- Email: [email protected]
The Future for Snarc Could Be Stark…
VIEWPOINT THE FUTURE FOR SNARC COULD BE STARK… Martin H. Fischer (Psychology Department, University of Dundee, Dundee, Scotland, UK)
ABSTRACT The present report by Wood et al. (2006, this issue) invites us to reconsider what we should believe about the cognitive representation of numbers. Researchers interested in numerical cognition have quickly embraced the idea that systematic spatial performance biases in number-related tasks must reflect an inherent spatial attribute of the underlying cognitive representation of numbers. The association between numbers and space (SNARC – spatial numerical association of response codes – effect) has effectively been used to augment the “mental number line” metaphor. Here I discuss the possibility that the SNARC effect is merely an instance of strategic problem solving. Key words: mental number line, SNARC effect, spatial coding, spatial strategy
The concept of a “mental number line” was initially introduced to capture two robust performance patterns in simple numerical tasks: specifically, in a magnitude classification task it is more difficult to discriminate two numbers when they reflect similar as opposed to different magnitudes (the distance effect), and also when they both reflect larger as opposed to smaller magnitudes (the size effect; Moyer and Landauer, 1967; Restle, 1970). By assuming that numbers are represented in our minds as sequentially ordered entries, with increasing overlap between entries for larger numbers, these two effects can be explained. The more recently discovered spatial numerical association of response codes (SNARC) effect (Dehaene et al., 1993) was interpreted as indicating that small number entries are somehow represented “to the left” of larger number entries, which are in turn “to the right”, and that these relative spatial codes are activated whenever we process number meanings. Thus, the mental number line had been transformed from a metaphorical concept into a spatially oriented line. A corollary of this notion of an environmentcentred spatial representation of numbers is that the spatial association should not be sensitive to manipulations of body posture, such as crossing the hands. This was already demonstrated by Dehaene et al. (1993), who found the same spatial biases with crossed and uncrossed hands when participants classified single digits by parity. In sharp contrast to this, Wood et al. (2006, this issue) now show that responding with crossed hands abolishes a previously present spatial association. Yet, other studies have replicated Dehaene et al.’s (1993) original crossed-hands result (e.g., Fischer and Hill, 2004; Müller and Schwarz, in press), and many Cortex, (2006) 42, 1066-1068
more mere replications were probably just not newsworthy. Scientific progress relies on the valid interpretation of reliable facts. A failure to replicate previously reported findings sometimes draws the attention of researchers to important inconsistencies: the inability to reproduce a basic result can even topple a theory by undermining its empirical foundations. So what is going on with the SNARC effect? Logically, the parity task (and many other numerical tasks) can be performed without having to associate numbers with space at all. This is what the present result of Wood et al. (2006, this issue) actually shows. A striking aspect of their report is that the authors were compelled to infer multiple competing spatial representations to account for the null result. Wood et al. (2006, this issue) postulate that, in addition to the environment-centred link between numbers and response keys, another, hand-centred representation has influenced their participants’ performance. Specifically, the present experimental procedure is believed to have placed these two hypothetical reference frames into misalignment, thus leading to competing spatial associations for any given number that cancel each other out. In the available data there is, however, no evidence for such a conflict: response latencies and error rates were not reliably higher when the crossed-hands data were compared to those of the Nuerk et al. (2005) precursor study, where the two reference frames had been spatially aligned in the same participants. Moreover, the proposed dual coding of numbers now requires an elaborate discussion of why the new, hand-based reference frame was more influential in the present experiment compared to several previous studies (see Müller and Schwarz, in press, for a similar
How STARC is SNARC?
