Elsevier

Brain and Language

Volume 92, Issue 3, March 2005, Pages 262-277
Brain and Language

Language effects in magnitude comparison: Small, but not irrelevant

https://doi.org/10.1016/j.bandl.2004.06.107Get rights and content

Abstract

It is assumed that number magnitude comparison is performed by assessing magnitude representation on a single analog mental number line. However, we have observed a unit–decade–compatibility effect in German which is inconsistent with this assumption (Nuerk, Weger, & Willmes, 2001). Incompatible magnitude comparisons in which decade and unit comparisons lead to different responses (e.g., 37_52 for which 3 < 5, but 7 > 2) are slower and less accurately responded to than compatible trials in which decade and unit comparisons lead to the same response (e.g., 42_57, for which 4 < 5 and 2 < 7). As overall distance was held constant, a single holistic magnitude representation could not account for this compatibility effect. However, because of the inversion property of the corresponding German two-digit number words (“einundzwanzig” <one-and-twenty>), the language-generality of the effect is questionable. We have therefore examined the compatibility effect with native English speakers. We were able to replicate the compatibility effect using Arabic notation. Thus, the compatibility effect is not language-specific. However, in cross-linguistic analyses language-specific modulations were observed not only for number words but also for Arabic numbers. The constraints imposed on current models by the verbal mediation of Arabic number comparison are discussed.

Introduction

Virtually every model of number processing assumes some sort of magnitude representation (Campbell, 1994; Dehaene, 1992; Dehaene & Cohen, 1995; McCloskey, 1992; Noel & Seron, 1993). However, the nature of this magnitude representation remains controversial. Dehaene et al.’s suggested the metaphor of a mental number line (Dehaene, 1989; Dehaene, Bossini, & Giraux, 1993; Dehaene, Dupoux, & Mehler, 1990; see Moyer & Landauer, 1967; Restle, 1970, for early suggestions) which is thought to be logarithmically compressed. Logarithmic compression means that the distance between two numbers corresponds to their logarithmic value rather than their linear numerical value (see also Brysbaert, 1995; Dehaene, 2001). Thus, the distance between the numbers 10 and 20 (101 and 2 × 101) on this logarithmically compressed number line would be the same as the distance between the numbers 100 and 200 (102 and 2 × 102). In contrast to the holistic view, McCloskey (1992) contended that the representation of magnitude is arranged into a place-value system consisting of the powers of 10 much like the Arabic number system itself with (abstract) powers of ten and symbolic unit values.

Major evidence for the analog view comes from Dehaene et al.’s (1990) two-digit magnitude comparison experiments using the distance effect. The distance effect refers to the fact that for one-digit numbers number comparison is faster for large distances (e.g., 5 and 9) than for small distances (e.g., 5 and 6). In Dehaene’s study, the comparison of 51 with 65 was faster than that of 59 with 65 (subsequently abbreviated as 51_65 and 59_65, respectively). If participants had been able to focus on the decade number only, the above two comparisons should not differ because the decade digits were the same. Instead, when Dehaene et al. fitted the data over the whole range of numbers (see also Dehaene, 1989), overall logarithmic distance seemed to account best for the data. Thus, in a magnitude comparison it is not the linear absolute numerical distance which best explained performance but the logarithmically compressed distance. Because the logarithmic compression has stronger effects on larger distances, the RT differences for relatively larger numerical distances (e.g., 31_65 vs. 33_65) are much smaller than for relatively smaller numerical distances (e.g., 61_65 vs. 63_65; see Dehaene, 1989; Dehaene et al., 1990, for logarithmically compressed RT curves). Dehaene et al. concluded that magnitude is represented on an analog logarithmically compressed mental number line without special reference to tens and units.

