Abstract Thinking

Cognition and Culrurc: i\Cross-Cultural Approach to Psychology J. Alurriha (Editor) 1903 Elscvicr Scicncc Puhlishcrs B.V.

317

Abstract Thinking Fons J. R. van de Vijver Madde E. Willemsen Tilburg University, Tilburg, The Netherlands Abstract Theoretical conceptualizations and empirical applications of formal and informal models of intergroup differences on abstract thinking are discussed. A model of abstract thinking is then described that attempts to synthesize formal and informal models. Introduction Cross-cultural ps chologists and cultural anthropologists share a longstanding interest in universal a n d culturally specific aspects of abstract thinking. During the last few decades, there has been a remarkable chan e in the orientation of studies of cultural differences on abstract thinking. Other fiebs of sychology have witnessed a similar historical shift. In personality and social psycho ogy, for example, the shift has centered on the question of whether persons or situations are more powerful in determining behavioral outcomes. McGuire (1985) has noted that these changes of paradigm are like pendulum swings with a period of 25 to 30 ears. Until recently this kind of paradi m shift did not seem to occur in the field ozabstract thinking. In the past, abstract tiinking was mainly studied in a decontextualized way; intelligence and cognitive aptitudes were studied as personalit traits that were taken to be stable across time and situations. More recently, lowever, situational and cultural characteristics have come to the fore. It is increasingly ap reciated that intellectual functioning should be studied in its cultural context (e. ., damenang, 1992). In the study of cognition, the dichotomy between the old and new approaches has become known under labels such as formal versus everyday reasoning (Galloti, 1989), formal versus informal reasoning (Miller-Jones, 1991), formal versus reasoning (Van de Vijver, 199l ) , and deductivism versus contextualism ( haskhar, 1984; Butterworth, 1992).

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In the first art of this chapter, the concept of abstract thinking will be discussed. F o r m J a n d informal models of reasoning can be discerned in the literature; both models will be described in the second part of the paper. Crosscultural studies using formal and informal models will be overviewed in the third part. In the final part, an attempt is made to delineate a model of abstract thinking in cross-cultural perspective that synthesizes formal and informal models. The Concept of Abstract Thinking Abstract thinkin can be considered to form the core of various kinds of complex reasoning suck as inductive thinking, deductive thinking, logical thinking, probabilistic thinking, proportional thinking, and conditional thinlung. In the past, various conceptualizations of abstract thinking have been proposed. Spearman (1923) was amon the first to recognize the importance of abstractions in intellectual functioning. +here are two rinciples underlying eneral intelligence, namely "noegenesis" and "abstractness. " !he latter refers to (agstract) phenomena that are not perceivable by the senses, while the former refers to a set of cognitive transformations to solve complex problems such as series continuations and the recognition of relationships such as equality or opposition.

F.J.R. van de Vijver and M.E. Willemsen

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In Pia etian psychology, abstract reasoning is studied in the context of formaloperational Anking, which is assumed to begin in Western cultures when the.child is somewhere between twelve. and fifteen years .of a e (e.g., Inhelder & Piaget, 1958). Formal-o rational thinking is characterrzed y, among other things, the child's ability to t ink in terms of hypotheses (i.e., assumed states of reality) rather than in terms of reali the possibility to " la around" with h otheses, the separation of form anTcontent, "scientific &&ng " and the fl?f mastery of concepts of probability and rportional$y. Inhelde; and Piaget (1958) present various procedures to assess a stract thinlung. In most of these tasks the subject has to "discover" a simple law of physics such as the e uilibnum in a balance or the oscillation time of a pendulum. In the latter, a pendium is attached to a table and the subject is told that different variables such as the length of the pendulum arm and the weight at the end of the ndulum may influence oscillation time; he or she should experimentally verify w ich variables really matter.

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More recent conce tualizations of abstract thinking could be added here; thus, Fischer (1980) has a r g u d that abstract thinkin is characterized b .the manipulation of abstract sets. In most of the proposals proffered, abstract t h i d n g involves what Piaget has called "secondorder operations." These involve cognitive o rations on the results of revious cognitive operations. The slim lest example can found in series extra o&ons such as "1, 2, 4, 7, 11 !wo cognitive transformations are requirdin the solution rocess. In the drst the series, "1, 2, 4, 7, 11"has to be transformed to a series of Zfferences of subsequent numbers, that is, "1(= 2 - l), 2(= 4 - 2), 3(= 7 - 4), 4(5 11 - 7)." The new series "1, 2, 3, 4, .It should then be extrapolated ("5, 6, ) and the results of the extra olation should be applied to 5), 22 [= 16 6), 29 (= 22 the onginal senes, producin "16 (= 11 7), . .." Other examples o tasks assessing abstract thinkin are verbal analogies (e "night : day = black : .'I), exclusion tests (e.g., "wkch animal does not b e . k g in the followin row: whale, shark, dolphin, hon?") and syllogisms (e.g., "Is Socrates mortal b i t is true that all men are mortal and that Socrates is a man?").

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Formal and Informal Models of Thought There are two traditions in co nitive ps chology to study abstract thinkin designated here as the formal and in ormal tra2tions (see Table 1). In the form tradition, there is an emphasis on problem solving in a possibly large but closed, finite problem space. An exam le would be the senes continuation problem presented above. All problems otthe test have a correct solution, which can be typically found by the application of some algorithm. The problems resented are usually not derived from daily life. There is no attempt to maximize tKe ecological validity of the stimulus material; ideal1 the difficulty of the test content (the psychological complexity; cf. Ceci, 1940) should be based exclusively on the cognitive transformations of interest (i.e., the second order operations).

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Problem solvin in the formal tradition has more than a casual resemblance to scientific activities. Tie scientist is often seen as the implicit normative model of the problem solver as scientists often emphasize generality in the solution process. A good problem solution should be based on an algorithm that can be formalized and generalized to new problems. In the informal tradition there is more emphasis on the natural context of cognitive rocesses. The stimulus material is usually derived from daily life. Scribner (f986) describes informal thinking as instrumental to the achievement of the goals of activities in daily life and it "stands in contrast to the type of thinking

Abstract Thinking

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Table 1 Differences Between Formal and Informal Traditions Formal Tradition

Informal Tradition

Closed problem spaces

Open problem spaces

Deterministic problems

Probabilistic problems

Formalized, artificial problems

Problems with a high ecological validity

Focus on correctness of solution

Focus on practical value of the solution (uncertainty reduction)

Context independence

Context dependence

The scientist as the model of the problem solver

The bricoleur as the model of the problem solver

Product oriented (psychometric a proach) or process oriented &agetian approach)

Process oriented

Cross-cultural comparisons of test performances

Intra-cultural studies

Algorithmic solutions

Heuristic solutions

Brief assessment of many problems

In-depth assessment of a single problem

Solution re uires conceptual, theoretical Rowledge

Solution requires procedural, practical knowledge

involved in performance of isolated mental tasks undertaken as ends in themselves" (p. 15). When compared with the problems of the formal tradition, there is more ambi uity with res ct to the initial state of the problem and the admissible transformations of tEinitial situation. The bricoleur rather-than the scientisf is fhe implicit ideal problem solver (Uvi-Strauss, 1976). The bncoleur is not pnmanly interested in solving a theoretical problem in a logically correct way but in the alleviation of a prachcal problem: Faced with the task, sa of repairing a faulty machine, [the bricoleur] looks over the materia7s at hand and improvises a solution. If the materials do not suffice, he may try to mod1 them in some way; but he is unlikely to seek new tools or to re-define t e problem (Gardner, 1981, p. 139).

