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Age, schooling and conditional reasoning
Cognitive Development 21 (2006) 131–145
Age, schooling and conditional reasoning Lavee Artman a , Sorel Cahan b,∗ , Dinah Avni-Babad b b
a The David Yellin College, Jerusalem, Israel School of Education, The Hebrew University, Jerusalem, Israel
Abstract This study estimated the independent effects of age and schooling in grades 7–9 on scores obtained on invalid conditional and class syllogisms. The results, which point to a negative, albeit small, effect of outof-school experience and to a sizeable positive effect of schooling, replicate previous findings in a different age range and support the counterintuitive hypothesis that accumulated daily experience with conditionals has a negative effect on the development of conditional reasoning, and that improved performance on invalid problems with age is entirely attributable to schooling. Contrary to most cognitive tasks, therefore, in which schooling operates in the same direction as out-of-school experience, in this case, schooling breaks daily-life interpretational habits, and therefore, is critical for development. © 2006 Elsevier Inc. All rights reserved. Keywords: Conditional reasoning; Schooling; Cognitive development
Invalid conditional syllogisms include two argument patterns: denying the antecedent (DA; if p then q, not p, not q) and affirming the consequent (AC; if p then q, q, p). The classical fallacies regarding these invalid argument forms are to accept “not q” or “p”, respectively, as true, which would be in line with biconditional interpretation (“if and only if”) of the conditional (Braine & O’Brien, 1991; Braine & Rumain, 1983; O’Brien, 1987). These fallacies are common in early childhood and their frequency decreases with the increase of age (e.g., Evans, Newstead, & Byrne, 1993). Two major approaches have been used to explain the origin of these fallacies and the developmental mechanism involved in their decrease with age: logical competency models (e.g., Markovits, 1993; Markovits & Barrouillet, 2002) and comprehension theory (e.g., Braine & O’Brien, 1998). According to the first, the fallacies are due to inadequate logical ability or constraints in premise representation, and their decrease with age reflects the development of conditional reasoning abilities. Traditional competency models (e.g., Overton, 1990; Overton, ∗
Corresponding author. Tel.: +972 2 5882083; fax: +972 2 5882098. E-mail address: [email protected] (S. Cahan).
0885-2014/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.cogdev.2006.01.004
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Ward, Black, Noveck, & O’brien, 1987; Piaget, 1972) consider children to be limited interpreters of the conditional to cases in which both elements (p and q) are either present or absent. Consequently, increased success rates with age reflect continuous maturation of logical competence. A more recent view (e.g., Markovits & Barrouillet, 2002) attributes the fallacies to children’s inability to represent the case of not p and q due to limited capacity of working memory or restricted world knowledge, and the decrease in their frequency with age to the development of working memory capacity and accumulation of world knowledge. In contrast, comprehension theory (e.g., Rumain, Connel, & Braine, 1983) attributes logical errors to the tendency to apply everyday discourse rules to reasoning tasks (Begg & Harris, 1982; Braine & O’Brien, 1991; Braine & Rumain, 1983; Rumain et al., 1983). According to this view, on problems in which neither the direct-reasoning routine nor any obvious available strategy leads to a solution, people will often base inferences on other resources, especially, conversational implicatures (O’Brien, Dias, & Roazzi, 1998). These generalized implicatures follow from extra logical meanings that are common to how a natural-language particle is construed (Grice, 1989) and are commonly made unless there are circumstances to inhibit them (Grice, 1975). Comprehension theory is especially relevant to the explanation of the classical fallacies in conditional reasoning. In natural language, the word ‘if’ is ambiguously represented as a conditional or a biconditional according to context (Evans et al., 1993, chap. 2). However, the common interpretation of a typical (i.e., circumstantial) ‘if’ is biconditional, that is, if not p then not q (e.g., Geis & Zwicky, 1971). This is because these conditionals express either a sufficient condition (in which the not p case is irrelevant and no further interpretation is required)1 or a necessary condition (e.g., threats, intentions and promises) for which the biconditional interpretation is appropriate. Consequently, inferring the obverse of a conditional (i.e., “if not p then not q”) is in fact a correct conclusion and should be considered rational in daily context (e.g., Evans, 1993; Fillenbaum, 1986). Indeed, Scholnick and Wing (1991, 1992) found that 70–90% of the conclusions derived by adults from invalid conditional arguments in daily conversations were biconditional. This also holds true with respect to everyday class statements of the form “all are”. Because the speaker in everyday life is expected to be as informative as possible, the correct interpretation of such statements is biconditional, that is “All A’s and only A’s” (see Cahan and Artman (1997) for a comprehensive discussion). In contrast to the logic of conversation, in propositional logic the meaning of a conditional statement is decontextualized from its intended meaning in everyday discourse. In logic there is only one correct interpretation of a conditional, namely, material implication: the meaning common to all conditional statements. Material implication between two propositions (P → Q) is a relation that is true in all cases but p and not q. That is, p is sufficient (but not necessary) to q, and q is necessary (but not sufficient) to p. Obviously, for this interpretation, the DA and AC argument forms are invalid by definition. The fallacies in conditional and class syllogisms can thus be ascribed to overgeneralization of the logic of discourse. Subjects apply a biconditional interpretation to the major premise because this is the only one they know to be appropriate for those everyday situations in which the addressee is required to make inferences. Hence, the generalization of the biconditional interpretation to laboratory conditional inference tasks should not be a surprise: 1
Indeed, Newstead, Ellis, Evans, and Dennis (1997) found that on truth table tasks, FT cases are judged by 92–96% of the subjects as false or irrelevant for a variety of everyday conditionals (temporal, causal, promise, threat, tip, warning, universal, intentional).
