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

$: Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task$
PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [Univ Catholique de Louvain UCL Bibliotheque de Psychologie &]On: 29 September 2009Access details: Access Details: [subscription number 910628705]Publisher Psychology PressInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

The Quarterly Journal of Experimental PsychologyPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t716100704

Rational numbers: Componential versus holistic representation of fractions in amagnitude comparison taskGaëlle 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 del'Education, Université Catholique de Louvain, Louvain-la-Neuve, Belgium b Unité Cognition etDéveloppement, Faculté de Psychologie et des Sciences de l'Education, Université Catholique de Louvain,Louvain-la-Neuve, Belgium

First Published on: 27 December 2008

To cite this Article Meert, Gaëlle, Grégoire, Jacques and Noël, Marie-Pascale(2008)'Rational numbers: Componential versus holisticrepresentation of fractions in a magnitude comparison task',The Quarterly Journal of Experimental Psychology,62:8,1598 — 1616

To link to this Article: DOI: 10.1080/17470210802511162

URL: http://dx.doi.org/10.1080/17470210802511162

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t716100704

http://dx.doi.org/10.1080/17470210802511162

http://www.informaworld.com/terms-and-conditions-of-access.pdf

Rational numbers: Componential versusholistic representation of fractions in a magnitude

comparison task

Gaelle Meert and Jacques GregoireUnite de Psychologie de l’Education et du Developpement, Faculte de Psychologie et des Sciences de l’Education, Universite

Catholique de Louvain, Louvain-la-Neuve, Belgium

Marie-Pascale NoelUnite Cognition et Developpement, Faculte de Psychologie et des Sciences de l’Education, Universite Catholique de Louvain,

Louvain-la-Neuve, Belgium

This study investigated whether the mental representation of the fraction magnitude was componen-tial and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the com-parison 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 withoutcommon components) were added to reduce the regularity of the stimuli. In both experiments, dis-tance effects indicated that participants compared the numerators for a/x_b/x fractions, but thatthe magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. Thepriming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denomi-nator magnitude was controlled during the comparison of these fractions. These results suggested ahybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magni-tude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude ofthe denominator and the magnitude of the numerator. However, adults might prefer to rely on themagnitudes of the components and compare the magnitudes of the whole fractions only when theuse of a componential strategy is made difficult.

Keywords: Numerical cognition; Fractions; Magnitude comparison; Distance effect; Congruity effect.

Interest in numerical cognition has motivatednumerous studies on the processing of positiveintegers (i.e., natural numbers), while othernumerical categories, such as rational numbers,

have not been greatly investigated. This studyinvestigated cognitive processing in adults of themagnitude of rational numbers represented byfractions. Our first aim was to better understand

Correspondence should be addressed to Gaelle Meert, Unite de Psychologie de l’Education et du Developpement, Faculte de

Psychologie et des Sciences de l’Education, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. E-mail:

[email protected]

This study was supported by a grant from Actions de Recherche Concertees 05/10–327 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).

1598 # 2008 The Experimental Psychology Society

http://www.psypress.com/qjep DOI:10.1080/17470210802511162

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY

2009, 62 (8), 1598–1616

Downloaded By: [Univ Catholique de Louvain UCL Bibliotheque de Psychologie &] At: 09:18 29 September 2009

how adults deal with fractions. The processing of afraction’s magnitude is crucial in mathematics,since fractions are used in algebraic, probabilistic,and proportional reasoning. This processing isalso required in many situations of everyday life,such as the calculation of drug dosage, the esti-mation of a discount, or the estimation of thechances of winning in a lottery. Fractions consistof a quotient between two integer numbers (e.g.,1/7) where the denominator is not equal to0. With regard to the structure of this symbolicrepresentation, we investigated whether themental magnitude of the whole fraction (i.e., hol-istic representation) or only the magnitudes of thefraction components (i.e., componential represen-tation) were accessed during the processing of thefraction magnitude in a numerical comparisontask.

Gallistel, Gelman, and Cordes (2005; see alsoGallistel & Gelman, 2000) have suggested thatthe cognitive foundation of magnitude processingis an innate system that can represent realnumbers, a numerical category larger than theinteger numbers, and the rational numbers (i.e.,one that includes them). Their theory emergesfrom a large body of empirical data showing thatanimals are able to process continuous quantities(e.g., duration) and discrete quantities (e.g.,flashes) as well as to perform arithmetical oper-ations combining both discrete and continuousquantities. Both countable quantities (discrete)and uncountable quantities (continuous) wouldbe represented in the same system by continuousmental magnitudes that would be approximate(i.e., with a scalar variability). Studies with adultshave suggested that the same mental magnitudesare accessed not only from analogical represen-tations (e.g., a collection of dots) but also fromsymbols (i.e., Arabic numerals or verbal numberwords) through the learning of a mapping (e.g.,Whalen, Gallistel, & Gelman, 1999). Whileinteger numbers refer only to discrete quantities,real numbers refer to both discrete and continuousquantities. Therefore, the mental magnitudeswould be isomorphic to real numbers and supportthe processing of their magnitude. Following thistheory, the magnitude processing system should

allow the representation of the magnitudes ofwhole fractions as they represent symbolicallyrational numbers, a part of the category of realnumbers.

Yet, in the cultural history of numbers, integernumbers are the foundation of the mathematicalthought from which mathematicians have devel-oped other categories of numbers. Gallistel et al.(2005) have suggested that integer numbers arethe foundation of the cultural history of numbersdue to the relative ease in establishing one-to-one mapping between the counted objects (i.e.,discrete quantities) and the sequence of symbols(i.e., verbal number words) by counting. The realnumber system was developed only later becausesuch counting was not appropriate for continuousquantities due to their density. Indeed, the use ofa reference unit (e.g., for the measurement of alength or a duration) only provides a roughmeasurement of the continuous quantity. The his-torical development of numbers would be therefore“a Platonic rediscovery of what the non-verbalbrain was doing all along” (Gallistel & Gelman,2000, p. 60).

To our knowledge, only Bonato, Fabbri,Umilta, and Zorzi (2007) have investigated theprocessing of the fraction magnitude in adults.Four experiments were reported in which adultscompared fractions to a fixed standard (1/5, 0.2,or 1). Two classical effects in numerical cognition,distance and SNARC (spatial-numerical associ-ation of response codes) effects, were used to testwhether the mental magnitude of the whole frac-tion was accessed during the task. In a numericalcomparison task, the distance effect refers to theimprovement in performance with the increase ofthe numerical distance between the targetnumber and the fixed standard (Moyer &Landauer, 1967). In other words, responses arefaster and more accurate when the distance islarge (e.g., 1_5) than when the distance is small(e.g., 4_5). The SNARC effect is an interactionbetween the response code and the number magni-tude: faster response to large numbers if theresponse is on the right side of space and tosmall numbers if the response is on the left sideof space (e.g., Dehaene, Bossini, & Giraux,

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8) 1599

RATIONAL NUMBER PROCESSING


1993; for reviews: Gevers & Lammertyn, 2005;Hubbard, Piazza, Pinel, & Dehaene, 2005). Theresults reported by Bonato et al. have shown thatresponse times (RTs) were better predicted bythe distance between the components of thefractions than by the distance between the wholefractions themselves. When the fixed standardwas not a fraction (e.g., 0.2), the participants con-verted it into a fraction in order to directlycompare the denominators. The SNARC effectwas less consistent across the various experiments.The most interesting result was the interactionbetween the response code and the magnitude ofthe denominators (reversed SNARC effect)during the comparison of fractions with numerator1 (e.g., 1/5_1/3). The authors concluded thatthe participants used strategies in order toprocess the fraction magnitude componentiallyand that the mental magnitude of the wholefraction was not accessed.

