Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task

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1 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY 2009, 62 (8), Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task Gaëlle Meert and Jacques Grégoire Unité de Psychologie de l Education et du Développement, Faculté de Psychologie et des Sciences de l Education, Université Catholique de Louvain, Louvain-la-Neuve, Belgium Marie-Pascale Noël Unité Cognition et Développement, Faculté de Psychologie et des Sciences de l Education, Université Catholique de Louvain, Louvain-la-Neuve, Belgium This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult. Keywords: Numerical cognition; Fractions; Magnitude comparison; Distance effect; Congruity effect. Interest in numerical cognition has motivated numerous studies on the processing of positive integers (i.e., natural numbers), while other numerical categories, such as rational numbers, have not been greatly investigated. This study investigated cognitive processing in adults of the magnitude of rational numbers represented by fractions. Our first aim was to better understand Correspondence should be addressed to Gaëlle Meert, Unité de Psychologie de l Education et du Développement, Faculté de Psychologie et des Sciences de l Education, Université Catholique de Louvain, Louvain-la-Neuve, Belgium. gaelle.meert@uclouvain.be This study was supported by a grant from Actions de Recherche Concertées 05/ of the French-Speaking Community of Belgium. G.M. and M.P.N are supported by the Fund for Scientific Research of the French-Speaking Community of Belgium (FRS-FNRS) # 2008 The Experimental Psychology Society DOI: /

2 RATIONAL NUMBER PROCESSING how adults deal with fractions. The processing of a fraction s magnitude is crucial in mathematics, since fractions are used in algebraic, probabilistic, and proportional reasoning. This processing is also required in many situations of everyday life, such as the calculation of drug dosage, the estimation of a discount, or the estimation of the chances of winning in a lottery. Fractions consist of a quotient between two integer numbers (e.g., 1/7) where the denominator is not equal to 0. With regard to the structure of this symbolic representation, we investigated whether the mental magnitude of the whole fraction (i.e., holistic representation) or only the magnitudes of the fraction components (i.e., componential representation) were accessed during the processing of the fraction magnitude in a numerical comparison task. Gallistel, Gelman, and Cordes (2005; see also Gallistel & Gelman, 2000) have suggested that the cognitive foundation of magnitude processing is an innate system that can represent real numbers, a numerical category larger than the integer numbers, and the rational numbers (i.e., one that includes them). Their theory emerges from a large body of empirical data showing that animals are able to process continuous quantities (e.g., duration) and discrete quantities (e.g., flashes) as well as to perform arithmetical operations combining both discrete and continuous quantities. Both countable quantities (discrete) and uncountable quantities (continuous) would be represented in the same system by continuous mental magnitudes that would be approximate (i.e., with a scalar variability). Studies with adults have suggested that the same mental magnitudes are accessed not only from analogical representations (e.g., a collection of dots) but also from symbols (i.e., Arabic numerals or verbal number words) through the learning of a mapping (e.g., Whalen, Gallistel, & Gelman, 1999). While integer numbers refer only to discrete quantities, real numbers refer to both discrete and continuous quantities. Therefore, the mental magnitudes would be isomorphic to real numbers and support the processing of their magnitude. Following this theory, the magnitude processing system should allow the representation of the magnitudes of whole fractions as they represent symbolically rational numbers, a part of the category of real numbers. Yet, in the cultural history of numbers, integer numbers are the foundation of the mathematical thought from which mathematicians have developed other categories of numbers. Gallistel et al. (2005) have suggested that integer numbers are the foundation of the cultural history of numbers due to the relative ease in establishing one-toone mapping between the counted objects (i.e., discrete quantities) and the sequence of symbols (i.e., verbal number words) by counting. The real number system was developed only later because such counting was not appropriate for continuous quantities due to their density. Indeed, the use of a reference unit (e.g., for the measurement of a length or a duration) only provides a rough measurement of the continuous quantity. The historical development of numbers would be therefore a Platonic rediscovery of what the non-verbal brain was doing all along (Gallistel & Gelman, 2000, p. 60). To our knowledge, only Bonato, Fabbri, Umiltà, and Zorzi (2007) have investigated the processing of the fraction magnitude in adults. Four experiments were reported in which adults compared fractions to a fixed standard (1/5, 0.2, or 1). Two classical effects in numerical cognition, distance and SNARC (spatial-numerical association of response codes) effects, were used to test whether the mental magnitude of the whole fraction was accessed during the task. In a numerical comparison task, the distance effect refers to the improvement in performance with the increase of the numerical distance between the target number and the fixed standard (Moyer & Landauer, 1967). In other words, responses are faster and more accurate when the distance is large (e.g., 1_5) than when the distance is small (e.g., 4_5). The SNARC effect is an interaction between the response code and the number magnitude: faster response to large numbers if the response is on the right side of space and to small numbers if the response is on the left side of space (e.g., Dehaene, Bossini, & Giraux, THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8) 1599