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. . . how is a subject to understand the meaning of an utterance in an artificial context such as this [experimental reasoning tasks], except by reference to its normal everyday usage? . . . [T]he DA inference is reasonably . . . made in most real-world contexts in which it appears. (Evans, 1993, p. 18) The decrease in erroneous responses to invalid conditional syllogisms with age is explained by comprehension theory as a growing dissociation from the everyday biconditional interpretation which, in turn, is attributed to an improved ability to set aside the “logic of conversation” when solving laboratory reasoning problems (Rumain et al., 1983). If this is the case, the following question arises: How, without assuming growth in logical competence, can we explain the gradual abandonment of the biconditional interpretation, and hence the decline in classical fallacies with age? Unfortunately and unsurprisingly, comprehension theory is not explicit regarding the causal model underlying this developmental process, due to the difficulties associated with its identification. That is, there is no specific explanation for how adults become “more able than children to set aside the logic of conversation when confronted by laboratory reasoning problems” (Rumain et al., 1983, p. 421). In order to succeed on laboratory tasks, a na¨ıve reasoner must know in advance that in the laboratory reasoning context she/he is expected to adopt a new and different interpretation to the premises, which is neither explicit in his/her instructions (Evans, 2002) nor compatible with the correct everyday usage of the ‘if’ particles. That is, successful performance on laboratory tasks requires children to (a) become acquainted with the unnatural demands of the ‘logic game’ (decontextualization and implicative-like interpretation) and (b) learn to adapt interpretations to contexts (it will be inadequate if not bizarre, to subject everyday communication to the partial meaning common to all conditionals). Indeed, Politzer (1986) noted that the development of logical thought always competes with previously acquired, contradictory laws; it tends toward a dual system and it depends on differentiation between and coordination of the two subsystems: language communication laws and propositional logic. Clearly, everyday life cannot be expected to provide the experiences necessary for this learning process. Daily experience cannot be expected to acquaint children with the implicative, asymmetric, interpretation of the conditional, nor can it be expected to provide them with practice in its use. In fact, given that the only relevant everyday life interpretation is biconditional, the accumulation of daily conversational experience should reinforce its usage rather than weaken it. That is, everyday experience should be dysfunctional to the disassociation from the biconditional interpretation and, therefore, also to success on invalid conditional syllogisms. The differentiation between and the coordination of language communication laws and propositional logic necessarily requires a different kind of experience, in which children have the opportunity to encounter and practice the ‘logic game’. According to Cahan and Artman (1997) this experience is provided by schooling. Mainly through mathematics and science, schooling indirectly offers children what is perhaps their first encounter with the “weak”, implicative, interpretation of conditional and class statements and provides them with practice in recognizing the context in which this asymmetric interpretation is appropriate. The directionality of class and conditional statements is introduced explicitly in geometry, through the distinction between direct theorems and inverse theorems. In addition, school science involves systematic analysis of the necessary and sufficient conditions that determine the occurrence of the phenomenon under study (Taylor, 1967). Many such scientific activities (e.g., testing the properties of materials using diagnostic indicators) require students to make conditional inferences, the correct formulation of which are constrained by the neces-
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sity/sufficiency status of particular cause-effect relations (e.g., “If the bulb lights up, the material is a conductor, but it is not true that if it is a conductor, the bulb will necessarily light up”). In addition, schooling fosters the development of secondary skills (Braine, 1990): Children learn that different situations require different interpretations. Literal analysis of texts, learning in settings isolated from the context of the objects referred to, and the taxonomic-like organization of knowledge teach the child to confine interpretation to the information explicitly stated in the text (Hildyard & Olson, 1978), to distinguish between what the sentence itself means and what one means by the same sentence (Braine, 1990), to construct verbal-hypothetical worlds (Applebee, 1984; Braine, 1990; Donaldson, 1987; Olson, 1977; Scribner & Cole, 1973), and organize classes of objects through hierarchical classification (Ward, Byrnes, & Overton, 1990). A reasoner equipped with this knowledge should notice that the first premise of an invalid conditional argument does not include information related to the second premise, to suspend judgments regarding conversational implicatures (i.e., to adopt ‘minimal commitment’ to the conditional in the major premise) and to envision and represent the possibility of “not p and q” (Cahan & Artman, 1997). In sum, according to this hypothesis, the decreased rates of the classical fallacies with age are entirely attributable to a culture-specific experience, extrinsic to the developing individual, namely, schooling. The effect of aging and out-of-school experiences is expected to be negative. These predictions regarding the effects of schooling and out-of-school experiences clearly contradict those derived by the logical competency models, according to which the decrease in the fallacies rates with age is considered to be a basic and universal process, resulting from the child’s transactions with any normal environment, schooled or not. According to this approach, the effect of schooling is negligible and development is mainly due to aging and the associated accumulation of experience. To date, Cahan and Artman’s (1997) study is the only available empirical evidence relevant to this issue. In their study, the amount of everyday experience was measured by chronological age and schooling was measured by grade level. The independent effects of age and schooling on scores obtained on invalid conditional and class syllogisms were estimated in grades 4–6 (age range 9.5–12.5). The results pointed to a negative, albeit very small, effect of chronological age and to a sizeable positive effect of schooling. The theoretical significance of these results is clearly dependent on their replicability across age range and tests. This study is a first step in this direction. The study estimated the unique effects of schooling (the student’s grade level at the time of test administration) and out-of-school discourse experience (i.e., chronological age) on success in solving invalid conditional and class problems among adolescents (grades 7–9), using a broad range of uncertain forms of conditional syllogisms, including all possible combinations of premise type (if the; if a; all are) and argument pattern.
1. Method 1.1. Participants The sample included 1885 seventh, eighth and ninth graders in three Israeli junior high schools (about 600 in each grade) (see Table 1). Age means were 12.3,13.3, and 14.3, respectively.
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Table 1 Sample by grade and age status (N) Age status
Grade 7
Underage “Normal” age January–October November–Decembera Overage Total a
8
9
6
13
9
465 62
495 63
510 70
58
67
67
591
638
656
These students were excluded from the data analysis.