proposal). Clearly, a more economical conclusion would have been that no spatial representations were involved at all in the Wood et al. (2006, this issue) experiment. One reason for the authors’ decision to propose their less economical conflict hypothesis is the fastaccumulating set of observations of systematic spatial performance biases in a diverse range of numerical tasks, all showing a magnitude-related gradient of spatial association (see Fias and Fischer, 2005, or Hubbard et al., 2005, for recent reviews). In the light of this evidence, a spatial representation was expected to be obligatorily involved in the task. A second reason for the authors’ decision to infer multiple spatial representations from a null result is the fact that their present data constitute only the second half of a larger study in which the very same group of participants had just provided clear evidence for their use of the typical spatial mapping of numbers (Nuerk et al., 2005). This information about sequentially administered task instructions provides an important clue, the implications of which have been overlooked: it is possible that presence or absence of an association between numbers and space is the result of an individual’s strategic decision in the light of both recent and current task demands, and not a reflection of their mental representation of numbers. This proposal of a strategic origin of the SNARC effect rests on two observations in the literature. First, the association between numbers and space is surprisingly flexible. Dehaene et al.’s (1993) original report already showed that the same numbers can become associated with left or right space, depending on whether they are the smallest or largest in the range of numbers used (their Experiment 3). This flexibility of the spatial associations for numbers was further highlighted by Bächtold et al. (1998) who reversed the SNARC effect by giving their participants different imagery instructions: numbers were either distances on a ruler or times on a clock face. Related to this, Fischer and Coeman (2005) modified their participants’ SNARC effect (in the vertical dimension) by letting them use either calculator or telephone keyboards for only a few minutes. The second relevant observation is that there are individual differences in the way numbers are associated with space. Again, evidence for this was already provided in Dehaene et al.’s (1993, Experiment 7) plot of the spontaneous spatial associations of a group of twenty participants, some of whom had an inverted SNARC effect. The authors’ interpretation was that the variable direction and strength of the number-to-space mapping reflected two spatially conflicting reading and writing habits (a proposal that is reminiscent of the present dual-process assumption of Wood et al., 2006, this issue). However, this idea of a culturally modulated SNARC effect is in conflict with more
1067
recent results (Ito and Hatta, 2004). Note that Wood et al. (2006, this issue) also point out the considerable inter-individual variability in the extent to which the “standard” spatial mapping of numbers was present in previous work, even among culturally homogeneous sets of participants. Taking these observations about both intra- and inter-individual variability of the SNARC effect seriously, it becomes possible to explain the pattern of findings without reference to a competition between multiple spatial mappings of numbers. Instead of invoking obligatorily activated spatial codes, the SNARC effect (regardless of size and direction) might just be the result of an individual’s strategic decision. In this view, the participants in Wood et al.’s (2006, this issue) study had adopted their initial spatial mapping (small numbers-left hand, large numbers-right hand, as documented in Nuerk et al., 2005) not as a result of the way numbers are represented in the mind, but because of an abundant experience with such mappings in real life. Following the crossing over of their hands in the Wood et al. (2006, this issue) experiment, they abandoned this previously useful mapping because this mapping was now counter-productive. The idea of a strategic association of response codes (SNARC = STARC) can also explain why the average response times in the Wood et al. (2006, this issue) study are not systematically slower than those of the preceding study, despite the alleged interference between spatial codes: there simply were no conflicting spatial codes involved. The fact that participants did not systematically (as a group) adopt any new mappings could be due to a number of reasons, from declining motivation to extensive practice from the previous conditions. The latter hypothesis is in line with the observation that numerically more skilled participants tend to have weaker spatial associations (Dehaene et al., 1993, Experiment 1; Fischer and Rottmann, 2005, Experiment 1). At first glance, the notion of strategic control over number-to-space mappings might appear as resignation in the light of inconsistent findings and recent failures to replicate SNARC-related effects. But taking the STARC hypothesis seriously will lead to important methodological improvements. One such improvement will be to compare, for each individual, the strength of their SNARC effect before and after an experimental manipulation. From this we can learn whether null results truly reflect an absence of spatial associations, or instead constitute an artefact from averaging across differential strategies. Tracking spatial associations across experimental manipulations will also tell us how individuals go about mapping their numbers onto space: those with the strongest environmentcentered SNARC effect might decide either to abandon any spatial mapping (to avoid the remapping cost), or to resort to an equally strong hand-centered spatial reference (because they
1068
Martin H. Fischer
benefit from either spatial strategy). Similarly, those with weak spatial strategies may be unaffected by either mapping. In addition, visuospatial and numerical abilities of participants in future SNARC studies should be assessed because these might constitute important predictors of spatial strategies in general and of the SNARC effect in particular (e.g., Bachot et al., 2005). In a similar vein, controlling for individual participants’ finger counting habits should provide useful evidence to re-evaluate the proposed role for handbased coordinate frames in number-related tasks, as suggested by Wood et al. (2006, this issue) as well as Müller and Schwarz (in press). In addition, hand-based number concepts may also prove to be a viable alternative explanation for the origin of SNARC effects (e.g., Di Luca et al., in press). Clearly, individual-oriented analyses can tell us a lot about the reliability, the validity, and thus the theoretical starkness of the SNARC effect. REFERENCES BACHOT J, GEVERS W, FIAS W and ROEYERS H. Number sense in children with visuo-spatial disabilities: Orientation of the mental number line. Psychology Science, 47: 172-183, 2005. Freely available at http://psychology-science.com/1-2005/ BÄCHTOLD D, BAUMÜLLER M and BRUGGER P. Stimulus-response compatibility in representational space. Neuropsychologia, 36: 731-735, 1998. DEHAENE S, BOSSINI S and GIRAUX, P. The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122: 371-396, 1993.
