Journal of Experimental Child Psychology 107 (2010) 244–259

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Journal of Experimental ChildPsychology

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Comparing the magnitude of two fractions with commoncomponents: Which representations are used by 10- and12-year-olds?

Gaëlle Meert a,*, Jacques Grégoire a, Marie-Pascale Noël b

a Unité de Psychologie de l’Education et du Développement, Faculté de Psychologie et des Sciences de l’Education,Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgiumb Unité Cognition et Développement, Faculté de Psychologie et des Sciences de l’Education, Université Catholique de Louvain,1348 Louvain-la-Neuve, Belgium

a r t i c l e i n f o

Article history:Received 9 April 2009Revised 9 April 2010Available online 2 June 2010

Keywords:Magnitude processingFractionsHolistic vs. componential representationNumerical distance effectNegative primingInterference

0022-0965/$ - see front matter � 2010 Elsevier Indoi:10.1016/j.jecp.2010.04.008

* Corresponding author. Fax: +32 (0) 10 47 85 8E-mail address: gaelle.meert@uclouvain.be (G. M

a b s t r a c t

This study tested whether 10- and 12-year-olds who can correctlycompare the magnitudes of fractions with common componentsaccess the magnitudes of the whole fractions rather than only com-pare the magnitudes of their components. Time for comparing twofractions was predicted by the numerical distance between thewhole fractions, suggesting an access to their magnitude. In addi-tion, we tested whether the relative magnitude of the denominatorinterferes with the processing of the fraction magnitude and, thus,needs to be inhibited. Response times were slower for fractionswith common numerators than for fractions with common denom-inators, indicating an interference of the magnitude of the denom-inators with the selection of the larger fraction. A negative primingeffect was shown for the comparison of natural numbers primed byfractions with common numerators, suggesting an inhibition of theselection of the larger denominator during the comparison of frac-tions. In conclusion, children who can correctly compare fractionswith common components can access the magnitude of the wholefractions but remain sensitive to the interference of the relativemagnitude of the denominators. This study highlights the fact thatbeyond the interference of natural number knowledge at the con-ceptual level (called the ‘‘whole number bias” by Ni & Zhou, 2005),children need to manage the interference of the magnitude of thedenominators (Stroop-like effect).

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9.eert).

G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259 245

Introduction

Research on fractions has mainly investigated the development of the conceptual understanding offractions and the obstacles to this learning. Yet, the representation of fraction magnitude has beentested by only a few studies, all of them testing adults. The current study aimed to test whether chil-dren who are able to correctly compare fractions with common denominators or common numeratorsaccess the magnitude of the whole fractions rather than just the magnitude of their components. Wealso tested whether the relative magnitude of the denominators interferes with the processing of thefraction magnitude by comparing performance on fractions with common numerators versus withcommon denominators. Moreover, the use of a priming paradigm allowed us to test whether the pro-cessing of the relative magnitude of the denominators, or the response associated with that compar-ison, is inhibited during the correct comparison of fractions.

Fractions denote rational numbers as the ratio of two integers. Therefore, they represent continu-ous magnitudes through the ratio of two discrete magnitudes. There is a wide consensus that one ofthe main sources of difficulty in learning fractions is the interference of previous counting and naturalnumber knowledge. Children tend to interpret fractions by referring to this knowledge, which is calledthe ‘‘whole number bias” (Ni & Zhou, 2005). Because the properties of natural numbers differ from theproperties of rational numbers, this bias interferes with fraction learning and leads to errors and mis-conceptions (e.g., Behr, Harel, Post, & Lesh, 1992; English & Halford, 1995; Grégoire & Meert, 2005;Hartnett & Gelman, 1998; Mack, 1993; Sophian, 1996; Stafylidou & Vosniadou, 2004). In Stafylidouand Vosniadou’s (2004) study, 37.5% of the fifth-graders considered fractions as two independent nat-ural numbers. This misconception led children to assume that the value of a fraction increases as thevalue of either the numerator or denominator increases or that the value of a fraction increases as thevalue of either the numerator or denominator decreases. The first conception is consistent with chil-dren’s existing knowledge of numbers (i.e., knowledge of natural numbers). The second one is a tran-sitional phase in the process of understanding fractions, a result of children’s attempts to integrate thenew information into their existing numerical knowledge. Stafylidou and Vosniadou underlined thatstudents from 10 to 17 years of age gradually overcome this whole number bias and change their con-ceptual understanding of numbers before their understanding of fractions matches the scientific con-cept. For instance, students need to understand that numbers are not only made from counting andcan be represented by a ratio between two numbers. In addition, they need to understand that eachnumber has not one and only one successor (notion of density).

This study focused on the interference of the prior knowledge (i.e., the concept of natural numbers)in the acquisition of the concept of fraction. To our knowledge, no study has dealt with the mental rep-resentation activated when processing fractions in children who have enough knowledge of the con-cept of fractions to take into account the function of the natural numbers composing the fractions. Inadults, the representation of the fraction magnitude has been investigated by using the numerical dis-tance effect, which is the decrease in response times (RTs) with increases of the numerical distancebetween the numbers being compared (Moyer & Landauer, 1967). Bonato, Fabbri, Umiltà, and Zorzi(2007) asked adults to compare a target fraction with a fixed standard (1/5, 0.2, or 1). When the fixedstandard was a fraction (i.e., 1/5), it always shared the numerator with the target fractions (e.g., 1/1, 1/2,1/3). The distance between the denominators predicted RTs better than the distance between the wholefractions, suggesting that participants compared these components directly. This was also true in exper-iments where the fixed standard was not a fraction (i.e., 0.2 or 1). In that case, participants converted thestandard into a fraction (e.g., 1/5, 5/5) to compare the components directly. Bonato and colleagues con-cluded that participants used a componential strategy (i.e., direct comparison of the denominators ornumerators) to select the larger fraction.

Meert, Grégoire, and Noël (2009) asked adults to compare both fractions with common denomina-tors (i.e., a/x_b/x; e.g., 3/7_5/7) and fractions with common numerators (i.e., x/a_x/b; e.g., 2/3_2/5) inthe same experiment. In that context, RTs for fractions with common numerators (i.e., x/a_x/b) werebest predicted by the distance between the whole fractions, suggesting an access to the magnitude ofthe whole fractions. On the other hand, RTs for fractions with common denominators (i.e., a/x_b/x)

246 G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259

were best predicted by the distance between the numerators, indicating that adults compared theirmagnitude directly. Thus, our magnitude processing system can represent the magnitude of the ratioof two natural numbers, but adults do not always access this representation. In some experimentalconditions, they compare the magnitude of the denominators or numerators instead. These compo-nential strategies are also appropriate as the comparison of the whole fractions in the case of fractionswith common components provided that participants select the fraction with the larger numerator orsmaller denominator as the larger one. The factors influencing the type of processing (componentialvs. holistic) should be further investigated. The relative efficiency of each processing in the experimen-tal conditions could partly determine their use. Indeed, accessing the magnitude of the whole fractioncould be made by accessing the magnitude of the fraction components and processing their ratio.Therefore, it could be more time-consuming than the direct comparison of the denominators ornumerators when only one componential strategy can be easily identified and used to correctly pro-cess all of the pairs of fractions through the experiment (see Bonato et al., 2007). On the other hand,when different componential strategies should be used to correctly process all of the pairs and, there-fore, could lead to a greater probability of giving a wrong answer, access to the magnitude of the wholefractions could be used, especially for fractions where the relative magnitude of the components isincongruent with the relative magnitude of the whole fractions such as in fractions with commonnumerators (see Meert et al., 2009).