1.2. Measures The conditional syllogism test consisted of (a) test items – 12 randomly-ordered DA and AC class and conditional problems and (b) camouflage items – 6 MP or MT items, 4 in the “if-then” form (2 MP and 2 MT) and 2 in the “all are” form (1 MP and 1 MT). These items (item nos. 1, 6, 8, 9, 13, 17 in Appendix A) were included in the test in order to minimize the possibility of “response set” and were excluded from the data analysis. Each of the 12 test items presented an argument composed of two premises and a conclusion which the subject had to evaluate as necessarily true, necessarily false or indeterminate (no way to tell). For example:
The major premises included three types of sentence (four items each): “if the” (e.g., if the A is . . . then it has to be a . . .), “if a” (if an A is . . . then it has to be a . . .) and “all . . . are” (all the A’s are F’s). For each type of premise, two items denied the antecedent and two affirmed the consequent. The test language was Hebrew (see Appendix A for an English translation of the original Hebrew version). 1.3. Procedure The test was administered on a classroom basis at the beginning of the school year. Each participant responded to all the 18 items. Two testers (university students who had received special instruction) were present in each classroom. Before the start of the test, the children were presented with two examples: one with an “all are” premise (MP), and the other with an invalid disjunctive syllogism (A or B or both, A, is B true?). Before revealing the correct answers, the tester explained how the premises should be interpreted in order to arrive at the proper conclusion.
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The rationale for this short training was to acquaint subjects with the test format and appropriate use of the “no way to tell” response (i.e., as a judgment of indeterminacy rather than of subjective uncertainty). The test was administered without any time limitation. 1.4. Scoring Each item was scored dichotomously (1 = correct, 0 = incorrect or no response). The percentage of “no response” answers ranged between 1 and 4% across items with negligible differences between grades. The proportion was higher, on the average, the higher the serial position of the item, perhaps due to fatigue. Ninety percent of the subjects in each grade level responded to all the items and 95% in each grade level reached the last item. Among subjects who skipped at least one item, 50% did not respond to two items at the most. Three total scores (percent correct answers), one for each of the three types of premises, were calculated for each examinee. 1.5. Design and methodological considerations 1.5.1. The between-grades paradigm A between-grade regression discontinuity approach (Cahan & Cohen, 1989; Cahan & Davis, 1987) was employed to disentangle the independent effects of schooling and age on success in conditional reasoning tasks. This approach involves administering the same test to at least two adjacent grade levels. It relies on the following assumptions: (1) the “allocation” of children to birth dates is random; (2) grade level is solely a function of chronological age; that is, admission to school is based on chronological age only according to some arbitrary cut-off point, and progression through grades is automatic (i.e., there are no drop-outs and children are neither kept back nor advanced a grade). On the basis of these assumptions, the net effect of chronological age and schooling are estimated by means of a regression discontinuity design (Cook & Campbell, 1979), in which test scores are regressed on chronological age within grades. In this design, the effect of age is reflected in the slope of the within-grade regression of test scores on chronological age, and the effect of schooling is reflected in the discontinuity between these regressions (see Fig. 1).
Fig. 1. The effects of age and schooling in the regression discontinuity design. Note: The dotted section in each regression line represents extrapolation of the regression line for the months of birth which were excluded due to their high proportion of selective grade misplacement.
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Specifically, the estimated effect of a one-year difference in chronological age in a given grade equals the difference between the oldest and youngest students in that grade in mean predicted scores (see dotted arrows in Fig. 1), and the estimated effect of one year of schooling equals the differences in mean predicted scores between the youngest children in any given grade (X) and the oldest children in the lower adjacent grade (X−1). 1.5.2. The truth of the assumptions As already stated, the between-grades paradigm makes two assumptions: (1) children are randomly allocated to birth dates; (2) grade level is solely a function of chronological age. Certain reservations regarding the truth of these assumptions, that have been raised before (Cahan & Cohen, 1989; Cahan & Davis, 1987), should be reiterated. The truth of the first assumption cannot be empirically tested. However, since we tested students in only three adjacent grades within a relatively homogeneous population, this assumption seems reasonable with respect to between-grades variability. As far as within-grade randomization is concerned, exceptions to this assumption can affect the estimation of age and schooling effects only if they are monotonically related to birth date, which is very unlikely. The second assumption of the model is only partially true. While grade retention and grade skipping are seldom practiced in the Israeli school system (Cahan & Cohen, 1989), admission to school is sometimes delayed and sometimes accelerated. Consequently, in any given grade there are “overage” and “underage” children, that is, children whose age should place them in a higher or lower grade, and there are also “missing” children (i.e., children who are learning in a higher or lower grade) (see Table 1). More important, the delay or acceleration of school admission is not random. The children whose admission was delayed are likely to be less developed intellectually than the other children in their age group, and those whose admission was accelerated are likely to be more developed. Hence, the children who remain in the “appropriate” grade are also selective (in the opposite direction). In addition, the relative frequency of grade misplacement is related to the month of birth, being particularly high near the cut-off point: Delays are especially frequent among the youngest children in each cohort and accelerations among the oldest (see Table 1). In particular, the relative frequency of delays is much higher (about 25%) among children born in November and December relative to the other months (2–8%) (see Table 1). Cook and Campbell (1979) refer to this possibility as a “fuzzy cutting point”. There are two ways in which selective misplacement may affect the within-grade regression slopes: (1) Due to the existence of underage and overage children in each grade. The direction of this effect cannot be established a priori since age and selection counteract each other in this case: the underage children are also brighter, while the overage ones are generally duller. (2) Due to the missing children in each grade. At the lower extreme of the age range, the missing children are those that have been delayed; hence, the mean test score of the remaining children in the youngest groups is higher than the true one. At the higher extreme of the age range, selection operates in the opposite direction: the missing children are the brightest ones, whose admittance to school has been accelerated. Consequently, the mean test score of the remaining children in the oldest group are lower than the true ones. Thus, the missing children at both extremities of the age range affect the within-grade regression slopes in the same direction: The empirically obtained slope is attenuated, that is, smaller than the true one, thus leading to an underestimated age effect and an overestimated schooling effect. In order to cope with this problem, two groups of subjects were excluded from the computation of the within-grade regressions: (a) students who were under- or overaged, and (b) students born
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Fig. 2. Within-grade regression lines of total scores on age (standardized scores).
in November or December, that is, the birth dates with the highest proportion of missing students (see Table 1). Thus, each within-grade regression was based only on children born between January and October of the appropriate year for that grade. While in each of these months of birth there still was a small proportion of missing students (see Table 1), this proportion did not vary considerably between the months of birth and, therefore, was not likely to affect the within-grade slope. 1.5.3. Linearity of the within-grade regressions The use of a linear model to examine development, which is generally asymptotic, may seem surprising. However, the deviation from linearity occurs over a span of several years, whereas the regressions in this research cover three one-year segments. Undoubtedly, even if development over several years is curvilinear, it can be satisfactorily approximated by short linear segments.
