In the presented paper, we aimed at investigating the relation between the SNARC effect and mathematical proficiency including a group of professional mathematicians. We recruited three groups of participants, professional mathematicians; professionals who use arithmetic in their everyday work, but who do not conduct research using mathematical reasoning itself; and controls, who are not or hardly ever required to use math in their everyday work.
Most importantly, in contrast to most previous studies, we found a significant between-group difference with respect to the SNARC effect, which was mainly driven by the professional mathematicians, whose SNARC effects have—to the best of our knowledge—never been studied before. Professional mathematicians did not reveal a significant SNARC effect, while the other two groups did. Professional mathematicians significantly differed from the other two control groups, while those two control groups with more or less arithmetic expertise did not differ from each other; this replicates earlier results of most studies before. This difference between mathematicians and control groups still held when various covariates were controlled for, such as RT characteristics (mean RT as well as intraindividual variability in RT). Within groups, fluid intelligence did not correlate with the SNARC, so that between-group differences in fluid intelligence cannot explain the SNARC effect, because if fluid intelligence determines the SNARC effect, it should do so within groups as well. The correlation at the sample level between fluid intelligence and SNARC was also not significant. The ANCOVA results controlling for fluid intelligence brought inconsistent findings, nevertheless they must be interpreted with great caution because ANCOVA assumptions were strongly violated (e.g., the covariate was not independent from the factor underlying group assignment) and the sample size was relatively small. In similar cases, several authors refrained from ANCOVA usage, when there is no zero-order correlation between a potential covariate and the dependent variable (e.g. Göbel, Moeller, Pixner, Kaufmann, & Nuerk,
2013).
The results also did not change substantially, when alternative methods of computing SNARC effects were used. We found the same pattern of results when magnitude slopes from multiple regression (i.e., controlling for the MARC effect) were analyzed. When we analyzed standardized SNARC slopes, the general pattern of results was similar: namely when compared to 0, professional mathematicians did not reveal a significant SNARC effect (contrary to the E and C groups). A notable difference between the analyses using non-standardized vs standardized slopes is that in the latter the between-group difference failed to reach significance. Standardized SNARC slopes consider the intraindividual variability within a subject, especially, how much dRT points are dispersed around the regression slope. So in principle the non-standardized regression slope could be almost 0; however, when all data points would be located almost exactly on the regression slope, the standardized slope would be very high. In essence, it is an index for how good the prediction of space-number associations by number magnitude is. However, it does not tell us much about how pronounced this association is. This is coded by the non-standardized slope, which reveals how many milliseconds faster a congruent spatial response is. In our data, the most likely explanation for the slight differences are high intraindividual variances in some participants. If those participants have high non-standardized SNARC slopes (e.g., in the C group), their non-standardized slopes might differ considerably from other groups but their predictions indexed by the standardized slopes might only be slightly different because they are corrected for their higher intraindividual variability.
Apart from group differences, the SNARC effect was related to response time characteristics (mean RT and intraindividual variability in RT). Nevertheless, this correlation was present only when non-standardized SNARC slopes were analyzed. The relation between SNARC and RT characteristics may therefore just be an artifact originating from a difference measure (i.e., dRT being the result of subtracting two RT) used to calculate SNARC (see Tzelgov, Zohar-Shai, & Nuerk,
2013, for a methodological critique of using non-standardized SNARC slopes). In the model proposed by Gevers et al. (
2006), the relationship between mean RT and SNARC is explained in terms of the cognitive mechanisms underlying the SNARC effect. As we have shown (and as already pointed out in Pinhas et al.,
2012; Tzelgov et al.,
2013), this relationship may largely originate from the way slopes are calculated, not from the nature of the SNARC effect itself. Therefore, it seems that the relationship between SNARC and mean RT (as well as variability in RT) may not be a consequence of the cognitive processes underlying the SNARC effect (e.g., Gevers et al.,
2006), but depend on the measure employed. As outlined above, fluid intelligence (Raven Matrices) scores differed between the
M group and the other two control groups. Nevertheless, fluid intelligence did not correlate with SNARC slopes nor standardized SNARC slopes; so individual and group differences in SNARC are not driven by fluid intelligence. However, fluid intelligence correlated moderately with RT characteristics. This observation is in line with the results showing a correlation between intelligence and chronometric tasks in general (Deary, Der, & Ford,
2001).
For the linguistic MARC effect, the different groups did not differ from one another. The MARC effect did not correlate with any other measure. Thus, diverging effects for professional mathematicians were specific to the SNARC effect per se and could not be generalized to another effect in the study.