Data in favor of an access to the magnitude of the whole fractions support the hypothesis that ourmagnitude system can represent the magnitude of rational numbers. This hypothesis is congruentwith the theory of Gallistel, Gelman, and Cordes (2006) (see also Gallistel and Gelman, 2000). Accord-ing to them, we inherit, through evolution, a system to represent both discrete and continuous quan-tities by continuous mental magnitudes with a scalar variability. These magnitudes would beisomorphic to real numbers (that include rational numbers) and support the processing of theirmagnitude because real numbers also refer to both discrete and continuous quantities. Therefore,the magnitude processing system should allow the holistic representation of rational numbers be-cause they are part of the category of real numbers. To support their theory, Gallistel and colleaguesreviewed a large body of empirical data showing that animals are able to process continuous quanti-ties (e.g., duration) and discrete quantities (e.g., number of flashes) as well as perform arithmeticaloperations combining both discrete and continuous quantities (e.g., calculating the return rate of a for-aging patch as the ratio between the number of rewards and the time spent multiplied by the averagesize of the rewards). Nevertheless, this model does not describe the mechanism that generates mentalmagnitudes for ratios of two numerosities. As mentioned previously, we could hypothesize that accessto the magnitude of the whole fractions is made through access to the magnitude of the fraction com-ponents and processing of their ratio. This indirect access could be achieved only by considering thenumerator and denominator in relation.

The representation of the magnitude of fractions could be hybrid (both componential and holistic)rather than purely holistic. Indeed, Meert and colleagues (2009) reported data supporting that the rel-ative magnitude of the components was also processed when the magnitude of the whole fractionswas accessed during the comparison of fractions with common numerators (i.e., x/a_x/b). They useda priming paradigm in which this comparison primed the comparison of natural numbers that wereidentical to or different from the denominators of the previous fractions (e.g., 3_5 primed by 2/5_2/3 or by 10/11_10/13, respectively). The comparison of fractions with common numerators (i.e.,x/a_x/b) negatively primed the subsequent comparison of natural numbers when they were identicalto the previous denominators. This negative priming effect was explained by residual inhibition andindicated that inhibition operated at the level of the response activated by the comparison of thedenominators (i.e., the denominator selected as the larger one) during the comparison of fractions.This response was inhibited because it interfered with the selection of the larger fraction. Indeed,the larger fraction was made up of the smaller denominator for fractions with common numerators(i.e., x/a_x/b).

Houdé (2000) had already underlined the role of inhibition in cognitive tasks performed by chil-dren and adults when competing strategies or knowledge create interference. Houdé and Guichart(2001) adapted Piaget’s conservation of number task into a priming paradigm to test whether9-year-olds who succeed in this task inhibit the visuospatial bias ‘‘length equals number.” In their

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paradigm, this bias needed to be inhibited on the prime (item with number/length interference) butbecame congruent in the probe (item with number/length covariation). A negative priming effect wasshown, suggesting the involvement of an inhibitory process in children who succeeded in this task.This study was replicated in adults with event-related potential (ERP) measures (Daurignac, Houdé,& Jouvent, 2006). An enhanced N200 indicated that adults still needed to inhibit this bias. Moreover,this ERP signature supports the explanation of the negative priming effect in terms of inhibition(Tipper, 1985, 2001).

The purpose of the current study was to identify the mental representations activated during mag-nitude comparison of two fractions with common components in children who succeed in performingthis task. A paradigm similar to the one used by Meert and colleagues (2009) in adults was used. Twotypes of pairs of fractions were presented: fractions with common denominators (a/x_b/x) and frac-tions with common numerators (x/a_x/b). The numerical distance effect was used to test whether chil-dren rely on the magnitudes of the components or on the magnitudes of the whole fractions to dealwith the task. If children compare only the denominators or numerators, the componential distance(i.e., the distance between the numerators for a/x_b/x fractions and the distance between the denom-inators for x/a_x/b fractions) should influence performance. Alternatively, if the magnitudes of thewhole fractions are accessed to perform the comparison, performance should vary with the distancebetween the whole fractions.

The pairs of fractions varied according to the congruity of the magnitude of the fraction componentswith the magnitude of the whole fractions. 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 thewhole fraction because the larger fraction (e.g., 5/7) is made up of the larger numerator (e.g., 5). In a pairof fractions with common numerators (i.e., x/a_x/b; e.g., 2/3_2/5), the relative magnitude of the denom-inator is incongruent with the relative magnitude of the whole fraction because the larger fraction (e.g.,2/3) is made up of the smaller denominator (e.g., 3). In children who succeed in comparing these frac-tions, we could hypothesize an interference of the magnitude of the natural numbers composing thedenominators with the magnitude of the whole fractions (i.e., a Stroop-like effect between the magni-tude of the fraction components and the magnitude of the whole fractions). This interference should leadto slower RTs for fractions with common numerators (e.g., 2/3_2/5) than for fractions with commondenominators (e.g., 3/7_5/7). Moreover, a priming paradigm was used to test whether the selection ofthe larger denominator or the response activated by this processing is inhibited during the comparisonof fractions to correctly select the larger one. The comparison of fractions primed the comparison of nat-ural numbers. The natural numbers were identical to the denominators or numerators of the previousfractions (specific priming) or differed from all of the fraction components (unspecific priming). If inhi-bition operates on the process of selection of the larger denominator, a negative priming effect on thesubsequent comparison of natural numbers should be shown in the specific and unspecific priming con-ditions relative to the control condition (where natural numbers were primed by a non-numerical stim-ulus).1 If inhibition operates on the response activated by the selection of the larger denominator,performance should be poorer for natural numbers identical to the denominators of fractions with commonnumerators (specific priming) compared with natural numbers differing from the denominators of the pre-vious fractions with common numerators (unspecific priming) and with natural numbers in the controlcondition. The performance deterioration for the comparison of natural numbers would be due to residualinhibition because this comparison should require the activation of the same response or processing as theone inhibited during the processing of the previous fractions.

Method

Participants

A total of 99 children in fifth grade (primary school) and 73 children in seventh grade (secondaryschool) were first tested by a paper-and-pencil questionnaire filled out in class. Children attended reg-

1 This last condition was not used in the study reported by Meert and colleagues (2009).

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ular classes from four schools. The questionnaire was used to select children with sufficient knowl-edge of fractions to perform the computerized task of magnitude comparison and the priming para-digm. They were selected if they correctly responded to at least four of five comparisons offractions with common denominators (a/x_b/x) and four of five comparisons of fractions with commonnumerators (x/a_x/b). This criterion was met by 51% of the fifth-graders and 81% of the seventh-grad-ers. In Belgium, according to the school curriculum, children should acquire the ability to comparefractions with common components between 8 and 10 years of age. Children were tested duringthe summer term. A total of 42 fifth-graders (mean age = 10 years 8 months, 20 boys and 22 girls)and 44 seventh-graders (mean age = 12 years 11 months, 28 boys and 16 girls) were asked to take partin the computerized experiment. Parents gave their informed consent to the participation of theirchildren.

