Introduction
The Spatial Numerical Association of Response Codes (SNARC) effect (Dehaene, Bossini, & Giraux,
1993) is a classic paradigm used in experimental psychology to probe number–space interactions. Typically, when asked to classify a number according to its magnitude (smaller vs. larger than a reference) or its parity (odd vs. even), participants are faster and more accurate when responding to smaller numbers in the left side of space, and to larger numbers in the right side of space (see Wood, Willmes, Nuerk, & Fischer,
2008, for a review). The hypothesis that number–space interactions tap an intrinsic property of the mental representation of numbers is still debated (e.g., Gevers, Verguts, Reynvoet, Caessens, & Fias,
2006; Gevers et al.,
2010), even though supported by converging evidence from behavioural, neuropsychological (Zorzi et al.,
2012), and neuroimaging studies (Cutini, Scarpa, Scatturin, Dell’Acqua, & Zorzi,
2012). It has been proposed that numerical magnitudes are mentally represented in a spatially ordered manner along a continuum, referred to as the Mental Number Line (MNL; Dehaene,
1992; Restle,
1970; Zorzi, Priftis, & Umiltà,
2002), and that number processing involves orienting of attention in this “number space” (Hubbard, Piazza, Pinel, & Dehaene,
2005; Umiltà, Priftis, & Zorzi,
2009; Zorzi et al.,
2012;
2002). The link has been traced back to infancy in the form of spontaneous association between numerosity and spatial extent (de Hevia & Spelke,
2010; Lourenco & Longo,
2010). A spatial mapping of symbolic numbers emerges during the early preschool period and it appears to be crucial for understanding magnitude relationships for exact numbers (Sella, Berteletti, Lucangeli, & Zorzi,
2016). However, the specific spatial layout for the ordering of magnitudes is known to be strongly influenced by cultural habits such as reading/writing direction (Dehaene et al.,
1993; Göbel, Shaki, & Fischer,
2011; Shaki, Fischer, & Petrusic,
2009). According to a recent theoretical framework (Fischer,
2012; Myachykov, Scheepers, Fischer, & Kessler,
2014), at the origins of this spatial arrangement may reside “situated” aspects of cognition, reflecting flexible representations rapidly changing as a function of task demands, experimental context, or available resources (e.g., Bächtold, Baumüller, & Brugger,
1998; Fischer, Mills, & Shaki,
2010; Pfister, Schroeder, & Kunde,
2013; Vuilleumier, Ortigue, & Brugger,
2004). In contrast, a more deeply rooted aspect of numerical cognition, reflecting “grounded” aspects and hence physical invariants of the surrounding environment (Fischer,
2012; sometimes referred to “tropisms”, Myachykov et al.,
2014), is the universal tendency to associate small magnitudes with lower space and large magnitudes with upper space (Ito & Hatta,
2004; Shaki & Fischer,
2012). This implies that vertical spatial–numerical associations should be more robust than horizontal ones, a prediction which has received some initial support (Fischer & Brugger,
2011; Wiemers, Bekkering, & Lindemann,
2014). Note that this discussion is cast into the broader theoretical framework of embodied cognition, which opposes the traditional view of cognition as abstract symbol manipulation (Barsalou,
2008) and inspired the notion of embodied number processing (Fischer,
2012; Fischer & Brugger,
2011; Lakoff & Núñez,
2000; Marghetis, Núñez, & Bergen,
2014).
