Spatial grounding of symbolic arithmetic: an investigation with optokinetic stimulation

The first experiment assessed the effect of horizontal OKS on mental calculation. Participants performed both additions and subtractions with auditorily presented two-digit numbers.

Experiment 1 provided evidence for a link between Operation Type and oculomotor behaviour in the vertical dimension, with additions leading to upward movements of the eyes with respect to subtractions. In Experiment 2, we administered vertical OKS to explore whether the explicit manipulation of spatial attention along the vertical axis may affect mental calculation and modulate the distribution of responses.

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).