Elsevier

Cognition

Volume 106, Issue 3, March 2008, Pages 1525-1536
Cognition

Brief article
Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task,☆☆

https://doi.org/10.1016/j.cognition.2007.06.003Get rights and content

Abstract

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.

Introduction

One of the most important issues in the area of numerical processing is how numbers are represented in the memory. Up to now, three effects have been extensively investigated to reveal the nature of mental representation of numbers: the distance effect, the problem-size or magnitude effect, and the SNARC (Spatial-Numerical Association of Response Codes) effect (Brysbaert, 1995, Dehaene et al., 1993, Dehaene et al., 1990, Fias et al., 1996). The distance effect (e.g., Moyer & Landauer, 1967) is revealed when subjects take longer and/or make more errors to process (e.g., to compare) two numbers close to each other in magnitude (e.g., 5 vs. 6) than to process two numbers farther away from each other (e.g., 3 vs. 8). The problem-size or magnitude effect means that subjects have more difficulty processing larger numbers than smaller ones (e.g., Brysbaert, 1995). Because these two effects rely on the magnitude or semantic representation of numbers, they have been used as an index of semantic processing of numbers. That is, their existence in a particular task would be considered as evidence of semantic processing, whereas their absence has been interpreted as a lack of semantic processing (Zhou et al., in press).

In contrast with the distance and problem-size effects, the SNARC effect reveals not only semantic processing, but also the shape of mental representation of numbers. The SNARC effect is revealed when subjects respond more quickly to smaller numbers with the left hand, and more quickly to larger numbers with the right hand on a number processing task (e.g., parity-judgment – whether a number is odd or even) (Dehaene et al., 1993). The SNARC effect suggests that numbers are stored in some type of a mental number line with smaller numbers on the left side and larger numbers on the right side (e.g., Dehaene et al., 1993, Restle, 1970, Seron et al., 1992, Zorzi et al., 2002).

Most studies of semantic or spatial representation of numbers have focused on one-digit numbers. Recently, researchers have debated about the nature of mental representation of two-digit numbers (see a review by Nuerk & Willmes, 2005). There are two representative models. The first model is the holistic model: That is, two-digit numbers are represented holistically (Brysbaert, 1995, Dehaene et al., 1990, Reynvoet and Brysbaert, 1999). For example, Reynvoet and Brysbaert (1999) found that the priming effect is constant for a given numerical distance regardless of whether the priming number is single- or two-digit number. For example, the priming effect of 7 on the target 9 is the same as the priming effect of 11 on the target 9, suggesting that 11 is indeed as close to 9 as 7 is on the mental number line. If 11 had been represented compositionally, either the decade or the unit digit (1) would reduce its priming effect because of its large distance from 9.

The second model is the compositional model in which the decade and unit digits of two-digit numbers are represented separately, perhaps with an additional holistic representation (Nuerk, Weger, & Willmes, 2001). The key evidence in support of this model comes from the congruence effect during the comparison of two-digit numbers (Nuerk et al., 2001). A number pair is defined as congruent or compatible when both the decade and unit digits of one number are greater than those of the other number (e.g., 67 and 52). The incongruent pairs have one number with a greater decade digit and the other number with a greater unit digit (e.g., 62 and 47). The congruence effect occurs when it is more difficult for subjects to respond to incongruent number pairs than to congruent pairs after controlling for overall numerical distance and size. The existence of the congruence effect suggests that subjects pay attention to the value of the compositions, not just the whole number. Further support for the compositional model came from two recent studies (Verguts and De Moor, 2005, Zhang and Wang, 2005). Verguts and De Moor (2005) found that the numerical distance effect was evident for unit digits when the decade digits were the same (e.g., 54 vs. 55, 51 vs. 59). When the decade digits differed by 1 (e.g., 59 vs. 61, 57 vs. 64), which was the only other condition, there was no distance effect for the whole numbers. Finally, Zhou et al. (in press) found that reaction times in a number-comparison task were a function of both the whole numbers and the unit digits.

Given the conflicting evidence as reviewed above, it remains unresolved whether two-digit numbers are processed holistically or compositionally (Fias and Fischer, 2005, Gevers and Lammertyn, 2005). The present study aimed to clarify the nature of mental representation of two-digit numbers by making three improvements on the previous research. First, most previous studies used either number-comparison (Dehaene et al., 1990, Nuerk et al., 2001, Verguts and De Moor, 2005, Zhang and Wang, 2005) or parity-judgment tasks (Dehaene, 1993) to investigate the nature of representation of two-digit numbers. Such tasks focus the subjects’ attention on the compositions, rather than the entirety, of the two-digit numbers. In a parity-judgment task, subjects would only need to focus their attention on the unit digits. In a number-comparison task, subjects are likely to pay most attention to the decade digits. Only when the decade digits were the same, do the subjects pay attention to the unit digits. In our study, to ensure that subjects pay attention to both decade and unit digits, we used a number-matching paradigm. When the two two-digit numbers were matched or the same (e.g., 12 vs. 12), subjects had to match both decade and unit digits.