An intriguing extension to the literature on number–space interactions comes from the study of spatial biases in mental arithmetic (Fischer & Shaki,
2014, for review). The Operational Momentum (OM) effect, first documented by McCrink et al., (
2007), refers to the tendency to overestimate the result of additions and underestimate the result of subtractions. The authors suggested that the OM might be ascribed to attentional movements along an internal (mental) numerical continuum, thereby mirroring other forms of perceptual and representational momentum (see Hubbard,
2014, for a review; for an overview of different accounts see Knops, Zitzmann, & McCrink,
2013; McCrink & Wynn,
2009). The OM has been replicated in a variety of experimental conditions, although most typically when the task involves nonsymbolic operands (e.g., Knops, Dehaene, Berteletti, & Zorzi,
2014; Knops, Viarouge, & Dehaene,
2009; McCrink et al.,
2007; McCrink & Wynn,
2009). The OM has been also observed using a variety of response modalities, such as multiple choice (e.g., Knops et al.,
2014; McCrink et al.,
2007), pointing to a location along a visual segment (Klein, Huber, Nuerk, & Moeller,
2014; Pinhas & Fischer,
2008), and dot production (Lindemann & Tira,
2011). Notably, Pinhas, Shaki, and Fischer (
2014) observed spatial associations even for the operation symbol: the addition operator (plus) was associated to faster right-sided responses, whereas the subtraction operator (minus) was associated to faster left-sided responses. Neuroimaging evidence suggests that cortical structures devoted to attention/eye movements are recruited during mental arithmetic (Knops, Thirion, Hubbard, Michel, & Dehaene,
2009). Knops et al. (
2009) observed that the pattern of brain activity in parietal cortex related to right-sided eye movements was also present during addition. Finally, addition and subtraction have been shown to produce spatial biases (leftwards and rightward, respectively) on attention orienting (Masson & Pesenti,
2014; Mathieu, Gourjon, Couderc, Thevenot, & Prado,
2016) and oculomotor behaviour (Hartmann, Mast, & Fischer,
2015; Holmes, Ayzenberg, & Lourenco,
2016; Werner & Raab,
2014). Note that the latter effects are in line with those triggered by isolated digits (e.g., Blini, Cattaneo, & Vallar,
2013; Casarotti, Michielin, Zorzi, & Umiltà,
2007; Fischer, Castel, Dodd, & Pratt,
2003; Klein et al.,
2014; but see Fattorini, Pinto, Rotondaro, & Doricchi,
2015; Zanolie & Pecher,
2014).
The hypothesis that mental arithmetic might be rooted in sensorimotor mechanisms is very intriguing, but it remains unclear which specific processes might be affected and to what extent. In this regard, an important contribution towards a better understanding of the spatial underpinnings of mental arithmetic comes from neuropsychological studies of patients with Unilateral Spatial Neglect (USN), a disorder consisting in the failure to report, respond to or orient towards stimuli in the contralesional side of space (more commonly the left one, Heilman, Watson, & Valenstein,
1985; Vallar,
1998). The initial finding that USN patients show deficits in number processing that are readily interpreted in terms of neglect for the number space (Zorzi et al.,
2002; see Umiltà, Priftis, & Zorzi,
2009, for a review) has been followed-up by many studies (e.g. Bonato, Priftis, Marenzi, & Zorzi,
2008; Priftis et al.,
2008; Priftis, Zorzi, Meneghello, Marenzi, & Umiltà,
2006; Rossetti et al.,
2004; van Dijck, Gevers, Lafosse, Doricchi, & Fias,
2011; Vuilleumier, Ortigue, & Brugger,
2004; Zorzi et al.,
2012; Zorzi, Priftis, Meneghello, Marenzi, & Umiltà,
2006). The recent extension to mental arithmetic has revealed that USN patients present an increased error rate when performing subtractions with respect to additions (Benavides-Varela et al.,
2014; Dormal, Schuller, Nihoul, Pesenti, & Andres,
2014). Dormal et al. (
2014) observed that performance in subtractions was particularly affected when the second operand was large (and a borrowing operation was involved); the authors thus suggested that patients could present difficulties in accessing numerical representations that are located to the left (contralesional) side of the first operand along a continuum transiently built in working memory. A recent single-case report also described a patient with right-sided neglect who showed a specific impairment for additions (Masson, Pesenti, Coyette, Andres, & Dormal,
2017). This is in line with several neuropsychological studies that—starting with the seminal study by Hécaen, Angelergues, & Houillier (
1961), who introduced the term “spatial acalculia”—describe calculation deficits that are secondary to visuospatial ones (Ardila & Rosselli,
1994; Boller & Grafman,
1983; de Hevia, Vallar, & Girelli,
2008, for review). Typical errors of these patients include the incorrect alignment of numbers in column, difficulties in maintaining the decimal places, and so on, thus being at least partly dissociable from non-strategic, approximation-related errors (Benavides-Varela et al.,
2016; de Hevia et al.,
2008).