3 MEERT, GRÉGOIRE, NOËL 1993; for reviews: Gevers & Lammertyn, 2005; Hubbard, Piazza, Pinel, & Dehaene, 2005). The results reported by Bonato et al. have shown that response times (RTs) were better predicted by the distance between the components of the fractions than by the distance between the whole fractions themselves. When the fixed standard was not a fraction (e.g., 0.2), the participants converted it into a fraction in order to directly compare the denominators. The SNARC effect was less consistent across the various experiments. The most interesting result was the interaction between the response code and the magnitude of the denominators (reversed SNARC effect) during the comparison of fractions with numerator 1 (e.g., 1/5_1/3). The authors concluded that the participants used strategies in order to process the fraction magnitude componentially and that the mental magnitude of the whole fraction was not accessed. Contrary to the lack of studies on fraction processing in adults, a considerable amount of studies concerns fraction learning in children. However, these studies have investigated the conceptual understanding of fractions rather than the cognitive processing of their magnitude. In the Stafylidou and Vosniadou study (2004), 11-year-old children often saw fractions as two independent natural numbers. Among other misconceptions observed in a numerical ordering task of fractions without common denominators, children claimed that the value of a fraction increases as the value of the numerator increases (e.g., 4/3, 5/6) or that the value of a fraction increases as the value of the numerator decreases (e.g., 4/3, 1/7) without taking into account the magnitudes of the denominators. In this study, students aged from 10 to 17 gradually overcame the interference of natural numbers and counting knowledge before their understanding of fractions matched the scientific concept. This process implies a conceptual change following several stages. The most frequent errors and misconceptions noticed in children learning fractions would result from the tendency to interpret fractions by referring to previous counting and natural number knowledge (called the whole number bias, Ni & Zhou, 2005). As the properties of natural numbers and counting differ from the properties of rational numbers, this bias interferes with fraction learning and leads to errors and misconceptions (e.g., Behr, Harel, Post, & Lesh, 1992; English & Halford, 1995; Grégoire & Meert, 2005; Hartnett & Gelman, 1998; Mack, 1993; Sophian, 1996; Stafylidou & Vosniadou, 2004). Bonato et al. (2007) suggested that the whole number bias observed in children is consistent with the componential strategies used to process the fraction magnitude. According to these authors, the understanding of fractions might be rooted in the ability to represent discrete quantities rather than continuous quantities. Adults might differ from children in their ability to use flexible and appropriate strategies of componential processing. Nevertheless, the whole number bias could simply reflect the misunderstanding of the complex symbol that fractions represent, at least at the beginning of fraction learning, and the interference of previous numerical knowledge. This bias does not preclude the magnitudes of whole fractions from being represented by mental magnitudes. As studies on fraction learning were seeking to investigate the understanding of fractions, they did not provide any direct evidence in favour of either holistic or componential representation in children who understand, at least in part, the symbolic representation of rational numbers in fraction form. The issue of the nature of magnitude processing (holistic vs. componential) has also been debated for other complex numerical symbols, such as two-digit numbers or negative numbers. For two-digit numbers, some authors have reported evidence in favour of a holistic representation of their magnitude (e.g., Dehaene, Dupoux, & Mehler, 1990; Reynvoet & Brysbaert, 1999) whereas others showed effects in favour of a componential representation (e.g., Nuerk, Weger, & Willmes, 2001; Ratinckx, Brysbaert, & Fias, 2005; Verguts & De Moor, 2005). To reconcile these conflicting results, Zhou, Chen, Chen, and Dong (2008) showed that the mode of presentation (simultaneous vs. sequential) could determine the processing of two-digit numbers (see also Zhang & Wang, 2005). On the other hand, 1600 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8)

4 RATIONAL NUMBER PROCESSING Nuerk and Willmes (2005) suggested that the holistic magnitudes could be activated in parallel with the magnitudes of tens and of units. For negative numbers, the nature of their processing seems to depend on the task. In a parity judgement task, the magnitudes of the digits are processed regardless of the sign (i.e., componential representation; Fischer & Rottmann, 2005; Nuerk, Iversen, & Willmes, 2004). In a magnitude comparison task, Shaki and Petrusic (2005) demonstrated componential representation when the task was performed only on negative numbers but holistic representation when the task was performed on both positive and negative number pairs. These studies suggest that the processing of the magnitude of Arabic numbers with a complex structure could depend on the experiment s conditions. The purpose of the current study was to identify the mental magnitudes activated during magnitude comparison of two fractions. We tested whether the processing of the fraction magnitude depends on the congruity between the magnitude of the whole fraction and the magnitudes of its components and on the ability to identify this congruity in the experimental context. In the first experiment, only pairs of fractions with common components were presented. In a pair of fractions with common denominators (i.e., a/x_b/x; e.g., 3/7_5/7), the relative magnitude of the numerator is congruent with the relative magnitude of the whole fraction since the larger fraction (e.g., 5/7) is made up of the larger numerator (e.g., 5). In a pair of fractions with common numerators (i.e., x/a_x/b; e.g., 2/3_2/5), the relative magnitude of the denominator is incongruent with the relative magnitude of the whole fraction since the larger fraction (e.g., 2/3) is made up of the smaller denominator (e.g., 3). In the first case, the congruity could lead participants to directly compare the numerators in order to select the larger fraction, without accessing the magnitudes of the whole fractions. In the second case, the