Contrary to the lack of studies on fraction pro-cessing in adults, a considerable amount of studiesconcerns fraction learning in children. However,these studies have investigated the conceptualunderstanding of fractions rather than the cognitiveprocessing of their magnitude. In the Stafylidouand Vosniadou study (2004), 11-year-old childrenoften saw fractions as two independent naturalnumbers. Among other misconceptions observedin a numerical ordering task of fractions withoutcommon denominators, children claimed that thevalue of a fraction increases as the value ofthe numerator increases (e.g., 4/3 , 5/6) or thatthe value of a fraction increases as the value of thenumerator decreases (e.g., 4/3 , 1/7) withouttaking into account the magnitudes of the denomi-nators. In this study, students aged from 10 to 17gradually overcame the interference of naturalnumbers and counting knowledge before theirunderstanding of fractions matched the scientificconcept. This process implies a conceptual changefollowing several stages. The most frequent errorsand misconceptions noticed in children learningfractions would result from the tendency to inter-pret fractions by referring to previous countingand natural number knowledge (called “the wholenumber bias”, Ni &Zhou, 2005). As the properties

of natural numbers and counting differ from theproperties of rational numbers, this bias interfereswith fraction learning and leads to errors and mis-conceptions (e.g., Behr, Harel, Post, & Lesh,1992; English & Halford, 1995; Gregoire &Meert, 2005; Hartnett & Gelman, 1998; Mack,1993; Sophian, 1996; Stafylidou & Vosniadou,2004). Bonato et al. (2007) suggested that thewhole number bias observed in children is consist-ent with the componential strategies used toprocess the fraction magnitude. According tothese authors, the understanding of fractionsmight be rooted in the ability to represent discretequantities rather than continuous quantities.Adults might differ from children in their abilityto use flexible and appropriate strategies of compo-nential processing.Nevertheless, thewhole numberbias could simply reflect the misunderstanding ofthe complex symbol that fractions represent, atleast at the beginning of fraction learning, and theinterference of previous numerical knowledge.This bias does not preclude the magnitudes ofwhole fractions from being represented by mentalmagnitudes. As studies on fraction learning wereseeking to investigate the understanding of frac-tions, they did not provide any direct evidence infavour of either holistic or componential represen-tation in children who understand, at least in part,the symbolic representation of rational numbersin fraction form.

The issue of the nature of magnitude processing(holistic vs. componential) has also been debatedfor other complex numerical symbols, such astwo-digit numbers or negative numbers. Fortwo-digit numbers, some authors have reportedevidence in favour of a holistic representation oftheir magnitude (e.g., Dehaene, Dupoux, &Mehler, 1990; Reynvoet & Brysbaert, 1999)whereas others showed effects in favour of a com-ponential representation (e.g., Nuerk, Weger, &Willmes, 2001; Ratinckx, Brysbaert, & Fias,2005; Verguts & De Moor, 2005). To reconcilethese conflicting results, Zhou, Chen, Chen, andDong (2008) showed that the mode of presen-tation (simultaneous vs. sequential) could deter-mine the processing of two-digit numbers (seealso Zhang & Wang, 2005). On the other hand,

1600 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2009, 62 (8)

MEERT, GREGOIRE, NOEL


https://www.researchgate.net/publication/5762684_The_Mental_Representation_of_Numerical_Fractions_Real_or_Integer?el=1_x_8&enrichId=rgreq-cf6408d9-660a-469d-88c6-fb5d9134ebfa&enrichSource=Y292ZXJQYWdlOzIzNzIwNzk1O0FTOjEwMzI0MTU1NTY0NDQyMUAxNDAxNjI2MTEzMTgx

https://www.researchgate.net/publication/12746605_Single-digit_and_two-digit_Arabic_numerals_address_the_same_semantic_number_line?el=1_x_8&enrichId=rgreq-cf6408d9-660a-469d-88c6-fb5d9134ebfa&enrichSource=Y292ZXJQYWdlOzIzNzIwNzk1O0FTOjEwMzI0MTU1NTY0NDQyMUAxNDAxNjI2MTEzMTgx

Nuerk and Willmes (2005) suggested that theholistic magnitudes could be activated in parallelwith the magnitudes of tens and of units. Fornegative numbers, the nature of their processingseems to depend on the task. In a parity judgementtask, the magnitudes of the digits are processedregardless of the sign (i.e., componential represen-tation; Fischer & Rottmann, 2005; Nuerk,Iversen, & Willmes, 2004). In a magnitude com-parison task, Shaki and Petrusic (2005) demon-strated componential representation when thetask was performed only on negative numbersbut holistic representation when the task was per-formed on both positive and negative numberpairs. These studies suggest that the processingof the magnitude of Arabic numbers with acomplex structure could depend on the exper-iment’s conditions.

The purpose of the current study was to identifythe mental magnitudes activated during magni-tude comparison of two fractions. We testedwhether the processing of the fraction magnitudedepends on the congruity between the magnitudeof the whole fraction and the magnitudes of itscomponents and on the ability to identify this con-gruity in the experimental context. In the firstexperiment, only pairs of fractions with commoncomponents were presented. In a pair of fractionswith common denominators (i.e., a/x_b/x; e.g.,3/7_5/7), the relative magnitude of the numeratoris congruent with the relative magnitude of thewhole fraction since the larger fraction (e.g., 5/7)is made up of the larger numerator (e.g., 5). In apair of fractions with common numerators (i.e.,x/a_x/b; e.g., 2/3_2/5), the relative magnitudeof the denominator is incongruent with the rela-tive magnitude of the whole fraction since thelarger fraction (e.g., 2/3) is made up of thesmaller denominator (e.g., 3). In the first case,the congruity could lead participants to directlycompare the numerators in order to select thelarger fraction, without accessing the magnitudesof the whole fractions. In the second case, theincongruity could lead the participants tocompare the magnitudes of the whole fractions,which could be accessed by estimating the ratiobetween the magnitude of the numerator and the

magnitude of the denominator for each fraction.Nevertheless, the processing of the denominatorscould interfere with this holistic processing asthe larger denominator does not make up thelarger fraction. Alternatively, participants couldcompare the denominators by taking intoaccount their incongruity with the fraction magni-tude (i.e., choosing the smaller denominator tochoose the larger fraction). In the second exper-iment, the same stimuli were mixed with pairs offractions without common components to reducethe regularity of the stimuli in the experiment.For these pairs of fillers, the congruity of the com-ponent magnitude cannot be identified without anaccess to the fraction magnitude. Therefore, accessto holistic representation of the fraction magni-tude would be required for these pairs and thencould be enhanced for fractions with commoncomponents as this processing is appropriate forall pairs.

The numerical distance effect was used to testwhether the participants rely on the magnitudesof the components or on the magnitudes of thewhole fractions to compare fractions. If partici-pants compare only the components of the twofractions, performance should be influenced bythe componential distance (i.e., the distancebetween the numerators for a/x_b/x fractionsand the distance between the denominators forx/a_x/b fractions). Alternatively, if the magni-tudes of the whole fractions are accessed beforeperforming the comparison, performance shouldvary with the distance between the whole fractions(i.e., the overall distance). Moreover, a primingparadigm was used. The comparison of fractionsprimed the comparison of natural numbers. Ifthe magnitudes of the whole fractions are accessedfor x/a_x/b fractions, the denominator magnitudecould interfere with the selection of the larger frac-tion, and the selection of the larger denominatormight be inhibited in order to select the largerfraction. Therefore, if natural numbers identicalto the denominators of x/a_x/b fractions (i.e.,a_b) are presented after these fractions, their com-parison should require the activation of the sameresponse as the response inhibited during the pro-cessing of the previous fractions. In this case, the




https://www.researchgate.net/publication/7254324_On_the_mental_representation_of_negative_numbers_Context-dependent_SNARC_effects_with_comparative_judgments?el=1_x_8&enrichId=rgreq-cf6408d9-660a-469d-88c6-fb5d9134ebfa&enrichSource=Y292ZXJQYWdlOzIzNzIwNzk1O0FTOjEwMzI0MTU1NTY0NDQyMUAxNDAxNjI2MTEzMTgx

residual inhibition on the response should leadto slower RTs than those to natural numbersdiffering from the denominators of the previousx/a_x/b fractions. On the other hand, RTs tonatural numbers primed by a/x_b/x fractionsshould be faster for natural numbers identical tothe numerators of these fractions (i.e., a_b) thanfor natural numbers differing from these numer-ators due to residual activation on the selectionof the larger numerator during the processing ofa/x_b/x fractions.

