Elsevier

Cognition

Volume 119, Issue 1, April 2011, Pages 114-119
Cognition

A working memory account for spatial–numerical associations

https://doi.org/10.1016/j.cognition.2010.12.013Get rights and content

Abstract

Several psychophysical and neuropsychological investigations have suggested that the mental representation of numbers takes the form of a number line along which magnitude is positioned in ascending order according to our reading habits. A longstanding debate is whether this spatial frame is triggered automatically as intrinsic part of the number semantics or whether it constitutes a short-term representation constructed during task execution. Although several observations clearly favor the working memory account, its causal involvement has not yet been demonstrated. In two experiments we show that information stored in working memory get spatially coded in function of its ordinal position in the sequence and that the spatial–numerical associations typically observed in number categorization tasks draw upon this mechanism.

Introduction

It is generally accepted that there is a tight link between numbers and space. This idea is fed by reports of mathematicians who describe the use of visuo-spatial imagery during mathematical reasoning (e.g. Fitzgerald & James, 2007). The fact that many people experience vivid mental number lines when imagining numbers suggests that such spatial representations are not confined to complex mathematical reasoning but also constitute the basis of basic number representations (Galton, 1880, Seron et al., 1992). From systematic behavioral studies that go beyond introspection, there is clear evidence that spatial processing is involved even during basic number processing tasks like number comparison or parity judgment. The most characteristic and robust finding is that in such tasks small numbers are preferentially responded to with the left hand and large numbers with the right hand. This Spatial Numerical Association of Response Codes (SNARC; Dehaene, Bossini, & Giraux, 1993) effect has been observed under diverse manipulations of stimulus, task and design parameters (for a review, see Fias and Fischer (2005)).

The classic conceptualization of the relation between numbers and space is that of the mental number line. According to this idea, semantic memory of numerical value takes the form of a horizontally oriented number line along which magnitude is organized from left-to-right or from right-to-left, depending on culturally determined reading habits (Dehaene et al., 1993). Some authors argue that the coordinate system defining this number line is based on the same metric as the coordinate system used to represent physical space (Umilta et al., 2009, Zorzi et al., 2002).

From such a conceptualization of semantic memory of number meaning, a stable association between a given number and a spatial association is expected. However, a number of reports show that the associations between numbers and space are more flexible than one would expect from a long-term memory representation. First, it has been shown that the SNARC effect is range-dependent (Dehaene et al., 1993, Fias et al., 1996). Numbers 4 and 5 elicit faster left than right responses when the numbers ranged from 4 to 9, but elicit faster right than left responses when numbers ranged from 1 to 5. In a similar vein, Ben Nathan, Shaki, Salti, and Algom (2009) demonstrated that the SNARC effect is determined by the relative instead of absolute numerical magnitude by changing the referent for the execution of a number comparison task from trial to trial. Second, Bachtold, Baumuller, and Brugger (1998) extended this idea of context-dependency to mental imagery and showed that asking subjects to imagine the numbers on a clock face reversed the spatial congruency effects that are typically observed. Third, it has been shown that Russian–Hebrew bilinguals associate small numbers with left and right numbers with large when they had read a Russian text shortly before, but showed a reversed SNARC effect when having read a Hebrew text before (Shaki & Fischer, 2008). Similarly, Fischer, Mills, and Shaki (2010) found that the SNARC effect changed after reading of several short texts where small numbers were spatially located on the right and large numbers on the left of the line in English and in Hebrew. Forth, Lindemann, Abolafia, Pratt, and Bekkering (2008) demonstrated that when the SNARC effect was measured after manipulating the sequential coding strategies by means of a concomitant working memory task, the size of the SNARC effect was modulated accordingly.

These observations might indicate that the spatial coding is not inherently associated to number but that it is constructed during task execution (see also Fischer, 2006, Fischer and Shaki, 2010; but see Treccani and Umilta (2010)), suggesting a crucial role of working memory (WM). This hypothesis led recently to a series of studies that showed that the SNARC effect disappeared under WM load (Herrera et al., 2008, van Dijck et al., 2009), suggesting that the availability of WM resources is necessary for the spatial coding of numbers to occur. We reasoned that a plausible mechanism would be that it is the position in a sequence of numbers in WM that determines the association between numbers and space, with items towards the beginning of the sequence associated with left and towards the end with right. The dilution of the SNARC effect when no WM slots are available to temporally represent the numbers used in the task, is consistent with this hypothesis, but is by itself not sufficient to conclude that the long-term semantic memory for number is not what determines the SNARC effect.

To distinguish between position in working memory and long-term memory associations as determinants of the SNARC effect we designed two experiments. In the first experiment, subjects were instructed to perform a parity judgment task on digits that subjects were maintaining in random order in WM. If the spatial coding of numbers has its origin in a long-term representation, a regular SNARC effect is expected. If on the contrary, the SNARC effect is the result of positional coding in WM, numbers should be spatially coded according to their position in the WM sequence irrespective of their numerical magnitude. Next, we reasoned that, if position in WM drives the SNARC effect, associations between WM position and spatially lateralized responses should occur for whatever stimuli maintained in WM. Moreover, this non-numerical effect should correlate with the numerical SNARC effect obtained from the same subjects using the traditional procedure for measuring the SNARC effect. This was tested in Experiment 2.

Section snippets

Experiment 1

To pit the long-term representation account directly against the WM account, subjects were instructed to perform a parity judgment task during the maintenance interval of a verbal WM task in which randomly ordered sequences of digits had to be remembered in correct order.

Experiment 2

The results of the Experiment 1 are clearly in line with the WM account for the SNARC effect and suggest that the spatial–numerical associations typically observed with numbers have their origin in the positional coding in WM. A further prediction is that such spatial–positional associations are not limited to numerical information, but can also observed with any type of information stored in WM. To test this prediction, we repeated the design of Experiment 1 with digits replaced by words not

General discussion

In two experiments we demonstrated that position in working memory (WM) is associated to space: Items from the beginning of a verbal WM sequence are responded to faster and more accurately with the left than with the right hand during the maintenance interval and the opposite is true for items towards the end of the sequence. In Experiment 1 the position-space associations were observed with number stimuli, without any differential effect of numerical value on left or right responses. In

Acknowledgements

This work was supported by the Belgian Science Policy, Interuniversity Attraction Poles program (P6/29).

References (32)

  • S. Shaki et al.

    Reading space into numbers – a cross-linguistic comparison of the SNARC effect

    Cognition

    (2008)
  • J.-P. van Dijck et al.

    Numbers are associated with different types of spatial information depending on the task

    Cognition

    (2009)
  • F. Van Opstal et al.

    The neural representation of extensively trained ordered sequences

    Neuroimage

    (2009)
  • V. Walsh

    A theory of magnitude: common cortical metrics of time, space and quantity

    Trends in Cognitive Sciences

    (2003)
  • M. Ben Nathan et al.

    Numbers and space: Associations and dissociations

    Psychon Bulletin & Review

    (2009)
  • S. Dehaene et al.

    The mental representation of parity and number magnitude

    Journal of Experimental Psychology-General

    (1993)
  • Cited by (0)

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