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Table 2 Estimated effects of one year of age (grades 7–9) and one year of schooling (grades 8–9) on DA and AC problem scores (S.D. Units)a Problem type
Schooling
Age
“If the” “If a” “All are”
.19 .15 .23
−.07 −.04 −.09
a
Standard errors range from .05 to .6.
1.6. Estimation of effects This study examined the effects of schooling in two grade levels (eighth and ninth), and the effects of age in three grade levels (seventh, eighth and ninth). To cope with random differences between cohorts and the resulting possibility of age-grade interactions, the true effects of one year of age and one year of schooling were estimated (following Cahan & Cohen, 1989) by averaging the three grade-specific age effects and the two grade-specific schooling effects, respectively. These averages are equivalent to the coefficients of age and grade level in the across-grade multiple regression of test scores on age and grade level. This is a conservative approach, which attributes between-grade variability of the estimates to random error and is based on the generally incorrect assumption that the true effect does not vary between grades (i.e., that the within-grade regressions are parallel and equidistant). We believe, nonetheless, that its adoption is the best course of action, because the information that these averages may fail to reveal is relatively minor, considering that we are dealing with only three adjacent grades. 2. Results Fig. 2 presents the within-grade regression lines of total scores (per invalid problem type) on chronological age. The within-grade slopes are indicative of a slight negative effect of chronological age on the scores on all three problem types, whereas the discontinuity between regression lines in adjacent grades is indicative of a marked schooling effect. The quantitative estimates of the two effects (Table 2) confirm the visual impression: Schooling has a positive net effect on development, whereas age has a negative, albeit very small, effect. The consistency of this pattern of results across the three problem types (“If the”, “If a” and “all are”) helps increase our confidence in their validity and generalizability across tasks. 3. Discussion The findings of this study – the minor negative effect of age, and the major positive effect of schooling – replicate the results of Cahan and Artman (1997) in a different age range. The results are thus consistent with the hypothesis of comprehension theory—namely, that daily conversational experience is dysfunctional in the development of conditional and class reasoning. At the same time, they are inconsistent with “intrinsic” models of the development of reasoning (e.g., Markovits & Barrouillet, 2002), according to which conditional reasoning abilities develop
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mainly as a consequence of psycho-physiological maturational factors, and constitute a universal accomplishment, characterizing the natural course of the developmental process per se, regardless of schooling. On the contrary, our results support the hypothesis that the negative effect of everyday discourse experience is compensated for by the much stronger positive effect of schooling. In other words, the results support the claim that the observed increase in success rates on DA and AC problems as a function of age, documented in many studies, is entirely attributable to school learning. Thus, they support a non-universal, constructivist view, according to which cognitive development cannot be characterized independently of a particular culture. In this view, conditional reasoning is actively constructed in certain kinds of (non-universal) cultural contexts, of the sort associated with formal schooling (Gellalty, 1987; Morris & Sloutsky, 1998; for a discussion of related issues, see Vygotsky (1987), and Larivee, Normandeau, and Parent (2000)). Furthermore, schooling has a unique status in the context of conditional reasoning. Contrary to other cognitive tasks, in which schooling operates in the same direction as out-of-school experience (for a review, see Ceci (1991)), in this case, schooling apparently breaks daily-life interpretational habits, and therefore, is critical for development. Some reservations regarding our results as well as Cahan and Artman’s (1997) should be kept in mind. First, because schooling in Western societies starts at a specific age (usually, 6–7), the schooling effect estimated by these studies is, in fact, the effect of the specific age × schooling combination rather than the main effect of schooling per se, regardless of age. Theoretically, this effect may, and is in fact likely to vary between different combinations of age and amount of schooling. Hence, it will be erroneous to conclude that much younger children (e.g., 3 years old) or unschooled adults would benefit from schooling like those who started school at the “normal” age. There are of course psycho-physiological prerequisites to receptiveness of schooling inputs. Similarly, schooling may be less effective during adulthood. Second, the critical contribution of schooling to the development of conditional reasoning found in our study may be specific to “Western-style” schooling systems. In view of the possibility of considerable variability between school systems, the issue of the universality of the central role of schooling in this realm is an empirical one, to be answered by additional studies. Third, our results, as well as those of Cahan and Artman (1997), are based on a specific population in a specific educational system (Hebrew speaking Israelis). The theoretical significance of these results is clearly dependent on their replicability in other populations and educational systems.