Design

Participants were asked to compare the magnitude of two fractions or two natural numbers. Twotypes of fraction were presented: fractions with common numerators (i.e., x/a_x/b; e.g., 2/5_2/3) andfractions with common denominators (i.e., a/x_b/x; e.g., 3/7_5/7). Two types of numerical distancewere computed for each pair of fractions: the distance between the whole fractions and the distancebetween the denominators or numerators (e.g., 0.29 and 2 for 3/7_5/7, respectively). These distancesdid not correlate perfectly for fractions with common numerators (r = .79) or for fractions with com-mon denominators (r = .77) because the magnitude of the components also influenced the distance be-tween the whole fractions. Therefore, two pairs of fractions can have the same distance between thenumerators but a different distance between the whole fractions. For instance, both 3/7_5/7 and15/19_17/19 have a distance of 2 between the numerators, but the distances between the whole frac-tions are equal to 0.29 and 0.11, respectively. This absence of perfect correlation allowed us to test therespective contribution of each distance to explain children’s performance.

Each pair of natural numbers was primed in four experimental conditions resulting from the crossingof two within-participant variables: type of prime (x/a_x/b vs. a/x_b/x) and priming specificity (specificvs. unspecific). The priming was specific when the numerators of fractions with common denominators(a/x_b/x) or the denominators of fractions with common numerators (x/a_x/b) were identical to the sub-sequent natural numbers (e.g., 3/7_5/7 primed 3_5). The priming was unspecific when none of the num-bers making up the fractions in the prime matched either of the natural numbers in the probe (e.g., 8/11_9/11 primed 3_5). A control condition in which the natural numbers were primed by a non-numer-ical stimulus (i.e., # #) was also presented. Participants needed to select the underlined symbol for thisstimulus. The control condition was used as a baseline to assess the effect of the experimental primingconditions. Fig. 1 shows an example for each of the five priming conditions.

Stimuli

Among the 64 pairs of fractions, 32 pairs consisted of fractions with common numerators (x/a_x/b)and 32 pairs consisted of fractions with common denominators (a/x_b/x) (see Appendix A). The samepairs of natural numbers were used for the numerators of fractions with common denominators(a/x_b/x) and for the denominators of fractions with common numerators (x/a_x/b), leading to a strictmatching of the numerical distance between these components. Both types of fraction were balancedfor the numerical distance between the whole fractions. For the fractions with common denominators(a/x_b/x), this distance ranged from 0.05 to 0.50 with a mean of 0.18 (SD = 0.11). For the fractions withcommon numerators (x/a_x/b), the distance ranged from 0.04 to 0.38 with a mean of 0.17 (SD = 0.09).Pairs were made up of irreducible fractions without multiple denominators to prevent participantsfrom simplifying or transforming fractions into fractions with common denominators. The magnitudeof all fractions was smaller than 1. Denominators differed from 10 to avoid transformation into dec-imal numbers. For each type of fraction, the larger fraction was presented on the left for half of thepairs. The fraction components were presented vertically and separated by the fraction bar. The 32pairs of natural numbers were identical to the denominators of fractions with common numerators(x/a_x/b) and to the numerators of fractions with common denominators (a/x_b/x). The larger natural

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Fig. 1. The five priming conditions with the time course of a trial are shown. From left to right: (1) specific priming by fractionswith common denominators (a/x_b/x), (2) unspecific priming by fractions with common denominators (a/x_b/x), (3) specificpriming by fractions with common numerators (x/a_x/b), (4) unspecific priming by fractions with common numerators (x/a_x/b), and (5) control priming by non-numerical symbols.

G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259 249

number was presented on the left in half of the pairs. Two pairs of non-numerical symbols were alsocreated. One of the symbols was underlined and was presented on the left in one pair and on the rightin the other. All of the stimuli were presented using SuperLab Pro 4 (Cedrus, San Pedro, CA, USA). Theviewing distance was approximately 60 cm. The numbers or symbols were centered on the screen andwere presented 5.3� from each other. The height and width were approximately 2.7 and 1.4�, respec-tively, for a fraction and were 0.9 and 0.8�, respectively, for a natural number or the symbol #. Stimuliwere presented in white printed characters (Arial font, normal) on a black background.

Procedure

Each pair of fractions was presented as the prime in both specific and unspecific priming condi-tions. Each pair of natural numbers was presented as the probe in all five priming conditions. In spe-cific priming, they were primed by the pairs of fractions with common numerators (x/a_x/b) and thepairs of fractions with common denominators (a/x_b/x) that had respective denominators and numer-ators identical to these natural numbers. In unspecific priming, the natural numbers were primed by apair of fractions with common numerators (x/a_x/b) and a pair of fractions with common denomina-tors (a/x_b/x) so that all of the fraction components differed from these natural numbers.

Each of the five priming conditions consisted of 32 trials with a prime and a probe, leading to a totalof 160 trials in the experiment. These trials were divided into four blocks. In each block, 8 trials of eachof the five conditions were presented so that a given comparison was not presented more than twotimes with at least 40 comparisons between them. A trial was never followed by a trial of the samecondition. The response side was counterbalanced for the pairs of fractions, for the pairs of naturalnumbers, and for the pairs of non-numerical symbols. The response side was also counterbalanced be-tween the prime and the probe in each priming condition.

Participants were tested individually in a quiet room in their school. The experiment lasted approx-imately 40 min. The instructions were given with examples. Participants were asked to indicate thelarger fraction, the larger natural number, or the underlined symbol by using a response box withtwo lateralized keys. They needed to press, as quickly and accurately as possible, the key on the sameside as the larger number or the underlined symbol. They performed a training block with four trials ofeach of the five priming conditions and then the four blocks in a random order. The time course of atrial is illustrated in Fig. 1. A fixation cross appeared at the center of the screen for 300 ms, followed bya black screen for 500 ms. Then the prime (i.e., a pair of fractions or a pair of non-numerical symbols)was presented until participants’ response. A black screen was presented for 500 ms before thepresentation of the probe (i.e., a pair of natural numbers) that remained until participants’ response.Finally, a black screen was presented for 1500 ms before the next trial. Responses were recorded bySuperLab Pro 4.

250 G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259

Statistical analyses

Analyses were run with SPSS 16. RTs associated with a correct response to fractions were analyzedafter suppressing RTs outside the range delimited by the mean ± 3 standard deviations (percentage ofoutlier RTs: M = 1.4%, SD = 0.9) and averaging RTs to the two presentations of the same pair. A linearmixed model (LMM) was run with grade (fifth vs. seventh), distance between the fractions, distancebetween the fraction components (i.e., the denominators of x/a_x/b fractions and the numerators ofa/x_b/x fractions), and type of fraction (a/x_b/x vs. x/a_x/b) as fixed main effects. The model alsoincluded interactions between grade and each distance, between type of fraction and each distance,and between grade and type of fraction.