However, we have published data which contrast with a simple holistic view (Nuerk et al., 2001, Nuerk et al., 2002). The manipulation we have used to investigate decade effects is unit–decade–compatibility (for short also coined compatibility). In a two-digit number comparison task (e.g., 42_57) a given number comparison is defined as unit–decade–compatible if both comparisons between tens and units lead to the same decision (e.g., for 42_57, both 4  <  5 and 2  <  7). A number comparison is defined as unit–decade–incompatible, if the two comparisons for units and tens lead to different decisions (e.g., for 47_62; 4  <  6, but 7  >  2). In both above comparisons the absolute overall distance is 15. Logarithmic distance was also the same for compatible and incompatible comparisons. Therefore, there is no substantial difference between compatible and incompatible trials with regard to the holistic representations of the involved numbers. Thus, according to the holistic view, no compatibility effect should be observable.

If, however, one assumes separate additional mental number line representations for tens and units (cf. McCloskey, 1992), tens and units may be separately processed in two-digit number comparison. If this is the case, unit–decade–compatibility effects may play a role (Nuerk et al., 2001, Nuerk et al., 2002; Nuerk, Weger, & Willmes, 2004; Nuerk & Willmes, in press; for an elaboration of that view). We have consistently found compatibility effects for the Arabic notation in German language (Nuerk et al., 2001, Nuerk et al., 2004): incompatible number pairs were responded to more slowly than compatible number pairs in both a participant-based and an item-based ANOVA. We concluded that tens and units were to some extent separately processed in a two-digit number comparison task.

One possible interpretation is that the compatibility effect is a response interference effect due to a possible response conflict between the responses triggered by the decade digit and unit digit comparisons. Thus one could presume that the compatibility effect does not rely on the semantic magnitude representations of tens and units themselves. The compatibility effect would then be a general interference effect not specific to numerical representations.

However, this response interference account is not consistent with the data. Nuerk et al. had manipulated the absolute distance of units in addition to compatibility. The absolute unit distance of the two numbers could be large, e.g., an absolute unit distance of seven in the trials 31_48 (8−1 = 7, compatible) and 38_51 (1−8 = −7, incompatible). In another orthogonally manipulated subset of trials, the absolute units distance could be small, 2, e.g., an absolute unit distance of one in trials 42_63 (3−2 = 1, compatible) and 43_62 (2−3 = −1, incompatible). We found an interaction between absolute unit distance and compatibility. The compatibility effect was larger for a large unit distance. A pure response interference account is not consistent with this interaction because the general response conflict is the same for all incompatible trials. Instead, the semantic magnitude of unit digits influences the compatibility effect indicating that this effect relies at least partially on an access to unit digit magnitude on the mental number line.

Another possible explanation is that the compatibility effect is due to perceptual specifics of the stimulus arrangement. In the original experiment, the Arabic numbers were presented in a column-wise decade–decade, unit–unit configuration, which might have facilitated comparisons of the two unit digits in addition to decade digit comparison. However, in the compatibility effect was replicated in a control experiment in which the two Arabic numbers were not presented above each other (Nuerk et al., 2004). Thus, the compatibility effect is not due to perceptual characteristics of the original experiment.

To summarize, the compatibility effect presents a challenge for the holistic model of two-digit magnitude representation (cf. Dehaene et al., 1990). It rather lends support to the notion of decomposed processing of tens and units (cf. McCloskey, 1992; Nuerk et al., 2001). A simple perceptual or response interference account cannot fully explain the compatibility effect and its interaction with unit distance.

One alternative explanation is that the compatibility effect is a language-specific effect: in German, the corresponding number words for all two-digit numbers between 21 and 98 are inverted (“einundzwanzig” = <one-and-twenty>). This specific property of verbal number representation starting with the irrelevant unit could be responsible for the compatibility effect. In languages in which the verbal representations of numbers in the range [21–98] start with the relevant decade number word (“twenty-one”), the interference of unit and decade may simply disappear. Such language specificities have been shown to affect number processing performance in other domains (e.g., Brysbaert, Fias, & Noel, 1998). It is therefore possible that the compatibility effect in the magnitude comparison task is either driven or at least modulated by language properties. Three Hypotheses can be distinguished:

  • 1.