In sum, professional mathematicians differed in the SNARC effect from the control groups in virtually all analyses. This group difference could not be explained by different RT characteristics or fluid intelligence. It was specific to the SNARC effect, but did not generalize to the MARC effect. Between the two non-professional mathematicians groups, no significant differences in the SNARC effect were observed, despite strong differences in daily arithmetic experience. This replicates earlier results (e.g., Cipora & Nuerk,
2013). SNARC effects do not seem to vary (much) with arithmetic proficiency in the normal range. Only when relatively large (
n > 35), gender-balanced samples are examined, one may expect to have significant statistical test results for relatively small effects (Hoffmann et al.,
2014b). The probability of finding a relationship between math proficiency and the SNARC effect increases when extreme groups are recruited (professional mathematicians vs people with math difficulties, as in Hoffmann et al.,
2014b).
Reasons for lack of/significantly reduced SNARC in professional mathematicians
There may be several reasons for a null or significantly reduced SNARC effect in mathematicians. Here we focus on possible differences in (1) domain-general cognitive abilities, (2) the nature of number representations, and (3) the embodied cognition perspective.
Inhibition and/or cognitive control capabilities Tasks measuring the SNARC effect are at some point influenced by inhibition processes. In incongruent trials (a smaller magnitude number has to be responded to with the right hand and a bigger magnitude number with the left hand), the natural spatial mapping (according to some views because of the number location on the Mental Number Line) has to be overcome by task instructions (Gevers et al.,
2006). Recent data show that the efficiency of inhibition correlates with the SNARC effect (Hoffmann et al.,
2014a). It is possible that mathematicians (already characterized by higher fluid intelligence) may also have better inhibition and cognitive control capacities, because not jumping to (i.e., inhibiting) premature conclusions without proof is what their daily work is about (see Embretson,
1995; for recent evidence on the relationship between cognitive control, working memory, and fluid intelligence see Chuderski, Taraday, Nęcka & Smoleń,
2012). According to this line of explanation, a directional spatial-numerical mapping as indexed by the SNARC effect may just be masked by effective cognitive control of interference, but not be absent in mathematicians per se.
Such a cognitive control account is supported by various related findings. First, cognitive control plays a major role in other number processing effects (Macizo & Herrera,
2011,
2013; Huber, Moeller, Nuerk, & Willmes,
2013; Huber, Klein, Willmes, Nuerk, & Moeller,
2014a; Huber, Mann, Nuerk, & Moeller,
2014c; Huber, Moeller, & Nuerk,
2014d; see also Nuerk, Moeller, Klein, Willmes, & Fischer,
2011; Nuerk, Moeller, & Willmes,
2015, for overviews). It would not be surprising if this extends to other numerical effects such as the SNARC effect as well. Second, selective attention has been shown to be a prerequisite for the SNARC effect (Nuerk et al.,
2005b): Even though the magnitude of distractors was processed in an Eriksen task, there was no SNARC effect for those distractors, only for the targets being attended. Third, inhibition is related to other numerical effects (Gilmore et al.,
2013) and the SNARC effect as well (Hoffmann et al.,
2014a).
More abstract processing in professional mathematicians This account does not refer to domain-general characteristics as above, but is rather related to the more domain-specific characteristics of mathematicians. In other words, their numerical representations might differ from those in non-mathematicians in that they may just be more abstract.
It has been argued that the SNARC effect is strongly influenced by cultural and embodied experience, such as reading direction (Shaki et al.,
2009), finger counting (Fischer,
2008; for a thorough discussion on factors influencing spatial-numerical associations see also: Fischer & Brugger,
2011; Göbel, Shaki, & Fischer,
2011). Possibly mathematicians—because of their daily routine with highly abstract concepts—have just overcome the cultural and embodied experiences which drive our default spatial directionality of magnitude. This could be tested in the future by examining other instances and paradigms of spatial-numerical association: Professional mathematicians with neglect may neglect smaller numbers, commonly on the left side of the number line, to a lesser extent.