EXPERIMENT 1

Method

ParticipantsA total of 40 university undergraduate psychologystudents (39 women; 34 right-handed) took partin this experiment and received course credit.The average age was 19 years (ranging from 18to 23 years).

DesignParticipants were asked to compare the numericalvalue of two fractions or two natural numbers.Two types of fraction were presented: fractionswith 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 naturalnumbers was primed in four priming conditionsresulting from the crossing of two within-subjectvariables: the type of prime (x/a_x/b vs. a/x_b/x) and the priming specificity (specific vs. unspeci-fic). The priming was specific when the numer-ators of a/x_b/x fractions or the denominators ofx/a_x/b fractions were identical to the subsequentnatural numbers. The priming was unspecificwhen all the fraction components differed fromthe subsequent natural numbers. The unspecific

priming was used as a baseline to assess theeffect of the specific priming. Figure 1 shows anexample of a trial (prime and probe) for each ofthe four conditions.

StimuliThe 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 pairof x/a_x/b fractions (e.g., 3 and 7 in 2/7_2/3)were the same natural numbers as the numeratorsof a given pair of a/x_b/x fractions (e.g., 3 and 7in 3/8_7/8). In this way, the numerators of a/x_b/x fractions and the denominators of x/a_x/bfractions were strictly matched for numerical dis-tance.1 Both types of fraction were balanced forthe numerical distance between the whole frac-tions. For a/x_b/x fractions, the overall distanceranged from 0.05 to 0.55 with a mean distance of0.20 (SD ¼ 0.13). For x/a_x/b fractions, therange was from 0.04 to 0.44 with a mean distanceof 0.18 (SD ¼ 0.11). Pairs were made up of irredu-cible fractions that were smaller than the unit andwhose components were numbers ranging from 2to 19 (excluding 10 as denominator). The use ofirreducible fractions prevented variability betweenand within participants due to the possibility ofsimplification. Indeed, simplification could leadto slower RTs and to a larger load in workingmemory associated with a higher probability ofmaking mistakes. This simplification would maskthe numerical distance effect as well as thepriming effect, preventing the identification ofthe numbers that primed the subsequent pair ofnatural numbers. Fractions with denominator 10were excluded as they might be easily transformedinto decimal numbers. For each type of fraction (a/x_b/x vs. x/a_x/b), the larger fraction was pre-sented on the left in half of the pairs in order tocounterbalance 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.




presented as the prime both in the specific primingand in the unspecific priming.

The pairs of natural numbers were identical tothe pairs of denominators in x/a_x/b fractionsand so to the pairs of numerators in a/x_b/x frac-tions. Each of these 32 pairs was presented as theprobe in all four priming conditions. In the specificpriming, they were primed by the pair of x/a_x/bfractions and by the pair of a/x_b/x fractionswhere a and b were identical to these naturalnumbers. In the unspecific priming, they wereprimed by one pair of x/a_x/b fractions and byone pair of a/x_b/x fractions so that all the frac-tion components differed from these naturalnumbers. The larger natural number was presentedon the left in half of the pairs in order to counter-balance the response side. The response side wasalso counterbalanced between the prime (i.e., frac-tions) and the probe (i.e., natural numbers) in eachpriming condition.

All the stimuli were presented on a screen usingSuperlab Pro 1.77 (Cedrus Corporation, SanPedro, California). The viewing distance wasabout 60 cm. In a display, the fractions or thenatural numbers were presented horizontally, 6.78away from each other and 3.38 away from thescreen’s centre. The fraction components were pre-sented vertically and separated by the fraction bar(as illustrated in Figure 1). The height and the

width of a fraction were, respectively, 3.38 and1.48. The height and the width of a naturalnumber were, respectively, 1.48 and 0.98. Arabicsymbols were presented in white printed characters(Times font, normal) on a black background.

ProcedureEach priming condition included 32 trials with aprime (a pair of fractions) and a probe (a pair ofnatural numbers). Four blocks were created.Within a block, 8 trials of each condition were pre-sented so that a trial was never followed by a trialof the same priming condition. Moreover, a givenpair of natural numbers or fractions was not pre-sented more than twice in a block, with at least40 stimuli between the two presentations.

Participants were tested individually, and theexperiment lasted about 30 minutes. They wereasked to indicate the larger number (i.e., thelarger fraction for fraction pairs and the largernatural number for natural numbers pairs) asquickly and as accurately as possible by using aresponse box with two lateralized keys. They hadto press the left-hand key or the right-hand keywhen the larger number was presented, respect-ively, on the left or on the right of the pair.After the instructions had been given, participantsperformed a training block of eight trials (primeand probe) and then the four blocks presented in

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.




random order. Responses were recorded bySuperlab Pro 1.77 (Cedrus Corporation, SanPedro, CA). The time sequence of a trial isshown in Figure 1. A cross appeared at thecentre of the screen for 300 ms. A black screenwas then presented for 500 ms, before the presen-tation of the prime, which remained on the screenuntil the participant’s response. A black screen wasshown for 500 ms before the presentation of theprobe, which disappeared at the participant’sresponse. Finally, a black screen lasting for1,500 ms separated the response to the probefrom the cross announcing the subsequent trial.

Results

The .05 significance level was chosen for all thestatistical analyses made with SPSS 16. Datafrom 6 participants were excluded because theirerror rate was higher than 35% for at least onetype of fraction, and their performance was there-fore too close or even poorer than the level ofsuccess by chance (50%). Among the systematicerrors (more than 80%), 2 participants chose thefraction with the smaller numerator for a/x_b/xfractions, and 1 participant chose the fractionwith the larger denominator for x/a_x/b fractions,probably due to conceptual misunderstanding ofthe function of the denominator or the numerator.The 3 remaining participants were close to thelevel of success by chance (from 40 to 60% oferrors) 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 34participants whose RTs to fractions and tonatural numbers were in the range delimitedby the mean of the sample +3 standard devi-ations. RTs and error rates were expressed,respectively, in milliseconds and in percentages.

FractionsAnalyses were run on the error rates and on themedians of RTs for correct responses computedfor each participant and for each pair. To test theeffect of congruity between the magnitude ofthe whole fraction and the magnitudes of itscomponents, 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 theRTs to a/x_b/x fractions (M ¼ 1,186,SD ¼ 222), F(1, 33) ¼ 20.82, p , .01, h2 ¼ .39.Moreover, the participants made significantlymore 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, h2 ¼ .31.The poorer performance on x/a_x/b fractionssuggested an interference of the denominatormagnitude with the selection of the larger fraction.

Median RTs and error rates for the pairs offractions (see Appendix) were then analysed bylinear regressions in order to test whether theperformance varied with the componential dis-tance (i.e., the distance between the two com-ponents that varied between the fractions; e.g., 2for 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 thefollowing way: (a) identification of the significantpredictor(s) by simple regressions for each typeof fraction and (b) test of the relative contributionof each predictor in multiple regression when bothpredictors were significant. RTs and error rates fora/x_b/x fractions were significantly predicted bythe 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 didnot 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 responsesto a/x_b/x fractions were faster and more accurateas the distance between the numerators increased,indicating that participants relied on the magni-tude of the numerators to compare the fractionswith common denominators (see Table 1).