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F.J.R. van de Vijver and M.E. Willemsen

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In the field of abstract thinkin , this amounts to the use of heuristics rather than algorithms. In Simon's (1847) terms, a bricoleur "satisfices" rather than "optimizes. I'

In the informal tradition, data are often collected by natural observation of "cognition in action." This cognitive behavior obviously has a high ecological validity. Finally, both traditions have studied cross-culgral similarities and differences in their own way. Whereas in the formal tradihon most studies are cross-cultural, studies in the informal Fadition tend to be intra-cultural. Both kinds of studies are described in the next sechon. Cross-cultural Studies in the Formal Tradition Cross-cultural studies of abstract thinking that have been carried out in the formal tradition can be divided into ps chometric studies on the one hand, and Piagetian studies on the other. In a dkscription of Western studies of abstract thinking, a third kind should be added--information-processing studies (e.g., Sternberg, 1977). Remarkabl no cross-cultural studies of abstract thinlung have been carried out in this researctline. Psvchometric studies. These studies t picall apply paper-and- ncil test that were developed in a Western context, to bo& a d s t e r n and a non- estern group. In an article on the assessment of abstract thinking in non-Western groups published in 1956, Jahoda noted that

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cross-cultural studies of test intelli ence (...) abound in the literature (...), but abstract behavior or its allegA absence more often forms the subject of impressionisticjudgments. (p. 237) Jahoda's complaint about the lack of empirical scrutiny and the often ill founded, sweeping generalizations about the minds of non-Western individuals has become somewhat obsolete since its publication. Many tests have been a lied in various cultural grou s, for instance Raven's Progressive Matrices, Catte I s Culture Fair Intelligence est, series continuations, figure exclusion, letter series, verbal analogies, and syllogisms. It would be impossible to give an overview of all crosscultural investi ations in which tests of abstract thinking were administered (overviews can%e found in Andor, 1983; Irvine, 1979; Jensen, 1980; Ord, 1970; Shuey,. 1958; and Vernon, 1979). Therefore, only one study will be presented involvin inductive thinkin (see Politzer's chapter, this volume, for examples of cross-cu turd studies of d uctive reasoning).

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Van de Vijver (1988, 1991) presented a test of inductive thinking to pupils from the two highest grades of rimary schools and the two lowest grades from secondary schools in Turkey, Zamtia, and the Netherlands; in the latter country two grou s were studied: a grou of native youngsters (N = 632) and a group of chiden of Turkish parents (I?= 135 , designated as the "Turlush-Dutch group. roups of six letters An exclusion test was administered to t!I ese pupils in which were presented; the sub'ect had to find the odd one out. ' he items had been athered on the basis of!five generatmg rules, which were presented to the pupils L i n g the test instruction: 1. Each group of letters has the same number of vowels. The vowels used in the test are A, E, I, 0, and U. In the test instructions the Y was also mentioned as a vowel. As the status of the letter can easily create confusion in English and in Dutch where it can be both a consonant and a

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Abstract Thinking

vowel, the letter was never used in connection to the first item generatin rule. The number of items to which the rule ap lied (i.e., the number of vowels per group of six letters of the item) cou d range from one to six. Five vowels are included in the following example (the correct answer is printed in boldface):

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QIAEOU WEIAVC EIOUAZ OUIEPA WUEIOA. 2. Each group of letters has an equal number of identical letters which are the same across groups. The rule could refer to 1 up to 6 letters. Example:

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3. Each group of letters has an equal number of identical letters which are not the same across groups. Example: GHHHHZ KVKKKM RRRSTT MMMPMT VVVVXY. Each roup of letters has a number of letters which appear the 2, 3, or 4) number of positions after each other in the same (i.e., alphabet. The letters A and B are said to have a difference of one position, the letters A and C a difference of two positions, etc. There is a difference of two positions in the following example: 4.

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5. Each group of letters has a number of letters which appear the same (i.e., 1, 2, 3, or 4 number of positions before each other in the alphabet. The number o letters in the last three rules could vary from 2 to 6. There is a difference of one position in the following example:

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MLKIIH GFEDCB UTSRQP Oh'MLKH XWVUTS. Various measures were taken to ensure the minimization of unwanted factors that could affect intergroup differences. Thus, the alphabet was printed at the top of the test pages with the vowels underlined in order to reduce intergroup differences in familianty with the alphabet and the concept of vowels. Also, the test administrations were preceded by lengthy instructions in which each of the item generating rules was explained and illustrated. This was followed by exercises in which, again, all item generating rules were used. The ma'or results are presented in Table 2. As can be seen there, the intergroup diderences on avera e scores were fairly large and (even though not further reported here) statistic& y high1 significant; the (native) Dutch subjects showed the highest scores. The standarJdeviations were farly constant across the groups; the same was found for the reliabilities (cf. Table 2). The systematic use of item generatin rules allowed for an interesting extension of the analyses aimed at gaining insigtt into the relative difficulty level of the facets. As can be seen in Figure 1, there was a strong negative relahonship in each cultural roup between the relative difficulty of the item and the number of letters involvs in the rule behind an item. With respect to the relative difficulty of the five item generating rules, it was found that the rules about equal letters (the second and the third rule) were the easiest in all cultural groups while the rules about

F.J.R. v a n de Vijver and M . E . Willemsen

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Table 2 Test Results Der Cultural Group Cultural group

Mean (max. =45) ~~

Standard deviation

Cronbach' s alpha

S-ample Slze

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29.1

6.3

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632

Turkish

25.7

6.4

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877

Turkish-Dutch

24.4

6.1

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135

Zambian

25.4

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.81

703

letters that come before or after each other in the alphabet (the fourth and the fifth rule) were the most complex. The latter item generating rules constituted a final facet, involving the difference of position in the al habet. It was found that lar er differences were more difficult to find (e.g., "A-b-C" was easier than "A-Cd", which was easier than "A-D-G", etc.). One of the most striking aspects of this pattern of findings was the absence of cross-cultural differences: the relative difficulty of various facet levels was invariant across cultural groups. Psychological complexity of the items was highly comparable across these groups. The inte retation of these cross-cultural differences creates a far from trivial problem. In a dition to genuine cross-cultural differences in inductive thinking, there are various alternative explanations such as differential stimulus familiarity, translation roblems (the tests were administere in En lish in Zambia, in Turkish in the Turkist! and Turkish-Dutch oups and in Dutcf in the Dutch group), and ex rimenter effects (each culturfroup had its own experimenter). In general, it is digcult to provide compelling evifence for one of the inte retahons and to falsify all other ones; the intergroup differences might have t o T e accounted for by a combination of the explanations mentioned. In the present study, evidence was found that the inter roup differences were related to educational differences. Various data on scho8 uality had been collected in the schools from which pupils were recruited. It was found that no significant intergroup differences in the test of inductive thinking remained after a statistical correctlon for the intercultural differences in terms of school qualit by means of a hierarchical re ession analysis (see Poortinga & Van de Vijver, lh7, for a description and an &stration of the statistical procedure in a cross-cultural context).