The crucial test for the hypothesis that attentional movements are functionally involved in mental arithmetic is to assess whether the latter can be affected by explicit experimental manipulations of visuospatial attention in healthy participants (in contrast to “nature’s experiments” based on patients with brain damage). Notably, the embodied cognition framework predicts that conceptual processing should be influenced by the sensorimotor mechanisms they rely on (Barsalou,
1999; Lakoff & Johnson,
1999) and that the influences between semantic and sensorimotor processing should be bi-directional (Gentilucci & Gangitano,
1998). For example, Stoianov, Kramer, Umiltà, and Zorzi (
2008) observed that lateralized, irrelevant spatial cues can modulate symbolic number comparison performance when they temporally overlap with the processing of numerical stimuli (also see Kramer, Stoianov, Umiltà, & Zorzi,
2011). Masson and Pesenti (
2016) recently extended this observation to mental arithmetic by showing that lateralized distracters interfered with calculation in a side-specific manner (i.e., left distracters affected performance in subtraction, whereas right distracters affected additions).
Another way to explicitly manipulate spatial attention is to exploit optokinetic stimulation (OKS) (Pizzamiglio, Frasca, Guariglia, Incoccia, & Antonucci,
1990) and thus eye movements (e.g., Casarotti, Lisi, Umiltà, & Zorzi,
2012; Moore, Armstrong, & Fallah,
2003). OKS consists of full-field visual stimuli (e.g., vertical stripes) moving coherently towards one specific direction. Such stimulation imposes eye movements by means of a peculiar physiological reflex (the optokinetic nystagmus) consisting in two alternating phases: first, the eyes start following the movement of the stimulation (pursuit phase); second, after a variable amount of time a compensatory saccade is made in the opposite direction, allowing the return to the initial position. The mean position of the eyes during OKS is usually shifted towards the side of saccadic eye movements (beating field), possibly as a compensatory attempt to re-orient the eyes in the direction which is implied by the optic flow (Watanabe,
2001). The direction of slow, pursuit eye movements, on the other hand, triggers spatial attention shifts (e.g., leftward for leftward coherent motion) that have been widely exploited for neglect rehabilitation (e.g., Kerkhoff, Keller, Ritter, & Marquardt,
2006). When applied to USN patients, alongside with other rehabilitation techniques, number processing has been found to improve together with visuospatial processing (Priftis, Pitteri, Meneghello, Umiltà, & Zorzi,
2012; Rossetti et al.,
2004; Salillas, Granà, Juncadella, Rico, & Semenza,
2009; Vuilleumier et al.,
2004; but see Pitteri et al.,
2015). Notably, the OKS technique is also effective in modulating number processing in healthy participants. Ranzini et al. (
2015) found that rightward OKS during number comparison abolished the typical response time disadvantage for large numbers compared to smaller numbers. Ranzini, Lisi, & Zorzi (
2016) found that eye movements (both saccades and pursuit) modulated comparison response times for large numbers as a function of motion direction. In the unique previous study that investigated the effect of horizontal OKS on arithmetic performance, Masson, Pesenti, and Dormal (
2016) found that rightward OKS speeded addition problems, but only those involving carry operations. OKS did not influence subtraction problems. Moreover, there was no effect of OKS on errors, possibly because the error rate was low (about 6% across conditions) and it was not broken down into procedural vs. estimation-related errors. In contrast, in the present study we investigated whether the strong manipulation of eye movements and spatial attention provided by OKS may affect the distribution of error responses to challenging mental arithmetic problems (i.e., addition and subtraction of pairs of two-digit numbers).