Two random factors were also included in the model because of their significant contribution to thevariance tested by the likelihood ratio test (Norusis, 2008). These factors allowed us to take into ac-count the dependence between our observations due to repeated measures. The random interceptfor participants allowed the intercept to vary between participants, whereas the random interceptfor the pairs of fractions allowed by-item adjustment of the intercept (Baayen, Davidson, & Bates,2008; Quené & van den Bergh, 2008). This last adjustment is suggested when the predictors are tiedto the items, such as the numerical distances computed from the 64 pairs of fractions used in thisexperiment, and the same list of items is used for all participants. If the fixed effects did not captureall of the properties of the items (e.g., familiarity of fractions, magnitude of fractions), the residualsmight still be correlated and the observed results might be limited to the selected items.

Traditional analysis approaches—ordinary least squares regression on the means of items com-puted by aggregating data over participants (by-item regression) and individual regressions (by-par-ticipant regression [see Lorch & Myers, 1990])—were not used for several reasons. First, none of theseanalyses allowed us to take into account both the variability induced by participants and the variabil-ity induced by items in the same analysis. In addition, a simulation study recently reported by Baayenand colleagues (2008) showed that, when the measures were repeated for participants and items andwhen the predictors were tied to the items, ignoring item variability that was actually present in thedata could lead to unacceptable high Type 1 error rates. Because the by-participant regression did nottake into account this variability, the Type 1 error rate was very high. On the other hand, the by-itemregression and the LMM with crossed random effects for items and participants led to acceptable Type1 error rates. Nevertheless, the power of the LMM was greater than that of the by-item regression.

In addition, LMM is less sensitive to the presence of missing data than a traditional analysis (Brown& Prescott, 1999; Van den Noortgate & Onghena, 2006). Due to errors or extreme RTs, data were miss-ing in our study. Finally, LMM is very comprehensive. The effect of both within- and between-partic-ipant variables, as well as the interactions, can be tested in one model, in contrast to the approachesrecommended by Lorch and Myers (1990) (Van den Noortgate & Onghena, 2006). For all of these rea-sons, we chose to run LMM with crossed random effects for participants and items despite LMM notyet being a common form of analysis in the field of numerical cognition.

The accuracy for fractions was analyzed by entering the percentage of participants who correctlycompared each pair of fractions in a multiple linear regression with the same predictors as those usedin the LMM on RTs. For both of these analyses, the distance between fractions and the distancebetween components were centered at their mean. In that way, the effect of type of fraction andthe effect of grade were tested at meaningful values of the distances. In addition, this transformationhas been recommended to reduce the correlations between the interaction terms and the independentvariables involved in these interactions that could lead to collinearity problems (e.g., Howell, 1997/1998, pp. 614–615; Judd & McClelland, 1989, p. 263). In the current study, all of the variance inflationfactors were smaller than 10 in the model with the centering of the continuous predictors.

Finally, to test the priming effect on natural numbers, the RTs to natural numbers associated with acorrect response and preceded by a correct response to the prime were averaged by priming conditionin each participant after the suppression of RTs outside the range delimited by the mean ± 3 standarddeviations (percentage of outlier RTs: M = 1.6%, SD = 0.8). Next, RTs were analyzed by an analysis ofvariance (ANOVA) with priming condition (five levels) as a within-participant variable and grade(two levels) as a between-participant variable. Due to violation of sphericity, the Huynh–Feldt correc-tion was applied to degrees of freedom. Finally, the mean RT in each experimental priming condition

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was compared with the mean RT in the control priming condition by paired t tests. The same analyseswere applied to the percentages of correct responses to natural numbers.

Results

Selection of participants by paper-and-pencil questionnaire

In the paper-and-pencil task of comparison of fractions, 39.4% of the fifth-graders and 72.6% of theseventh-graders correctly compared all of the pairs of fractions (i.e., five pairs of fractions with com-mon denominators and five pairs of fractions with common numerators). Among children who com-mitted systematic errors, the patterns of errors suggested strategies such as ‘‘the larger fraction hasthe larger component” and ‘‘the larger fraction has the smaller component” (see Appendix B). In thefirst case, children correctly compared fractions with common denominators (a/x_b/x) but not frac-tions with common numerators (x/a_x/b), suggesting that children considered fractions as two inde-pendent natural numbers and applied their knowledge of natural numbers to fractions. In the secondcase, children correctly compared fractions with common numerators (x/a_x/b) but not fractions withcommon denominators (a/x_b/x). Stafylidou and Vosniadou (2004) described this pattern as a syn-thetic model reflecting children’s effort to accommodate knowledge of natural numbers to fractions.

Among children who correctly compared at least four of five pairs for each type of fraction (51% ofthe fifth-graders and 81% of the seventh-graders), 42 fifth-graders and 44 seventh-graders were askedto take part in the computerized experiment. Despite the use of a paper-and-pencil questionnaire toselect participants, data from 5 fifth-graders and 3 seventh-graders were excluded because the chil-dren’s percentage of correct responses was lower than 65 and, hence, near the level of success bychance (i.e., 50%) for at least one type of fraction in the computerized task. Thus, analyses were runon data from 37 fifth-graders and 41 seventh-graders.

Performance on fractions in computerized task

RTsA LMM was run on RTs with grade (fifth vs. seventh), distance between the fractions, distance be-

tween the fraction components (i.e., the denominators of x/a_x/b fractions and the numerators ofa/x_b/x fractions), and type of fraction (a/x_b/x vs. x/a_x/b) as fixed main effects. The interactions be-tween grade and each distance, between type of fraction and each distance, and between grade andtype of fraction were also included in the model. Results are presented in Table 1. Only the significanteffects are described in the text.

The effect of grade was significant. Fifth-graders (M = 1867 ms, SD = 645) responded more slowlythan seventh-graders (M = 1548 ms, SD = 361). The effect of type of fraction was also significant. Chil-dren compared x/a_x/b fractions (M = 1747 ms, SD = 555) more slowly than a/x_b/x fractions

Table 1Linear mixed model on response times to fractions.

Source df1 df2 F p

Grade 1 75.99 15.45* .00Type.F 1 4134.13 68.27* .00Dist.C 1 42.42 0.01 .94Dist.F 1 66.39 10.95* .00Type.F � Grade 1 4823.01 3.20 .07Dist.C � Grade 1 4822.91 1.62 .20Dist.C � Type.F 1 4814.08 0.13 .72Dist.F � Grade 1 4822.91 1.40 .24Dist.F � Type.F 1 3752.66 8.40* .00

Note. Degrees of freedom were approximated by the Satterthwaite method. Type.F, type of fraction (x/a_x/b vs. a/x_b/x); Dist.C,distance between components; Dist.F, distance between fractions.* p < .01.