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The previous exam le of a cross-cultural study in the formal tradition was chosen to highlight a fairy consistent pattern of findin s in the area of abstract thinking. First, there is a profusion of cross-cultural difkrences in average scores. Thepattern is often consistent: if Western tests are used, Western sub'ects typically obtrun higher scores than nowWestern sub'ects; literate subjects neariy always get higher scores than illiterates. Whereas tke cross-cultural differences in average scores on tests of abstract thinking abound, it is common to find similarities in standard deviations and reliabilities. Factor analyses often show high degrees of invariance across cultural groups; the size of the eigenvalues and of the loadings on

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Figure 1. Proportion of correct scores per facet in each cultural group. the factors of the rotated solutions tend to travel well across cultures. Tests of abstract thinking often have their highest loadings on either a separate reasoning factor or on a general intelligence factor (cf. Irvine, 1979). It should be emphasized that these results are often hard to interpret despite their consistenc across assessment instruments and cultural groups. The actual inte retation o infergroup differences often, regrettably, rests on unsubstantiated, S ~ O C consideratlons such as intergroup djfferqnces on tesbwiseness or stimulus kiliarity: Cross-cultural studies gain jn valid1 if evidence in favor of a articular inte retatlon of intergroup differences is offere?(Poortinga & Malpass, 19i6). The vali ation of a particular interpretation often amounts to an extension of t+e original research design by the adophon of measures that presumabl induce intergroup differences in scores. An example is the previously menuondstudy of intergroup differences in inductive thinking in which various measures of school quality were included.

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F.J.R. van de Vijver and M.E. Willemsen

324

and of the ossibility of cross-cultural variability in cognitive development, he has expressed tfe need for cross-cultural research: Psychology elaborated in our [i.e., Genevan] environment, which is charactenzed by a certain culture and a certain language, remains essentially conjectural as lon as the necessary cross-cultural material has not been gathered as a contro . (Piaget, 1974, p. 309)

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Many psycholo@sts shared this view. Ap arently, not alI Piagetian stages were deemed equally interesting for cross-cultur research. The transition from preoperational to concrete operational thought has been wide1 examined, whereas studies into formal operational thinking are rare (cf. Dasen, 19&, 1977). The scarcit of studies did not reclude consistency in findin s if tests as described by Inhefder and Piaget (19&) were administere in non-bestern, often illiterate or oorly educated groups, then no subjects attiuned the sta e of formal operational tkinking. An investigation carried out by Kelly (1977) wilf serve as an example. This author administered a slightly adapted version of the pendulum roblem to Papuans (New Guinea). Half of the sample had at least one ear of formal schoolin Kelly found that not a single subject passed the norms ofyformal operational thirfiang The contrast of these findings with research in Western cultures should not b e overrated. There is ample evidence that a nonnegligible roportion of the adult opulation in Western cultures performs poorly on the Pnhelder and Piaget tasks &g., Neimark, 1975). The consistency of the negative findings has sparked a debate as to whether or not formal operational thinking is. universally attained. Both universalist and relativist claims can be found in Pia et's own wntings. In the beginning era of cross-cultural research, Piaget (1966f1974) seemed to interpret the cross-cultural data at face value: "It is quite possible (and it is the im ression given by the known ethnogra hic literature) that in numerous cultures thinkin does not proceed beyond we level of concrete operations, and does not reach g a t of propositional operations, elaborated between 12 and 15 ears of age in our culture" @. 309). Later Piaget 1972) favored a more universdst position; he ar ued that even though this may not e revealed in the testin situation, "all normal s8jects attain the sta e of formal operations if not between fl-12 to 14-15 years, in any case between f5 and 20 years" @. 10).

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Buck-Morss (1975) is one of the few authors who disclaims the universality of formal operational thinking on theoretical rather than empirical grounds; she argues that formal operational thinking is restricted to Western, industrialized countries. There is a causal relation between the occurrence of formal operational thinking and industrialization: "Abstract, formal reco nition may reflect a particular social structure, embodying the principles of ex&ange value, reification, and alienation which govern roduction and exchange in the industrialized West" @. 35). The separahon of 8 r m and content, a characteristic of the final phase of the cognitive development in Piaget's the0 is considered a cognitive consequence of the "reification" of commodities, o the completely separated context of production and consumption of an industrialized country. The separation on an economic level leads to a separation on a cognitive level.

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In a similar vein, Chapman (1988) has criticized the assumed universalism of formal operations by ar uing that cognitive development cannot be construed as unidirecbonal and teleofogical pro ress towards a universal end state, formal operational thinking. Ontogenetic 8velopment starts from a universal initial state

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(of the newborn) from which it adually moves away. The direction of development is determined by the cu turd context. Develo mental endpoints will differ across cultures. In Western cultures formal operationafthinkin is the highest stage of cognitive development. Referring to Northro (1987), Chapman distinguishes "theoretical knowing" fostered in Western cu tures and ''aesthetic knowing" more prevalent in Eastern cultures. The latter "refers to the direct a prehension of experienced qualities, whereas [the former] refers to the postulation of entities or processes that account for certain regularities in experience without themselves being given in experience'' @. 102). We know "aesthetically" the heat of the sun and "theoretically" that the heat is generated by the sun.

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Although no empirical test of either Buck-Morss' or Chapman's positions have been carried out, there seem to be strong em incal ar uments against the overriding impact of economical factors. First, as il ustrated a ove, cross-cultural differences on tests of abstract thinking are influenced by formal education. Therefore, it seems that the relation between formal o rational thinkin and industrialization postulated by Buck-Morss is inflated by krmal education. Also, the failure to display formal o rational Xinking on the tasks of individuals in non-Western groups contrasts wid? the often reported ethnographic evidence that illiterates show formal o rattonal thinkin outside the test -sitpation (see, .e.g., Hutchins, 1980; Jahqda,. E80, pp. 117-11%). Therefore, it is not surpnsing that most authors m u n h n an expectation of umversality of formal operational thinking (e.g., Dasen, 1977; Jahoda, 1982).

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It should be acknowledged that the assumed universality is not without its problems; there is a need to reconcile the theoretical expectation with the empirically observed cultural specificit One of the most interesting suggestions has been made by Piaget (1972) himself: He has offered three hypotheses intended to close the gap. The first one refers to the speed of develo .merit. Cross-cultural differences are assumed to be caused by an intercultural ifference in speed of cognitive development. Secondly, formad operahonal. thinking may. be a conse uence of a particular cognitive specialization not obtluned by each individual. Formi o rational thinking is then not a natural succession to the concrete operationafe hase induced by the inherent short cominp of the latter phase, but "a structural a&ancement in the direction .of specialization @.. 9), just as an ability like painting is onl present in some gifted individuals. The ophon which Piaget actually chooses is a tiird one in which the universality of formal operational thinkin is retained. Though not necessarily at the same a e, all individuals are assumd to "reach this stage according to their aptitudes an their professional specializations" @. 10). For exam le, a lawyer will be capable of formal operational thinking in the a plication of juri ical conce ts, whereas a car mechanic does the same in reasorung a out car engines. In enerJ the transfer between these application domains will be limited; for many incfviduals the availability.of formal operational structures is limited to their area of specialization. In reasorung about car engines the lawyer may only dispose of concrete operational structures.