The second improvement of our study over the previous research is that we examined several effects. In addition to the distance and magnitude effects examined in previous research, we also examined the SNARC effect. Researchers have demonstrated the SNARC effect for two-digit numbers, but have not used it to investigate whether the numbers are holistically or compositionally represented. For example, Dehaene et al. (1990) asked their participants to press a left or right key to indicate whether visually presented two-digit numbers (from 31 to 99, excluding 65) were smaller or larger than a fixed reference number (65). They found a significant SNARC effect. That is, if the target number was smaller than 65, it was faster for subjects to press the left key than the right key. Conversely, when the target number was larger than 65, it was to press the right key than the left key. These results were also confirmed (although only at a trend level) with a parity-judgment task (Dehaene et al., 1993) and a simultaneously-presented, number-comparison task (Brysbaert, 1995). Brysbaert (1995) asked subjects to compare two simultaneously presented two-digit numbers and found that subjects responded more quickly when the smaller number of the number pair was on the left side and the larger one on the right side than when it was the other way around.

Although these results demonstrated the SNARC effect for two-digit numbers, they did not examine whether two-digit numbers were processed holistically or compositionally. To address that issue, we need to decompose the SNARC effect into those parts that are associated with the whole number, the decade digits, and the unit digits.

Third, we also included the stage of processing as a variable. It is possible that representation of two-digit numbers may depend on the stage of processing.1 In our first experiment, we presented the two two-digit numbers serially to ensure that the subjects processed the first two-digit number and kept it in their short-term memory before they attempted to match it with the second number. In our second experiment, two two-digit numbers were presented simultaneously so that both numbers are processed simultaneously in the working memory.

Regression analyses were used to examine the distance, magnitude, and SNARC effects (Fias et al., 1996). For the distance effect, reaction times for the non-matched numbers were regressed on the numerical distance based on either the decade digits or the unit digits. For the magnitude effect, the matching trials were used to ensure that subjects paid attention to the entire two-digit numbers. Reaction times for these matching trials were regressed on the whole numbers, decade digits, and unit digits. For the SNARC effect, the differences between the left and the right hands were regressed on the whole numbers, decade digits, and unit digits. Again only trials with matched number pairs were used in these analyses to ensure that subjects paid attention to both decade and unit digits and to avoid other potential confounding effects introduced by number pairs of different sizes. Based on the previous research that showed the distance, magnitude, and SNARC effects (see above for references), we expected the distance, magnitude, and SNARC effects to be significant when regressed on the whole numbers. Because decade digits and the whole numbers had very high shared variance (r = .995), decade digits are expected to show the same effects. Thus the critical results involved only the unit digits. According to the compositional model, unit digits would be a significant and unique predictor (after controlling for either the whole numbers or the decade digits) of the three types of number effects, whereas, according to the holistic model, they would not be.

Section snippets

Subjects

Twenty undergraduates (ten males and ten females) from Beijing Normal University were recruited. The average age was 21.5 years, ranging from 18.3 to 23.6 years. All participants had normal or corrected-to-normal eyesight. They did not participate in any experiments with number tasks during the past half a year prior to this study. They gave written informed consent before the experiment. After the experiment, each participant was paid RMB 20 yuan (about US$2.5).

Materials

Seventy-two two-digit numbers

Subjects

Twenty undergraduates (ten males and ten females) from Beijing Normal University were recruited. The average age was 20.1 years, ranging from 18.0 to 21.9 years. The other characteristics for these subjects matched those for the subjects in Experiment 1.

Materials

The materials were the same as those in Experiment 1.

Procedure

The procedure used in present experiment was the same as that used in Experiment 1 with the following exceptions. The number pairs were presented simultaneously for 800 ms in this experiment.

General discussion

The present study aimed to investigate whether two-digit numbers showed holistic or compositional representation. By using a number-matching paradigm, this study found that the mental representation of two-digit numbers depended on the stage of processing. The initial processing seemed to involve the compositions of the two-digit numbers as shown by the significant distance and magnitude effects for unit digits in Experiment 2 (simultaneous presentation). Once the numbers are stored in the

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This manuscript was accepted under the editorship of Jacques Mehler.

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This study was supported by the National 973 Project (2003CB716803) and the National Pandeng Project (95).

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