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(M = 1652 ms, SD = 520). This effect tended to interact with grade. Although it was significant in bothgrades, the difference between the two types of fraction tended to be larger in fifth grade (mean dif-ference of 111 ms) than in seventh grade (mean difference of 80 ms). The effect of distance betweenthe whole fractions was significant and interacted with type of fraction. This interaction indicated thatthe effect of distance between the whole fractions was stronger for x/a_x/b fractions than for a/x_b/xfractions. The reduced model (including only the significant main effects and the significant interac-tion) confirmed these results and provided better estimates of the effect of distance between the frac-tions according to the type of fraction (see Brown & Prescott, 1999); for the x/a_x/b fractions, it wasb = �856.13 (SE = 141.02), t(62.21) = �6.07, p < .01, and for the a/x_b/x fractions, it was b = �280.78(SE = 117.64), t(61.96) = �2.39, p = .02.

In summary, younger children responded more slowly than older children, and responses wereslower for fractions with common numerators (x/a_x/b) than for fractions with common denominators(a/x_b/x). Moreover, RTs decreased as the distance between the whole fractions increased, with thiseffect being significant for both types of fraction but stronger for fractions with common numerators(x/a_x/b) (see Fig. 2).

AccuracyA multiple linear regression was run on the pairs of fractions, with the percentage of participants

who correctly compared each of them as the dependent variable and the same predictors as thoseused in the analysis of RTs. Results are presented in Table 2. Only the significant results are reportedin the text. The effects of grade and type of fraction were significant. More children correctly comparedthe fractions in seventh grade (M = 92.9%) than in fifth grade (M = 85.3%) and for a/x_b/x fractions(M = 90.9%) than for x/a_x/b fractions (M = 87.2%). The effect of distance between the whole fractionsinteracted significantly with type of fraction. Its effect was not significant for a/x_b/x fractions,b = �1.55 (SE = 8.19), t = �0.19, p > .10, but was significant for x/a_x/b fractions, b = 28.36 (SE = 9.58),t = 2.96, p < .01. The results of the reduced model, including only the previous significant effects, con-firmed the results.

In summary, more children answered correctly in seventh grade than in fifth grade and for fractionswith common denominators (a/x_b/x) than for fractions with common numerators (x/a_x/b). More-over, the number of participants who correctly compared fractions increased as the distance betweenfractions with common numerators (x/a_x/b) increased.

Priming effect on natural numbers

RTsRTs to natural numbers (see Table 3) were analyzed by an ANOVA with priming condition (five lev-

els) as a within-participant variable and grade (two levels) as a between-participant variable. The ef-

Fig. 2. Response times decreased as the distance between fractions with common numerators (x/a_x/b) (left panel) andfractions with common denominators (a/x_b/x) (right panel) increased in fifth grade (white circles) and seventh grade (blacktriangles).

Table 2Multiple regression on percentage of correct responses to fractions.

Variable b SE b t p

Grade 7.16 0.88 .63 8.11* .00Type.F �4.00 0.89 �.35 �4.51* .00Dist.C 0.51 0.56 .13 0.91 .36Dist.F �1.55 8.19 �.03 �0.19 .85Type.F* Grade 0.82 1.25 .06 0.65 .51Dist.C* Grade 0.87 0.65 .16 1.33 .18Dist.C* Type.F �1.24 0.66 �.23 �1.88 .06Dist.F* Grade �3.81 10.08 �.05 �0.38 .71Dist.F* Type.F 29.91 10.40 .33 2.88* .00

Note. R2 = .65 and MSE = 12.38. Type.F, type of fraction (x/a_x/b vs. a/x_b/x); Dist.C, distance between components; Dist.F,distance between fractions.* p < .01.

G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259 253

fect of grade was significant, F(1, 76) = 6.91, p = .01, MSE = 208029, g2p = .08. Fifth-graders responded

more slowly than seventh-graders. The effect of condition was significant, F(2.37, 179.97) = 12.44,p < .01, MSE = 9995, g2

p = .14, and did not interact significantly with grade, F(2.37, 179.97) = 2.76,p = .06. Relative to the control condition, responses were slower for specific and unspecific primingsby x/a_x/b fractions, t(77) = �4.33, p < .01, and t(77) = �3.33, p < .01, respectively, and faster for spe-cific priming by a/x_b/x fractions, t(77) = 3.16, p < .01 (see Fig. 3). The difference between unspecificpriming by a/x_b/x fractions and the control condition was not significant, t(77) = �0.71, p > .10.

AccuracyThe same analyses were run on the percentage of correct responses (see Table 3 for the means). An

ANOVA with condition (five levels) as a within-participant variable and grade (two levels) as a be-tween-participant variable was run. The effect of grade was not significant, F(1, 76) = 0.63, p > .10.The effect of condition was significant, F(3.38, 256.82) = 26.51, p < .01, MSE = 13.24, g2

p = .26, and didnot interact with grade, F(3.38, 256.82) = 2.06, p = .10. Children made more errors in specific primingby x/a_x/b fractions than in the control condition, t(77) = 6.86, p < .01. The other experimental condi-tions did not differ significantly from the control condition (all ps > .10).

In summary, relative to the control condition, children compared natural numbers more quicklywhen natural numbers were primed by fractions with common denominators (a/x_b/x) and withnumerators equal to these natural numbers (positive priming effect). Conversely, children comparednatural numbers more slowly in both specific and unspecific primings by fractions with commonnumerators (x/a_x/b) (negative priming effect). In specific priming by these fractions, children alsomade more errors.

Discussion

This study aimed to test the magnitude representations activated during the comparison of frac-tions with common components. In particular, we examined whether the magnitude of the whole

Table 3Mean response time and mean percentage of correct responses in each of the five priming conditions.

Prime Specificity RT (ms) RC (%)

Non-numerical symbols N/A 954 (192) 97.2 (3.2)a/x_b/x fractions Specific 925 (185) 97.6 (2.9)

Unspecific 962 (244) 97.0 (3.3)x/a_x/b fractions Specific 993 (213) 92.9 (5.5)

Unspecific 999 (269) 96.7 (4.0)

Note. Standard deviations are in parentheses. RC, mean correct responses, N/A, not applicable.

Fig. 3. The priming effect on natural numbers is illustrated by the response time difference between each experimentalcondition and the control condition. Error bars represent the 95% confidence interval. From left to right: specific priming byfractions with common denominators [S(a/x_b/x)], unspecific priming by fractions with common denominators [Uns(a/x_b/x)],unspecific priming by fractions with common numerators [Uns(x/a_x/b)], and specific priming by fractions with commonnumerators [S(x/a_x/b)].

254 G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259

fractions, rather than just the magnitude of the fraction components, was accessed in fifth-graders andseventh-graders who could correctly compare these fractions. Moreover, we investigated whether theprocessing of the denominators interfered with the selection of the larger fraction in these children. Toaddress these questions, the effect of numerical distances and the effect of congruity of the relativemagnitude of the components with the magnitude of the whole fractions were tested on the perfor-mance on fractions. These analyses were combined with an analysis of the priming effect of the com-parison of fractions on the comparison of natural numbers.