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It has been pointed out by Dasen (1977) that the option Piaget chooses creates a aradox. Formal operational thinking is considered in the Piagetian a proach the ot$y way of thinking in which the logical operations are fully separab e from the context of a plication. Form and content are separable entihes in formal operahonal thinking. $ a pears now tha! this kind of thinking is dom+n-s cific and hence, stimulus-bounk: Only in a limited cognitive domain can individua srmanipulate form and content independently.

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To the three hypotheses offered by Piaget, a fourth one can be added. The

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F.J.R. van de Vijver and M . E . Willc.msen

assessment procedures in the Piagetian tradition can be criticized for a number of methodological reasons. The score validity is endangered by, amon other things, a low interrater reliability, heavy reliance on the subject's verb izations, a low ecological validity of the tasks (the oscillation time of a pendulum is not an everyday phenomenon for many subjects, even in Western cultures) and a differential mastery of ex rimental manipulation as a way to separate the impact of potentially relevant v a r i a E s in a problem solving situation. There is a need to devise new instruments with a higher stimulus familiarity for the groups studied: "New tasks have to be devised which test the same cognitive structures, but which are directly relevant to the daily activities and interest of the subjects" (Dasen, 1977, p. 197).

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In former da s it was rather common to reconcile the observed cultural s cificity of formar operational thinking on Inhelder and Piaget's tasks and the tEoretically assumed universalism b referring to the competence-performance distinction (e.g., Dasen & Heron, 19iO). The alleged poor performance of nonWestern subjects is then thought to differ substantially from their competence, which is considered to be a universal attainment. The distinction has a severe drawback in that competence is a concept without clear empirical referents (cf. Cole, Gay, .Click, & Sharp, 1971). The introduction of the corn tence-performance distinction renders the universality issue untestable. The cpothesis that an individual's rformance cannot be falsified competence is higher than his or her observed without assumptions which relate both concepts. Fp" or example, it could be assumed that the performance is equal to the competence when, apart from irrelevant fluctuations in motivation, no score changes are observed on repeated test administrations. It will be difficult to find empirical support for the plausibility of such an assumption. Cross-cultural Studies in the Informal Tradition

The informal tradition has become known in cross-cultural ps cholo y under names such as "eveyday cognition" (Rogoff & Lave, 1984; Segall, dasen, b e g , & ally Poortin a, 1990), practical intelligence" (Wagner & Sternberg, 1986), k n o w l d ie" (Dasen & Bossel-Lagos, 1989), and "indigenous co nition" (Berry, Irvine, Hunt, 1988). In this tradition, data are often collectef by naturalistic observation of "cognition in action." Cole, Gay, Click, and Sharp (1971) analyzed the way in which Kpelle (Liberia) construct arguments and draw conclusions from data in a divorce law case. In the case there were four participants: the paramount chief, before whom the case was heard; Tuang, the woman who was filing for divorce; Baawei, her mother who had received the bride price; and Baa, her husband. Tuang's line of argument focussed on unmet social obligations and malevolence by her husband. She complained that her husband had not built a farm for her. But she did not talk about how she had left her husband and had gone to her parental home for one and a half years. Her husband Baa described the sexual infidelity of Tuang and he argued that it was impossible for him to build a farm for her, since she was the one who had disappeared for a long time. The paramount chief concluded that it was Tuang who was wrong and that her arguments were illogical: Woman, it is you who are wrong. (. . .) You said you sued him because of his ways. But I didn't see any of these things you described here. (. . .) If a woman s ends a year and a half (away), can the man stay there and start a farm for er? (p. 181)

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The analyses showed how Tuang in this case selected information, used it to

Abstract

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underline her arguments and deliberately left out ce&n information. This resumabl illustrates universal aspects of co nitive functioning in such a context. PnterestingyG Cole, Gay, Glick, and Sh (1871) point out that most divorce cases "py indicating that the paramount chiefs too, among the pelle are won by men, possib deliberately omit particular aspects of the case and focus on others.

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Another eve day situation which these authors observed among the. K Ile was the playing of Malan Game. This game is played on a board with SIX Eles on the left and on the right side. Each layer starts with four seeds in each hole at his or her side of the board. On his or i e r turn the player has to move all the seeds in any hole. The player who has acquired all the seeds wins. The seeds have to be moved counterclockwise by dropping one seed in each successive hole or by collecting them in the hand. Capturing seeds can be done by placing the last seed in a hole with one or two seeds in it, on the opponent's side. Thirty games were recorded and analyzed in detail to discover the strate y of the layers. The good players used hy othetical rules about the capture of t e seeds $.f., "If I play the seeds from this Role, and the opponent plays the seeds from that ho e, then I can win two seeds on the next move.') Also, they used a clear and consistent set 0; strategies (e.g., "waitin,p until the opponent has made the first capturing move, "redistribution of forces and "keepin lar e numbers of seeds in certain holes in the middle of the own side of the board'5. fccording to Cole, Gay, Glick, and Sharp (1971), these two examples of traditional Kpelle problem solving situations can be analyzed in terms of psychological processes like "selectivity in the use of information" and "use of strategies."

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Mathematical activities in everyday situations have been a recurrent topic of stud in the informal tradition. Observations of behavior in natural settings are often comgined with (formal) tests of the same psychological rocesses in these studies. A ood example is the Adult Math Project (Lave, 1988 Lave, Murtaugh, & de la kocha, 1984). This roject was designed to investigate arithmetic rocesses durin grocery shopping. &e data were obtained by following grocery sYloppers throug the supermarket and maintaining a conversation with them about how they "determined the best buy" when comparing similar grocery items. For example, a r compares two boxes of sugar, one priced at $2 for 5 pounds and the other ikir-f or 10 ounds, and decides to bu the 10 pound box, since two 5 pound boxes Furthermore, some o r t h e observed behavior was translated into a would cost simulation experiment in which they had to determine the best bu of bottles, jars, boxes, and cans with different quantities and different prices. A few examples are given in Table 3 .