The same analyses were run for x/a_x/bfractions. RTs and error rates were significantlypredicted 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 distancebetween the denominators [respectively, b¼ –.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/b fractions




were faster and more accurate as the distancebetween the whole fractions increased, suggestingthat the magnitudes of the whole fractions wereaccessed and compared (see Table 1).3

Natural numbersThe median of RTs associated with a correctresponse and preceded by a correct response tofractions was computed for each participant andfor each pair. The mean error rate was lowerthan 5% in the four priming conditions and there-fore was not analysed. To test whether the selec-tion of the larger denominator was inhibitedduring the processing of x/a_x/b fractions, RTsto natural numbers were analysed by anANOVA, 2 (prime: x/a_x/b vs. a/x_b/x) � 2(priming specificity: specific vs. unspecific).4 Themain effect of prime was significant, F(1,33) ¼ 22.84, p , .01, h2 ¼ .41. The main effect

of priming specificity was not significant, F(1,33) ¼ 0.67, p . .10. The prime and the primingspecificity interacted significantly, F(1, 33) ¼ 8.01,p , .01, h2 ¼ .19 (see Figure 2). In the specificpriming, RTs were significantly slower whennatural numbers followed x/a_x/b fractions(M ¼ 631, SD ¼ 103) than when they followeda/x_b/x fractions (M ¼ 593, SD ¼ 94), t(33) ¼–4.47, p , .01. This effect was not significant inthe unspecific priming, t(33) ¼ –1.25, p . .10,supporting the use of performance in the unspeci-fic priming as a baseline. When the prime was x/a_x/b fractions, RTs were slower in the specificpriming (M ¼ 631, SD ¼ 103) than in theunspecific priming (M ¼ 612, SD ¼ 100),t(33) ¼ 2 2.97, p , .01. On the other hand,when the prime was a/x_b/x fractions, RTs didnot differ significantly in the specific priming(M ¼ 593, SD ¼ 94) and in the unspecific

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 – .40� –.27 –.51�� – .61�� –.46�� –.24

Errors – –.17 –.35� – –.36� –.27

Overall – .78�� – .82��

Componential – –

2 With RTs – .56�� –.47��a –.46��a – .51�� –.62�� –.39�b

Errors – –.36�b –.52�� – –.40� –.30b

Overall – .78�� – .82��

Componential – –

Note: RT ¼ response time.aRedundant predictors in multiple regression. bSuppressor variable in multiple regression.�p � .05; ��p � .01, two-tailed.

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 sup-

pressor 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 R2 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).




priming (M ¼ 605, SD ¼ 88), t(33) ¼ 1.70,p ¼ .10.5

Discussion

Performance on a/x_b/x fractions varied accordingto the numerical distance between the numerators,suggesting that participants compared the magni-tudes of the numerators that were congruent withthe magnitudes of the whole fractions to selectthe larger fraction. Conversely, performance onx/a_x/b fractions varied according to the distancebetween the whole fractions. This result suggeststhat the larger fraction was selected by accessingthe magnitudes of the whole fractions and bycomparing them. Additionally, participants wereslower and made more errors for the comparisonof x/a_x/b fractions than for the comparison ofa/x_b/x fractions. The difference of performance

between the two types of fraction could be due tothe congruity effect between the magnitude of thecomponents that varied between the two fractionsand the magnitude of the whole fractions.However, the access to the magnitudes of thewhole fractions and their comparison could alsoexplain poorer performance on fractions x/a_x/b.Indeed, this processing could be slower and lessaccurate than the comparison of the numerators ifwe hypothesize that the holistic representationrequires access to themental magnitudes of the frac-tion components and the estimation of their ratio.

The presence of a priming effect induced by frac-tion comparison over a natural number comparisonspeaks in favour of the congruity effect. Indeed, RTswere slower in specific priming by x/a_x/b fractionsthan in unspecific priming by these fractions. Thiscost can be explained by the residual inhibition onselection of the larger denominator. Indeed, whilethe participants were accessing the magnitudes ofthe whole fractions and comparing them, thedenominators could be compared and the largerdenominator selected automatically. The selectionof the larger denominator could interfere with thatprocessing and had to be inhibited so that thelarger fraction was correctly selected. Therefore,when the subsequent natural numbers were identicalto the denominators, the previously inhibitedresponse had to be activated, leading to slowerresponses. The comparison of the denominatorscould be automatic since access to a holistic rep-resentation does not require the comparison of thedenominators and since the absence of an effect ofthe distance between the denominators suggeststhat participants did not select the smaller denomi-nator to select the larger fraction.

It could be surprising that participants com-pared the whole fractions whereas comparison ofthe denominators would have been sufficient for

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).

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.




x/a_x/b fractions (i.e., the choice of the smallerdenominator). A possible explanation is that theuse of a componential strategy for all pairs of frac-tions would have introduced a cognitive cost for allpairs of fractions. Indeed, it would have involved atask switching between the choice of the smallercomponent for x/a_x/b fractions and the choiceof the larger component for a/x_b/x fractions.

EXPERIMENT 2

When the task was performed on a/x_b/x fractionsand on x/a_x/b fractions, participants compared thenumerators of a/x_b/x fractions without accessingthe magnitudes of the whole fractions and com-pared the whole fractions for x/a_x/b fractions.However, the regularity of these fractions (i.e.,with common denominators or with commonnumerators) might have facilitated the use of a com-ponential strategy for a/x_b/x fractions. Indeed, theparticipants compared the numerators of the a/x_b/x fractions probably because they could easily ident-ify the pairs of fractions with congruity between themagnitudes of the components and the magnitudesof the whole fractions. In Experiment 2, fillers wereincluded to reduce the regularity of the stimuli andto discourage the use of componential strategies.The filler pairs were made up of fractions withoutcommon components so that the magnitude ofthe denominator and the magnitude of the numer-ator could be congruent or incongruent with themagnitude of the whole fraction. Therefore, in theabsence of common denominators or numerators,the congruity between the fraction componentsand the whole fraction could not be identifiedwithout an access to the magnitudes of the wholefractions. The presence of fillers might then leadto holistic processing for all pairs of fractions,even for a/x_b/x fractions, as this processing isappropriate for all pairs of fractions.

Method

ParticipantsA total of 41 university undergraduate psychologystudents (34 women; 34 right-handed) participated

for course credit. The average age was 20 years(ranging from 19 to 23 years).

DesignThe design was the same as that in Experiment 1,except that new pairs of fractions, each followed bya pair of natural numbers, were added as fillers.They were made up of two fractions with differingdenominators and numerators (e.g., 6/11_3/8)and varied according to the level of congruitybetween the magnitude of the whole fraction andthe magnitudes of the components. The threepossible levels of congruity were used to reducethe regularity of the stimuli and to discouragethe use of componential strategies. The larger frac-tion was made up (a) of the smaller denominatorand the smaller numerator (e.g., 5/12 , 4/7),(b) of the smaller denominator and the largernumerator (e.g., 4/13 , 6/11), or (c) of thelarger denominator and the larger numerator(e.g., 3/8 , 6/11).

StimuliThe experimental pairs of fractions and of naturalnumbers were those used in Experiment 1. A totalof 60 filler pairs of fractions were created for eachlevel of congruity. In these pairs, fractions wereirreducible, below the unit, and with a denomi-nator from 2 to 14 (except 10) and a numeratorfrom 1 to 11. Moreover, one denominator wasnever the multiple of the other denominator inorder to avoid encouraging participants to searchfor a common denominator. For each category offractions presented as fillers, the larger fractionwas presented on the left in half of the trials inorder to counterbalance the response side. Atotal of 60 filler pairs of natural numbers werecreated and differed from the experimental pairsof natural numbers. They were presented as theprobes for each category of filler pairs of fractionsand therefore were presented three times. Theresponse side was counterbalanced within thefiller pairs of natural numbers as well as betweenthe filler pairs of fractions and the subsequentpairs of natural numbers.




ProcedureAmong the 308 trials made up of a prime and aprobe, 128 were experimental trials, and 180were filler trials. Trials were divided in eightblocks. Each block was made up of 4 experimentaltrials of each priming condition and 7 or 8 fillertrials of each level of congruity. A trial never fol-lowed a trial of the same condition. The timesequence of a trial was the same as that inExperiment 1. After the instruction was given,participants initially performed a training blockof 20 trials and then the eight blocks in arandom order varying for each participant. Theexperiment lasted about 50 minutes.