Results showed that the RTs decreased as the distance between the whole fractions increasedduring their comparison. This distance effect was stronger for fractions with common numerators(x/a_x/b) than for fractions with common denominators (a/x_b/x). These results suggest an access tothe magnitude of the whole fractions for both types of fraction. Nevertheless, the smaller distance ef-fect for fractions with common denominators (a/x_b/x) could indicate that even if children preferen-tially accessed the magnitude of the whole fractions, some of them, at least for some trials of thiscondition, accessed only the magnitude of the numerators. On the other hand, children respondedmore slowly and made more errors for fractions with common numerators (x/a_x/b) than for fractionswith common denominators (a/x_b/x), suggesting an interference of the relative magnitude of thedenominators. The effect of the distance between the whole fractions and the effect of the congruitybetween the magnitude of fractions and the magnitude of their components together suggest that therepresentation of the fraction magnitude is hybrid (i.e., both holistic and componential) rather thanpurely holistic.

A priming paradigm allowed us to test whether the processing of the relative magnitude of thedenominators was inhibited to correctly compare the fractions. In both grades, children compared nat-ural numbers more slowly when natural numbers were primed by fractions with common numerators(x/a_x/b) no matter whether the denominators were identical to or different from the natural num-bers. This negative priming effect suggests a cost affecting the comparison of natural numbers dueto residual inhibition; the processing of natural numbers required a process or response that had beeninhibited during the processing of the previous fractions. Because the negative priming effect wasshown for natural numbers identical to or different from the previous denominators, inhibition oper-ated on the comparison of the denominators. Thus, inhibition allowed us to prevent the selection ofthe larger denominator and its mapping with the motor response. Indeed, the selection of the largerdenominator would have interfered with the selection of the larger fraction for fractions with commonnumerators (x/a_x/b).

The performance on fractions and the priming effect did not interact significantly with grade. Nev-ertheless, even if it was not significant, the difference in performance between fractions with commonnumerators (x/a_x/b) and fractions with common denominators (a/x_b/x) tended to differ between the

G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259 255

two grades, with this difference being larger in fifth grade than in seventh grade. It could suggest agreater sensitivity to the interference in fifth grade. The absence of a significant interaction couldbe due to our selection procedure of children. Indeed, to test the effect of a correct comparison of frac-tions on the subsequent comparison of natural numbers, we selected children who could correctlycompare both types of fraction (i.e., fractions with common denominators and fractions with commonnumerators). Nevertheless, we should keep in mind that the percentage of children who met the selec-tion criterion differed in the two grades: 51% of fifth-graders and 81% of seventh-graders.

The processing of the magnitude of fractions differs in children and adults, as suggested by the dis-tance effect as well as the priming data. First, children seem to compare the magnitudes of the wholefractions for both types of fractions with common components, whereas adults compare the numer-ators for fractions with common denominators (a/x_b/x) and the whole fractions for fractions withcommon numerators (x/a_x/b) (Meert et al., 2009). Nevertheless, we cannot exclude that some chil-dren could compare the magnitudes of the numerators for some pairs of fractions with commondenominators (a/x_b/x) because the effect of the distance between fractions was smaller for these frac-tions than for fractions with common numerators (x/a_x/b) in our study. Children could tend to apply astrategy that is appropriate for all pairs (i.e., selecting the larger fraction) because they could be lessable than adults to flexibly use several strategies (see also Bonato et al., 2007). It would be interestingto test whether children also access the magnitude of the whole fractions if only fractions with com-mon denominators (a/x_b/x) are presented or whether children directly compare the numerators inthat context to test the impact of the presence of fractions with common numerators (x/a_x/b) inthe task. In addition, using retrospective verbal reports after each trial combined with records ofRTs and accuracy could be helpful to test whether children use different strategies for the same typeof fraction and whether one of them is predominant. This method is widely used in studies on arith-metic and has shown its validity and reliability to identify the strategies used by participants to solveoperations and the factors influencing their use, including problem-related characteristics, individualdifference variables, and relative efficiency of a strategy relative to available strategies (e.g., Campbell& Alberts, 2009; Imbo & Vandierendonck, 2008; Robinson et al., 2006). According to Siegler (1999),participants have several strategies available and will try to choose the fastest and most accuratestrategy among all available strategies for each problem. The variability of strategies and the adaptivechoice among them could also apply to the processing of fractions.

Second, the level of processing at which inhibition operates seems to differ between children andadults. The negative priming effect was shown in children for natural numbers identical to or differentfrom the previous denominators (e.g., 2/3_2/5 primed 3_5 or 8_9). In adults, the negative primingeffect was shown only when the natural numbers were identical to the previous denominators(e.g., 2/3_2/5 primed 3_5) (Meert et al., 2009). When comparing fractions with common numerators(x/a_x/b), the relative magnitude of the fractions and denominators was processed (process of compar-ison). Inhibition operated on the process of comparison of the denominators (i.e., ‘‘select the largernatural number”) in children and on the response activated by this process (e.g., ‘‘5”) in adults. Thus,inhibition operated before the mapping of this response with the motor response (i.e., left vs. righthand) because the motor response was counterbalanced between the prime and the probe in eachpriming condition. Inhibition could occur at a later stage of processing in adults than in children be-cause adults used the selection of the larger natural number for fractions with common denominators(a/x_b/x). Therefore, inhibition occurred on the response (i.e., the component selected as the largerone) only when this response was in conflict with the selection of the larger fraction (i.e., for x/a_x/b fractions).

Our study is the first to highlight ability of accessing the magnitude of the whole ratios expressedsymbolically (i.e., fractions) in children. This access could be surprising in view of the acknowledgedcomplexity of fraction learning. Further research should investigate access to the magnitude of thewhole fractions and the properties of this representation. More specifically, we should further testwhether this access is made (a) through access to the magnitude of the fraction components by alearned mapping and the processing of their ratio or (b) by a learned mapping that directly links frac-tions to mental magnitudes. Except for some very common fractions (e.g., 1/2, 1/3, 1/4), learned map-ping between fractions and mental magnitudes seems implausible because there is an infinite numberof equivalent fractions for each ratio. Because our research used fractions with components as large as

256 G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259

19, our results suggest rather that children are able to process the magnitude of a ratio between twomagnitudes accessed from Arabic digits.