incongruity could lead the participants to compare the magnitudes of the whole fractions, which could be accessed by estimating the ratio between the magnitude of the numerator and the magnitude of the denominator for each fraction. Nevertheless, the processing of the denominators could interfere with this holistic processing as the larger denominator does not make up the larger fraction. Alternatively, participants could compare the denominators by taking into account their incongruity with the fraction magnitude (i.e., choosing the smaller denominator to choose the larger fraction). In the second experiment, the same stimuli were mixed with pairs of fractions without common components to reduce the regularity of the stimuli in the experiment. For these pairs of fillers, the congruity of the component magnitude cannot be identified without an access to the fraction magnitude. Therefore, access to holistic representation of the fraction magnitude would be required for these pairs and then could be enhanced for fractions with common components as this processing is appropriate for all pairs. The numerical distance effect was used to test whether the participants rely on the magnitudes of the components or on the magnitudes of the whole fractions to compare fractions. If participants compare only the components of the two fractions, performance should be influenced by the componential distance (i.e., the distance between the numerators for a/x_b/x fractions and the distance between the denominators for x/a_x/b fractions). Alternatively, if the magnitudes of the whole fractions are accessed before performing the comparison, performance should vary with the distance between the whole fractions (i.e., the overall distance). Moreover, a priming paradigm was used. The comparison of fractions primed the comparison of natural numbers. If the magnitudes of the whole fractions are accessed for x/a_x/b fractions, the denominator magnitude could interfere with the selection of the larger fraction, and the selection of the larger denominator might be inhibited in order to select the larger fraction. Therefore, if natural numbers identical to the denominators of x/a_x/b fractions (i.e., a_b) are presented after these fractions, their comparison should require the activation of the same response as the response inhibited during the processing of the previous fractions. In this case, the THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8) 1601

5 MEERT, GRÉGOIRE, NOËL residual inhibition on the response should lead to slower RTs than those to natural numbers differing from the denominators of the previous x/a_x/b fractions. On the other hand, RTs to natural numbers primed by a/x_b/x fractions should be faster for natural numbers identical to the numerators of these fractions (i.e., a_b) than for natural numbers differing from these numerators due to residual activation on the selection of the larger numerator during the processing of a/x_b/x fractions. EXPERIMENT 1 Method Participants A total of 40 university undergraduate psychology students (39 women; 34 right-handed) took part in this experiment and received course credit. The average age was 19 years (ranging from 18 to 23 years). Design Participants were asked to compare the numerical value of two fractions or two natural numbers. Two types of fraction were presented: fractions with common numerators (x/a_x/b, e.g., 2/7_2/ 3) and fractions with common denominators (a/ x_b/x, e.g., 3/8_7/8). Each pair of natural numbers was primed in four priming conditions resulting from the crossing of two within-subject variables: the type of prime (x/a_x/b vs. a/x_b/ x) and the priming specificity (specific vs. unspecific). The priming was specific when the numerators of a/x_b/x fractions or the denominators of x/a_x/b fractions were identical to the subsequent natural numbers. The priming was unspecific when all the fraction components differed from the subsequent natural numbers. The unspecific priming was used as a baseline to assess the effect of the specific priming. Figure 1 shows an example of a trial (prime and probe) for each of the four conditions. Stimuli The pairs of fractions consisted of 32 pairs of x/ a_x/b fractions and 32 pairs of a/x_b/x fractions (see Appendix). The denominators of a given pair of x/a_x/b fractions (e.g., 3 and 7 in 2/7_2/3) were the same natural numbers as the numerators of a given pair of a/x_b/x fractions (e.g., 3 and 7 in 3/8_7/8). In this way, the numerators of a/ x_b/x fractions and the denominators of x/a_x/b fractions were strictly matched for numerical distance. 1 Both types of fraction were balanced for the numerical distance between the whole fractions. For a/x_b/x fractions, the overall distance ranged from 0.05 to 0.55 with a mean distance of 0.20 (SD ¼ 0.13). For x/a_x/b fractions, the range was from 0.04 to 0.44 with a mean distance of 0.18 (SD ¼ 0.11). Pairs were made up of irreducible fractions that were smaller than the unit and whose components were numbers ranging from 2 to 19 (excluding 10 as denominator). The use of irreducible fractions prevented variability between and within participants due to the possibility of simplification. Indeed, simplification could lead to slower RTs and to a larger load in working memory associated with a higher probability of making mistakes. This simplification would mask the numerical distance effect as well as the priming effect, preventing the identification of the numbers that primed the subsequent pair of natural numbers. Fractions with denominator 10 were excluded as they might be easily transformed into decimal numbers. For each type of fraction (a/ x_b/x vs. x/a_x/b), the larger fraction was presented on the left in half of the pairs in order to counterbalance the response side. 2 Each pair was 1 The magnitude of the common numerators in x/a_x/b fractions was smaller than the magnitude of the common denominators in a/x_b/x fractions. It was due to the use of irreducible fractions and therefore of fractions whose numerical value is smaller than 1 (i.e., with the denominator larger than the numerator). Control of simplification took priority over control of the size of the common components as the simplification could have introduced between- and within-participant variability. 2 A given pair of fractions was not presented in the two left right orders, and the pairs with a left response were not matched to the pairs with a right response. Therefore, we did not test for the SNARC effect THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8)