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The everyday math activities were compared with scores on isomorph math problems. These are problems that are structurally similar but use different stimulus domains. An exam le of an isomorph problem for the first buy problems of Table 3 would be: "WhicR ratio is larger: 90145 or 10/4?" A huge difference was observed between the performances on the math test on the one hand and the performances in the supermarket and the best buy ex eriment on the other hand; the percentages of correct scores were 59%, 98%, and 3 % , respectively), though the arithmetic problems were formally similar. Moreover, the supermarket and best buy tasks revealed a different pattern of relations with schooling and aging compared with the formal math task. According to Lave (1988) the results of this project indicate that:

B

There may not be just better and worse performances or more and less successful realizations of some basic arithmetic competence. Rather, there appear to be ualitatively different practices of arithmetic in different settings. (p. 63)

F.J.R. van de Vijver and M.E. Willemsen

328

Table 3 Best Buy Simulation Problems Price (P)

Best buy

Ratio

A

211 PalPb

A

312 QalQb

B

116 QbIPb

Peanuts can A

9oc

10 02.

can B

45c

4 oz.

jar A

$1.50

18 02.

jar B

$1.05

12 oz.

can A

21c

3 oz.

can B

30c

5 oz.

Jam

Olives

Pa stands for the price of the frrst item and Pb tU.

the pnce ot the second item. Qa stands for $e quantity of the first item and b for the quantity of the second item. The rahos gwen are those around which e problem was designed (Lave, 1988, p. 104).

Scribner (1984, 1986) has studied eve day mathematics in several settings, for example in a dairy factory in Baltimore. #is study, hke the Adult Math Pro ect, started with naturalistic observations of job performance (i.e., roduct assem 1 ). The assemblers are responsible for sending out dairy products o&red by drivers for their daily routes. These products are stored and handled in standard size cases that hold a certain number of units of a given size (4 gallons, 8 half-gallons, 16 uarts, 32 pints, 48 half-pints). If an order requests a quantity not evenly divisib e into cases, the order is represented in the number of cases plus or minus a number of units. For example, quarts come 16 to a case, so orders for 17 to 24 uarts are 1 unit (expressed on the load-out form as "1 1 3 up to 1 expressed as 1 case case 8 units ("1 8"). Orders for 25 to 31 quarts are eTpressed as 2 cases - 7 units ("2 7") up to 2 cases - 1 unit ("2 1l'). Observations of the preloaders showed that they had worked out interpretive procedures for the number representations and often departed from the literal instructions. They often choose a way of order iilling, which satisfied the order in the fewest moves and with the least effort, b using partially full cases (e ., if an order asks for 1-6 quarts, 2 units can be a d d d t o 8, or 4 units to 6). For gese least effort solutions the assembler had to switch from a decimal to a hexadecimal number system, which they most of the time did with speed and accuracy.

i

P

+ -

+ +

-

+

Abstract Thinking

319

The observations of the product assemblers were, together with the observations of wholesale delivery drivers, invento men and office clerks, translated in job simulations and experimental tasks. Arworkers were given all job simulations, im lying that on some of the tasks a worker was an expert and on some a novice. A scl!ool mathematics task was included as well. Students of a junior high school received several job simulation tasks and a formal math test. This test was also given to the workers of the dai factory. One of the most outstanding findin s was that the difference in strategyyy dairy workers and students even when .t e com arisons were restricted to those students and dairy workers that achieved similar For example, only experienced assemblers consistently ]eve[ of accurac employed least e f z r t strategies, while the students solved the assembly roblems exact1 in the way the order was presented to them (i.e., the students d i g not use partidy full cases). This difference was also found between novices and experts among the dairy workers:

i

the problem solving process is restructured by the knowledge and strategy re rtoire available to the expert in comparison to the novice. (Scribner, I!%, p. 38) Another example in which arithmetic behavior was observed in a daily settin can be found in the work of Carraher (1986) with construction foremen in Brazif This study investigated how different experiences with mathematics between construction foremen and students influenced their strategies in solving scalar transformation problems. In daily life, foremen who often work with blueprints are familiar with scalar transformations. In the experiment, the sub'ects were iven four blueprints drawn to different scales, namely 1:100, 150, 1:4$, and 1:35.3. The first two are commonly used in construction and the last two are never used. The scale used was not specified on the blueprint. For most of the walls on each blueprint the measurements were indicated, but for some of the walls there were no measurements. The subjects received one measure from the blueprint and the corresponding real life measure for the same wall. This first pair of values was necessary to determine the scale. In the other airs of values they received, the real life measure was unknown. For example, for t e 1 :100 scale the first pair was 3 cm : 3 m, and the other known values were 4 cm, 2.8 cm, and 3.2 cm. The subjects had to figure out the missing measurements by using the information from the blueprint. It was found that although students used a more generalizable problem solving strategy, the foremen were significantly more accurate in their solutions on1 for the familiar scales. Carraher concludes that job ex rience seems to enric measurement with meaning. Schliemann and Carraher (1K2) describe studies with street vendors and lottery bookies in Brazil which revealed that performance on everyday mathematical tasks was nearly always correct and not correlated with school experiences.

g

h

The cognitive behavior of racetrack handicappers was studied by Ceci and Liker (1986). These authors assumed that the process of handicapping would be quite complex, since it involves between ten and twenty variables (with multiple levels of each) that are combined into some sort of multiplicative model. Ceci and Liker designed their own racing program with fifty constructed races, which were given to excellent handicappers (experts) and to men who also attended races, but were inferior with regard to handicappin (non-experts). The analyses showed that experts assigned "weights" to each variabfe, combined the variables systematically in complex, nonadditive ways, and computed a rough odds/probability equivalent for each horse. Non-experts did not use such a complex reasoning process. Furthermore, the results showed no correlation between IQ and the measures of handicapping skill.

FJ.R, van de Vijver and M.E. Willemsen

330

Childs and Greenfield (1980) gave unschooled female weavers in Zinancanteco in Mexico a series of problems to find out whether the weavers could generalize their weaving skills to [email protected]. The subjects had to lace w e e n sticks of different colors and widths into. a-wooden frame-to make di erent str~pedpatterns, varying in complextty and f a d i a n t y to the weaving patterns. The same task was given to unschooled male non-weavers, to schooled males and to female American college students. The results indicatfi that the subjects with experience in weavin or schooling showed a more analyhc representahon of woven patterns than unschoo ed non-weavers. On the other hand, the weavers were not as successful as the schooled and unschooled non-weavers in working with unfamiliar patterns:

2

!

cognitive effects of a craft skill need not generalize beyond the requirements of the craft itself. @. 310) Similarly, Lave (see Greenfield & Lave, 1982), working with tailors in Liberia, concluded that "the ability to generalize co nitive skills to unfamiliar but related situations may be heavily constrained" @. 14%). Another cross-cu!tura! com arison was made by Cole,. Gay and Glick (1974) in their study about eshmahon o quanhhes of nce, for which h e Kpelle have an exact measure system. The sub'ects in this experiment were illiterate Kpelle adults stimuli were four household carusters containing as well as American adults. dry rice in amounts of 1.5, 3, 4.5, and 6 pint cans of rice. The canisters varied in size for each value and were approximately half filled. The subjects had to estimate how man cans of nce there were in each caruskr (a standard s u e can was shown to them). &e results pointed out that the Kpelle adults were much more accurate in their estimation than American adults.