DI LUCA S, GRANA A, SEMENZA C, SERON X and PESENTI M. Finger-digit compatibility in Arabic numeral processing. The Quarterly Journal of Experimental Psychology (in press). FIAS W and FISCHER MH. Spatial representation of numbers. In Campbell JI (Ed), Handbook of Mathematical Cognition. New York: Psychology Press, 2005. FISCHER MH and COEMAN P. Moving the Mental Number Line: Rapid Effects of Training. Poster at the European Summer School on “Neuroscience of number processing” in Erice, Italy, 3-10 July 2005. FISCHER MH and HILL RL. A SNARC in the Dark: Input Modality Affects Number Representation. Poster at the 22nd Workshop on “Cognitive Neuropsychology” in Bressanone, Italy, 25-30 January 2004. FISCHER MH and ROTTMANN J. Do negative numbers have a place on the mental number line? Psychology Science, 47: 22-32, 2005. Freely available at http://psychology-science.com/12005/ HUBBARD EM, PIAZZA M, PINEL P and DEHAENE S. Interactions between numbers and space in parietal cortex. Nature Reviews Neuroscience, 6: 435-448, 2005. ITO Y and HATTA T. Spatial structure of quantitative representation of numbers: Evidence from the SNARC effect. Memory and Cognition, 32: 662-673, 2004. MOYER RS and LANDAUER TK. Time required for judgments of numerical inequality. Nature, 215: 1519-1520, 1967. MÜLLER D and SCHWARZ W. Is there an internal association of numbers to hands? A hierarchical model of the SNARC effect. Memory and Cognition (in press). NUERK HC, WOOD G and WILLMES K. The universal SNARC effect: The association between number magnitude and space is amodal. Experimental Psychology, 52: 187-194, 2005. RESTLE F. Speed of adding and comparing numbers. Journal of Experimental Psychology, 83: 274-278, 1970. WOOD G, NUERK, HC and WILLMES K. Crossed hands and the SNARC effect: A failure to replicate Dehaene, Bossini and Giraux (1993). Cortex, 42: 1078-1088, 2006.
Martin H. Fischer, Psychology Department, University of Dundee, DD1 4HN, Dundee, Scotland, UK. e-mail: [email protected]
(Received 17 February 2006; accepted 8 March 2006; action editor: Yves Rossetti)
ABSTRACT The present report by Wood et al. (2006, this issue) invites us to reconsider what we should believe about the cognitive representation of numbers. Researchers interested in numerical cognition have quickly embraced the idea that systematic spatial performance biases in number-related tasks must reflect an inherent spatial attribute of the underlying cognitive representation of numbers. The association between numbers and space (SNARC – spatial numerical association of response codes – effect) has effectively been used to augment the “mental number line” metaphor. Here I discuss the possibility that the SNARC effect is merely an instance of strategic problem solving. Key words: mental number line, SNARC effect, spatial coding, spatial strategy
The concept of a “mental number line” was initially introduced to capture two robust performance patterns in simple numerical tasks: specifically, in a magnitude classification task it is more difficult to discriminate two numbers when they reflect similar as opposed to different magnitudes (the distance effect), and also when they both reflect larger as opposed to smaller magnitudes (the size effect; Moyer and Landauer, 1967; Restle, 1970). By assuming that numbers are represented in our minds as sequentially ordered entries, with increasing overlap between entries for larger numbers, these two effects can be explained. The more recently discovered spatial numerical association of response codes (SNARC) effect (Dehaene et al., 1993) was interpreted as indicating that small number entries are somehow represented “to the left” of larger number entries, which are in turn “to the right”, and that these relative spatial codes are activated whenever we process number meanings. Thus, the mental number line had been transformed from a metaphorical concept into a spatially oriented line. A corollary of this notion of an environmentcentred spatial representation of numbers is that the spatial association should not be sensitive to manipulations of body posture, such as crossing the hands. This was already demonstrated by Dehaene et al. (1993), who found the same spatial biases with crossed and uncrossed hands when participants classified single digits by parity. In sharp contrast to this, Wood et al. (2006, this issue) now show that responding with crossed hands abolishes a previously present spatial association. Yet, other studies have replicated Dehaene et al.’s (1993) original crossed-hands result (e.g., Fischer and Hill, 2004; Müller and Schwarz, in press), and many Cortex, (2006) 42, 1066-1068
more mere replications were probably just not newsworthy. Scientific progress relies on the valid interpretation of reliable facts. A failure to replicate previously reported findings sometimes draws the attention of researchers to important inconsistencies: the inability to reproduce a basic result can even topple a theory by undermining its empirical foundations. So what is going on with the SNARC effect? Logically, the parity task (and many other numerical tasks) can be performed without having to associate numbers with space at all. This is what the present result of Wood et al. (2006, this issue) actually shows. A striking aspect of their report is that the authors were compelled to infer multiple competing spatial representations to account for the null result. Wood et al. (2006, this issue) postulate that, in addition to the environment-centred link between numbers and response keys, another, hand-centred representation has influenced their participants’ performance. Specifically, the present experimental procedure is believed to have placed these two hypothetical reference frames into misalignment, thus leading to competing spatial associations for any given number that cancel each other out. In the available data there is, however, no evidence for such a conflict: response latencies and error rates were not reliably higher when the crossed-hands data were compared to those of the Nuerk et al. (2005) precursor study, where the two reference frames had been spatially aligned in the same participants. Moreover, the proposed dual coding of numbers now requires an elaborate discussion of why the new, hand-based reference frame was more influential in the present experiment compared to several previous studies (see Müller and Schwarz, in press, for a similar
How STARC is SNARC?