6 RATIONAL NUMBER PROCESSING Figure 1. A trial with its time course for each condition: (a) specific priming by a/x_b/x fractions, (b) unspecific priming by a/x_b/x fractions, (c) specific priming by x/a_x/b fractions, (d) unspecific priming by x/a_x/b fractions. presented as the prime both in the specific priming and in the unspecific priming. The pairs of natural numbers were identical to the pairs of denominators in x/a_x/b fractions and so to the pairs of numerators in a/x_b/x fractions. Each of these 32 pairs was presented as the probe in all four priming conditions. In the specific priming, they were primed by the pair of x/a_x/b fractions and by the pair of a/x_b/x fractions where a and b were identical to these natural numbers. In the unspecific priming, they were primed by one pair of x/a_x/b fractions and by one pair of a/x_b/x fractions so that all the fraction components differed from these natural numbers. The larger natural number was presented on the left in half of the pairs in order to counterbalance the response side. The response side was also counterbalanced between the prime (i.e., fractions) and the probe (i.e., natural numbers) in each priming condition. All the stimuli were presented on a screen using Superlab Pro 1.77 (Cedrus Corporation, San Pedro, California). The viewing distance was about 60 cm. In a display, the fractions or the natural numbers were presented horizontally, 6.78 away from each other and 3.38 away from the screen s centre. The fraction components were presented vertically and separated by the fraction bar (as illustrated in Figure 1). The height and the width of a fraction were, respectively, 3.38 and The height and the width of a natural number were, respectively, 1.48 and Arabic symbols were presented in white printed characters (Times font, normal) on a black background. Procedure Each priming condition included 32 trials with a prime (a pair of fractions) and a probe (a pair of natural numbers). Four blocks were created. Within a block, 8 trials of each condition were presented so that a trial was never followed by a trial of the same priming condition. Moreover, a given pair of natural numbers or fractions was not presented more than twice in a block, with at least 40 stimuli between the two presentations. Participants were tested individually, and the experiment lasted about 30 minutes. They were asked to indicate the larger number (i.e., the larger fraction for fraction pairs and the larger natural number for natural numbers pairs) as quickly and as accurately as possible by using a response box with two lateralized keys. They had to press the left-hand key or the right-hand key when the larger number was presented, respectively, on the left or on the right of the pair. After the instructions had been given, participants performed a training block of eight trials (prime and probe) and then the four blocks presented in THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8) 1603

7 MEERT, GRÉGOIRE, NOËL random order. Responses were recorded by Superlab Pro 1.77 (Cedrus Corporation, San Pedro, CA). The time sequence of a trial is shown in Figure 1. A cross appeared at the centre of the screen for 300 ms. A black screen was then presented for 500 ms, before the presentation of the prime, which remained on the screen until the participant s response. A black screen was shown for 500 ms before the presentation of the probe, which disappeared at the participant s response. Finally, a black screen lasting for 1,500 ms separated the response to the probe from the cross announcing the subsequent trial. Results The.05 significance level was chosen for all the statistical analyses made with SPSS 16. Data from 6 participants were excluded because their error rate was higher than 35% for at least one type of fraction, and their performance was therefore too close or even poorer than the level of success by chance (50%). Among the systematic errors (more than 80%), 2 participants chose the fraction with the smaller numerator for a/x_b/x fractions, and 1 participant chose the fraction with the larger denominator for x/a_x/b fractions, probably due to conceptual misunderstanding of the function of the denominator or the numerator. The 3 remaining participants were close to the level of success by chance (from 40 to 60% of errors) respectively for a/x_b/x fractions, for x/ a_x/b fractions, and for both types of fraction. Therefore, analyses were run on the data from 34 participants whose RTs to fractions and to natural numbers were in the range delimited by the mean of the sample +3 standard deviations. RTs and error rates were expressed, respectively, in milliseconds and in percentages. Fractions Analyses were run on the error rates and on the medians of RTs for correct responses computed for each participant and for each pair. To test the effect of congruity between the magnitude of the whole fraction and the magnitudes of its components, a one-way analysis of variance (ANOVA) was run with the type of fraction (a/ x_b/x vs. x/a_x/b) as a within-subject variable. The RTs to x/a_x/b fractions (M ¼ 1,297, SD ¼ 256) were significantly slower than the RTs to a/x_b/x fractions (M ¼ 1,186, SD ¼ 222), F(1, 33) ¼ 20.82, p,.01, h 2 ¼.39. Moreover, the participants made significantly more errors with x/a_x/b fractions (M ¼ 6.9, SD ¼ 7.3) than with a/x_b/x fractions (M ¼ 3.4, SD ¼ 3.6), F(1, 33) ¼ 14.54, p,.01, h 2 ¼.31. The poorer performance on x/a_x/b fractions suggested an interference of the denominator magnitude with the selection of the larger fraction. Median RTs and error rates for the pairs of fractions (see Appendix) were then analysed by linear regressions in order to test whether the performance varied with the componential distance (i.e., the distance between the two components that varied between the fractions; e.g., 2 for 3/7_5/7) or with the overall distance (i.e., the distance between the two whole fractions; e.g., 0.29 for 3/7_5/7). Analyses were run in the following way: (a) identification of the significant predictor(s) by simple regressions for each type of