On the Mental Representation of Negative Numbers: Context-Dependent SNARC Effects with Comparative Judgments

Article excerpt

In one condition, positive and negative number pairs were compared in separate blocks of trials. In another condition, the positive and the negative number pairs were intermixed. In the intermixed condition, comparisons involving negative numbers were faster with the left hand than with the right, and comparisons were faster with the right hand than with the left hand with the positive numbers; that is, a spatial numerical association of response codes (SNARC) effect was obtained, in which the mental number line was extended leftward with the negative numbers. On the other hand, in the blocked condition, a reverse SNARC effect was obtained with the negative numbers; that is, negative number pairs have the same underlying spatial representation as the positive numbers in this context. Nongraded semantic congruity effects, obtained in both the blocked and the intermixed conditions, are consistent with the idea that magnitude information is extracted prior to the generation of discrete semantic codes.

At the outset, in the teaching of number concepts to elementary school children, numbers are depicted in spatial terms. For example, Del Grande, Jones, Lowe, and Morrow (1974), in their text for the sixth grade, introduce integers in terms of walks. As such, integers represent both length and direction. They write the following:

Here is Larry on a sidewalk. The sidewalk is marked off at regular intervals. If Larry walks 3 spaces to the right, we call it a +3 walk (positive 3). If he walks 4 spaces to the left, we call it a -4 walk (negative 4). (p. 60)

Although the concept of integers as points on the number line is the basis for their mathematical representation, is there evidence for a mental representation of integers in terms of length and direction-that is, a mental number line!

Indeed, two lines of evidence, taken together, suggest that the mental representation of numbers is in terms of extents. First, following on Moyer and Landauer's (1967) seminal work, a split, or distance, effect is now a hallmark of comparative judgments with numbers (and symbolic comparison generally): The larger the difference between the numbers compared, the shorter the response time (RT). second, depending on the range of numbers used, a magnitude effect can be obtained: With split held constant at, say, one, as the numbers compared become larger, RTs increase, reflecting a Weber-law-like property. The distance effect and the magnitude effect, taken together, have led researchers (e.g., Dehaene, 1992) to postulate a compressive analogue mental representation for numbers; that is, numbers are represented in terms of length.

The most striking and influential demonstration of a spatial basis for the mental representation of a number line, with numbers arranged from left to right, was obtained by Dehaene, Bossini, and Giraux (1993). Dehaene et al. required their participants to classify the digits from zero to nine as odd or even (i.e., to make parity judgments). They found that parity judgments with the relatively small numbers in the set were faster with the left hand than with the right hand. However, when the numbers were relatively large, the participants responded more quickly with their right hand. The authors argued that since large numbers are represented to the right on the number line, they preferentially elicit a rightward response. Similarly, since small numbers are represented to the left on the number line, they become more readily associated with the left hand than with the right hand. Capturing the association between the spatial component of the mental representation of numbers and the hand of the response, Dehaene et al. labeled their finding as the spatial numerical association of response codes (SNARC) effect.

A clear extension of the SNARC effect (and the teaching of arithmetic) suggests that the number line extends leftward to include the negative numbers (see Fischer, 2003a). On the other hand, it is possible that negative numbers do not have a psychological reality. …