proposal). Clearly, a more economical conclusion would have been that no spatial representations were involved at all in the Wood et al. (2006, this issue) experiment. One reason for the authors’ decision to propose their less economical conflict hypothesis is the fastaccumulating set of observations of systematic spatial performance biases in a diverse range of numerical tasks, all showing a magnitude-related gradient of spatial association (see Fias and Fischer, 2005, or Hubbard et al., 2005, for recent reviews). In the light of this evidence, a spatial representation was expected to be obligatorily involved in the task. A second reason for the authors’ decision to infer multiple spatial representations from a null result is the fact that their present data constitute only the second half of a larger study in which the very same group of participants had just provided clear evidence for their use of the typical spatial mapping of numbers (Nuerk et al., 2005). This information about sequentially administered task instructions provides an important clue, the implications of which have been overlooked: it is possible that presence or absence of an association between numbers and space is the result of an individual’s strategic decision in the light of both recent and current task demands, and not a reflection of their mental representation of numbers. This proposal of a strategic origin of the SNARC effect rests on two observations in the literature. First, the association between numbers and space is surprisingly flexible. Dehaene et al.’s (1993) original report already showed that the same numbers can become associated with left or right space, depending on whether they are the smallest or largest in the range of numbers used (their Experiment 3). This flexibility of the spatial associations for numbers was further highlighted by Bächtold et al. (1998) who reversed the SNARC effect by giving their participants different imagery instructions: numbers were either distances on a ruler or times on a clock face. Related to this, Fischer and Coeman (2005) modified their participants’ SNARC effect (in the vertical dimension) by letting them use either calculator or telephone keyboards for only a few minutes. The second relevant observation is that there are individual differences in the way numbers are associated with space. Again, evidence for this was already provided in Dehaene et al.’s (1993, Experiment 7) plot of the spontaneous spatial associations of a group of twenty participants, some of whom had an inverted SNARC effect. The authors’ interpretation was that the variable direction and strength of the number-to-space mapping reflected two spatially conflicting reading and writing habits (a proposal that is reminiscent of the present dual-process assumption of Wood et al., 2006, this issue). However, this idea of a culturally modulated SNARC effect is in conflict with more
1067
recent results (Ito and Hatta, 2004). Note that Wood et al. (2006, this issue) also point out the considerable inter-individual variability in the extent to which the “standard” spatial mapping of numbers was present in previous work, even among culturally homogeneous sets of participants. Taking these observations about both intra- and inter-individual variability of the SNARC effect seriously, it becomes possible to explain the pattern of findings without reference to a competition between multiple spatial mappings of numbers. Instead of invoking obligatorily activated spatial codes, the SNARC effect (regardless of size and direction) might just be the result of an individual’s strategic decision. In this view, the participants in Wood et al.’s (2006, this issue) study had adopted their initial spatial mapping (small numbers-left hand, large numbers-right hand, as documented in Nuerk et al., 2005) not as a result of the way numbers are represented in the mind, but because of an abundant experience with such mappings in real life. Following the crossing over of their hands in the Wood et al. (2006, this issue) experiment, they abandoned this previously useful mapping because this mapping was now counter-productive. The idea of a strategic association of response codes (SNARC = STARC) can also explain why the average response times in the Wood et al. (2006, this issue) study are not systematically slower than those of the preceding study, despite the alleged interference between spatial codes: there simply were no conflicting spatial codes involved. The fact that participants did not systematically (as a group) adopt any new mappings could be due to a number of reasons, from declining motivation to extensive practice from the previous conditions. The latter hypothesis is in line