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Appendix A. The Syllogism Test
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Overton, W. F., Ward, S. L., Black, J., Noveck, I. A., & O’Brien, D. P. (1987). Form and content in the development of deductive reasoning. Developmental Psychology, 23, 22–30. Piaget, J. (1972). Intellectual evolution from adolescence to adulthood. Human Development, 15, 1–12. Politzer, G. (1986). Laws of language use and formal logic. Journal of Psycholinguistic Research, 15, 47–92. Rumain, B., Connel, J., & Braine, M. D. S. (1983). Conversational comprehension processes are responsible for reasoning fallacies in children as well as adults: If is not the biconditional. Developmental Psychology, 19, 471–481. Scholnick, E. K., & Wing, C. S. (1991). Speaking deductively: Pre-schooler’s use of if in conversation and conditional inference. Development Psychology, 27, 249–258. Scholnick, E. K., & Wing, C. S. (1992). Speaking deductively: Using conversation to trace origins of conditional thought in children. Merril-Palmer Quarterly, 38, 1–20. Scribner, S., & Cole, M. (1973). Cognitive consequences of formal and informal education. Science, 182, 553–559. Taylor, R. (1967). Causation. In P. Edwards (Ed.), The encyclopedia of philosophy (Vol. 2, pp. 56–66). New York: Macmillan. Vygotsky, L. S. (1987). The development of scientific concepts in childhood. In R. W. Rieber & A. S. Carton (Eds.), The collected works of L.S. Vygotsky: Volume 1. Problems of general psychology (pp. 167–241). New York: Plenum Press. Ward, S. L., Byrnes, J. P., & Overton, W. F. (1990). Organization of knowledge and conditional reasoning. Journal of Educational Psychology, 82, 832–837.
Age, schooling and conditional reasoning Lavee Artman a , Sorel Cahan b,∗ , Dinah Avni-Babad b b
a The David Yellin College, Jerusalem, Israel School of Education, The Hebrew University, Jerusalem, Israel
Abstract This study estimated the independent effects of age and schooling in grades 7–9 on scores obtained on invalid conditional and class syllogisms. The results, which point to a negative, albeit small, effect of outof-school experience and to a sizeable positive effect of schooling, replicate previous findings in a different age range and support the counterintuitive hypothesis that accumulated daily experience with conditionals has a negative effect on the development of conditional reasoning, and that improved performance on invalid problems with age is entirely attributable to schooling. Contrary to most cognitive tasks, therefore, in which schooling operates in the same direction as out-of-school experience, in this case, schooling breaks daily-life interpretational habits, and therefore, is critical for development. © 2006 Elsevier Inc. All rights reserved. Keywords: Conditional reasoning; Schooling; Cognitive development
Invalid conditional syllogisms include two argument patterns: denying the antecedent (DA; if p then q, not p, not q) and affirming the consequent (AC; if p then q, q, p). The classical fallacies regarding these invalid argument forms are to accept “not q” or “p”, respectively, as true, which would be in line with biconditional interpretation (“if and only if”) of the conditional (Braine & O’Brien, 1991; Braine & Rumain, 1983; O’Brien, 1987). These fallacies are common in early childhood and their frequency decreases with the increase of age (e.g., Evans, Newstead, & Byrne, 1993). Two major approaches have been used to explain the origin of these fallacies and the developmental mechanism involved in their decrease with age: logical competency models (e.g., Markovits, 1993; Markovits & Barrouillet, 2002) and comprehension theory (e.g., Braine & O’Brien, 1998). According to the first, the fallacies are due to inadequate logical ability or constraints in premise representation, and their decrease with age reflects the development of conditional reasoning abilities. Traditional competency models (e.g., Overton, 1990; Overton, ∗
Corresponding author. Tel.: +972 2 5882083; fax: +972 2 5882098. E-mail address: [email protected] (S. Cahan).
0885-2014/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.cogdev.2006.01.004
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Ward, Black, Noveck, & O’brien, 1987; Piaget, 1972) consider children to be limited interpreters of the conditional to cases in which both elements (p and q) are either present or absent. Consequently, increased success rates with age reflect continuous maturation of logical competence. A more recent view (e.g., Markovits & Barrouillet, 2002) attributes the fallacies to children’s inability to represent the case of not p and q due to limited capacity of working memory or restricted world knowledge, and the decrease in their frequency with age to the development of working memory capacity and accumulation of world knowledge. In contrast, comprehension theory (e.g., Rumain, Connel, & Braine, 1983) attributes logical errors to the tendency to apply everyday discourse rules to reasoning tasks (Begg & Harris, 1982; Braine & O’Brien, 1991; Braine & Rumain, 1983; Rumain et al., 1983). According to this view, on problems in which neither the direct-reasoning routine nor any obvious available strategy leads to a solution, people will often base inferences on other resources, especially, conversational implicatures (O’Brien, Dias, & Roazzi, 1998). These generalized implicatures follow from extra logical meanings that are common to how a natural-language particle is construed (Grice, 1989) and are commonly made unless there are circumstances to inhibit them (Grice, 1975). Comprehension theory is especially relevant to the explanation of the classical fallacies in conditional reasoning. In natural language, the word ‘if’ is ambiguously represented as a conditional or a biconditional according to context (Evans et al., 1993, chap. 2). However, the common interpretation of a typical (i.e., circumstantial) ‘if’ is biconditional, that is, if not p then not q (e.g., Geis & Zwicky, 1971). This is because these conditionals express either a sufficient condition (in which the not p case is irrelevant and no further interpretation is required)1 or a necessary condition (e.g., threats, intentions and promises) for which the biconditional interpretation is appropriate. Consequently, inferring the obverse of a conditional (i.e., “if not p then not q”) is in fact a correct conclusion and should be considered rational in daily context (e.g., Evans, 1993; Fillenbaum, 1986). Indeed, Scholnick and Wing (1991, 1992) found that 70–90% of the conclusions derived by adults from invalid conditional arguments in daily conversations were biconditional. This also holds true with respect to everyday class statements of the form “all are”. Because the speaker in everyday life is expected to be as informative as possible, the correct interpretation of such statements is biconditional, that is “All A’s and only A’s” (see Cahan and Artman (1997) for a comprehensive discussion). In contrast to the logic of conversation, in propositional logic the meaning of a conditional statement is decontextualized from its intended meaning in everyday discourse. In logic there is only one correct interpretation of a conditional, namely, material implication: the meaning common to all conditional statements. Material implication between two propositions (P → Q) is a relation that is true in all cases but p and not q. That is, p is sufficient (but not necessary) to q, and q is necessary (but not sufficient) to p. Obviously, for this interpretation, the DA and AC argument forms are invalid by definition. The fallacies in conditional and class syllogisms can thus be ascribed to overgeneralization of the logic of discourse. Subjects apply a biconditional interpretation to the major premise because this is the only one they know to be appropriate for those everyday situations in which the addressee is required to make inferences. Hence, the generalization of the biconditional interpretation to laboratory conditional inference tasks should not be a surprise: 1
Indeed, Newstead, Ellis, Evans, and Dennis (1997) found that on truth table tasks, FT cases are judged by 92–96% of the subjects as false or irrelevant for a variety of everyday conditionals (temporal, causal, promise, threat, tip, warning, universal, intentional).