The current study does not allow us to conclude whether children access the magnitude of thewhole fraction from the magnitude of its components and by which mechanism. This issue couldbe addressed by using retrospective verbal reports after each trial combined with records of responsetimes and accuracy. Nevertheless, several strategies could be hypothesized for the current task (i.e., amagnitude comparison task on fractions with common components). Children could construct a vis-uospatial representation such as a portion of a pie or a bar. They could also solve operations on thecomponents. Literature on simple arithmetic, and in particular on multiplication and division fromArabic digits, could be enlightening for this issue. Indeed, the relation between the denominatorand the numerator is multiplicative. Multiplication is mainly solved by direct retrieval from long-termmemory in children from second grade to sixth grade (Imbo & Vandierendonck, 2008), whereas divi-sion is mainly solved by multiplication reference in children from fifth grade to seventh grade (e.g.,2 � ? = 8 for 8 � 2 = ?) (Robinson et al., 2006). Children could divide the numerator by the denomina-tor (e.g., 4 � 9 for 4/9) or divide 1 by the denominator and multiply the result by the numerator [e.g.,(1 � 9) � 4 for 4/9]. However, retrieving arithmetic facts could be inadequate for fractions most of thetime. Indeed, fractions representing natural numbers (e.g., 4/2), and so corresponding to a knownarithmetic fact (e.g., 4 � 2 = ? or 2 � ? = 4) form a small set of fractions. In our study, we used onlyfractions with a magnitude between 0 and 1, corresponding to arithmetic facts probably not storedin long-term memory (e.g., 4 � 9 = ? or 9 � ? = 4 for 4/9) except perhaps for very familiar fractions(e.g., 1 � 2 = ? or 2 � ? = 1 for 1/2). Instead, approximate arithmetic could be used in combination withbenchmarks such as 1/2, 1/3, and 1/4 (e.g., 4 is less than half of 9 but larger than its third for 4/9). Fi-nally, in the current study, because fractions always had either common numerators or commondenominators, it remains possible that children processed the difference between the denominatorand numerator for each fraction and selected the fraction with the smaller difference (i.e., the one thatwas closest to 1). If this is the case, further studies should use fractions with the same numerator–denominator distance but varying in their magnitude (e.g., 4/5 vs. 11/12) to test whether children pro-cess the distance to 1 in relative terms (e.g., 1/5 vs. 1/12) or in absolute terms (e.g., 1 vs. 1).

The link between the ability to access the magnitude of a ratio from two Arabic digits and the abil-ity to process nonsymbolic ratios or perform multiplicative transformations on nonsymbolic stimulishould also be investigated. Several studies have reported that children can process ratios betweencontinuous quantities before receiving systematic instruction on ratios and their symbolization (Jeong,Levine, & Huttenlocher, 2007; Mix, Levine, & Huttenlocher, 1999; Singer-Freeman & Goswami, 2001).As examples, 6-year-olds can choose the game with the greater probability of winning when this isdetermined by a ratio of two continuous quantities (i.e., a doughnut divided into a red part and a bluepart) (Jeong et al., 2007), and 4-year-olds are able to perform simple operations on fractional amountsor to match figures according to visuospatial ratios (Mix et al., 1999; Sophian, 2000). Barth, Baron,Spelke, and Carey (2009) showed that 6- and 7-year-olds are able to apply multiplicative transforma-tions to discrete quantities and, in particular, to represent the half of a set and compare it with a thirdset. Their performance varies with the ratio between the to-be-compared quantities, suggesting thatchildren operate on mental magnitudes. In the light of these results, the access to the magnitude of thewhole fractions described in our study is less surprising if we hypothesize that it relies on the ability toprocess nonsymbolic ratios or multiplicative relations. This hypothesis does not exclude that childrenneed to understand the relation between magnitudes denoted by symbolic ratios (i.e., fractions),which could be submitted to slow learning and the interference of the previous symbolic learning(i.e., natural numbers). Indeed, in our study, children were 10 and 12 years old, had already receivedsystematic instructions on nonsymbolic and symbolic ratios, and represented 51% and 81% of the fifth-and the seventh-graders, respectively.

The ability to process ratios could even be preverbal, as suggested by a recent study. Using a habit-uation paradigm, McCrink and Wynn (2007) showed that 6-month-olds were able to extract the ratioof two discrete collections in much the same way as they could compare two numerosities. The infantscould detect a change in the ratio if the new ratio differed from the habituated one by a factor of 2 butnot by a factor of 1.5. The magnitude estimation system involved in the processing of numerositieswould also support the ability to discriminate two ratios. These results, combined with the data in

G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259 257

animals reviewed by Gallistel and colleagues (2006), are consistent with the theory suggested by theseauthors. Our numerical magnitude system would be inherited from the evolution and would supportthe representation not only of natural numbers referring to discrete magnitudes but also of realnumbers (including rational numbers expressed by ratios) referring to both discrete and continuousmagnitudes. Indeed, the properties of our magnitude system (e.g., the representation by continuousmental magnitudes on which operations can be applied) make this hypothesis plausible. Further stud-ies are needed to test whether the ability to process ratios is language independent and whether thisability supports the latter ability to process fractions.

In conclusion, children who have enough understanding of the function of fraction components tocorrectly compare fractions with common components can access the magnitude of the whole frac-tions but are sensitive to the interference of the relative magnitude of denominators. This study hashighlighted that, beyond the interference of natural number knowledge at the conceptual level (i.e.,the whole number bias), research should also investigate the Stroop-like effect between the magni-tude of the fraction components and the magnitude of the whole fractions. In addition to a conceptualunderstanding of fractions, children need to manage conflict between the relative magnitude of thefraction components and the relative magnitude of the whole fractions.

Acknowledgments

G.M. and M-P.N. are supported by the Fund for Scientific Research of the French-Speaking Commu-nity of Belgium (FRS–FNRS). This study was supported by a Grant from Actions de Recherche Concertées(05/10-327) of the French-Speaking Community of Belgium. We thank Christine Nossent and NicolasMichaux for their help with the data collection, and we thank the schools and children who took partin this research.

Appendix A. List of experimental stimuli

Natural numbers

Fractions a/x_b/x Fractions x/a_x/b

3

5 3/7 5/7 2/5 2/3 7 3 3/8 7/8 2/7 2/3 4 5 5/7 4/7 3/4 3/5 7 4 7/9 4/9 3/4 3/7 7 5 5/9 7/9 4/7 4/5 7 8 8/9 7/9 5/7 5/8 8 9 8/11 9/11 7/9 7/8 5 9 9/11 5/11 2/5 2/9

15

11 15/17 11/17 7/11 7/15 13 11 13/14 11/14 10/11 10/13 11 17 11/19 17/19 10/17 10/11 13 12 12/17 13/17 11/13 11/12 12 14 14/17 12/17 11/12 11/14 15 12 15/17 12/17 11/12 11/15 12 16 12/17 16/17 11/16 11/12 17 12 12/19 17/19 11/17 11/12 12 18 18/19 12/19 11/12 11/18 14 13 14/15 13/15 11/13 11/14 13 15 13/16 15/16 11/15 11/13 16 13 13/17 16/17 11/16 11/13 13 17 17/19 13/19 12/13 12/17 18 13 18/19 13/19 11/13 11/18

(continued on next page)

258 G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259

Appendix A (continued)

Natural numbers

Fractions a/x_b/x Fractions x/a_x/b

14

15 14/17 15/17 13/15 13/14 16 14 14/17 16/17 13/16 13/14 14 17 17/19 14/19 13/14 13/17 18 14 18/19 14/19 13/14 13/18 15 16 15/17 16/17 13/16 13/15 17 15 15/19 17/19 14/17 14/15 15 18 18/19 15/19 13/15 13/18 17 16 17/19 16/19 15/16 15/17 16 18 16/19 18/19 13/18 13/16 18 17 17/19 18/19 13/18 13/17