with the observation that numerically more skilled participants tend to have weaker spatial associations (Dehaene et al., 1993, Experiment 1; Fischer and Rottmann, 2005, Experiment 1). At first glance, the notion of strategic control over number-to-space mappings might appear as resignation in the light of inconsistent findings and recent failures to replicate SNARC-related effects. But taking the STARC hypothesis seriously will lead to important methodological improvements. One such improvement will be to compare, for each individual, the strength of their SNARC effect before and after an experimental manipulation. From this we can learn whether null results truly reflect an absence of spatial associations, or instead constitute an artefact from averaging across differential strategies. Tracking spatial associations across experimental manipulations will also tell us how individuals go about mapping their numbers onto space: those with the strongest environmentcentered SNARC effect might decide either to abandon any spatial mapping (to avoid the remapping cost), or to resort to an equally strong hand-centered spatial reference (because they
1068
Martin H. Fischer
benefit from either spatial strategy). Similarly, those with weak spatial strategies may be unaffected by either mapping. In addition, visuospatial and numerical abilities of participants in future SNARC studies should be assessed because these might constitute important predictors of spatial strategies in general and of the SNARC effect in particular (e.g., Bachot et al., 2005). In a similar vein, controlling for individual participants’ finger counting habits should provide useful evidence to re-evaluate the proposed role for handbased coordinate frames in number-related tasks, as suggested by Wood et al. (2006, this issue) as well as Müller and Schwarz (in press). In addition, hand-based number concepts may also prove to be a viable alternative explanation for the origin of SNARC effects (e.g., Di Luca et al., in press). Clearly, individual-oriented analyses can tell us a lot about the reliability, the validity, and thus the theoretical starkness of the SNARC effect. REFERENCES BACHOT J, GEVERS W, FIAS W and ROEYERS H. Number sense in children with visuo-spatial disabilities: Orientation of the mental number line. Psychology Science, 47: 172-183, 2005. Freely available at http://psychology-science.com/1-2005/ BÄCHTOLD D, BAUMÜLLER M and BRUGGER P. Stimulus-response compatibility in representational space. Neuropsychologia, 36: 731-735, 1998. DEHAENE S, BOSSINI S and GIRAUX, P. The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122: 371-396, 1993.
DI LUCA S, GRANA A, SEMENZA C, SERON X and PESENTI M. Finger-digit compatibility in Arabic numeral processing. The Quarterly Journal of Experimental Psychology (in press). FIAS W and FISCHER MH. Spatial representation of numbers. In Campbell JI (Ed), Handbook of Mathematical Cognition. New York: Psychology Press, 2005. FISCHER MH and COEMAN P. Moving the Mental Number Line: Rapid Effects of Training. Poster at the European Summer School on “Neuroscience of number processing” in Erice, Italy, 3-10 July 2005. FISCHER MH and HILL RL. A SNARC in the Dark: Input Modality Affects Number Representation. Poster at the 22nd Workshop on “Cognitive Neuropsychology” in Bressanone, Italy, 25-30 January 2004. FISCHER MH and ROTTMANN J. Do negative numbers have a place on the mental number line? Psychology Science, 47: 22-32, 2005. Freely available at http://psychology-science.com/12005/ HUBBARD EM, PIAZZA M, PINEL P and DEHAENE S. Interactions between numbers and space in parietal cortex. Nature Reviews Neuroscience, 6: 435-448, 2005. ITO Y and HATTA T. Spatial structure of quantitative representation of numbers: Evidence from the SNARC effect. Memory and Cognition, 32: 662-673, 2004. MOYER RS and LANDAUER TK. Time required for judgments of numerical inequality. Nature, 215: 1519-1520, 1967. MÜLLER D and SCHWARZ W. Is there an internal association of numbers to hands? A hierarchical model of the SNARC effect. Memory and Cognition (in press). NUERK HC, WOOD G and WILLMES K. The universal SNARC effect: The association between number magnitude and space is amodal. Experimental Psychology, 52: 187-194, 2005. RESTLE F. Speed of adding and comparing numbers. Journal of Experimental Psychology, 83: 274-278, 1970. WOOD G, NUERK, HC and WILLMES K. Crossed hands and the SNARC effect: A failure to replicate Dehaene, Bossini and Giraux (1993). Cortex, 42: 1078-1088, 2006.
Martin H. Fischer, Psychology Department, University of Dundee, DD1 4HN, Dundee, Scotland, UK. e-mail: [email protected]
(Received 17 February 2006; accepted 8 March 2006; action editor: Yves Rossetti)