Results

Data from 11 participants were excluded becausetheir error rate was higher than 35% in at leastone experimental category of fractions (a/x_b/xvs. x/a_x/b) or one filler category of fractions. Atotal of 4 participants committed systematicerrors (more than 80% of errors). These errorsreflected the choice of the fraction with thesmaller numerator for all pairs without commonnumerators (n ¼ 1), the choice of the fractionwith the larger denominator for all pairs withoutcommon denominators (n ¼ 1), the choice of thefraction with the smaller numerator when thedenominators were common (i.e., a/x_b/x) andwith the smaller denominator for all other pairs(n ¼ 1), and the choice of the fraction with thelarger denominator when the numerators werecommon (n ¼ 1). These errors could be at leastin part due to conceptual misunderstanding ofthe function of the denominator and/or thenumerator as they concerned even the fractionswith common components. For 6 participants,the percentage of correct responses was close tothe level of success by chance when the larger frac-tion had the larger denominator (n ¼ 4), when thelarger fraction had the smaller numerator (n ¼ 1),and when the fractions had common numerators(n ¼ 1). Finally, 1 participant responded randomlyfor all categories.

Analyses were therefore run on data from 30participants whose RTs were in the range

delimited by the mean of the sample + 3 stan-dard deviations for fractions and for naturalnumbers.

FractionsAnalyses were run on the error rates and on themedians of RTs for correct responses computedfor each participant and for each pair. AnANOVA 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 fractionssignificantly slower (M ¼ 1,540, SD ¼ 306) thanthey did a/x_b/x fractions (M ¼ 1,389,SD ¼ 222), F(1, 29) ¼ 39.21, p , .01, h2 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,h2 ¼ .20. These differences replicated thoseobtained in Experiment 1 and suggested a cogni-tive cost for x/a_x/b fractions due to the incongru-ity between the magnitude of the denominator andthe magnitude of the whole fraction. To directlycompare these results with the results of the firstexperiment, an ANOVA was run with the exper-iment as a between-subject variable (withoutfillers vs. with fillers) and the type of fraction asa within-subject variable. The effect of the typeof fraction did not vary with the experiment forthe RTs, F(1, 62) ¼ 1.39, p . .10, and for theerror rates, F(1, 62) ¼ 0.02, p . .10. On theother hand, the effect of the experiment was sig-nificant for RTs, F(1, 62) ¼ 13.32, p , .01,h2 ¼ .18, but not for the error rates, F(1,62) ¼ 1.60, p . .10. In short, participants in theexperiment with fillers (M ¼ 1,465, SD ¼ 317)were significantly slower than participants in theexperiment without fillers (M ¼ 1,242,SD ¼ 244), whatever the type of fraction.

To test whether RTs and error rates for thepairs of a/x_b/x fractions (see Appendix) variedaccording to the distance between the numeratorsor according to the overall distance, simple linearregressions were first run to identify the significantpredictor(s). Then significant predictors weresimultaneously entered in a regression to assesstheir relative contribution. Given their intercorre-lation, the method suggested by Tzelgov and




Henik (1991) was used to detect redundancy effectand suppression effect. Simple regressions onRTs showed that the overall distance, R2 ¼ .23,b ¼ –.47, t(30)¼ –2.96, p , .01, and the distancebetween numerators, R2 ¼ .21, b ¼ –.46,t(30)¼ –2.85, p , .01, were significant predictors(see Table 1). The correlation between RTs andcomponential distance did not differ significantlyfrom the correlation between RTs and overall dis-tance, t(29)¼ –0.09, p . .10. When both dis-tances were simultaneously entered in a regression(R2 ¼ .25), the overall distance and the distancebetween the numerators were no longer significantpredictors: 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 predictorsaccounted for the same variance (i.e., redundancy),making it impossible to assess which distance wasthe important factor (see Field, 2005, p. 175;Hair, Anderson, Tatham, & Black, 1998, p. 188).Simple regressions on error rates showed also a sig-nificant effect of the overall distance, R2 ¼ .13,b ¼ –.36, t(30)¼ –2.10, p , .05, and of the dis-tance between numerators, R2 ¼ .27, b ¼ –.52,t(30)¼ –3.33, p , .01. In the multiple regressionon error rates (R2 ¼ .27), the effect of the distancebetween the numerators remained significant,b ¼ –.61, t(29) ¼ –2.43, p ¼ .02, whereas theeffect of the overall distance was not any more sig-nificant, b ¼ .12, t(29)¼ 0.48, p . .10. Theinflation of the first coefficient and the change ofsign of the second one were due to suppressioneffect. The overall distance acted as a negative sup-pressor variable by suppressing residual variance inthe prediction of error rates from the distancebetween the numerators. However, this contri-bution was not significant. In short, responseswere more accurate as the distance between thenumerators increased, suggesting that participantsrelied on the component magnitude to comparea/x_b/x fractions. RTs were equally predicted bythe overall distance and by the distance betweenthe numerators. The redundancy effect between thedistances preventedus fromconcludingwhether par-ticipants compared only the numerators or whetherthey compared also the magnitudes of the wholefractions.

The same analyses were made for x/a_x/b frac-tions. In simple regressions, the overall distanceand the componential distance significantly pre-dicted RTs: respectively, R2 ¼ .38, b ¼ –.62,t(30) ¼ –2.96, p , .01, and R2 ¼ .15, b ¼ –.39,t(30) ¼ –2.29, p ¼ .03 (see Table 1). When bothdistances were simultaneously entered in aregression (R2 ¼ .42), the effect of the overalldistance remained significant, b ¼ –.90, t(29)¼ –3.67, p , .01, whereas the effect of the distancebetween the denominators was not anymore signifi-cant, b ¼ .35, t(29)¼ 1.42, p . .10. The distancebetween the denominators was a suppressor variablebut its contribution was not significant. Simpleregressions showed that the error rates were signifi-cantly predicted by the overall distance, R2 ¼ .16,b ¼ –.40, t(30) ¼ –2.40, p ¼ .02, and tended tobe significantly predicted by the componentialdistance, R2 ¼ .09, b ¼ –.30, t(30)¼ –1.74,p ¼ .09. However, in the multiple regression(R2 ¼ .16), neither the overall distance nor the dis-tance between the denominators (suppressor vari-able) 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 participantsrelied mainly on the magnitudes of the whole frac-tions, as suggested by the effect of the overall dis-tance on RTs.

The introduction of fillers intended to discou-rage the use of componential strategies, as it wasassumed that component congruity could not bedetermined without access to the magnitudes ofthe whole fractions for these pairs. Analyses wererun to check that the overall distance influencedthe RTs and the error rates for the filler pairs offractions. 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 fillerswere analysed according to the overall distance,the distance between the numerators, and the dis-tance between the denominators (e.g., respectively,0.15, 1, and 5 for 5/12_4/7). When the largerfraction had the larger numerator and the largerdenominator (e.g., 3/8_6/11), RTs were signifi-cantly predicted by the overall distance,R2 ¼ .24, b ¼ –.49, t(58) ¼ –4.24, p , .01, and




by the distance between the numerators, R2 ¼ .09,b ¼ –.29, t(58) ¼ –2.35, p ¼ .02. When bothpredictors were simultaneously entered in aregression (R2 ¼ .24), the effect of the overalldistance remained significant, b ¼ –.54,t(57) ¼ –3.38, p , .01, whereas the effect of thedistance between the numerators was not signifi-cant, b ¼ .08, t(57) ¼ 0.48, p . .10. The coeffi-cient of the overall distance was inflated as thedistance between the numerators was a suppressorvariable. Simple regressions on error rates showedthat the effect of the overall distance was signifi-cant, R2 ¼ .44, b ¼ –.66, t(58) ¼ –6.74,p , .01, as well as the effect of the distancebetween the numerators, R2 ¼ .09, b ¼ –.30,t(58) ¼ –2.37, p ¼ .02. In the multiple regression(R2 ¼ .49), the distance between the numerators,b ¼ .30, t(57) ¼ 2.33, p ¼ .02, mainly contributedto the prediction of error rates by suppressingresidual variance from the overall distance,b ¼ –.87, t(57) ¼ –6.67, p , .01. In short, par-ticipants mainly compared the magnitude of thewhole fractions for this category of fillers. Whenthe larger fraction had the smaller denominatorand the larger numerator (e.g., 4/13_6/11), theeffect of the overall distance on RTs was signifi-cant, R2 ¼ .49, b ¼ –.70, t(58) ¼ –7.40,p , .01, as well as the effect of the distancebetween the numerators, R2 ¼ .11, b ¼ –.33,t(58) ¼ –2.69, p , .01. When both predictorswere entered in a regression (R2 ¼ .54), the dis-tance between the numerators, b ¼ .35,t(57) ¼ 2.69, p , .01, mainly contributed by clear-ing out irrelevant variance from the overall dis-tance, b ¼ –.94, t(57) ¼ –7.35, p , .01. Onlythe overall distance significantly predicted theerror rates, b ¼ –.44, t(58) ¼ –3.75, p , .01. Inshort, participants compared the magnitude ofthe whole fractions for this category of fillers.Finally, when the larger fraction had the smallerdenominator and the smaller numerator (e.g., 5/12_4/7), only the distance between the numer-ators tended to significantly predict the RTs,R2 ¼ .07, b ¼ –.24, t(58) ¼ –1.86, p ¼ .07. The