P

de

These examples of studies in the informal tradition show some consistent findings. First, there is little or no correlation between performances in informal tasks and formal tests. Second, there is little transfer of co nitive skills from a familiar to an unfamiliar problem solving situation. Third, here are remarkable differences between experts and novices with regard to accuracy and strategies they use in problem solving tasks. Towards a Model of IntervrouD Differences in Abstract Thinking

P

Models that try to accommodate inter roup differences in abstract thinking should cover evidence from the formal and in o v a l traditions. More enerally, the relationship between formal and informal thinlun should be estabkhed. This amounts to the question of whether there is essenti one kind of abstract thinking that is studied in the two traditions in a somewhat di erent-way or whether abstract thinkin as displayed in formal and informal domams involves dissimilar psychofogical processes. The debate is not unlike the Spearman-Thurstone controversy about the number of factors in intelhgence (Spearman, 1904, 1923, 1927; Thurstone, 1935, 1938; Thurstone & Thurstone, 1941).

4

The (S armanian) osition that there is onl .one kind of abstract thinking is not popular t ese da s. h e of the most com ling arguments a amst a umtary view of abstract t h i d n g is taken to be roviderby studies on prob em isomorphs Wason's (1968) famous "four card proI! lem" is probably the most widely stud14 example. In the onfinal froblem four cards are shown to a subject contamin the followin symbols: A", D", "4", and "7", respectively. The subject is tolcfthat each cari has a letter on one side and a number on the other side. He or she should

f

331

Abstract Thinking

indicate which (and only which) card need to be turned in order to determine the validity of the rule "If there is a vowel on one $de of a card then there is an even number on the other side." The correct soluhon is the card with the "A" and the card with the "7". This solution is given infrequentl despite the fairly simple structure of the problem. It has been found repeat& that the performance is strongly influenced b the nature of the stimulus domain &ee Girotto & Light, 1992 for an overview). Xhen instead of the rule about vowels and letters, the rule is phrased as "If a person is drinking beer, then the person must be over 19," there is an increase in the proportion of correct answers of over 50% in comparison with the original vowel-number problem (e.g., Rips, 1990). A second line of evidence against the unitary view is provided b studies in which the performance on (formal) intelligence tasks and (informal) intelrectual tasks derived from daily life are com wed, as described in the previous section. Low or zero correlahons between tasL of everyda reasoning abilities and (formal) measures of intelligence have been reported &g., Ceci & Liker, 1986; Galloti, 1989; Wagner & Sternberg, 1986). In sum, the empirical evidence does not seem to support the unitary view. The current opinion is well captured b Galloti (1989) who ar ues that "formal and everyday reasonin abilities ma be ort ogonal or at least may e influenced by ve different kinds of actors" @. 3i;8). The present authofs do not concur entirely wi this view. First, the differences between the paradigms are exaggerated. It is insufficiently realized that fairly trivial methodological aspects influence the outcome of a stud For instance, the tasks as employed in the informal padition afe often more difzcult than in the formal tradition. Thus analogies as studied in the informal tradition tend to be more complex than in the formal tradition. The differences can be illustrated with Duncker's (1945, uoted in.Holyoak & Nisbet!, 1988) radiation problem, an analogy problem that is o en used in the informal tradihon:

K

F

%

3

8

Suppose that you are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to o rate on the atient, but There is a b n d of ray unless the tumor is destroyed the patient will that at a sufficiently hi h intensity can destroy the tumor. Unfortunately, at this intensit the heafthy tissue that the rays pass through on the way to the tumor w i i also be destroyed. At lower intensihes the rays are harmless to healthy tissue, but will not affect the tumor either. How can the rays be used to destro the tumor without injuring the healthy tissue? (quoted in Holyoak & Nisgett, 1988, p. 82)

g.

Gick and Holyoak (1980), administering the radiation problem, provided their subjects with a potentiall useful analogy in the form of a story. A general wanted to ca ture a fortress in center of the country. Many roads went to the fortress, but t ese were mined so that onl small rou s could pass without detonatin the mines. The difficulty of this anE iogy .tas di fers markedly from the more c assit aper-and-pencil assessment involving items such as "night : day = black : e differences in difficulty level in problem studies in the formal and inforpal trafition could easily though incorrectly convey the impression that dissimilar psychological processes are involved.

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bu

de

& P

.f. ..

There is another methodological caveat. The study of the cognitive behavior of experts in the informal tradition can be revealing but it should be realized that by defirution ex rts show little variation in their area of expertise as compared to the general opugtion. This restriction of ran e can have severe consequences on the statistid relationships with other tasks; tasfs that are uncorrelated in the restncted

F.J.R. van de Vijver and M.E. Willemsen

332

population may be strongly related in an unrestricted population. The old debate whether intelligence is unifactorial or multifactonal has initiated much research but was rather sterile from a theoretical point of view (Van de Vijver, in press); there appeared to be no simple right and wrong. The same may hold true for abstract thinkin . It is more fruitful to develop models to describe rformances on both forma f and informal tasks than to pursue a debate on the E m o eneous or heterogeneous nature of abstract thinking. Such a model will be d e s c n i later. Before our own model can be presented, we shall discuss the hypotheses on the cognitive come uences of schooling that have been put forward in the literature cf. Rogoff, 1981; Alviste, 1991). This detour is needed-becausethe hypotheses of ered are closely related to our own model of abstract thinlung.

6

Cognitive Consequences of Schooling The hypotheses to explain the ubiquitous performance differences between schooled and unschooled individuals can be divided into three kinds. The first one refers to factual knowledge. Many cognitive tasks, sometimes im licitly, require knowledge of facts tau ht in school. Inhelder and Piaget's peniulum problem provides a good examp e. Prior experience with or knowledge about a which is typically obkuned in a science curriculum will facilitate the peRendulum ormance. During their formal education, individuals acquire an expanding knowledge base that can be used in testing situabons.

B

The second kind of explanation holds that children acquire cognitive strategies in school that they would not have obtained otherwise (e.g., Tulviste, 1991). There is a watershed between literates and illiterates in this explanation.. It has been argued by Vygotsky that learning to read and write,hq a formatwe influence on hi hpr order cogmtive functions such as abstract thinlun (Tulviste, 1991). Vygotsfy s theory has led to interesbn studies of "unschoo ed literacy" among the Vai in Libena (Scribner & Cole, 1581) and Cree Indians in.Northern.Canada (Berry & Bennett, 1991). In both g$u s, peo le learn the tradihonal scnpt outside school, hence "unschooled literacy. t was ound in both studies that learning to read and write did not have a formative influence on abstract thinking.

f

P

P

Third, differences in performance of schooled and unschooled subjects may also be related to non-central aspects of the assessment procedure that q e inherent to a test but do not form its focus. Psychological tests are often :ather similar to school tasks; Lanc (1989) claims that children treat tests as school-like" @. 16). Followin dodnow (1976), he argues that there are various "information- rocessing tactics for in school which transfer to problems and tests devisdby crosscultural psycholopp" @. 16). For instance, school places a preTium on the manipulation of 'universals" such as "pairs", "sets", and "opposites. Also, the upil is taught in school that trial and error is better than m e n g no attemet at all. h e s e information-processing tactics are part of the "imphcit curriculum of the school; it ma be noted that the. first kind of explanation invqlved the "explicit curriculum" orthe school. Schoolin is geared towards a broademng of the range of ap lications of previously acquired s&,*Us. Transfer trainin , a vital element in many sciool curricula tends to enlarge the metacognitive skills o the pupil.

died

B

2

Serpell and Deregowski (1980 have given a similar rationale to explain inter roup differences in the area o picture perception. They ar ued that the inabifity of some non-Western r p l e to recognize 3D-cues in ictures oes not point to the absence of 3D-skills; rat er, pictures are not r e c o g d a s a context in which these skills should be applied. So, in the perceptual area, too, metacognition is taken to constitute a prominent source of cross-cultural differences.