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. . . how is a subject to understand the meaning of an utterance in an artificial context such as this [experimental reasoning tasks], except by reference to its normal everyday usage? . . . [T]he DA inference is reasonably . . . made in most real-world contexts in which it appears. (Evans, 1993, p. 18) The decrease in erroneous responses to invalid conditional syllogisms with age is explained by comprehension theory as a growing dissociation from the everyday biconditional interpretation which, in turn, is attributed to an improved ability to set aside the “logic of conversation” when solving laboratory reasoning problems (Rumain et al., 1983). If this is the case, the following question arises: How, without assuming growth in logical competence, can we explain the gradual abandonment of the biconditional interpretation, and hence the decline in classical fallacies with age? Unfortunately and unsurprisingly, comprehension theory is not explicit regarding the causal model underlying this developmental process, due to the difficulties associated with its identification. That is, there is no specific explanation for how adults become “more able than children to set aside the logic of conversation when confronted by laboratory reasoning problems” (Rumain et al., 1983, p. 421). In order to succeed on laboratory tasks, a na¨ıve reasoner must know in advance that in the laboratory reasoning context she/he is expected to adopt a new and different interpretation to the premises, which is neither explicit in his/her instructions (Evans, 2002) nor compatible with the correct everyday usage of the ‘if’ particles. That is, successful performance on laboratory tasks requires children to (a) become acquainted with the unnatural demands of the ‘logic game’ (decontextualization and implicative-like interpretation) and (b) learn to adapt interpretations to contexts (it will be inadequate if not bizarre, to subject everyday communication to the partial meaning common to all conditionals). Indeed, Politzer (1986) noted that the development of logical thought always competes with previously acquired, contradictory laws; it tends toward a dual system and it depends on differentiation between and coordination of the two subsystems: language communication laws and propositional logic. Clearly, everyday life cannot be expected to provide the experiences necessary for this learning process. Daily experience cannot be expected to acquaint children with the implicative, asymmetric, interpretation of the conditional, nor can it be expected to provide them with practice in its use. In fact, given that the only relevant everyday life interpretation is biconditional, the accumulation of daily conversational experience should reinforce its usage rather than weaken it. That is, everyday experience should be dysfunctional to the disassociation from the biconditional interpretation and, therefore, also to success on invalid conditional syllogisms. The differentiation between and the coordination of language communication laws and propositional logic necessarily requires a different kind of experience, in which children have the opportunity to encounter and practice the ‘logic game’. According to Cahan and Artman (1997) this experience is provided by schooling. Mainly through mathematics and science, schooling indirectly offers children what is perhaps their first encounter with the “weak”, implicative, interpretation of conditional and class statements and provides them with practice in recognizing the context in which this asymmetric interpretation is appropriate. The directionality of class and conditional statements is introduced explicitly in geometry, through the distinction between direct theorems and inverse theorems. In addition, school science involves systematic analysis of the necessary and sufficient conditions that determine the occurrence of the phenomenon under study (Taylor, 1967). Many such scientific activities (e.g., testing the properties of materials using diagnostic indicators) require students to make conditional inferences, the correct formulation of which are constrained by the neces-
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sity/sufficiency status of particular cause-effect relations (e.g., “If the bulb lights up, the material is a conductor, but it is not true that if it is a conductor, the bulb will necessarily light up”). In addition, schooling fosters the development of secondary skills (Braine, 1990): Children learn that different situations require different interpretations. Literal analysis of texts, learning in settings isolated from the context of the objects referred to, and the taxonomic-like organization of knowledge teach the child to confine interpretation to the information explicitly stated in the text (Hildyard & Olson, 1978), to distinguish between what the sentence itself means and what one means by the same sentence (Braine, 1990), to construct verbal-hypothetical worlds (Applebee, 1984; Braine, 1990; Donaldson, 1987; Olson, 1977; Scribner & Cole, 1973), and organize classes of objects through hierarchical classification (Ward, Byrnes, & Overton, 1990). A reasoner equipped with this knowledge should notice that the first premise of an invalid conditional argument does not include information related to the second premise, to suspend judgments regarding conversational implicatures (i.e., to adopt ‘minimal commitment’ to the conditional in the major premise) and to envision and represent the possibility of “not p and q” (Cahan & Artman, 1997). In sum, according to this hypothesis, the decreased rates of the classical fallacies with age are entirely attributable to a culture-specific experience, extrinsic to the developing individual, namely, schooling. The effect of aging and out-of-school experiences is expected to be negative. These predictions regarding the effects of schooling and out-of-school experiences clearly contradict those derived by the logical competency models, according to which the decrease in the fallacies rates with age is considered to be a basic and universal process, resulting from the child’s transactions with any normal environment, schooled or not. According to this approach, the effect of schooling is negligible and development is mainly due to aging and the associated accumulation of experience. To date, Cahan and Artman’s (1997) study is the only available empirical evidence relevant to this issue. In their study, the amount of everyday experience was measured by chronological age and schooling was measured by grade level. The independent effects of age and schooling on scores obtained on invalid conditional and class syllogisms were estimated in grades 4–6 (age range 9.5–12.5). The results pointed to a negative, albeit very small, effect of chronological age and to a sizeable positive effect of schooling. The theoretical significance of these results is clearly dependent on their replicability across age range and tests. This study is a first step in this direction. The study estimated the unique effects of schooling (the student’s grade level at the time of test administration) and out-of-school discourse experience (i.e., chronological age) on success in solving invalid conditional and class problems among adolescents (grades 7–9), using a broad range of uncertain forms of conditional syllogisms, including all possible combinations of premise type (if the; if a; all are) and argument pattern.