Appendix B. Percentages of fifth-graders and seventh-graders for each level of performanceexpressed as the number of correct comparisons (maximum = 5) for each type of fraction on thepaper-and-pencil comparison task

Grade

x/a_x/b a/x_b/x

0

1 2 3 4 5 Total

5

0 0.0 0.0 0.0 2.0 0.0 16.2 18.2 1 1.0 0.0 0.0 0.0 1.0 1.0 3.0 2 0.0 0.0 1.0 0.0 1.0 1.0 3.0 3 2.0 0.0 0.0 0.0 1.0 1.0 4.0 4 1.0 0.0 0.0 0.0 1.0 6.1 8.1 5 15.2 1.0 2.0 2.0 4.0 39.4 63.6

Total

19.2 1.0 3.0 4.0 8.0 64.7 100

7

0 0.0 0.0 0.0 0.0 0.0 4.1 4.1 1 1.4 0.0 0.0 0.0 0.0 0.0 1.4 2 0.0 0.0 0.0 0.0 0.0 1.4 1.4 3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4 0.0 0.0 1.4 0.0 0.0 4.1 5.5 5 6.8 0.0 2.7 1.4 4.1 72.6 87.6

Total

8.2 0.0 4.1 1.4 4.1 82.2 100

References

Baayen, R. H., Davidson, D. J., & Bates, D. M. (2008). Mixed-effects modeling with crossed random effects for subjects and items.Journal of Memory and Language, 59, 387–556.

Barth, H., Baron, A., Spelke, E., & Carey, S. (2009). Children’s multiplicative transformations of discrete and continuousquantities. Journal of Experimental Child Psychology, 103, 441–454.

Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook ofresearch on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 296–333). NewYork: Simon & Schuster/Macmillan.

Bonato, M., Fabbri, S., Umiltà, C., & Zorzi, M. (2007). The mental representation of numerical fractions: Real or integer? Journal ofExperimental Psychology: Human Perception and Performance, 33, 1410–1419.

Brown, H., & Prescott, R. (1999). Applied mixed models in medicine. Chichester, UK: John Wiley.Campbell, J. I. D., & Alberts, N. M. (2009). Operation-specific effects of numerical surface form on arithmetic strategy. Journal of

Experimental Psychology: Learning, Memory, and Cognition, 35, 999–1011.Daurignac, E., Houdé, O., & Jouvent, R. (2006). Negative priming in a numerical Piaget-like task as evidenced by ERP. Journal of

Cognitive Neuroscience, 18, 730–736.

G. Meert et al. / Journal of Experimental Child Psychology 107 (2010) 244–259 259

English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Hillsdale, NJ: Lawrence Erlbaum.Gallistel, C. R., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4,

59–65.Gallistel, C. R., Gelman, R., & Cordes, S. (2006). The cultural and evolutionary history of the real numbers. In P. Jaisson & S. C.

Levinson (Eds.). Evolution and culture: A Fyssen Foundation symposium (Vol. 17, pp. 247–274). Cambridge, MA: MIT Press.Grégoire, J., & Meert, G. (2005). L’apprentissage des nombres rationnels et ses obstacles [Rational number learning and its

obstacles]. In M. P. Noël (Ed.), La dyscalculia: Trouble du développement numérique chez l’enfant (pp. 223–251). Marseille,France: Solal.

Hartnett, P., & Gelman, R. (1998). Early understandings of numbers: Paths or barriers to the construction of newunderstandings? Learning and Instruction, 8, 341–374.

Houdé, O. (2000). Inhibition and cognitive development: Object, number, categorization, and reasoning. Cognitive Development,15, 63–73.

Houdé, O., & Guichart, E. (2001). Negative priming effect after inhibition of number/length interference in a Piaget-like task.Developmental Science, 4, 119–123.

Howell, D. C. (1998). Méthodes statistiques en sciences humaines [Statistical methods for psychology] (M. Rogier, Trans.). Brussels,Belgium: De Boeck [Original work published 1997].

Imbo, I., & Vandierendonck, A. (2008). Effects of problem size, operation, and working-memory span on simple-arithmeticstrategies: Differences between children and adults? Psychological Research, 72, 331–346.

Jeong, Y., Levine, S. C., & Huttenlocher, J. (2007). The development of proportional reasoning: Effect of continuous versus discretequantities. Journal of Cognition and Development, 8, 237–256.

Judd, C. M., & McClelland, G. H. (1989). Data analysis: A model comparison approach. San Diego: Harcourt, Brace, Jovanovich.Lorch, R. F., & Myers, J. L. (1990). Regression analyses of repeated measures data in cognitive research. Journal of Experimental

Psychology: Learning, Memory, and Cognition, 16, 149–157.Mack, N. K. (1993). Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E.

Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 85–105). Hillsdale, NJ: Lawrence Erlbaum.McCrink, K., & Wynn, K. (2007). Ratio abstraction by 6-month-old infants. Psychological Science, 18, 740–745.Meert, G., Grégoire, J., & Noël, M. P. (2009). Rational numbers: Componential vs. holistic representation of fractions in a

magnitude comparison task. Quarterly Journal of Experimental Psychology, 62, 1598–1616.Mix, K. S., Levine, S. C., & Huttenlocher, J. (1999). Early fraction calculation ability. Developmental Psychology, 35, 164–174.Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520.Ni, Y., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number

bias. Educational Psychologist, 40, 27–52.Norusis, M. J. (2008). SPSS 16.0 advanced statistical procedures companion. Upper Saddle River, NJ: Prentice Hall.Quené, H., & van den Bergh, H. (2008). Examples of mixed-effects modeling with crossed random effects and with binomial data.

Journal of Memory and Language, 59, 413–425.Robinson, K. M., Arbuthnott, K. D., Rose, D., McCarron, M. C., Globa, C. A., & Phonexay, S. D. (2006). Stability and change in

children’s division strategies. Journal of Experimental Child Psychology, 93, 224–238.Siegler, R. S. (1999). Strategic development. Trends in Cognitive Sciences, 3, 430–435.Singer-Freeman, K. E., & Goswami, U. (2001). Does half a pizza equal half a box of chocolates? Proportional matching in an

analogy task. Cognitive Development, 16, 811–829.Sophian, C. (1996). The trouble with fractions. In Children’s numbers (pp. 89–101). Boulder, CO: Westview.Sophian, C. (2000). Perceptions of proportionality in young children: Matching spatial ratios. Cognition, 75, 145–170.Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning

and Instruction, 14, 503–518.Tipper, S. P. (1985). The negative priming effect: Inhibitory priming by ignored objects. Quarterly Journal of Experimental

Psychology A, 37, 571–590.Tipper, S. P. (2001). Does negative priming reflect inhibitory mechanisms? A review and integration of conflicting views.

Quarterly Journal of Experimental Psychology A, 54, 321–343.Van den Noortgate, W., & Onghena, P. (2006). Analysing repeated measures data in cognitive research: A comment on regression

coefficient analyses. European Journal of Cognitive Psychology, 18, 937–952.