In one of his seminal works, Stanislas Dehaene (2005) suggested that processing numerical magnitudes, such as in simple arithmetic, may employ “recycled” neural mechanisms of spatial attention operating on the mental representations of numbers, which is probably intrinsically spatially organized in the form of a “mental number line.” Close affinity between the mental representations of numbers and space is established in the current state of knowledge. One of the most prolific demonstrations of this affinity was Dehaene et al. (1993) study on so-called “SNARC” effect (spatial-numerical association of response codes), which indicates quicker reactions to relatively small numbers with the left hand and relatively large numbers with the right hand. Hence, the mental number line seems to be left-to-right oriented, at least in Western cultures, where left-to-right script is used (c.f. Göbel et al., 2011, for discussion of cultural and contextual variability of the mental number line), and numbers automatically activate corresponding spatial positions on this line. If so, the addition of natural numbers may require moving internal attention rightward (from smaller to larger numerical magnitudes), while subtraction may require leftward motion of attention. Indeed, it is worth noting that careful analyses by Cipora et al. (2018) and Toomarian and Hubbard (2018) also showed several other factors that may affect spatial-numerical associations.

Perhaps one of the strongest proofs for the involvement of spatial attention in number processing and arithmetic comes from the research on hemineglect – a syndrome of attentional ignoring of one side of the visual field (typically the left) after a contralateral lesion in the parietal cortex (for more detailed analysis of the symptoms and the neural bases of hemineglect, see Vallar and Bolognini, 2014). The research showed that patients with left visual field neglect ignore small numbers (below five). For example, when asked to state the number lying in the middle between one and nine, they may provide the answer seven, despite five being the correct value. Moreover, the patients have preserved addition skills but make more subtraction errors compared to healthy subjects or to patients with lesions in the right hemisphere who do not have neglect (Dormal et al., 2014). Right visual field neglect is relatively rare; however, case studies by Masson et al. (2017) demonstrated deficiencies in both large number processing and addition in such patients.

Increasing numbers of studies with healthy subjects also show contrasting patterns of leftward vs. rightward cuing of attention on addition vs. subtraction. For example, Mathieu et al. (2016) demonstrated that presenting a second operand on the left side of the screen sped up participants’ reactions in one-digit subtraction, while placing a second operand on the right side quickened reactions in addition.

Other research suggests that arithmetic operation may prime attention to the left or to the right. For example, in Masson and Pesenti (2016), study participants solved arithmetic problems and were then asked to detect a star presented either on the left or on the right side of the screen. Solving subtraction problems enhanced target detection on the left, while addition improved reactions to targets presented on the right side. Similar effects of directing attention through arithmetic operation (subtraction to the left and addition to the right) have also been documented by other researchers (Liu et al., 2017; Zhu et al., 2018, 2019). In another study with dual-task procedure (Masson and Pesenti, 2016) a distractor placed on the left impaired subtraction, while one placed on the right impaired addition. Taken together, these studies show evidence toward bidirectional interference between numerical operations and spatial attention.

There is, however, another prominent bias in mental arithmetic. McCrink et al. (2007) demonstrated that when processing non-symbolic addition and subtraction (judging accuracy of the outcome of two summed or subtracted visual sets), people tend to accept overly large sets as outcomes of addition and overly small sets as outcomes of subtraction, which implies a tendency toward overestimating the results of addition and underestimation those of subtraction. The effect was later shown even in 9-month-old infants (McCrink and Wynn, 2009).

Knops et al. (2009a, b) integrated these two lines of research, demonstrating under-/overestimation biases in both symbolic and non-symbolic arithmetic, and additionally showed that both symbolic and non-symbolic arithmetic involve neural responses similar to those accompanying the saccadic eye-movement control processes. In particular, in the Knops et al. (2009a) study, a pattern classifier was trained in deciding between leftward and rightward saccades on the basis of the functional MRI scans of the parietal cortices of fifteen participants. Next, the classifier was presented with the scans of the brain activities of the same participants during arithmetic problem solving. The addition problems were classified as rightward saccades with above-random likelihood, while classification of the subtraction problems was at the random level (note that the participants in this study fixated their gaze centrally during arithmetic tasks, so no real saccades were involved). These results can suggest that the tendencies toward linking addition with the right-directed attention vs. linking subtraction with the left side and the tendencies toward overestimating the outcomes of addition vs. underestimating the outcomes of subtraction may be closely related and could involve the same mechanisms.