1. Method 1.1. Participants The sample included 1885 seventh, eighth and ninth graders in three Israeli junior high schools (about 600 in each grade) (see Table 1). Age means were 12.3,13.3, and 14.3, respectively.
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Table 1 Sample by grade and age status (N) Age status
Grade 7
Underage “Normal” age January–October November–Decembera Overage Total a
8
9
6
13
9
465 62
495 63
510 70
58
67
67
591
638
656
These students were excluded from the data analysis.
1.2. Measures The conditional syllogism test consisted of (a) test items – 12 randomly-ordered DA and AC class and conditional problems and (b) camouflage items – 6 MP or MT items, 4 in the “if-then” form (2 MP and 2 MT) and 2 in the “all are” form (1 MP and 1 MT). These items (item nos. 1, 6, 8, 9, 13, 17 in Appendix A) were included in the test in order to minimize the possibility of “response set” and were excluded from the data analysis. Each of the 12 test items presented an argument composed of two premises and a conclusion which the subject had to evaluate as necessarily true, necessarily false or indeterminate (no way to tell). For example:
The major premises included three types of sentence (four items each): “if the” (e.g., if the A is . . . then it has to be a . . .), “if a” (if an A is . . . then it has to be a . . .) and “all . . . are” (all the A’s are F’s). For each type of premise, two items denied the antecedent and two affirmed the consequent. The test language was Hebrew (see Appendix A for an English translation of the original Hebrew version). 1.3. Procedure The test was administered on a classroom basis at the beginning of the school year. Each participant responded to all the 18 items. Two testers (university students who had received special instruction) were present in each classroom. Before the start of the test, the children were presented with two examples: one with an “all are” premise (MP), and the other with an invalid disjunctive syllogism (A or B or both, A, is B true?). Before revealing the correct answers, the tester explained how the premises should be interpreted in order to arrive at the proper conclusion.
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The rationale for this short training was to acquaint subjects with the test format and appropriate use of the “no way to tell” response (i.e., as a judgment of indeterminacy rather than of subjective uncertainty). The test was administered without any time limitation. 1.4. Scoring Each item was scored dichotomously (1 = correct, 0 = incorrect or no response). The percentage of “no response” answers ranged between 1 and 4% across items with negligible differences between grades. The proportion was higher, on the average, the higher the serial position of the item, perhaps due to fatigue. Ninety percent of the subjects in each grade level responded to all the items and 95% in each grade level reached the last item. Among subjects who skipped at least one item, 50% did not respond to two items at the most. Three total scores (percent correct answers), one for each of the three types of premises, were calculated for each examinee. 1.5. Design and methodological considerations 1.5.1. The between-grades paradigm A between-grade regression discontinuity approach (Cahan & Cohen, 1989; Cahan & Davis, 1987) was employed to disentangle the independent effects of schooling and age on success in conditional reasoning tasks. This approach involves administering the same test to at least two adjacent grade levels. It relies on the following assumptions: (1) the “allocation” of children to birth dates is random; (2) grade level is solely a function of chronological age; that is, admission to school is based on chronological age only according to some arbitrary cut-off point, and progression through grades is automatic (i.e., there are no drop-outs and children are neither kept back nor advanced a grade). On the basis of these assumptions, the net effect of chronological age and schooling are estimated by means of a regression discontinuity design (Cook & Campbell, 1979), in which test scores are regressed on chronological age within grades. In this design, the effect of age is reflected in the slope of the within-grade regression of test scores on chronological age, and the effect of schooling is reflected in the discontinuity between these regressions (see Fig. 1).
Fig. 1. The effects of age and schooling in the regression discontinuity design. Note: The dotted section in each regression line represents extrapolation of the regression line for the months of birth which were excluded due to their high proportion of selective grade misplacement.
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Specifically, the estimated effect of a one-year difference in chronological age in a given grade equals the difference between the oldest and youngest students in that grade in mean predicted scores (see dotted arrows in Fig. 1), and the estimated effect of one year of schooling equals the differences in mean predicted scores between the youngest children in any given grade (X) and the oldest children in the lower adjacent grade (X−1). 1.5.2. The truth of the assumptions As already stated, the between-grades paradigm makes two assumptions: (1) children are randomly allocated to birth dates; (2) grade level is solely a function of chronological age. Certain reservations regarding the truth of these assumptions, that have been raised before (Cahan & Cohen, 1989; Cahan & Davis, 1987), should be reiterated. The truth of the first assumption cannot be empirically tested. However, since we tested students in only three adjacent grades within a relatively homogeneous population, this assumption seems reasonable with respect to between-grades variability. As far as within-grade randomization is concerned, exceptions to this assumption can affect the estimation of age and schooling effects only if they are monotonically related to birth date, which is very unlikely. The second assumption of the model is only partially true. While grade retention and grade skipping are seldom practiced in the Israeli school system (Cahan & Cohen, 1989), admission to school is sometimes delayed and sometimes accelerated. Consequently, in any given grade there are “overage” and “underage” children, that is, children whose age should place them in a higher or lower grade, and there are also “missing” children (i.e., children who are learning in a higher or lower grade) (see Table 1). More important, the delay or acceleration of school admission is not random. The children whose admission was delayed are likely to be less developed intellectually than the other children in their age group, and those whose admission was accelerated are likely to be more developed. Hence, the children who remain in the “appropriate” grade are also selective (in the opposite direction). In addition, the relative frequency of grade misplacement is related to the month of birth, being particularly high near the cut-off point: Delays are especially frequent among the youngest children in each cohort and accelerations among the oldest (see Table 1). In particular, the relative frequency of delays is much higher (about 25%) among children born in November and December relative to the other months (2–8%) (see Table 1). Cook and Campbell (1979) refer to this possibility as a “fuzzy cutting point”. There are two ways in which selective misplacement may affect the within-grade regression slopes: (1) Due to the existence of underage and overage children in each grade. The direction of this effect cannot be established a priori since age and selection counteract each other in this case: the underage children are also brighter, while the overage ones are generally duller. (2) Due to the missing children in each grade. At the lower extreme of the age range, the missing children are those that have been delayed; hence, the mean test score of the remaining children in the youngest groups is higher than the true one. At the higher extreme of the age range, selection operates in the opposite direction: the missing children are the brightest ones, whose admittance to school has been accelerated. Consequently, the mean test score of the remaining children in the oldest group are lower than the true ones. Thus, the missing children at both extremities of the age range affect the within-grade regression slopes in the same direction: The empirically obtained slope is attenuated, that is, smaller than the true one, thus leading to an underestimated age effect and an overestimated schooling effect. In order to cope with this problem, two groups of subjects were excluded from the computation of the within-grade regressions: (a) students who were under- or overaged, and (b) students born
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Fig. 2. Within-grade regression lines of total scores on age (standardized scores).