    Strong language-specificity Hypothesis: the compatibility effect is a language-specific effect. It is therefore observed in German (with its inverted two-digit number words), but not in English. Any conclusions about the properties of a magnitude representation based on the compatibility effect would not be language-specific.

  • 2.

    Strong language-independency Hypothesis: the compatibility effect is a translingual effect. It is not influenced by language in any way: the data for English participants mirror those observed for German participants. In this case, conclusions about the properties of magnitude representation may be applied generally.

  • 3.

    Weak language-specificity Hypothesis: the compatibility effect is modulated but not fully determined by language. This Hypothesis assumes that Hypotheses 1 and 2 are both valid to a certain extent. The compatibility effect may also appear in English (thus showing translingual properties), but the inversion property of the German two-digit number words may enhance the role of units (and thus the compatibility effect):

  • (a)

    This language modulation may be a pure encoding effect. In this case, language modulation should be found for number words, which differ between English and German, but not for Arabic numbers (because Arabic numbers are identical in German and English).

  • (b)

    This language modulation is located later in number processing at a representational level. In this case, language modulation should also be found for Arabic number comparison.

To elaborate on these Hypotheses, we will now review language-specific effects in other domains of number processing. Several studies indicate that difficulties arise when the sequence of tens and units in Arabic and verbal notation is inverted. In English, this is only true for teen numbers (‘14’  ‘fourteen’; the unit is in the second position for Arabic notation, but in the first one for verbal notation). In Chinese, no such inversion (and no special names for ‘11’ and ‘12’) exists. Teen numbers are simply coded as ‘ten-one’ (11), ten-two (12), ten-three (13), and so on (Butterworth, 1999). Taiwanese (Chinese-speaking) children perform better in number processing tasks than English children, but particularly so when they get to the teens (Miller & Stigler, 1987). When shopping is simulated Taiwanese children are particularly better in calculating when things cost 11 (‘ten-one’ rather than ‘eleven’) pence, if they possess 10 and 1 pence coins (Nunes & Bryant, 1996). Thus, consistent and transparent transcoding properties of the corresponding verbal notation seem to facilitate performance in children.

In German and Dutch, however, not only the teens but all two-digit numbers are inverted. Exactly this inversion has been shown to cause number processing problems in a German brain-damaged patient (Blanken, Dorn, & Sinn, 1997). Moreover, a trilingual patient has been tested for inversion problems (Proios, Weniger, & Willmes, 2002). She had major problems with German two-digit numbers: she transcoded ‘siebenundvierzig’ (‘47’ inverted as ‘seven-and-forty’) into 740. This additive composition error is sometimes made by children when they get to the hundreds (i.e., when three-hundred and twenty-seven becomes 30027; Power & Dal Martello, 1990). However, she had no such problems in Greek in which no inversion exists for two-digit numbers.

Language-specific effects in number processing are not restricted to children and patients: number processing in normal adult persons without brain-damage can also be affected by inversion properties. Brysbaert et al. (1998) have investigated addition tasks in a language with inversion (Dutch) and a language without inversion (French). They observed language-specific results. When the solution of 4 + 21 (in Arabic or verbal notation) had to be named in Dutch <‘five-and-twenty’> participants were faster than for 21 + 4. In contrast, for French <‘twenty-five’> 21 + 4 is named faster than 4 + 21 (see p. 63 in Brysbaert et al., 1998; ten-unit + unit problems without carry-over across decade boundaries). However, in their Experiment 2 in which the required response was to type in the solution on the keyboard (rather than to name the response) the language-specific effect disappeared for the above type of addition problem. Nevertheless, a language-specific effect prevailed in tasks of the type ‘twenty’ + ‘four’ in verbal notation. In French, ‘twenty’ + ‘four’ was responded to more quickly than ‘four’ + ‘twenty’ while in Dutch there was no such order effect. In Arabic notation, no such difference between languages was obtained for manual responses. In summary, this study shows moderate language-specific effects of the inversion property in the addition task which seem to depend on output, problem type and notation (i.e., Arabic or verbal notation).