McCrink et al. (2007) proposed three explanations for the under-/overestimation bias in mental arithmetic. According to the first explanation, the bias is an effect of a simple heuristic, “addition is more, subtraction is less.” Such a heuristic may be interpreted either as a metaphorical description of some unknown mechanism or as a selection bias at the decision stage rather than the computation stage.

The second explanation assumes that overestimation in addition and underestimation in subtraction results from the properties of the spatial representation of numbers. The mental number-line is compressed (logarithmically scaled), while the arithmetic requires operating in uncompressed space (probably because it uses “recycled” spatial processes that evolved to process real space; cf. Dehaene, 2005). Inaccuracy of rescaling a compressed to an uncompressed representation (and vice versa) may, under some assumptions, cause typical under- or overestimation biases. In some way, this may be an instantiation of the mechanism underlying the heuristic “addition is more, subtraction is less.” However, as demonstrated in some later studies, such a heuristic does not work in some cases. The tendency toward under-/overestimating the results of subtraction/addition may sometimes be neutralized or even reversed (see e.g., Knops et al., 2013; Charras et al., 2014; Shaki et al., 2015; Blini et al., 2019). This makes both the heuristic and compression mechanism hypotheses problematic. Knops et al. (2014) constructed a computational model based on the imperfect uncompression hypothesis and compared its predictions to data collected from fourteen subjects performing numerosity estimation, numerosity comparison, and arithmetic (symbolic, non-symbolic, and cross-notational) tasks. At the group level, they found expected under-/overestimation biases; however, there was considerable individual variability, and individual precision of numerical magnitude representation (which is supposed to be dependent on scaling and compression/uncompression processes) did not correlate with the individual bias sizes in arithmetic. Thus, the biases may not be fully explained by the properties of numerical representation (i.e., scaling and compression).

The third alternative explanation assumes that the estimation biases are due to involvement of the spatial-attentional processes in mental arithmetic. Using the mental number-line metaphor, when subtracting or adding two numbers, one has to focus internal attention on the position of the first operand on the number line. Next, she/he moves attention by the distance proportional to the second operand’s magnitude to reach the unknown position of the result magnitude (McCrink et al., 2007). Visual attention has built-in a mechanism of anticipatory overestimation of the expected position of a moving object in the direction of motion. Freyd and Finke (1984) demonstrated such an anticipatory mechanism of attention. They found that people watching a sequence of still images of a moving object overestimate its future position in the direction of apparent motion when asked to select the next frame. In reference to momentum-based theory of motion in human “naive physics” this phenomenon was named “representational momentum” (“RM”). This mechanism, adapted for the purpose of mental arithmetic, may lead to underestimation in subtraction (setting focus too far in the direction of decreasing numbers), while the opposite may occur in addition. Per analogiam, McCrink et al. (2007) named the under-/overestimation bias in mental arithmetic “operational momentum” (“OM”).

Momentum-like effects are common in attention and action. Hubbard (2015, 2017) analyzed different types of “momentum-like” effects in attentional processes and proposed that representational momentum is an important functional adaptation. When planning an action, e.g., reaching for an object, or planning a saccade directed at a moving object, we notice the object’s spatial position with a few tens of milliseconds of delay because the neural signal from the sensory input has to be transferred to the cortical level. Execution of the saccade or action would take another ten or even a few hundred milliseconds. Thus, at any given moment, the actual position of the object and the position represented in the brain are shifted relative to each other. To make the behavior efficient, the target’s position has to be anticipated. An overestimation of the target’s position proportional to the target’s speed may, therefore, provide optimal strategy in these cases, resulting in a representational momentum effect. Despite the fact that such a strategy may not be optimal in the case of arithmetic (usually, the precise outcome of arithmetic operation is desired), the same mechanism may produce the operational momentum effect (assuming the mechanisms of spatial attention are adapted or “recycled” for number processing). It should be noted, however, that this hypothesis also cannot easily explain the reversed operational momentum effect.

