering this question is the multisamplstructured means analysis (Bentler, 1993). This method enables one tdene at one and the same time the factors underlying performance on thtasks involved, and to compare two or more groups in the attainment of thconstruct represented by each of the factors identied. In the present casthe model shown in Fig. 1A was extended into a structured means mode

The reader is reminded that this model involved one rst-order factoassociated with all nine scores and three component-specic factors (one foarithmetic operations, one for algebraic reasoning, and one for proportionreasoning). Thus, this model allows one to partial out possible differencebetween the two groups in the general factor from differences in any of thother three factors.

The technical specications of this model need to be presented. First, th

paths from the intercept of each of the variables (i.e. a construct that denethe means of the variables) were specied as free parameters in each groubut they were also constrained to be equal across the two groups. In this wathe intercepts may be taken as a kind of baseline level for the variableTherefore, any differences in the means of the variables across groups muresult from other sources. Secondly, the loadings of all variables on each othe four factors were constrained to be equal across the two groups. Thirdl

the intercepts of the factors were xed at 0 in the Greek group, whereas thewere dened as free parameters to be estimated in the Indian group. Givethat in this kind of model the intercepts of the latent factors have an arbitrarorigin, they are interpretable only in a relative sense. Thus, in the presencase, nding that the Indian intercept for any of the factors is signicantldifferent from 0, which is the value of the corresponding Greek intercepwould imply that the Indian subjects have more (if the difference is positive

or less (if the difference is negative) of whatever is represented by this facto(see Bentler 1993)




GENERAL DISCUSSION

This study leads to a number of conclusions about the effects of culture o

the structure and development of complex thought abilities.

Structure. Regarding structure, this study has shown that thcomposition of quantitative-relational thought— that is, one of the ve SSSdescribed by our theory—is the same in two different and remote cultureThe reader is reminded that performance on the three batteries used waorganised into the component-specic factors (i.e. an arithmetic operationan algebraic reasoning, and a proportional reasoning factor) and the twgeneral factors specied by the theory. The sceptical reader might justiabraise a technical objection regarding our use of conrmatory factor analysiand this requires some discussion before going on to examine thdevelopmental implications of the study.

According to this objection, both the rst-order (model 1A) and thsecond-order (model 1B) (see Fig. 1) general factors we have identie

confound general processes pertaining to the two domain-general systemwith general quantitative thought processes common to all mathematicatasks. This is because there were no measures representing each of thesgeneral constructs separately. As a result, there is no way to specify in oumodels what variance of the performance attained on the three batteries associated with the quantitative-relational SSS as such and what variance associated with the general systems. Although technically correct, th

objection would undermine our conclusions only if the present study werseen in isolation. However, as reported at the beginning, this study is part oa larger programme. In the past, we have reported many studies whicnvolved tasks directly representing each of the two general systems togethe

with tasks representing several SSSs (e.g. Demetriou et al., 1993b, 199Gonida, Demetriou, & Efklides, 1994; Platsidou, Demetriou, & Zhan1994). These studies clearly showed that both within and across cultures th

SSSs and their components always stand as autonomous constructs over anabove both of the two general systems—that is, the processing system an




the general factor was considerably stronger than it was in the Greek groupThis effect was particularly pronounced in the case of the factor representin

algebraic reasoning; a factor very weak among Indian subjects anmoderately strong among Greek subjects. Interestingly enough, the relatiobetween the component-specic factors, although weak, proved to brelatively closer among the Indian subjects. Strange as they may seem, thesndings show nothing more than the two sides of the same coin: When thspecialised skills fall short of the demands of the task at hand, it is thresponsibility of the general systems to take control and cope with th

demands. Thus, in these cases the general systems do both: They intervene ithe problem-solving process (hence their high contribution to the variancon specic scores), and they orchestrate the application of the componensystems themselves (hence the higher correlation between these componensystems). When the general systems are weak, their effect becomes evidenn the functioning of the specialised structures which depend on them, eve

when the domain-specic skills are available to the subject. This seems to bthe meaning of the nding, provided by the structured means analysis, thathe Indians were lower than the Greeks on the general factor but they dinot differ on any of the component-specic factors.

Development. Like structure, development itself seems to involve botuniversal and culturally specic aspects. In so far as universals arconcerned, the study showed, rst, that developmental sequences, whic

nvolve levels mainly differentiated by the number of the operationnvolved, remain invariant across cultures. Levels involving more operationwere always found to scale higher than levels involving fewer operationThis might be taken to indicate that sequences dened by processing-loaspecications tend to be universal. Secondly, there is also strong universalitn the sequencing of the skills that lead from reality-referenced structures t

symbolic structures and which acquire their meaning via the inte

networking of symbols. This is suggested by the fact that the levels of thalgebraic thought battery were also found to scale in the same way in the twl l h h h l l diff i d f h h