The aim of this study was to investigate language-specific alterations of the compatibility effect respectively its modulation by notation (see Hypotheses 3a and b). Above, we have introduced the compatibility effect and its interaction with unit distance for Arabic notation in German participants. Since we also wish to study verbal notation in this study, we will now briefly consider on the compatibility effect for verbal notation in German language.

Compatibility has also played a role for the verbal notation: however, we did not observe a compatibility main effect but only an interaction with unit distance which was the same as for Arabic notation: large unit distance incompatibility tended to lead to greater deterioration in performance (Nuerk et al., 2002). However, for small incompatibilities we have observed a reversed effect: compatible magnitude comparisons were slower than incompatible ones. The reason for this finding may be simple. Overall distance was matched between compatible and incompatible stimulus groups. When computing the overall distance for incompatible trials the absolute unit distance (see above) must be subtracted from the decade distance; e.g., for 47_62, the decade distance is 6  4 = 2 and the unit distance is 7  2 = 5. The overall distance 15 is computed as 2 × 101  5 × 100 because the larger two-digit number has the smaller unit digit. When computing the overall distance for compatible trials the absolute unit distance (see above) must be added to the decade distance; e.g., for 42_57, the decade distance is 5  4 = 1 and the unit distance is 7  2 = 5. The overall distance 15 is computed as 1 × 101 + 5 × 100 because the larger number also has the larger unit digit. The constant overall distance is composed of 10 times decade distance and units distance. Because unit distance is always subtracted in incompatible trials, the decade distance must necessarily be larger for those incompatible trials (see Nuerk et al., 2002, for a detailed discussion and mathematical elaboration of this issue). So, if the German participants had only compared the decades they should have been faster in incompatible trials due to a common decade digit distance effect. This was not the case for large unit distance incompatibility. Only for a comparison of number words with small unit distance incompatibility was this decade effect not overcome.

Section snippets

Experiment: Number comparison for Arabic digits and English number words for native English speakers

In this experiment, Hypotheses 1 and 2 were tested for Arabic numbers and for English number words. If the compatibility effect is language-specific, no compatibility effect should be observed for a language such as English which has no inversion property for the stimulus numbers between 21 and 98. If the compatibility effect is translingual, the same results should be observed as in German: a main compatibility effect for Arabic numbers and an interaction of compatibility with unit distance

Joint analyses of native English and German speakers

The study with English speakers yielded qualitatively very similar effects for Arabic numbers, but very diverging effects for number words. However, some quantitative differences for Arabic numbers were also present. For example, the compatibility main effect was 31 ms for native German speakers, but only 18 ms for native English speakers. Some magnitude measures, for example problem size, tended to be predictors logarithmically in English, but absolutely in German. To examine common and

General discussion

The general discussion focuses on two points. First, we discuss how the attributes of different languages modulate the influence of tens and units in different notations and the consequences of such a modulation. Second, we evaluate current number processing models in two respects: (i) their potential to account for the unit–decade–compatibility effect, (ii) their predictions with regard to language-specific influences in the number comparison task we found even for Arabic notation.

Conclusions

The conclusions of this study are straightforward. The compatibility effect for Arabic numbers is not language-specific: The inversion property in German (<one-and-twenty>) is not responsible for the compatibility effect. The reliability of the unit–decade–compatibility effect across languages is not consistent with the notion of one single mental number line for all two-digit numbers. Instead, in our opinion, the results strengthen the conclusion of Nuerk et al., 2001, Nuerk et al., 2002 that

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    This research was supported by a grant of the Interdisciplinary Group for Clinical Research—CNS of the RWTH Aachen and a grant by the DFG (German Research Society, KFO 112-TP2) to Klaus Willmes supporting Hans-Christoph Nuerk. We are grateful to Stuart Fellows for his help with checking the English grammar and style of the paper.

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