To this aim, we examined the distribution of error responses to assess the prediction that leftward OKS would induce underestimation of the correct result, whereas rightward OKS would induce overestimation when compared to a neutral condition (i.e., static OKS). In the mental addition and subtraction of two-digits numbers, two main procedures have been documented (Beishuizen, Van Putten, & Van Mulken,
1997; Fuson,
1992). The “base-ten” procedure is based on decomposition of each operand into tens and ones, which are then separately processed and recombined only in the last step. The “sequential” procedure starts instead with counting by tens up or down from the first operand. As mental operations with the tens represent a crucial aspect in both procedures, we operationally considered decade errors (i.e., errors differing from the correct answers by a multiple of ten units) as a signature of procedural errors, which are known to be particularly affected by spatial deficits (Dormal et al.,
2014). Furthermore, we considered all remaining errors (i.e., in the unit range) as index of estimation errors, and also predicted a possible modulation by OKS. Concerning the direction of this modulation, both (voluntary) saccadic and pursuit eye movements have been found to impact number processing (Ranzini et al.,
2016). However, based on previous studies with OKS (Masson et al.,
2016; Ranzini et al.,
2015), we predicted shifts in the number space coherent with the OKS direction (i.e., in the direction of pursuit eye movements, and away from the beating field). Specifically, in keeping with the “moving along the mental number line” analogy (McCrink et al.,
2007), we hypothesized rightward OKS to induce overestimation of the correct result (i.e., more pronounced rightward movement along the line) and leftward OKS to induce underestimation, similarly for additions and subtractions and on top of the OM effect. Finally, we hypothesized that eye movements would be spatially biased as a function of the type of operation (addition vs. subtraction) (Hartmann, Mast, & Fischer,
2015; Klein et al.,
2014; Pinhas & Fischer,
2008), independently of OKS conditions. In particular, we predicted a shift in the mean position of the eyes (leftward for subtractions, rightward for additions), but only during the calculation phase (Liu, Cai, Verguts, & Chen,
2017; Masson, Letesson, & Pesenti,
2017).
To the best of our knowledge, no previous study has simultaneously probed the existence of bi-directional links between mental calculation and eye movements. Additionally, in the present study we sought to investigate the impact of a vertical, beside the more canonical horizontal, direction of OKS. Vertical OKS has never been used in previous studies on number processing or calculation, but assessing its effects is of primary theoretical significance in light of the hypothesis that changes in magnitude along the vertical axis (where “up” represents “more”) are more directly linked to human sensorimotor experience, as in the actions of stacking or removing objects from a pile (Fischer & Brugger,
2011). Vertical OKS might be more prone to affect participants’ performance than horizontal OKS because the former refers to a grounded aspect (as opposed to a situated) of numerical cognition (Fischer,
2012; Myachykov et al.,
2014). With respect to the predictions, we expected upward OKS to induce overestimation of the correct result (similarly to rightward OKS), and downward OKS to induce underestimation of the correct result (similarly to leftward OKS).
To summarize, we sought to concurrently assess: (1) the effect of overt shifts of attention/eye movements on mental calculation, as well as the reciprocal effect (i.e., the effect of solving arithmetic operations on oculomotor control); (2) differential effects of horizontal vs. vertical OKS; (3) differential effects of the spatial manipulation in terms of types of error.
General discussion
In the present study we explored the effects of OKS, a technique that induces shifts of spatial attention secondary to shifts in eye movements (optokinetic nystagmus), on mental calculation. Our aim was to investigate both the potential modulation of arithmetic performance by OKS (from spatial attention/eye movements to the calculation domain) and the complementary effects of operation type (subtractions vs. additions) on eye movements (from the calculation domain to eye movements/spatial attention). As noted in the introduction, vertical OKS was used for the first time in the present study because assessing its effect (also in comparison to horizontal OKS) is of primary theoretical significance for embodied accounts of number processing and calculation.