fraction and (b) test of the relative contribution of each predictor in multiple regression when both predictors were significant. RTs and error rates for a/x_b/x fractions were significantly predicted by the componential distance: respectively, b ¼.51, t(30) ¼ 3.23, p,.01, and b ¼.35, t(30) ¼ 2.06, p ¼.05. The overall distance did not significantly predict the RTs, b ¼.27, t(30) ¼ 1.56, p..10, and the error rates, b ¼.17, t(30) ¼ 0.92, p..10. The responses to a/x_b/x fractions were faster and more accurate as the distance between the numerators increased, indicating that participants relied on the magnitude of the numerators to compare the fractions with common denominators (see Table 1). The same analyses were run for x/a_x/b fractions. RTs and error rates were significantly predicted by the overall distance [respectively, b ¼.46, t(30) ¼ 2.82, p,.01, and b ¼.36, t(30) ¼ 2.13, p ¼.04], but not by the distance between the denominators [respectively, b ¼.24, t(30) ¼ 1.36, p..10, and b ¼.27, t(30) ¼ 1.55, p..10]. The responses to x/a_x/bfractions 1604 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8)

8 RATIONAL NUMBER PROCESSING Table 1. Intercorrelations between RTs, error rates, overall distance, and componential distance by type of fraction without and with fillers a/x_b/x fractions x/a_x/b fractions Experiment Fillers RTs Errors Overall Componential RTs Errors Overall Componential 1 Without RTs Errors Overall Componential 2 With RTs a.46 a b Errors.36 b b Overall Componential Note: RT ¼ response time. a Redundant predictors in multiple regression. b Suppressor variable in multiple regression. p.05; p.01, two-tailed. were faster and more accurate as the distance between the whole fractions increased, suggesting that the magnitudes of the whole fractions were accessed and compared (see Table 1). 3 Natural numbers The median of RTs associated with a correct response and preceded by a correct response to fractions was computed for each participant and for each pair. The mean error rate was lower than 5% in the four priming conditions and therefore was not analysed. To test whether the selection of the larger denominator was inhibited during the processing of x/a_x/b fractions, RTs to natural numbers were analysed by an ANOVA, 2 (prime: x/a_x/b vs. a/x_b/x) 2 (priming specificity: specific vs. unspecific). 4 The main effect of prime was significant, F(1, 33) ¼ 22.84, p,.01, h 2 ¼.41. The main effect of priming specificity was not significant, F(1, 33) ¼ 0.67, p..10. The prime and the priming specificity interacted significantly, F(1, 33) ¼ 8.01, p,.01, h 2 ¼.19 (see Figure 2). In the specific priming, RTs were significantly slower when natural numbers followed x/a_x/b fractions (M ¼ 631, SD ¼ 103) than when they followed a/x_b/x fractions (M ¼ 593, SD ¼ 94), t(33) ¼ 4.47, p,.01. This effect was not significant in the unspecific priming, t(33) ¼ 1.25, p..10, supporting the use of performance in the unspecific priming as a baseline. When the prime was x/ a_x/b fractions, RTs were slower in the specific priming (M ¼ 631, SD ¼ 103) than in the unspecific priming (M ¼ 612, SD ¼ 100), t(33) ¼ , p,.01. On the other hand, when the prime was a/x_b/x fractions, RTs did not differ significantly in the specific priming (M ¼ 593, SD ¼ 94) and in the unspecific 3 The distance uncorrelated with RTs and error rates (i.e., the overall distance for a/x_b/x fractions and the distance between the denominators for x/a_x/b fractions) was identified as a suppressor variable by the method suggested by Tzelgov and Henik (1991). A suppressor variable is a variable that clears out residual variance from the other predictor in multiple regression. Therefore, when both distances were simultaneously entered in a regression, the coefficient of the valid distance was inflated, and the coefficient of the suppressor distance changed sign (i.e., became positive). However, we did not report multiple regressions as the suppressor distance never contributed to the improvement of the prediction (all p..10 for R 2 change) and as its coefficient was never significant (all ps..10). 4 The same analyses were also run on the medians of RTs to the pairs associated with a correct response and preceded by a correct response to the prime in all four priming conditions to maintain a strict matching between conditions. This analysis led to the same significant effects in Experiment 1 and in Experiment 2. These results, however, were not reported because of the heavy loss of data for some participants. Moreover, this trimming could have induced an imbalance in the control of the lateralization of the motor response to the probe relative to the motor response to the prime, as a given trial could have the same response as the prime in a condition and a different response in another condition (this control was made within each priming condition). THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8) 1605

9 MEERT, GRÉGOIRE, NOËL Figure 2. Mean response time (RT, in ms) for the comparison of natural numbers in each priming condition in the experiment without fillers (i.e., Experiment 1; left panel) and in the experiment with fillers (i.e., Experiment 2; right panel). priming (M ¼ 605, SD ¼ 88), t(33) ¼ 1.70, p ¼ Discussion Performance on a/x_b/x fractions varied according to the numerical distance between the numerators, suggesting that participants compared the magnitudes of the numerators that were congruent with the magnitudes of the whole fractions to select the larger fraction. Conversely, performance on x/a_x/b fractions varied according to the distance between the whole fractions. This result suggests that the larger fraction was selected by accessing the magnitudes of the whole fractions and by comparing them. Additionally, participants were slower and made more errors for the comparison of x/a_x/b fractions than for the comparison of a/x_b/x fractions. The difference of performance between the two types of fraction could be due to the congruity effect between the magnitude of the components that varied between the two fractions and the magnitude of the whole fractions. However, the access to the magnitudes of the whole fractions and their comparison could also explain poorer performance on fractions x/a_x/b. Indeed, this processing could be slower and less accurate than the