Dissociation of the Distance Effect and Size Effect in One-Digit Numbers

Article excerpt

Magnitude comparison of single digits is robustly characterized by a distance effect (close numbers are more difficult to compare than numbers further apart) and a size effect (for a given distance, comparison difficulty increases with increasing size). The distance effect indicates access to the mental number line (Dehaene, 1997), and the size effect is usually interpreted as indicating that the mental number line represents larger numbers more vaguely than smaller ones. In contrast, we have argued earlier (Verguts, Fias, & Stevens, 2005) that for symbolic numbers (Arabic or verbal notation), the size effect does not originate from the mental number line but, instead, originates from mappings to relevant output components that are specific for magnitude comparison. If the latter is true, it should be possible to dissociate the distance effect from the size effect in tasks other than magnitude comparison. In two experiments, we observed a robust distance effect in same/different judgments, which implies access to the mental number line. Yet the size effect was absent. Consistent with our prediction, this finding establishes a dissociation between the size effect and the distance effect.

Numbers in Arabic and verbal notations are highly abstract: They bear no relation to their magnitudes whatsoever. Yet comparing two such numbers is subject to a distance effect, meaning that it is more difficult to compare two numbers if they are close (e.g., 1 and 2) than if they are far apart (e.g., 1 and 8). This is obtained both when the numerical magnitude is relevant (e.g., Moyer & Landauer, 1967; Schwarz & Stein, 1998) and when it is irrelevant (Dehaene & Akhavein, 1995). A distance effect is also robustly obtained in priming tasks, in the sense that smaller prime-target distances lead to shorter response times (RTs; e.g., Reynvoet & Brysbaert, 1999). The distance effect suggests that a number's magnitude is accessed immediately and that the number is placed on a mental number line (Dehaene, 1997), after which further processing can proceed.

In addition to a distance effect, magnitude comparison (which number is smaller/larger?) is subject to a size effect: Comparison of two numbers is easier for small than for large numbers (e.g., 1 and 2 vs. 8 and 9). The latter finding has led some authors to posit that the number line represents large numbers more vaguely than small numbers, so that discriminating between larger numbers is more difficult. Different implementations of this general idea have been proposed (e.g., logarithmic compression, Dehaene, 1992; scalar variability, Gallistel & Gelman, 1992; see also recent discussions in Carey, 2001, and Dehaene, 2001). Arithmetical operations (e.g., addition or multiplication) typically also exhibit a size effect, but it is controversial whether this size effect has the same origin as that observed in number comparison (Dehaene, Piazza, Pinel, & Cohen, 2003; Gallistel & Gelman, 1992). In the following, we will therefore ignore size effects in arithmetic.

We (Verguts, Fias, & Stevens, 2005) have recently argued that, at least in the range of one-digit numbers, large numbers (e.g., 8 or 9) are represented as exactly (or as vaguely) as small numbers (e.g., 1 or 2). One argument was that with symbolic numbers'-that is, numbers in Arabic or verbal notation-the size effect appears in magnitude comparison, but not in number naming or parity judgment. A second argument was that, in masked priming studies with number naming and parity judgment (e.g., Reynvoet & Brysbaert, 1999; Reynvoet, Caessens, & Brysbaert, 2002), there is a distance effect between the prime and the target, but no size effect (of either the prime or the target). We argued that the size effect in magnitude comparison originates from mappings (connections) from the number line to the task-relevant output component.

To substantiate this, we trained a neural network on magnitude comparison, number naming, and parity judgment. …

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