The results of the arithmetic task, in terms of accuracy of response, showed an overall pattern of underestimation for additions and overestimation for subtractions. Note that this is the opposite with respect to the classic OM effect, which consists in overestimation of the result during addition and underestimation during subtraction (McCrink et al.,
2007). The first important difference between our study and many others in literature is the modality of both stimuli presentation and response, which is auditory/verbal rather than visual/motor. Linguistic stimuli are thought to recruit partly different mechanisms and cognitive strategies (Hubbard,
2014); studies employing verbal stimuli often found “pseudoneglect” for mental number space (Göbel, Calabria, Farnè, & Rossetti,
2006; Loftus, Nicholls, Mattingley, Chapman, & Bradshaw,
2009; Lourenco & Longo,
2010), namely an overall tendency to underestimate the correct result. Interestingly, studies exploring the effect of scanning direction in visual line bisection indicate that “pseudoneglect” in physical space (leftward bisection error) is only found (or it is stronger) when the visual line is canonically explored from left to right, whereas forced leftward scanning typically leads to rightward bisection errors (or smaller “pseudoneglect”) (Brodie & Pettigrew,
1996; Chokron, Bartolomeo, Perenin, Helft, & Imbert,
1998; one of the largest performance modulator according to the meta-analysis by Jewell & McCourt,
2000). Given that subtractions could be characterized by a leftward movement along the MNL, the observed overestimation may be explained in light of this different “scanning” direction (right to left). This spatial account could provide a comprehensive description of our results, but it fails in describing cases in which the OM manifest itself in its canonical direction. Furthermore, the lack of modulation of this effect by OKS might also cast doubts on its spatial origin.
An alternative, non-spatial account of the results is offered by the anchoring effect (Tversky & Kahneman,
1975). In their seminal study, Tversky and Kahneman found that participants’ responses to a complex mental arithmetic problem are strongly modulated by the size of the first operand, yielding higher vs. lower estimates by simply swapping the order of the operands. Though the overall magnitude of the arithmetic problems was balanced across operations in our stimulus set, the first operand was more often larger for subtraction than for addition. As a result, anchoring on the first operand would lead to overestimation for subtraction (i.e., a result closer to the high number anchor) and underestimation for addition (i.e., a result closer to the low number anchor). Thus, the lack of a classic OM might eventually be due to the specific list of items adopted in our study. Note that at least one study showed the classic OM with a list of stimuli balanced for the magnitude of the result (Knops, et al.,
2009, experiment 2); this would seem to be at odds with the anchoring effect. However, a reverse OM effect has also been found in recent studies of symbolic arithmetic for non-zero addition and subtraction problems (Shaki, Pinhas, and Fischer,
2017; also see Shaki and Fischer,
2017). The authors framed the OM effect in the context of at least three competing mental biases and heuristics: the more-or-less heuristic reflects ecological (grounded) associations of additions with larger outcomes and subtractions with smaller outcomes; the sign–space association captures a culturally driven tendency to associate additions with the right space and subtractions to the left space; finally, non-spatial biases such as the anchoring effect also have a preeminent role and, under certain conditions, prevail on spatial biases (Shaki, Pinhas, and Fischer,
2017). Crucially, this framework specifically predicts a reverse OM, at least for non-zero second operands, in line with our results. The lack of a modulation of this effect by OKS, furthermore, corroborates the notion that spatial biases may only have a marginal role in symbolic arithmetic, overruled by non-spatial ones.
Though many errors were distributed within a few units from the correct result, as expected for numerical estimation, we also observed a sizable portion of procedural errors, that is, responses that differed from the correct ones at the tens position (decade errors). This is in line with the evidence that procedural knowledge is important for mental addition and subtraction with two-digit numbers. Most notably, we found that vertical OKS influenced the probability of procedural errors during mental calculation: positive decade errors were more frequent for subtraction with respect to addition, but downward OKS abolished this bias. These results suggest that the procedures triggered by multi-digit calculation rely, to some extent, on spatial/attentional processes, in agreement with neuropsychological observations (Ardila & Rosselli,
1994; de Hevia et al.,
2008, for review). Indeed, a spatial component has also been identified for procedural errors, for example, when carrying and borrowing procedures are required and numbers should be arranged in “mental columns”. Subtractions require borrowing; additions require carry. An error in the procedure requiring borrowing (e.g., leftovers are neglected, borrowing is not performed) yields more often positive decade errors; an error occurring in the carry procedure yields more often negative decade errors. Therefore, the similarity between the pattern of procedural (positive decade errors for subtraction and negative decade errors for addition) and the pattern of estimation errors (overestimation for subtractions, underestimation for additions) does not imply a shared origin, and does not necessarily relate to the MNL framework. Dissociations between procedural and estimation strategies have been reported in neuropsychological studies (Ardila & Rosselli,
1994; Boller & Grafman,
1983; de Hevia, Vallar, & Girelli,
2008, for review) and indeed we observed one in the present study in terms of the modulating effect of OKS.