comparison of the numerators if we hypothesize that the holistic representation requires access to the mental magnitudes of the fraction components and the estimation of their ratio. The presence of a priming effect induced by fraction comparison over a natural number comparison speaks in favour of the congruity effect. Indeed, RTs were slower in specific priming by x/a_x/b fractions than in unspecific priming by these fractions. This cost can be explained by the residual inhibition on selection of the larger denominator. Indeed, while the participants were accessing the magnitudes of the whole fractions and comparing them, the denominators could be compared and the larger denominator selected automatically. The selection of the larger denominator could interfere with that processing and had to be inhibited so that the larger fraction was correctly selected. Therefore, when the subsequent natural numbers were identical to the denominators, the previously inhibited response had to be activated, leading to slower responses. The comparison of the denominators could be automatic since access to a holistic representation does not require the comparison of the denominators and since the absence of an effect of the distance between the denominators suggests that participants did not select the smaller denominator to select the larger fraction. It could be surprising that participants compared the whole fractions whereas comparison of the denominators would have been sufficient for 5 The distance effect was tested on natural numbers to check that their magnitude was processed in all priming conditions. The distance effect was significant in the specific priming by x/a_x/b fractions, b ¼.60, t(30) ¼ 4.06, p,.01; in the unspecific priming by x/a_x/b fractions, b ¼.49, t(30) ¼ 3.12, p,.01; and in the unspecific priming by a/x_b/x fractions, b ¼.51, t(30) ¼ 3.28, p,.01; and it tended to be significant in specific priming by a/x_b/x fractions, b ¼.34, t(30) ¼ 1.98, p ¼.06. The correlations between RTs and distance did not differ significantly between the priming conditions with the t test suggested by Williams (1959, cited in Howell, 1997/1998; all ps..10). These distance effects suggested that the magnitude was processed in all priming conditions THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8)

10 RATIONAL NUMBER PROCESSING x/a_x/b fractions (i.e., the choice of the smaller denominator). A possible explanation is that the use of a componential strategy for all pairs of fractions would have introduced a cognitive cost for all pairs of fractions. Indeed, it would have involved a task switching between the choice of the smaller component for x/a_x/b fractions and the choice of the larger component for a/x_b/x fractions. EXPERIMENT 2 When the task was performed on a/x_b/x fractions and on x/a_x/b fractions, participants compared the numerators of a/x_b/x fractions without accessing the magnitudes of the whole fractions and compared the whole fractions for x/a_x/b fractions. However, the regularity of these fractions (i.e., with common denominators or with common numerators) might have facilitated the use of a componential strategy for a/x_b/x fractions. Indeed, the participants compared the numerators of the a/x_b/ x fractions probably because they could easily identify the pairs of fractions with congruity between the magnitudes of the components and the magnitudes of the whole fractions. In Experiment 2, fillers were included to reduce the regularity of the stimuli and to discourage the use of componential strategies. The filler pairs were made up of fractions without common components so that the magnitude of the denominator and the magnitude of the numerator could be congruent or incongruent with the magnitude of the whole fraction. Therefore, in the absence of common denominators or numerators, the congruity between the fraction components and the whole fraction could not be identified without an access to the magnitudes of the whole fractions. The presence of fillers might then lead to holistic processing for all pairs of fractions, even for a/x_b/x fractions, as this processing is appropriate for all pairs of fractions. Method Participants A total of 41 university undergraduate psychology students (34 women; 34 right-handed) participated for course credit. The average age was 20 years (ranging from 19 to 23 years). Design The design was the same as that in Experiment 1, except that new pairs of fractions, each followed by a pair of natural numbers, were added as fillers. They were made up of two fractions with differing denominators and numerators (e.g., 6/11_3/8) and varied according to the level of congruity between the magnitude of the whole fraction and the magnitudes of the components. The three possible levels of congruity were used to reduce the regularity of the stimuli and to discourage the use of componential strategies. The larger fraction was made up (a) of the smaller denominator and the smaller numerator (e.g., 5/12, 4/7), (b) of the smaller denominator and the larger numerator (e.g., 4/13, 6/11), or (c) of the larger denominator and the larger numerator (e.g., 3/8, 6/11). Stimuli The experimental pairs of fractions and of natural numbers were those used in Experiment 1. A total of 60 filler pairs of fractions were created for each level of congruity. In these pairs, fractions were irreducible, below the unit, and with a denominator from 2 to 14 (except 10) and a numerator from 1 to 11. Moreover, one denominator was never the multiple of the other denominator in order to avoid encouraging participants to search for a common denominator. For each category of fractions presented as fillers, the larger fraction was presented on the left in half of the trials in order to counterbalance the response side. A total of 60 filler pairs of natural numbers were created and differed from the experimental pairs of natural numbers. They were presented as the probes for each category of filler pairs of fractions and therefore were presented three times. The response side was counterbalanced within the filler pairs of natural numbers as well as between the filler pairs of fractions and the subsequent pairs of natural numbers. THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8) 1607