%

The Components of Abstract Thinking The current model of abstract thinking proposed here includes four components. The first three close1 correspond with the three hypotheses to explain the test performance differences o r illiterates and literates explained in the previous section. The first com onent is called pragmatic knowledge. It refers to the immense amount of Zclarative, factual knowledge about ourselves and the environment that is collected through informal and formal education and that is stored in semantic memory. The knowledge reflects both facts and (implicit) theories (i.e., shared beliefs about the world). Pragmatic knowledge is a summary label for the kind of information that is used in the process of abstract thinking; it is the content to which the reasoning is applied, the "carrier of the reasoning." Some of these theories are sound generalizations based on scientific knowledge, but most are inductive generalizations which, however impressively sup orted, are not necessarily valid. This does not render pragmatic knowledge use ess. On the contrary, ragmatic knowledge is indispensable in our daily dealings with uncertainties; after a1 , imperfect knowledge is superior to no knowledge at all.

P

P

The second element refers to reasoning schemes, which are stimulus transformation rules (cf. Holland, Holyoak, Nisbett, & Thagard, 1986). In the case of absIract thinking these are the algorithms and heuristics to deal with second-order operations. The schemes do not have any reference to a articular domain of application; stimulus particulars have been "scra d "o!f Examples are combinatorial analysis, statistical laws such as Bayes' tR",orem and the Law of large numbers, and logical schemes such as the modus ponens and the modus tollens. Mathematics, statistics and formal logic often provide formal descriptions of these rules. However, there are also man reasoning schemes that (currently?) lack a formal descri tion Examples of such Keuristics are everyday rules of thumb and the Law of smalfnumbers or other cognitive biases studied by Tversk and Kahneman (e.g., 1974). Algorithmic and heuristic problem solving are basical y identical in the present view; their only difference involves the kind of reasoning scheme that is used. Some of the reasoning schemes will ield valid conclusions, but other ones are not necessarily valid. Thus, the overcon8dence in the generalizabilit of results based on small sam les is apparently a pervasive heuristic in everyday li& (Tversky & Kahneman, 1974.f

r

%rids

The uestion arises whether it is possible to rovide a concise description of the various of reasoning schemes involved in aistract thinkiy. Is there a core set of cognitive transformations that together constitute second or er operations? It could be conjectured that all algorithms and heuristics involves in abstract thinking can be re resented as a mathematical roup. The most obvious candidate would be the INRZgroup proposed by Piaget (cf. Flavell, 1963, p. 216). This is a set of four Booleanlike operators, referring to a set of interrelated cognitive transformations. The four elements are identity, negation, reciprocit and correlation. The effect of reached or annihilated by a a particular member of the grou can always combination of other members o the group. A simple example is the identity relation, in which the original input remains unchanged on output. The same effect can be reached by usin a negation of a negation; more technicall "NN = I". It INRC roup can be used to model bot? algorithms and could be ar ued that heuristics. f n the latter not all refationships between the members of the groups are present. This would mean that heuristics can be described as an INRC group with partly irreversible transformations.

P

6

tk

The INRC group has a conjectural status as a characterization of abstract

334

FJ.R. van de Vijver and M.E. Willemsen

31

thinking. This group may turn out to be inade uate and may need alteration. Yet, this would not et invalidate the basic idea that algorithms and heuristics involved in abstract thidmg can be represented as a mathematd group. Reasoning schemes are often considered the core of abstract thinkin . Most studies of abstract thinking in the formal tradition focus on this as ct. Ifowever, there is a need to redress the imbalance. Even thou h it is acgowledged that reasoning schemes are the distinguishing feature of agstract thinkin their role should not be overestimated; they form only one aspect of abstract tknking. An interesting case in point is formed by problem isomorphs such as the various versions of Wason's task described above. Even though by definition problem isomorphs refer to a single reasonin scheme, empirical studies have convincingly shown the sometimes pervasive in uence of shmulus domain (i.e., knowledge) on test perfylrmance. As another exam le, verbal analogies SUC as ?lay : night white : may be structurally sim iL to "cathode : anode = white , but most likely, the will have highly dissimilar difficulty levels in a iormal opulation. The s chorogical com lexity of both items is identical but the item difficulty will be gderent. Ceci (p990) raises a similar point regarding intelligence tests:

. . . ..

. ..

a

Kragmatic

There is a growing body of evidence illustrating that com lex performance on ( . . . ) inferential reasoning tasks is p r e d i c d b y individual differences in knowled e at least as much, if not more so, than it is by differences in IQ scores. &. 26) This point will hold a fortiori in cross-cultural research in which large differences in educational background of testees often prevail. A crucial aspect in the process of abstract thinking is the establishment of a link between pragmatic knowled e and reasoning schemes. The link is designated as the metaco nitive component, k e third element in abstract thinking. Establishin a o!! the reco nition that the solution of a problem demands a particufar link startsm reasoning scheme. On f e basis of this recognition there will be an "assembly" of ra mahc knowledge and reasoning schemes into a single "plan of action" (Simon, !9&). In the area of memory research there has been a great deal of study of this component, commonly labeled as "metamernorial" or mnemonic skills" (e.g., Brown, 1975). Unfortunately, metacognition is an underdeveloped area in studies of abstract thinking. With the exce tion of Hol oak's work on factors affecting the reco nizability of analo ous probgms (e.g., d llan d , Holyoak Nisbett, & Thagard, 1984 and the work of fotovsky, Hayes and Simon (1985) on factors influencing the solubon time of various versions of the Tower of Hanoi problem, em irical work is scarce. Metacognitive research is also notably absent in the area o cross-cultural research. There is ample evidence that in cross-cultural research that metacognitive differences ma constitute an important source of cross-cultural differences in test performance. For instance, some subjects do not provide an answer to the problem as posed in the task. This effect has been frequently observed among unschooled o le using s 110 isms or other cognitive tasks with a counterfactual content (e. En%ner & Core, f981). If confronted with a question such as "What color did bear have my friend saw yesterday, if you assume that all bears are purple?" not all people are willing to make the assumption and to consider the question as a logcal task. The kind of answers encountered can be described as either logical or empirical. In the former the answer refers to "the world as described in the premises" (e.g., "The bear you saw was pu le because all bears are urple"), while empirical answers refer to the "actual worl8 ("I do not know; you Eetter ask your friend" or "There are no purple bears").