error rates were significantly predicted by theoverall distance, R2 ¼ .09, b ¼ –.30, t(58) ¼–2.44, p ¼ .02, and tended to be predicted bythe distance between the denominators,R2 ¼ .06, b ¼ –.24, t(58) ¼ –1.90, p ¼ .06. Inthe multiple regression on error rates (R2 ¼ .10),the effect of the overall distance only tended tobe significant, b ¼ –.24, t(57) ¼ –1.69, p ¼ .10,and the effect of the distance between the denomi-nators was no longer significant, b ¼ –.12,t(57) ¼ –0.83, p . .10. This pattern of resultssuggested that these distances were partiallyredundant and that the error rates tended to bemainly predicted by the overall distance. Theresults for this category of fillers suggested thatparticipants relied on the magnitude of the numer-ators and tended to rely on the magnitude of thewhole fractions. These results were not as clearas those from the two other categories and couldbe due to a smaller variability of the distanceswithin this category. In short, for at least two cat-egories of fillers, the overall distance was the bestpredictor of both RTs and error rates.

Natural numbersThe mean error rate was lower than 5% in the fourpriming conditions and therefore was not analysed.The median of the RTs for correct responses pre-ceded by a correct response to the prime was calcu-lated for each participant and for each pair in eachcondition. To test the priming effect of fractioncomparison over natural number comparison,RTs to natural numbers were analysed by anANOVA, 2 (prime: x/a_x/b vs. a/x_b/x) � 2(priming specificity: specific vs. unspecific).6 Thisanalysis showed a significant main effect of theprime, F(1, 29) ¼ 10.32, p , .01, h2 ¼ .26. Themain effect of the specificity was not significant,F(1, 29) ¼ 0.89, p . .10. The interactionbetween the prime and the priming specificitywas significant, F(1, 29) ¼ 9.90, p , .01,h2 ¼ .25 (see Figure 2). The effect of the primewas significant in the specific priming,t(29) ¼ –3.79, p , .01. The RTs to natural

6 See Footnote 4.




numbers were slower when they followed x/a_x/bfractions (M ¼ 874, SD ¼ 130) than when theyfollowed a/x_b/x fractions (M ¼ 834,SD ¼ 104). This effect was not significant in theunspecific priming, t(29) ¼ –0.34, p . .10.When the prime was a/x_b/x fractions, RTswere slower in the unspecific condition(M ¼ 858, SD ¼ 102) than in the specific con-dition (M ¼ 834, SD ¼ 104), t(29) ¼ 4.04,p , .01. Contrary to Experiment 1, the effect ofspecificity was not significant when the prime wasx/a_x/b fractions, t(29) ¼ –1.48, p . .10. Theseresults were in favour of facilitation in the specificpriming by a/x_b/x fractions.7

To directly compare the priming effect in thetwo experiments, an ANOVA was run with theexperiment as a between-subject variable(without fillers vs. with fillers). The main effectof the prime and the interaction between theprime and the specificity were significant: respect-ively, F(1, 62) ¼ 30.04, p , .01, h2 ¼ .33, andF(1, 62) ¼ 17.99, p , .01, h2 ¼ .22. The maineffect of the specificity was not significant, F(1,62) ¼ 0.06, p . .10. These effects did not interactsignificantly with the experiment (all p . .10).While the effect of the prime was not significantin the unspecific priming, this effect was significantin the specific priming, t(63) ¼ –5.86, p , .01.When the prime was a/x_b/x fractions, RTswere significantly faster in the specific condition(M ¼ 706, SD ¼ 156) than in the unspecific con-dition (M ¼ 724, SD ¼ 158), t(63) ¼ –3.73,p , .01. When the prime was x/a_x/b fractions,RTs were significantly slower in the specific con-dition (M ¼ 745, SD ¼ 168) than in the unspecificcondition (M ¼ 728, SD ¼ 162), t(29) ¼ –2.98,p , .01. On the other hand, the main effect ofthe experiment was significant, F(1, 62) ¼ 94.59,p , .01, h2 ¼ .60. The participants in the exper-

iment with fillers (M ¼ 857, SD ¼ 112) wereslower than the participants in the experimentwithout fillers (M ¼ 610, SD ¼ 96).

Discussion

Even if some participants produced systematicerrors probably due to conceptual misunderstand-ing of fractions or responded randomly to somecategories of fractions, the majority of participantsdealt relatively well with fractions even in theabsence of common components. However, par-ticipants were slower than the participants inExperiment 1, suggesting that the inclusion offillers made the task harder. Indeed, fillersincreased the complexity of the stimuli. Theabsence of common components and the variabil-ity in the congruity of the fraction componentsmade almost impossible the use of componentialstrategies to compare these fractions. Indeed, com-paring only the numerators or the denominatorswould not have allowed selection of the larger frac-tion for all the filler pairs of fractions as the con-gruity of the components could not have beenidentified without an access to the magnitudes ofthe whole fractions. Analyses confirmed anaccess to holistic representation for at least twoof the three categories of fillers. Nevertheless,slower responses could not be explained only byaccess to the magnitudes of the whole fractions.Indeed, responses to x/a_x/b fractions wereslower in Experiment 2 (with fillers) despite thatthe overall distance effect suggested access to hol-istic representations in Experiment 1 (withoutfillers). Moreover, responses to natural numberswere also slower in Experiment 2 than inExperiment 1. This cost could be explained bythe great variability of the congruity of the fractioncomponents in the experiment and could suggest

7 The distance effect was tested on natural numbers in order to exclude whether the facilitation effect was due to the episodic

retrieval of the response activated during the processing of the previous fractions rather than due to the residual activation from

the processing of the components during the fraction comparison. The distance effect on RTs was significant in all priming con-

ditions: in the specific priming by a/x_b/x fractions, b ¼ –.42, t(30) ¼ –2.57, p ¼ .02; in the unspecific priming by a/x_b/x frac-

tions, b ¼ –.43, t(30) ¼ –2.64, p ¼ .01; in the unspecific priming by x/a_x/b fractions, b ¼ –.36, t(30) ¼ –2.10, p ¼ .04; and in

the specific priming by x/a_x/b fractions, b ¼ –.40, t(30) ¼ –2.42, p ¼ .02. The correlations between the RTs and the distance did

not differ between the priming conditions (all ps . .10).




that the representation of fraction magnitude washybrid (i.e., componential and holistic) ratherthan purely holistic. In Experiment 1, even if hol-istic representations were accessed to comparex/a_x/b fractions, participants could also rely onthe magnitudes of the denominators. When thecongruity of the components could not be easilydetermined and varied from one trial to anothertrial as in Experiment 2, performance was poorer.

The introduction of filler pairs of fractions wasintended to discourage componential strategies fora/x_b/x fractions. Responses to these fractionswere less accurate as the distance between thenumerators increased, suggesting that participantsrelied on the magnitudes of the numerators.Unfortunately, the redundancy effect betweenthe overall distance and the distance between thenumerators in the analysis of RTs prevented usfrom assessing whether participants comparedalso the magnitudes of the whole fractions.