11 MEERT, GRÉGOIRE, NOËL Procedure Among the 308 trials made up of a prime and a probe, 128 were experimental trials, and 180 were filler trials. Trials were divided in eight blocks. Each block was made up of 4 experimental trials of each priming condition and 7 or 8 filler trials of each level of congruity. A trial never followed a trial of the same condition. The time sequence of a trial was the same as that in Experiment 1. After the instruction was given, participants initially performed a training block of 20 trials and then the eight blocks in a random order varying for each participant. The experiment lasted about 50 minutes. Results Data from 11 participants were excluded because their error rate was higher than 35% in at least one experimental category of fractions (a/x_b/x vs. x/a_x/b) or one filler category of fractions. A total of 4 participants committed systematic errors (more than 80% of errors). These errors reflected the choice of the fraction with the smaller numerator for all pairs without common numerators (n ¼ 1), the choice of the fraction with the larger denominator for all pairs without common denominators (n ¼ 1), the choice of the fraction with the smaller numerator when the denominators were common (i.e., a/x_b/x) and with the smaller denominator for all other pairs (n ¼ 1), and the choice of the fraction with the larger denominator when the numerators were common (n ¼ 1). These errors could be at least in part due to conceptual misunderstanding of the function of the denominator and/or the numerator as they concerned even the fractions with common components. For 6 participants, the percentage of correct responses was close to the level of success by chance when the larger fraction had the larger denominator (n ¼ 4), when the larger fraction had the smaller numerator (n ¼ 1), and when the fractions had common numerators (n ¼ 1). Finally, 1 participant responded randomly for all categories. Analyses were therefore run on data from 30 participants whose RTs were in the range delimited by the mean of the sample + 3 standard deviations for fractions and for natural numbers. Fractions Analyses were run on the error rates and on the medians of RTs for correct responses computed for each participant and for each pair. An ANOVA was run with the type of fraction (a/x_b/x vs. x/a_x/b) as a within-subject variable. The participants compared x/a_x/b fractions significantly slower (M ¼ 1,540, SD ¼ 306) than they did a/x_b/x fractions (M ¼ 1,389, SD ¼ 222), F(1, 29) ¼ 39.21, p,.01, h 2 ¼.57, and made more errors with x/a_x/b fractions (M ¼ 8.4, SD ¼ 7.1) than with a/x_b/x fractions (M ¼ 5.1, SD ¼ 5.1), F(1, 29) ¼ 7.48, p ¼.01, h 2 ¼.20. These differences replicated those obtained in Experiment 1 and suggested a cognitive cost for x/a_x/b fractions due to the incongruity between the magnitude of the denominator and the magnitude of the whole fraction. To directly compare these results with the results of the first experiment, an ANOVA was run with the experiment as a between-subject variable (without fillers vs. with fillers) and the type of fraction as a within-subject variable. The effect of the type of fraction did not vary with the experiment for the RTs, F(1, 62) ¼ 1.39, p..10, and for the error rates, F(1, 62) ¼ 0.02, p..10. On the other hand, the effect of the experiment was significant for RTs, F(1, 62) ¼ 13.32, p,.01, h 2 ¼.18, but not for the error rates, F(1, 62) ¼ 1.60, p..10. In short, participants in the experiment with fillers (M ¼ 1,465, SD ¼ 317) were significantly slower than participants in the experiment without fillers (M ¼ 1,242, SD ¼ 244), whatever the type of fraction. To test whether RTs and error rates for the pairs of a/x_b/x fractions (see Appendix) varied according to the distance between the numerators or according to the overall distance, simple linear regressions were first run to identify the significant predictor(s). Then significant predictors were simultaneously entered in a regression to assess their relative contribution. Given their intercorrelation, the method suggested by Tzelgov and 1608 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8)

12 RATIONAL NUMBER PROCESSING Henik (1991) was used to detect redundancy effect and suppression effect. Simple regressions on RTs showed that the overall distance, R 2 ¼.23, b ¼.47, t(30) ¼ 2.96, p,.01, and the distance between numerators, R 2 ¼.21, b ¼.46, t(30) ¼ 2.85, p,.01, were significant predictors (see Table 1). The correlation between RTs and componential distance did not differ significantly from the correlation between RTs and overall distance, t(29) ¼ 0.09, p..10. When both distances were simultaneously entered in a regression (R 2 ¼.25), the overall distance and the distance between the numerators were no longer significant predictors: respectively, b ¼.29, t(29) ¼ 1.15, p..10, and b ¼.23, t(29) ¼ 0.90, p..10. These results suggested that both predictors accounted for the same variance (i.e., redundancy), making it impossible to assess which distance was the important factor (see Field, 2005, p. 175; Hair, Anderson, Tatham, & Black, 1998, p. 188). Simple regressions on error rates showed also a significant effect of the overall distance, R 2 ¼.13, b ¼.36, t(30) ¼ 2.10, p,.05, and of the distance between numerators, R 2 ¼.27, b ¼.52, t(30) ¼ 3.33, p,.01. In the multiple regression on error rates (R 2 ¼.27), the effect of the distance between the numerators remained significant, b ¼.61, t(29) ¼ 2.43, p ¼.02, whereas the effect of the overall distance was not any more significant, b ¼.12, t(29) ¼ 0.48, p..10. The inflation of the first coefficient and the change of sign of the second one were due to suppression effect. The overall distance acted as a negative suppressor variable by suppressing residual variance in the prediction of error rates from the distance between the numerators. However, this contribution was not significant. In short, responses were more accurate as the distance between the numerators increased, suggesting that participants relied on the component magnitude to compare a/x_b/x fractions. RTs were equally predicted by the overall distance and by the distance between the numerators. The redundancy effect between the distances prevented us from concluding whether participants compared only the numerators or whether they compared also the magnitudes of the whole fractions. The