In the recent study of Masson et al. (
2016), rightward OKS was found to induce a relative speed-up of addition problems requiring a carrying procedure. Horizontal OKS had no effect in the present study, but it must be stressed that our calculation task was more difficult compared to that of Masson et al. (which involved adding/substracting a one-digit number to/from a two-digit number). As previously noted, there is variability in the procedures that can be used for addition and subtraction of two-digit numbers, which implies less reliability of response times; in addition, we explicitly favoured problems that were likely to yield error responses (more than half of the trials in some conditions, either estimation or decade errors), as to better assess error distribution, and as a consequence only a few trials per cell were left to compute response times. In our study, therefore, RTs should be interpreted with much caution because they are likely to be very noisy and unreliable. On top of this, additionally, our list of stimuli was not optimized to be split according to whether a carrying/borrowing procedure was involved (i.e., problems were unbalanced), and using this information as an additional factor in the analyses was simply not possible in light, again, of the insufficient number of observations per cell. Overall, it is our view that our study does not detract from that of Masson et al. because it is far from replicating its procedures, analyses, and experimental hypotheses. On the contrary, we suggest that both our results and those of Masson et al. represent a signature of the spatial/attentional processes that take place during mental calculation: attentional shifts occurring in a direction that is compatible to the putative movement along the MNL (e.g., downward for subtractions) may have facilitatory effects on mental calculation that extend to procedural components. Finally, it is conceivable that we observed an effect of vertical OKS, but not of horizontal OKS, because addition and subtraction might be grounded in early experiences with the physical world along the vertical axis, such as the universal (and culture-independent) actions of stacking and removing items from a pile (Fischer,
2012; Myachykov et al.,
2014; but see Liu, Verguts, Li, Ling, & Chen,
2017, for contrasting results).
The role of spatial attention in numerical cognition is still hotly debated. Several non-spatial accounts have been proposed as alternative explanations for number–space interactions (e.g., Aiello et al.,
2012; Fattorini, Pinto, Merola, D’Onofrio, & Doricchi,
2016; Fias, van Dijck, & Gevers,
2011; Proctor & Cho,
2006). However, there is now a wealth of studies showing that numbers or magnitudes (Blini et al.,
2013; Cattaneo, Fantino, Mancini, Mattioli, & Vallar,
2012; Fischer et al.,
2003; Ishihara, Jacquin-Courtois, Rode, Farnè, & Rossetti,
2013) or arithmetical operations (Hartmann, Mast, & Fischer,
2015; Klein et al.,
2014; Liu et al.,
2017; Masson & Pesenti,
2014; Pinhas & Fischer,
2008) may induce shifts of spatial attention (but see also Fattorini, Pinto, Rotondaro, & Doricchi,
2015; Zanolie & Pecher,
2014, for failed replication attempts). A second growing line of evidence is also contributing, in the other way around, by manipulating explicitly spatial attention to explore its potential influence on number processing. For example, Stoianov, Kramer, Umiltà, and Zorzi (
2008) found that irrelevant lateralized visuospatial cues can influence number processing, with faster response times (RTs) to small numbers following left cues compared to right cues and the opposite pattern for large numbers (also see Kramer, Stoianov, Umiltà, & Zorzi,
2011). Similar results were observed by Hartmann, Grabherr, and Mast (
2012, experiment 2) using a vestibular stimulation (which is known to affect the metric of spatial representations) during a magnitude comparison task (see also Loetscher, Schwarz, Schubiger, & Brugger,
2008). Ranzini et al. (
2015) administered optokinetic stimulation (OKS) to healthy participants during number processing and observed that number comparison RTs were modulated by OKS direction (see also Ranzini, Lisi, & Zorzi,
2016). These effects seem to extend to simple arithmetic (Masson & Pesenti,
2016; Masson et al.,
2016). Besides being in line with these observations, the present study joins for the first time the two lines of research by assessing concurrently both the effect of overt shifts of attention/eye movements on mental calculation and its reciprocal effect (i.e., the effect of solving arithmetic operations on oculomotor control), overall adding compelling evidence on the tight coupling between the number space and eye movements.