As in Experiment 1, performance on x/a_x/bfractions was poorer than performance ona/x_b/x fractions, suggesting that the magnitudeof the denominator interfered with the magnitudeof the whole fraction. However, contrary toExperiment 1, RTs to natural numbers were notsignificantly slower when they were identical tothe denominators of the preceding x/a_x/b frac-tions than when they were different from them.This lack of significance could be due to a TypeII error as the priming effect did not interact withthe experiment. The hybrid processing suggestedby poorer performance in the experiment withfillers led us to think that the comparison of thedenominators leading to a negative priming effectwas not necessarily automatic as initially discussedin Experiment 1. Nevertheless, the specificity ofthe negative priming suggested that the selectionof the larger denominator was inhibited. Thispriming effect could not be accounted for by taskswitching between the choice of the smallerdenominator and the choice of the larger naturalnumber as the effect of the prime was not signifi-cant in the unspecific priming conditions. On theother hand, in the priming by a/x_b/x fractions,the RTs to natural numbers were significantlyfaster when they were identical to the numerators

of these fractions than when they were differentfrom them. The facilitation in the specificpriming by a/x_b/x fractions could be due toresidual activation on the selection of the largernumerator during the processing of a/x_b/xfractions.

GENERAL DISCUSSION

This study tested the processing of the fractionmagnitude and the impact of the experimentalconditions on this processing. Whereas Bonatoet al. (2007) concluded in favour of a componentialrepresentation of the fraction magnitude, ourresults suggest that the magnitude of the wholefraction can be accessed in some conditions.

The processing of the magnitude of fractionswith common components seems to depend onthe congruity between the magnitude of the com-ponents and the magnitude of the whole fraction,at least when only fractions with common denomi-nators (i.e., a/x_b/x) and fractions with commonnumerators (i.e., x/a_x/b) are presented duringthe experiment. When the magnitude of the com-ponent is congruent with the magnitude of thewhole fraction, as in fractions with commondenominators, the numerical distance effectsuggests that only the numerators are compared,without an access to the magnitudes of thewhole fractions. When the magnitude of the com-ponent is incongruent with the magnitude of thewhole fraction, as in fractions with commonnumerators, the numerical distance effect suggestsan access to the magnitudes of the whole fractionsbefore performing the comparison. A cognitivecost is observed during the comparison of naturalnumbers primed by these fractions when theirdenominators are identical to the naturalnumbers. This cost could be due to residual inhi-bition on the selection of the larger denominator.It suggests that the denominators are also com-pared while the magnitudes of the whole fractionsare accessed and compared.

When participants are asked to compare notonly fractions with common components but alsofractions without common components, the




magnitudes of the whole fractions are accessed forfractions with common numerators and at leastfor two categories of fractions without commoncomponents. For these last fractions, the congruityof the fraction components cannot be determinedwithout access to themagnitudes of the whole frac-tions. For fractions with common denominators,participants compare the numerators, as suggestedby the effect of the distance between the numer-ators on response accuracy. Nevertheless, theredundancy effect in the analysis of RTs preventsus from assessing whether participants onlycompare the numerators due to the salience of thecommon denominators, or whether participantscompare also themagnitudes of the whole fractionsdue to the influence of fractions without commoncomponents. Performance is poorer with fillersthan without fillers even when the distance effectsuggests access to holistic representation in bothconditions (i.e., for fractions with common numer-ators). This result suggests that participantscompare the magnitudes of the whole fractions,which are probably accessed by estimating theratio between the magnitude of the denominatorand the magnitude of the numerator, but alsocompare the magnitudes of the denominatorsand/or the numerators. In further research, itmight be interesting to test the relative contri-bution of each type of processing in a way that over-comes the statistical limit due to the highcorrelation between the componential distanceand the overall distance, especially for fractionswith common denominators.

The processing of the fraction magnitude couldvary according to the experimental conditions. Ifonly fractions with common numerators had beenpresented in the first experiment, participantsmight have compared the denominators andchosen the smaller denominator as their magnitudeis incongruent with the magnitude of the wholefraction. In fact, Bonato et al. (2007) showed thatparticipants directly compared the denominatorswhen they were asked to compare only fractionswith numerator 1 to the fixed standard 1/5.Their experiments only produced evidence infavour of componential strategies, probablybecause their experimental conditions did not

discourage their use. Indeed, their stimuli werevery regular, and the congruity between the magni-tude of the components and the magnitude of thewhole fractions did not always vary in a givenexperiment. Therefore, their participants usedvery specific strategies in these experimental con-ditions. If we consider both our results and theresults reported by Bonato et al. (2007), partici-pants could prefer to use a componential strategy.When various pairs of fractions are presented,this strategy is difficult to apply as it requires flexi-bility and can even be impossible when the congru-ity of the fraction components cannot be identifiedwithout an access to the magnitudes of the wholefractions. Access to holistic representation ismore likely in these cases.

The effect of the numerical distance between thewhole fractions suggests that their magnitude canbe represented by our magnitude processingsystem, which is congruent with the theory ofGallistel and Gelman (2000). However, access toholistic representation seems to occur when theuse of a componential strategy is made difficult.Therefore, this access might not be automatic.The nature of mapping between fractions andmental magnitudes may be an important factordetermining this access. Indeed, a magnitudecan be represented by an infinite number ofequivalent fractions (e.g., 1/5, 2/10, 3/15, etc.).Consequently, learned mapping could be limitedto very frequent fractions (e.g., 1/2, 1/3, 1/4).Moreover, the structure of fractions may notfacilitate mapping between fractions and mentalmagnitudes as fractions represent magnitudesthrough the relation between two discrete magni-tudes. As already suggested, the processing of thefraction magnitude could be more an approxi-mation than the retrieval of a mental magnitudeassociated with a specific symbol.

In conclusion, our study provides evidence thatthe magnitude of the whole fractions can beaccessed in adults performing a numerical com-parison. However, adults may prefer to onlycompare the magnitudes of the fraction com-ponents and compare the magnitudes of thewhole fractions when the use of a componentialstrategy is made difficult by the experimental




conditions. Access to the magnitude of the wholefraction (i.e., holistic representation) could bemade through estimation of the ratio betweenthe magnitude of the denominator and the magni-tude of the numerator. While the mental magni-tudes of the whole fractions are accessed andcompared, the denominators or the numeratorsalso seem to be compared, suggesting that thecomparison of the fraction magnitudes is bothcomponential and holistic. Hybrid processing hasbeen also suggested for two-digit numbers.According to this hypothesis, the approximatemagnitude of the overall two-digit numberwould be activated in addition to the magnitudesof the tens and of the units (for a discussionabout this hypothesis, see Nuerk & Willmes,2005). Further research should investigate themechanism by which the magnitude of the wholefraction is accessed as well as the interactionbetween componential strategies and the accessto holistic representation.

Original manuscript received 6 February 2008

Accepted revision received 1 September 2008

First published online 27 December 2008

REFERENCES

Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992).Rational number, ratio, and proportion. In D.A. Grouws (Ed.), Handbook of research on mathemat-

ics teaching and learning. A project of the national

council of teachers of mathematics (pp. 296–333).New York: Simon and Schuster Macmillan.

Bonato, M., Fabbri, S., Umilta, C., & Zorzi, M.(2007). The mental representation of numericalfractions: Real or integer? Journal of Experimental

Psychology: Human Perception and Performance, 33,1410–1419.

Dehaene, S., Bossini, S., & Giraux, P. (1993). Themental representation of parity and number magni-tude. Journal of Experimental Psychology: General,122, 371–396.

Dehaene, S., Dupoux, E., & Mehler, J. (1990). Isnumerical comparison digital? Analogical and sym-bolic effects in two-digit number comparison.Journal of Experimental Psychology: Human

Perception and Performance, 16, 626–641.

English, L. D., & Halford, G. S. (1995). Mathematics

education: Models and processes. Mahwah, NJ:Lawrence Erlbaum Associates, Inc.

Field, A. (2005). Discovering statistics using SPSS.

London: Sage.Fischer, M. H., & Rottmann, J. (2005). Do negative

numbers have a place on the mental number line?Psychology Science, 47, 22–32.

Gallistel, C. R., & Gelman, R. (2000). Non-verbalnumerical cognition: From reals to integers. Trends

in Cognitive Sciences, 4, 59–65.Gallistel, C. R., Gelman, R., & Cordes, S. (2005). The

cultural and evolutionary history of the real numbers.In P. Jaisson & S. C. Levinson (Eds.), Culture and

evolution: A Fyssen Foundation symposium (pp. 247–274). Cambridge, MA: MIT Press.