More flexible spatial-
numerical representations Another related, albeit slightly different, account is that rather than having no spatial association with numbers (because they are abstract), mathematicians possess much more flexible representations. In the literature, it was usually claimed that the spatial code is automatically activated when numbers are perceived (Fias, Lauwereyns, & Lammertyn,
2001). Nevertheless, it was demonstrated that under particular conditions number magnitude (in case of distracter numbers) can be processed semantically, but the spatial code is not activated (Nuerk et al.,
2005a, see above). So, mathematicians may have strong spatial-numerical associations, however, they may map numbers to space in a highly flexible way. Therefore, default left-to-right mappings like in the SNARC effect may become weaker or disappear. This may be particularly the case in the parity judgment task, where relating numerical magnitude to space is by no means mandatory to accomplish the parity decision. Mathematicians possessing more flexible representations may simply not activate the spatial aspects that are irrelevant for the task demands. It is possible, however, that in a magnitude comparison task, when spatial coding of magnitude may be helpful, mathematicians also activate more spatial-numerical associations. Here we can only conclude that mathematicians do not activate them automatically, when magnitude is task irrelevant. Evidence for such an account comes from a recent unpublished study by Cohen Kadosh and colleagues.
4 They observed that mathematicians are better in a number line estimation task (cf. Siegler & Opfer,
2003) for positive numbers. So, mathematicians may well be able to map numbers to space, but they might do so less automatically in a default direction.
Stable but non-
linear/non-
horizontal numerical representations It is also possible that a considerable proportion of mathematicians possess relatively stable but non-linear monotone or even non-monotone or non-horizontal spatial-numerical representations (bent lines, circular or irregular forms, vertical or radial associations). These representations may resemble those reported for persons with number-form synesthesia. Synesthesia may influence elementary numerical processing (Cohen Kadosh & Henik,
2007), including the SNARC effect (Sagiv, Simner, Collins, Butterworth, & Ward,
2006). This may also be the case in mathematicians.
Embodied cognition explanation:
5 It is also possible that high math competence or arithmetic skills (characteristic of both professionals using math in their everyday work—the
E group and mathematicians) leads to a reduced SNARC when compared to controls. If this is correct,
a smaller SNARC effect should be found in the engineer group compared to the control group in the present study. This is in line with results described by Hoffmann et al.
(2014a), where a group consisting of mathematics and engineering students revealed a significantly smaller SNARC effect than controls.
However, an opposite effect can be expected from an embodied cognition perspective. Possibly, the professional work requirements of engineers relate more strongly to spatial properties of the environments as well as a higher propensity of motion in space. Such kinds of activity may even enhance the spatial mapping of numerical representations. If this is correct, a stronger SNARC effect should be found in the engineer group compared to the control group in the present study.
If both mechanisms are operative for engineers and influence the SNARC effect in opposite directions, they may cancel each other out. This may lead to a null difference between engineers and controls. If the mechanisms do not fully cancel each other out, some differences between engineers and controls may be observed.
Nevertheless, mathematicians differ from controls. The reason may be that they have less embodied experiences of space-number associations, because their daily work relates to abstract concepts. Therefore, embodiment does not lead to enhanced space-number associations in the M group. As a consequence, only their higher math skills may influence the SNARC effect and may lead to a reduced effect, as compared to controls.
Limitations of the presented study6
In the present study, we did not include objective measures of calculation skills or math expertise. Therefore, we cannot be sure whether the
M group did not differ from the
E group with respect to calculation skills. The pattern of possible differences in arithmetic performance between the
E and the
M group may be qualitatively different: engineers may practice calculation skills more, whereas mathematicians mostly focus on mental manipulations of abstract material. Several cases of double dissociations between mental calculation efficiency and math expertise have also been reported (for an overview, see Pesenti,
2005). So, while this study established that there exists a difference between professional engineers and professional mathematicians, it does not yet allow strong conclusions about the underlying nature of this difference.
Administration of tasks aimed at measuring flexibility and abstractness of representations would help answering such questions regarding the nature of representations in professional mathematicians. It would also be interesting to include measures of cognitive inhibition in order to directly test whether the
M group outperforms other groups in this respect and for measures of complex problem solving skills (see Sonnleitner et al.,
2013). These latter abilities seem to be particularly important for professional mathematicians and may also moderate spatial-numerical associations.
One must also keep in mind that the sample was not gender matched, precluding to test for an impact of gender on the SNARC effect. Males were reported to reveal a stronger SNARC effect (Bull et al.,
2013). However, since the proportion of male and female participants did not differ between groups, between-group SNARC differences cannot be attributed to gender differences. Nevertheless, with the current design, it was impossible to trace the interaction effects of gender and math skills on spatial-numerical associations. Testing this research question would also require larger sample sizes, since gender differences tend to be rather small. All these issues need to be addressed in future studies.
Possible differences between the
E and the
C group also deserve further investigation. Because the Bayesian analyses revealed only weak evidence in favor of a null effect, there is some probability that a between-group difference may still exist, especially since it was shown by Bull et al. (
2013) that between-group differences in the SNARC effect are relatively hard to detect (Cipora & Wood,
2012 for simulation data).