As Hubbard (2010) notes, there are a number of possible interpretations of the representational momentum mechanism, ranging from the low-level pereceptual-attentional mechanism to processes controlled by (implicit) conceptual knowledge. The explanatory status of representational momentum as an analog of operational momentum in arithmetic is then unclear. It may be either purely metaphorical, merely providing another description (like a model of a general attentional heuristic), or mechanistic, assuming use of the very same attentional processes in both spatial orienting and arithmetic (c.f. Hubbard, 2015, 2017). Under this more literal, mechanistic interpretation, RM and OM may share some computational and brain resources, which could lead to some interferences of attentional and numerical effects. Specifically, real motion of the attentionally tracked object may interfere with internal attention directed onto represented “motion” along the mental number-line and influence numerical processes, increasing or diminishing estimation biases in arithmetic. Some studies imply that this may indeed be the case. For example, perceiving leftward motion restores small numbers in space in left hemifield neglect (Salillas et al., 2009). It was also found that the direction of motion can cue attentional processes engaged in numerosity estimation. Schwiedrzik et al. (2016) used the neural adaptation paradigm in their experiment. Participants were adapted to leftward or rightward motion of a large cloud of dots. Adaptation to leftward motion led to overestimation of dot numerosity, while rightward motion led to underestimation biases. This suggests that the mechanism of attentional tracking leftward-moving objects sets attention to the small number side, while rightward motion moves attention to the larger number side (note that neural adaptation cancels direct effects of the stimuli, promoting the reversed pattern of reactions). The motion-induced operational momentum was also demonstrated in non-symbolic arithmetic in preschool children (Haman and Lipowska, 2021).

The interplay between representational and operational momentum mechanisms may, however, have further consequences. To effectively predict the future position of a tracked object, the representational momentum mechanism has to depend on the object’s velocity. Indeed, Freyd and Finke (1985), and Finke et al. (1986) documented velocity and acceleration effects on RM (see also Hubbard, 2015, 2017, for more extended review). However, as Hubbard states, the analog of velocity in operational momentum is unclear. Nevertheless, if the operational and representational momenta share a common mechanism, one may expect that not only motion itself but also velocity of motion may influence numerical processes and arithmetic. Thus, two possibilities may be considered. According to the first one, fast motion accompanying arithmetic operation may provide a cue for further shift of attention along the mental number-line in the direction of the operation, which should increase OM-related estimation biases. This hypothesis seems to be the most appealing one. We have noted above that directional attentional cuing (also involving moving stimuli) has a vivid effect on numerical processes and mental arithmetic. Typically, reaction times decrease in addition when attention is cued right and in subtraction when it is cued left. One may then expect that fast rightward motion should promote OM-based responses in addition, while fast leftward motion would enhance OM in subtraction. This expectation is based on well-established effects of priming and numerical distance. Comparing the size of two numbers at a greater distance on the number line is easier and requires less processing time than comparing two close numbers. In turn, priming the position on the mental number line facilitates access to a given number, as shown in both experimental studies and computer simulations (Zorzi et al., 2005).

There is, however, an alternative hypothesis that motion may be processed as a distractor competing for the same attentional resources. In this case, a fast rightward-moving stimulus would compete for attentional resources with numerical processes in addition, thus attenuating the internal attentional shift along the mental number-line and minimizing or even reversing the OM effect, while the same is expected for quick leftward motion in subtraction. At least one study (Masson and Pesenti, 2016) has demonstrated that presenting a distractor (albeit static) on the left side impairs subtraction, while a distractor located on the right impairs addition. Furthermore, the reversed effect of direction of motion on numerosity estimation, caused by neural adaptation induced by motion in the Schwiedrzik et al. (2016) study, may provide additional justification for this hypothesis. Nonetheless, this hypothesis seems to be less likely than the first one. Indeed, McCrink and Hubbard (2017) demonstrated that performing approximate arithmetic tasks under a loaded attention condition (in a divided attention task) even increased OM-related biases.

The experiment described below is the first attempt to investigate the influence of speed and direction of motion on arithmetic. Thus, it was one of the first direct tests of the hypothesis of the common attentional mechanism of the representational and operational momenta.