Gevers, W., & Lammertyn, J. (2005). The hunt forSNARC. Psychology Science, 47, 10–21.

Gregoire, J., & Meert, G. (2005). L’apprentissage desnombres rationnels et ses obstacles [Rationalnumber learning and its obstacles]. In M. P. Noel(Ed.), La dyscalculie. Trouble du developpement

numerique chez l’enfant (pp. 223–251). Marseille,France: Solal.

Hair, J. F., Anderson, R. E., Tatham, R. L., & Black,W. C. (1998). Multivariate data analysis. UpperSaddle River, NJ: Prentice Hall.

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

Howell, D. C. (1998). Methodes statistiques en sciences

humaines [Statistical methods for psychology] (M.Rogier, Trans.). Bruxelles: De Boeck. (Originalwork published 1997.)

Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S.(2005). Interactions between number and space inparietal cortex. Nature Reviews Neuroscience, 6,435–448.

Mack, N. K. (1993). Learning rational numbers withunderstanding: 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: LawrenceErlbaumAssociates, Inc.

Moyer, R. S., & Landauer, T. K. (1967). Time requiredfor judgements of numerical inequality. Nature, 215,1519–1520.

Ni, Y., & Zhou, Y. D. (2005). Teaching and learningfraction and rational numbers: The origins andimplications of whole number bias. Educational

Psychologist, 40, 27–52.




Nuerk, H. C., Iversen, W., & Willmes, K. (2004).Notational modulation of the SNARC and theMARC (linguistic markedness of response codes)effect. Quarterly Journal of Experimental Psychology

57A, 835–863.Nuerk, H. C., Weger, U., & Willmes, K. (2001).

Decade breaks in the mental number line? Puttingthe tens and units back in different bins. Cognition,82, B25–B33.

Nuerk, H. C., &Willmes, K. (2005). On the magnituderepresentations of two-digit numbers. Psychology

Science, 47, 52–72.Ratinckx, E., Brysbaert, M., & Fias, W. (2005).

Naming two-digit Arabic numerals: Evidence frommasked priming studies. Journal of Experimental

Psychology: Human Perception and Performance, 31,1150–1163.

Reynvoet, B., & Brysbaert, M. (1999). Single-digit andtwo-digit Arabic numerals address the same seman-tic number line. Cognition, 72, 191–201.

Shaki, S., & Petrusic,W.M. (2005). On the mental rep-resentation of negative numbers: Context-dependentSNARC effects with comparative judgments.Psychonomic Bulletin and Review, 12, 931–937.

Sophian, C. (1996). Children’s numbers. Boulder, CO:Westview Press.

Stafylidou, S., & Vosniadou, S. (2004). The develop-ment of students’ understanding of the numericalvalue of fractions. Learning and Instruction, 14,503–518.

Tzelgov, J., &Henik, A. (1991). Suppression situations inpsychological research: Definitions, implications, andapplications. Psychological Bulletin, 109, 524–536.

Verguts, T., & De Moor, W. (2005). Two-digitcomparison: Decomposed, holistic, or hybrid?Experimental Psychology, 52, 195–200.

Whalen, J., Gallistel, C. R., & Gelman, R. (1999).Nonverbal counting in humans: The psychophysicsof number representation. Psychological Science, 10,130–137.

Zhang, J., & Wang, H. (2005). The effect of externalrepresentations on numeric tasks. Quarterly Journal

of Experimental Psychology 58A, 817–838.Zhou, X., Chen, C., Chen, L., & Dong, R. (2008).

Holistic or compositional representation of two-digit numbers? Evidence from the distance, magni-tude, and SNARC effects in a number-matchingtask. Cognition, 106, 1525–1536.




APPENDIX

List of experimental pairs of fractions

The median RT and the error rate by type of fraction and by experiment

a/x_b/x fractions x/a_x/b fractions

Experiment 1 Experiment 2 Experiment 1 Experiment 2

Stimulus RT Err. RT Err. Stimulus RT Err. RT Err.

3/7 5/7 1,215 1.5 1,302 3.3 2/5 2/3 1,206 5.9 1,403 6.7

3/8 7/8 1,146 4.4 1,339 0.0 2/7 2/3 1,215 2.9 1,422 5.0

5/7 4/7 1,261 2.9 1,459 10.0 3/4 3/5 1,357 10.3 1,387 15.0

7/9 4/9 1,255 2.9 1,378 13.3 3/4 3/7 1,172 2.9 1,170 0.0

4/9 8/9 1,169 5.9 1,282 1.7 3/8 3/4 1,219 5.9 1,378 6.7

5/9 7/9 1,176 7.4 1,491 20.0 4/7 4/5 1,260 5.9 1,442 1.7

8/9 7/9 1,094 2.9 1,258 1.7 5/7 5/8 1,411 19.1 1,683 30.0

13/14 11/14 1,275 4.4 1,339 3.3 10/11 10/13 1,341 5.9 1,514 3.3

11/19 17/19 1,194 1.5 1,403 1.7 10/17 10/11 1,413 7.4 1,514 5.0

12/17 13/17 1,238 7.4 1,611 10.0 11/13 11/12 1,156 4.4 1,419 8.3

14/17 12/17 1,389 2.9 1,458 1.7 11/12 11/14 1,344 11.8 1,477 0.0

15/17 12/17 1,225 1.5 1,319 1.7 11/12 11/15 1,249 4.4 1,298 10.0

12/17 16/17 1,083 0.0 1,322 3.3 11/16 11/12 1,212 5.9 1,562 13.3

12/19 17/19 1,099 1.5 1,466 0.0 11/17 11/12 1,320 5.9 1,411 5.0

18/19 12/19 889 0.0 1,259 1.7 11/12 11/18 1,226 7.4 1,342 1.7

14/15 13/15 1,152 1.5 1,354 10.0 11/13 11/14 1,342 7.4 1,506 20.0

13/16 15/16 1,079 4.4 1,386 1.7 11/15 11/13 1,240 2.9 1,590 6.7

13/17 16/17 1,194 7.4 1,334 1.7 11/16 11/13 1,256 10.3 1,523 10.0

17/19 13/19 1,126 4.4 1,409 5.0 12/13 12/17 1,328 4.4 1,439 0.0

18/19 13/19 1,094 0.0 1,275 3.3 11/13 11/18 1,218 4.4 1,466 13.3

14/17 15/17 1,174 2.9 1,507 10.0 13/15 13/14 1,329 5.9 1,587 10.0

14/17 16/17 1,194 1.5 1,503 3.3 13/16 13/14 1,329 1.5 1,603 3.3

17/19 14/19 1,208 5.9 1,414 3.3 13/14 13/17 1,198 1.5 1,490 6.7

18/19 14/19 1,045 1.5 1,214 0.0 13/14 13/18 1,323 7.4 1,510 3.3

15/17 16/17 1,187 0.0 1,387 6.7 13/16 13/15 1,409 7.4 1,586 6.7

15/19 17/19 1,186 2.9 1,427 16.7 14/17 14/15 1,359 5.9 1,558 0.0

18/19 15/19 1,045 5.9 1,335 0.0 13/15 13/18 1,279 4.4 1,673 13.3

17/19 16/19 1,339 10.3 1,595 15.0 15/16 15/17 1,264 2.9 1,663 6.7

16/19 18/19 1,144 4.4 1,491 5.0 13/18 13/16 1,560 10.3 1,663 15.0

17/19 18/19 1,153 4.4 1,270 6.7 13/18 13/17 1,396 22.1 2,050 18.3

9/11 3/11 1,127 2.9 1,242 1.7 2/3 2/9 1,196 1.5 1,395 3.3

15/17 11/17 1,088 0.0 1,298 0.0 7/11 7/15 1,467 14.7 1,666 21.7

Note: Experiment 1: without fillers. Experiment 2: with fillers. RT ¼ response time (in ms). Err. ¼ error rate (in percentages).




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

Documents

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