In this regard, our results on the eye movements data are clear-cut, and corroborate recent studies (Hartmann, Mast, & Fischer,
2015), including the finding of Knops et al. (
2009) that saccade-related activity in the posterior parietal cortex was a good predictor of the type of mental operation participants were engaged into. In our study, subtractions, with respect to additions, were linked to a leftward displacement of gaze position, rightward for additions, when horizontal OKS was administered. Notably, this displacement was only observed during the response phase and not at other points of the trial, possibly because the MNL was activated only in the calculation phase (in line with Liu et al.,
2017, and Masson et al.,
2017). Moreover, the type of arithmetic operation had also an effect on the vertical dimension, with subtractions consistently linked to an overall downward displacement of the eyes with respect to additions. The latter effect was highly reliable—in contrast to that on the horizontal plane—and it was observed across experiments (independently of the type of OKS). These findings are in line with the predictions of a hierarchical model of number–space interactions (Fischer,
2012; Myachykov et al.,
2014), which suggests that the vertical dimension is grounded in universal (physical) constraints (as opposed to the horizontal dimension, which is situated and culturally dependent; but see Liu, Verguts, Li, Ling, & Chen,
2017, for contrasting results). One caveat of the present study is that we cannot disentangle whether the modulation of eye movements observed during the calculation reflect the intrinsic calculation process or the magnitude of the result. Both factors might play a role (Klein et al.,
2014; Loetscher, Bockisch, Nicholls, & Brugger,
2010), though recent studies suggest that the former has the major impact (Hartmann et al.,
2015). In this regard, it is worth noticing that we did not observe spatial displacement of eye position in the time window corresponding to the processing of the first operand, despite the fact that it was typically, albeit not always, larger for subtractions than for addition (Liu et al.,
2017; Masson, Letesson, et al.,
2017). This corroborates the view that numerical magnitude might retain only a marginal effect on oculomotor behaviour during calculation. Finally, one may wonder if task difficulty could be at the roots of the overall downward displacement of gaze position during mental subtraction. We are not in the position to provide an answer on the basis of our data, as in this study subtractions were more difficult than addition problems (i.e., the former presented lower accuracy and slower response times than the latter). Other studies using much easier problems sets (e.g., operands < 10, Hartmann et al.,
2015) or counting upward/downward by 3 (Hartmann, Mast, & Fischer,
2016) found similar results (i.e., additions, with respect to subtractions, leading to upward shifts of gaze position). However, behavioural measures revealed faster RTs for additions with respect to subtractions also in some of these cases (Hartmann et al.,
2015), and indeed this difference in difficulty is an established finding in cognitive arithmetic (Ashcraft,
1992). It is important for future studies to be aware of this possible confound, and possibly create stimuli set that are roughly balanced for difficulty.
In summary, overt orienting of spatial attention/eye gaze along the vertical axis influenced the participants’ responses to two-digit mental arithmetic problems. Conversely, gaze position shifted during the calculation phase in a direction consistent with the type of operation. These results highlight the pervasive nature of spatial processing in mental arithmetic which, as neuropsychological studies suggest, may manifest itself through a range of diverse processes (and errors, e.g., procedural). Here we also provide evidence for the number space to be grounded in common sensorimotor mechanisms (i.e., eye movements). More studies are needed to shed light on these complex interactions; as a first step, we urge researchers to concurrently assess both the effects of spatial manipulations on mental arithmetic and their complementary effects (i.e., from the number space to oculomotor control or spatial attention).