P

tk

335

Abstract Thinking

The execution of the process constitutes the final element. In the present conceptualization this refers to the execution of the plan of action (i.e., the application of a reasonin scheme to pragmatic knowledge, as organized by the metacognitive component.! Computer programming provides a good metaphor of the current model of abstract thinking. Both involve transformahons of incoming stimuli. Su pose that a statistical analysis has to be carried out by a computer. The an&gy of the pragmatic knowledge in the statistical analysis is the practical information about the data to be commumcated to the computer such as sample size, and kinds and number of variables. The reasoning schemes are comparable to the programs or modules of the statistical package that are needed (e.g., carrying out an analysis of variance or com uting a correlation). The metacognitive component has an ando ue in the esta4ishment of the link between the declarations and the stimulus transkrmations; it refers to the declarations in the pro am that the researcher wants the computer to carry out this particular rogram on 8,se articular variables. The final component refers to the execution orthe statistical mdule required on the data. Cross-cultural differences in abstract thinkinz. Each of the four elements presented in the previous section can give rise to cross-cultural differences. The allowance for a detailed interpretahon of cross-cultural differences in test performance is one of the major advantages of the model of abstract thinking proposed. However, cross-cultural studies in the formal tradition that have been carried out do not allow for a detailed interpretation in terms of the components of abstract thinking. The interpretation of the intergroup differences in test performance can be considered the Achilles heel of the formal and the informal approach. To ascribe intergroup differences in test performance to abstract thinking as such may be true but this interpretation, assuming that abstract thinking is a complex process including various components, lacks recision. Among other things, it could cover an interpretation of inter roup tfifferences in terms of a differential mask of reasoning schemes foften the im licitly assumed differential mastery of the ra matic knowredge re uired in interpretation) or the test (e.g., as a consequence of differential sc 00 'ng). From a psyc ological point of view the two interpretations are not interchangeable.

07,

1%

R

The lack of a detailed interpretation of intergroup differences in the past prohibits the presentation of a surve of studies of each of the four components of abstract thinlung. Therefore, the phsibilit y of cross-cultural differences on each component has to be evaluated on more indirect grounds. It is a well established fact that durin socialization people from different cultural backgrounds ather partly differenti3 knowledge. Socializahon implies specialization. Indivifuals wdl be most thorou hly trruned in culturally selected areas of everyday knowledge (e. ., how to deafwith extreme physical conditions such as heat), formal education pk.g., how to deal with arithmetic problems) and professional areas e.g., how to deal with a broken car). Specializations can lead to sizable cultural di ferences in pra matic knowledge. A strong capitalization in assessment on culture-specific knowfedge usually ives rise to substantial intergroup score differences; examples are given in Wagner (f981).

'f

Whether specialization is equally important in reasoning schemes is debatable. The longstanding focus on lin uishc specializations and derived psycholo ical differences (such as in the Sapir-bhorf h pothesis) has detracted the attention bom basic, universal elements of language suc as grammar. In basic aspects, languages do not vary greatly (Greenberg, 1963); a cogrutively complex phenomenon such as

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F.J.R. van de Vijver and M.E. Willernsen

constancy in word order in a sentence appears to be a universal characteristic of languages. It could well be ar ued that universal aspects such as the need to obey rules of word order and of in exions, and the separation of actor and action are much more si nificant in the development of abstract thinking than the areas of cultural speci ization.

9

a

It is unfortunate that virtually no data on cross-cultural differences in the maste of reasoning schemes are available. It can be conjectured that large crossculturx differences in their occurrence will be unlikely; the schemes can be taken to be part and parcel of "the human condition." Relevant empirical evidence comes from anthropological sources such as Klich's (1988) studies of Aboriginals. Whereas Western tests tend to produce low scores among these people, anthropological research points to the existence of highly complex reasoning schemes in other areas such as their mytholo y. Uvi-Strauss' work on "the savage mind" (1976, originally printed in 1962) ias amply demonstrated that highly complex reasoning schemes are universal attainments. The basic tenet of these and many other writings in cultural anthro ology is that a failure to detect com lex cognitive functioning tells more about t e oor quality of assessment an the lack of acquaintance of the researcher with tfe local culture than about the cultural group at hand.

2

R

For the third element, the metacognitive processes, the picture is probably most complicated. Analogous to the argument about pra matic knowledge, there will be areas of specialization in which individuals have t e highest metacognitive skills. Therefore, cross-cultural differences are quite likely. Metacognihve skdls are taught through informal and formal education, In particular, the latter rofoundl influences metacognitive skills; transfer traming, highly characteristic of Formal &cation can be seen as the enhancement of metaco nitive skills. "Testwiseness" is another important exam le of metacognitive know ed e that can lead to substantial intergroup differences. remains to be em irically ktermined to what extent metacognitive skills can account for intergroup di ferences.

a

P

7

Evidence on the occurrence of cross-cultural differences in execution is not available. As executional problems are probably rather unsystematic, they are not very interesting from a cross-cultural point of view unless this component would show systematic intergroup differences which seems unlikely. Conclusion

In future studies we should maximize our efforts to come up with inte retable data. If cross-cultural differences are found or to be expected, attempts shozd be made to rovide a detailed interpretation. Statements such as "Group A is better in abstract tginking than group B" need specification. The model proposed here in which abstract thinking is divided into four components (pragmatic knowledge, reasoning scheme, metacognitive knowledge, and execution) can provide a frame of reference. Furthermore, it is relevant to t to delineate factors in the environment that can be taken to be responsible for t e intergroup differences in performance.

x

The period in which the differences between the formal and informal traditions were highlighted should become history. The formal tradition, particularly the psychometric studies, em hasizes methodological sophistication though lacks awareness of the relevance o the cultural context of cognitive functioning. The informal tradition is just the opposite. It is obvious that both traditions can and should complement each other.

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References Andor, L. E. (1983). Psycholo ical and sociolo ical studies of the black eople of Afica, south .the fahara. 196@1&5. An annotated bibLgraphy. Johannesburg: ational Institute for Personnel Research.

$

Berry, J. W., & Bennett, J. A. (1991). Cree literacy: Cultural context and s chological consequences. Cross-cultural Psychology Monographs, No. 1. Firburg: Tilburg Umversity Press. Berry, J. W., Irvine, S. H., & Hunt, E. B. (Eds.) (1988). Indigenous cognition: Functioning in cultural context. Dordrecht: Nijhoff. Bhaskhar, R. (1984). Definitions of "Explanation" and of "Realism." In W. E. Bynum, E. L. Browne, & R. Porter (Eds.), Dictionary of the history of science. Princeton, NJ: Princeton University Press. Knowin knowing about Brown, A. L. (1975). The development of memo knowing, and knowing how to know. In H.%. Reese Advances in child development and behavior (Ml. 10). New York: Academic Press.

(b.),

Buck-Morss, S. (1975). Socio-economic bias in Piaget's the0 and its implications for cross-cultural studies. Human Development, 18, 35-79. Butterworth, G. (1992). Context and co nition in models of co nitive rowth. In P. Light & G. Butterworth (Eds.), Antext and cognition: %ays oflearning and knowing. New York: Harvester Wheatsheaf. Carraher, T. N. (1986). From drawings to buildings; working with mathematical scales. Intemtional Journal of Behavioral Development, 9, 527-544.

.

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