However, some interesting developmental differences were observedMost notable was the difference in the sequencing of the proportion

reasoning items. This nding is interesting in two respects. From the point oview of developmental theory, the nding indicates that culture-specistrategies may formulate their own developmental sequences once thstructure of problems is not dened by processing constraints or constrainspecic to the representation and handling of symbolic structures. From thpoint of view of the methodology of cross-cultural studies, the ndinndicates that this kind of developmental sequence may not be ver

appropriate as a means intended to test the presence of universals icognitive development. This kind of sequence biases the eld to viedevelopment as only culture-specic because it conceals its universaaspects. Our proposal is very clear in this regard. To be able to uncover thuniversals and the specicities of the organisation and development ohuman mind, cross-cultural researchers must include in their diagnostic ananalytical armoury tools able to capture both the universal and the speci

Manuscript received 12 August 199Revised manuscript received 28 February 199

REFERENCES

Bentler, P.M. (1993). EQS: Structural equations program manual. Los Angeles: BMDStatistical Software.

Case, R. (1992). The mind’s staircase. Hillsdale, NJ: Lawrence Erlbaum Associates Inc.Cheng, P.W., & Holyoak, K.J. (1985). Pragmatic reasoning schemas. Cognitive Psycholog

17 , 391–416.Dasen, P.R. (1977). Piagetian psychology: Cross-cultural contributions. New York: Gardn

(Halsted/Wiley).Dasen, P.R., & de Ribaupierre, A. (1988). Neo-Piagetian theories: Cross-cultural an

differential perspectives. In A. Demetriou (Ed.), The neo-Piagetian theories of cogniti

development: Toward an integration (pp. 287–326). Amsterdam: North-Holland.Demetriou, A. (1993) Cognitive development. Thessaloniki: Art of Text.Demetriou, A., & Efklides, A. (1988). Experiential Structuralism and neo-Piagetia

theories: Toward an integrated model. In A. Demetriou (Ed.), The neo-Piagetian theories cognitive development: Toward an integration (pp. 173–222). Amsterdam: North-Hollan

D i A & Efklid A (1989) Th ’ i f h




Demetriou, A., Efklides, A., & Platsidou, M. (1993c). Meta-Piagetian solutions epistemological problems: A reply to Campbell. Monographs of the Society for Research

Child Development , 58, 192–202 (Serial No. 234).

Demetriou, A., Makris, N., & Adecoya, J. (1994, July). Metacognitive awareness about t structure of mind in Greece and Nigeria. Paper presented at the thirteenth Biennial Meetinof ISSBD, Amsterdam.

Demetriou, A., Platsidou, M., Efklides, A., Metallidou, Y., & Shayer, M. (1991). Thdevelopment of quantitative-relational abilities from childhood to adolescence: Structurscaling, and individual differences. Learning and Instruction: The Journal of Europea

Association for Reseach in Learning and Instruction, 1, 19–43.Fischer, K.W., Hant, H.H., & Russell, S. (1984). The development of abstractions

adolescence and adulthood. In M. Commons, F. Richards, & C. Armon (Eds.), Beyon

formal operations. New York: Praeger.Gardner, H. (1983). Frames of mind: The theory of multiple intelligences. New York: Bas

Books.Gonida, E., Demetriou, A., & Efklides, A. (1994, July). Deductive and inductive reasoning

different thought domains: Cognitive and metacognitive dimensions. Paper presented at thThirteenth Biennial Meetings of ISSBD, Amsterdam.

Gustafsson, J.E. (1988). Broad and narrow abilities in research on learning and instructioIn R.J. Sternberg (Ed.), Advances in the psychology of intelligence (Vol. 4, pp. 35–71

Hillsdale, NJ: Lawrence Erlbaum Associates Inc.Halford, G.S. (1993). Children’s understanding: The development of mental mode

Hillsdale, NJ: Lawrence Erlbaum Associates Inc.Hartnett, P., & Gelman, R. (in press). Early understanding of number: Paths or barriers

the construction of new understandings. Learning and Instruction. The Journal of Europea

Association for Research in Learning and Instruction.

Karmiloff-Smith, A. (1992). Beyond modularity. Cambridge, MA: MIT Press.Küchemann, D. (1981). Algebra. In K.M. Hart (Ed.), Children’s understanding

mathematics. Oxford: Murray.

Noelting, G. (1980). The development of proportional reasoning and the ratio concepEducational Studies in Mathematics, 11, 217–253; 331–363.

Pascual-Leone, J., & Goodman, D. (1979). Intelligence and experience: A neo-Piagetiaapproach. Instructional Science, 8, 301–367.

Platsidou, M., Demetriou, A., & Zhang, X.Q. (1994, July). The structure and development

processing capacity in Greece and China. Paper presented at the Thirteenth BienniMeetings of ISSBD, Amsterdam.

Rasch, G. (1980). Probabilistic models for some intelligence and attainment. ChicagUniversity of Chicago Press.

Shayer, M., Demetriou, A., & Pervez, M. (1988). The structure and scaling of concreoperational thought: Three studies in four countries. Genetic, Social, and Gener

universals and specificities in the structure and of quantitative-relational thought

Documents