same analyses were made for x/a_x/b fractions. In simple regressions, the overall distance and the componential distance significantly predicted RTs: respectively, R 2 ¼.38, b ¼.62, t(30) ¼ 2.96, p,.01, and R 2 ¼.15, b ¼.39, t(30) ¼ 2.29, p ¼.03 (see Table 1). When both distances were simultaneously entered in a regression (R 2 ¼.42), the effect of the overall distance remained significant, b ¼.90, t(29) ¼ 3.67, p,.01, whereas the effect of the distance between the denominators was not any more significant, b ¼.35, t(29) ¼ 1.42, p..10. The distance between the denominators was a suppressor variable but its contribution was not significant. Simple regressions showed that the error rates were significantly predicted by the overall distance, R 2 ¼.16, b ¼.40, t(30) ¼ 2.40, p ¼.02, and tended to be significantly predicted by the componential distance, R 2 ¼.09, b ¼.30, t(30) ¼ 1.74, p ¼.09. However, in the multiple regression (R 2 ¼.16), neither the overall distance nor the distance between the denominators (suppressor variable) was a significant predictor: respectively, b ¼.46, t(29) ¼ 1.57, p..10, and b ¼.07, t(29) ¼ 0.25, p..10. In short, the participants relied mainly on the magnitudes of the whole fractions, as suggested by the effect of the overall distance on RTs. The introduction of fillers intended to discourage the use of componential strategies, as it was assumed that component congruity could not be determined without access to the magnitudes of the whole fractions for these pairs. Analyses were run to check that the overall distance influenced the RTs and the error rates for the filler pairs of fractions. The mean error rate was 9.8% (SD ¼ 13.1%), and the mean RT was 1,331 ms (SD ¼ 170 ms) for all filler pairs of fractions. RTs and error rates to each category of fillers were analysed according to the overall distance, the distance between the numerators, and the distance between the denominators (e.g., respectively, 0.15, 1, and 5 for 5/12_4/7). When the larger fraction had the larger numerator and the larger denominator (e.g., 3/8_6/11), RTs were significantly predicted by the overall distance, R 2 ¼.24, b ¼.49, t(58) ¼ 4.24, p,.01, and THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8) 1609

13 MEERT, GRÉGOIRE, NOËL by the distance between the numerators, R 2 ¼.09, b ¼.29, t(58) ¼ 2.35, p ¼.02. When both predictors were simultaneously entered in a regression (R 2 ¼.24), the effect of the overall distance remained significant, b ¼.54, t(57) ¼ 3.38, p,.01, whereas the effect of the distance between the numerators was not significant, b ¼.08, t(57) ¼ 0.48, p..10. The coefficient of the overall distance was inflated as the distance between the numerators was a suppressor variable. Simple regressions on error rates showed that the effect of the overall distance was significant, R 2 ¼.44, b ¼.66, t(58) ¼ 6.74, p,.01, as well as the effect of the distance between the numerators, R 2 ¼.09, b ¼.30, t(58) ¼ 2.37, p ¼.02. In the multiple regression (R 2 ¼.49), the distance between the numerators, b ¼.30, t(57) ¼ 2.33, p ¼.02, mainly contributed to the prediction of error rates by suppressing residual variance from the overall distance, b ¼.87, t(57) ¼ 6.67, p,.01. In short, participants mainly compared the magnitude of the whole fractions for this category of fillers. When the larger fraction had the smaller denominator and the larger numerator (e.g., 4/13_6/11), the effect of the overall distance on RTs was significant, R 2 ¼.49, b ¼.70, t(58) ¼ 7.40, p,.01, as well as the effect of the distance between the numerators, R 2 ¼.11, b ¼.33, t(58) ¼ 2.69, p,.01. When both predictors were entered in a regression (R 2 ¼.54), the distance between the numerators, b ¼.35, t(57) ¼ 2.69, p,.01, mainly contributed by clearing out irrelevant variance from the overall distance, b ¼.94, t(57) ¼ 7.35, p,.01. Only the overall distance significantly predicted the error rates, b ¼.44, t(58) ¼ 3.75, p,.01. In short, participants compared the magnitude of the whole fractions for this category of fillers. Finally, when the larger fraction had the smaller denominator and the smaller numerator (e.g., 5/ 12_4/7), only the distance between the numerators tended to significantly predict the RTs, R 2 ¼.07, b ¼.24, t(58) ¼ 1.86, p ¼.07. The error rates were significantly predicted by the overall distance, R 2 ¼.09, b ¼.30, t(58) ¼ 2.44, p ¼.02, and tended to be predicted by the distance between the denominators, R 2 ¼.06, b ¼.24, t(58) ¼ 1.90, p ¼.06. In the multiple regression on error rates (R 2 ¼.10), the effect of the overall distance only tended to be significant, b ¼.24, t(57) ¼ 1.69, p ¼.10, and the effect of the distance between the denominators was no longer significant, b ¼.12, t(57) ¼ 0.83, p..10. This pattern of results suggested that these distances were partially redundant and that the error rates tended to be mainly predicted by the overall distance. The results for this category of fillers suggested that participants relied on the magnitude of the numerators and tended to rely on the magnitude of the whole fractions. These results were not as clear as those from the two other categories and could be due to a smaller variability of the distances within this category. In short, for at least two categories of fillers, the overall distance was the best predictor of both RTs and error rates. Natural numbers The mean error rate was lower than 5% in the four priming conditions and therefore was not analysed. The median of the RTs for correct responses preceded by a correct response to the prime was calculated for each participant and for each pair in each condition. To test the priming effect of fraction comparison over natural number comparison, RTs to natural numbers were analysed by an ANOVA, 2 (prime: x/a_x/b vs. a/x_b/x) 2 (priming specificity: specific vs. unspecific). 6 This analysis showed a significant main effect of the prime, F(1, 29) ¼ 10.32, p,.01, h 2 ¼.26. The main effect of the specificity was not significant, F(1, 29) ¼ 0.89, p..10. The interaction between the prime and the priming specificity was significant, F(1, 29) ¼ 9.90, p,.01, h 2 ¼.25 (see Figure 2). The effect of the prime was significant in the specific priming, t(29) ¼ 3.79, p,.01. The RTs to natural 6 See Footnote THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8)