in November or December, that is, the birth dates with the highest proportion of missing students (see Table 1). Thus, each within-grade regression was based only on children born between January and October of the appropriate year for that grade. While in each of these months of birth there still was a small proportion of missing students (see Table 1), this proportion did not vary considerably between the months of birth and, therefore, was not likely to affect the within-grade slope. 1.5.3. Linearity of the within-grade regressions The use of a linear model to examine development, which is generally asymptotic, may seem surprising. However, the deviation from linearity occurs over a span of several years, whereas the regressions in this research cover three one-year segments. Undoubtedly, even if development over several years is curvilinear, it can be satisfactorily approximated by short linear segments.
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Table 2 Estimated effects of one year of age (grades 7–9) and one year of schooling (grades 8–9) on DA and AC problem scores (S.D. Units)a Problem type
Schooling
Age
“If the” “If a” “All are”
.19 .15 .23
−.07 −.04 −.09
a
Standard errors range from .05 to .6.
1.6. Estimation of effects This study examined the effects of schooling in two grade levels (eighth and ninth), and the effects of age in three grade levels (seventh, eighth and ninth). To cope with random differences between cohorts and the resulting possibility of age-grade interactions, the true effects of one year of age and one year of schooling were estimated (following Cahan & Cohen, 1989) by averaging the three grade-specific age effects and the two grade-specific schooling effects, respectively. These averages are equivalent to the coefficients of age and grade level in the across-grade multiple regression of test scores on age and grade level. This is a conservative approach, which attributes between-grade variability of the estimates to random error and is based on the generally incorrect assumption that the true effect does not vary between grades (i.e., that the within-grade regressions are parallel and equidistant). We believe, nonetheless, that its adoption is the best course of action, because the information that these averages may fail to reveal is relatively minor, considering that we are dealing with only three adjacent grades. 2. Results Fig. 2 presents the within-grade regression lines of total scores (per invalid problem type) on chronological age. The within-grade slopes are indicative of a slight negative effect of chronological age on the scores on all three problem types, whereas the discontinuity between regression lines in adjacent grades is indicative of a marked schooling effect. The quantitative estimates of the two effects (Table 2) confirm the visual impression: Schooling has a positive net effect on development, whereas age has a negative, albeit very small, effect. The consistency of this pattern of results across the three problem types (“If the”, “If a” and “all are”) helps increase our confidence in their validity and generalizability across tasks. 3. Discussion The findings of this study – the minor negative effect of age, and the major positive effect of schooling – replicate the results of Cahan and Artman (1997) in a different age range. The results are thus consistent with the hypothesis of comprehension theory—namely, that daily conversational experience is dysfunctional in the development of conditional and class reasoning. At the same time, they are inconsistent with “intrinsic” models of the development of reasoning (e.g., Markovits & Barrouillet, 2002), according to which conditional reasoning abilities develop
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mainly as a consequence of psycho-physiological maturational factors, and constitute a universal accomplishment, characterizing the natural course of the developmental process per se, regardless of schooling. On the contrary, our results support the hypothesis that the negative effect of everyday discourse experience is compensated for by the much stronger positive effect of schooling. In other words, the results support the claim that the observed increase in success rates on DA and AC problems as a function of age, documented in many studies, is entirely attributable to school learning. Thus, they support a non-universal, constructivist view, according to which cognitive development cannot be characterized independently of a particular culture. In this view, conditional reasoning is actively constructed in certain kinds of (non-universal) cultural contexts, of the sort associated with formal schooling (Gellalty, 1987; Morris & Sloutsky, 1998; for a discussion of related issues, see Vygotsky (1987), and Larivee, Normandeau, and Parent (2000)). Furthermore, schooling has a unique status in the context of conditional reasoning. Contrary to other cognitive tasks, in which schooling operates in the same direction as out-of-school experience (for a review, see Ceci (1991)), in this case, schooling apparently breaks daily-life interpretational habits, and therefore, is critical for development. Some reservations regarding our results as well as Cahan and Artman’s (1997) should be kept in mind. First, because schooling in Western societies starts at a specific age (usually, 6–7), the schooling effect estimated by these studies is, in fact, the effect of the specific age × schooling combination rather than the main effect of schooling per se, regardless of age. Theoretically, this effect may, and is in fact likely to vary between different combinations of age and amount of schooling. Hence, it will be erroneous to conclude that much younger children (e.g., 3 years old) or unschooled adults would benefit from schooling like those who started school at the “normal” age. There are of course psycho-physiological prerequisites to receptiveness of schooling inputs. Similarly, schooling may be less effective during adulthood. Second, the critical contribution of schooling to the development of conditional reasoning found in our study may be specific to “Western-style” schooling systems. In view of the possibility of considerable variability between school systems, the issue of the universality of the central role of schooling in this realm is an empirical one, to be answered by additional studies. Third, our results, as well as those of Cahan and Artman (1997), are based on a specific population in a specific educational system (Hebrew speaking Israelis). The theoretical significance of these results is clearly dependent on their replicability in other populations and educational systems.
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Appendix A. The Syllogism Test
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