We tested adult participants with two-digit addition and subtraction problems in which the operands moved across a computer screen either from the left to the right or vice versa. The participant’s task was to judge the accuracy of the proposed result. There were four levels of speed/velocity of the operand’s motion. According to the most prevalent hypothesis based on the common mechanisms for representational and operational momenta, we expected that increased velocity of moving numbers would also increase operational momentum. This can manifest either in decision times or increased acceptance of too-high outcomes in addition or too-low outcomes of subtraction, or in both. While we tested this hypothesis primarily, we also discuss alternative hypotheses in light of our results.

It should be clearly stated that such research design makes sense only if the mechanisms of both OM and RM are attentional. As mentioned above, interpretation of the momentum-like effects in terms of attentional processes is only one possible hypothesis in much wider range. In the case of a different nature of these mechanisms, our study does not lead to unambiguous predictions.

Twenty-four young adults (university students, 16 females, 8 males; 23 righthanded) with normal or corrected-to-normal vision voluntarily agreed to participate. They were informed only that the study concerns some aspects of numerical perception. All participants gave their written consent and received a small gift for participation. The procedure conformed to the requirements of the Declaration of Helsinki. Two additional participants did not complete the entire procedure and were replaced.

Each participant watched 288 computer-generated animations, half of which presented addition problems, and the other half subtraction problems. The participant sat in front of the monitor so that the distance between the eyes and the screen was about 60 cm, but some uncontrolled variance in this regard was inevitable due to the differences in the posture of the participants and head movements during testing. A computer equipped with a 17” panoramic (9:16) LCD screen was used to run the experiment. The operands (two-digit numbers, 96 pixel font) entered the display either from the left or the right side (both operands in the same trial entered from the same side). They were then visible as they moved toward the center of the screen with varied velocity, until they were concealed behind a centrally located occluder (approximately 150 × 150 pixels) with an operator sign (plus or minus) displayed on it (see Figure 1). In half of the trials, the path of the moving operands was shortened by black rectangles displayed on both sides of the screen (the operands were starting from an edge of one of them). For each of the distances, motion lasted either 0.8 s (short events) or 1.2 s (long events). Combining event duration and distance resulted in four levels of operand motion speed: approximately 17 cm/s, 20.5 cm/s, 25.5 cm/s, and 30.5 cm/s (assuming that the mean distance between participant’s eyes and the screen was approximately 60 cm, angular speed levels may be approximated as 15.8, 18.3, 23.8, and 27.5 degrees per second, respectively). Five hundred milliseconds after the second operand was hidden, the occluder disappeared, revealing a number suggested as the outcome option. Such a time course of the event seems to be well fitted to the combined time course of mental arithmetic (Restle, 1970), RM (Freyd and Johnson, 1987), and mapping the number representation to space (Toomarian and Hubbard, 2018). Figure 1 illustrates the display and the course of the trials.

The subject then had to decide whether the outcome was right or wrong and pressed using dominant hand an appropriate key (“z” or “m”) on the computer keyboard, marked with colored stickers. For half of the subjects, the “correct” and “incorrect” buttons were switched. The selected response and reaction times (counted from the disappearance of the occluder to the moment of key-pressing) were recorded. There was a pause of 500 ms (still display) before the onset of the first operand, another pause between the first operand hiding completely behind the occluder and the appearance of the second operand, and a final pause between the hiding of the second operand and the unveiling of the outcome. The experimental procedure was programmed in C++ and was run under Microsoft Windows XP OS.

The stimuli consisted of 4 test blocks of 72 addition or subtraction problems (two addition and two subtraction blocks). We chose to block arithmetic operations, as they might partially activate different brain resources (Rosenberg-Lee et al., 2011); thus, mixing them together could lead to additional switching costs. The problems were randomly selected from a large pool of problems meeting the following conditions: (1) both operands and the outcome are positive two-digit numbers, and (2) only no-carry and no-borrowing problems were used. One-third of the problems in each block had a correct result (“c”) as the outcome, one-third had outcomes underestimated by 2 (“c−2”), and one-third of the problems had outcomes overestimated by 2 (“c + 2”). In the case of under- and overestimated outcomes, the outcome always belonged to the same ten as the correct result. We only used no-carry, no-borrowing problems because some work suggests the possibility that numerical units and tens are processed separately. In the case of carry or borrowing problems, predictions concerning spatial-numerical associations may be contradictory, depending on if they were based on ten or unit digit magnitude (Verguts and De Moor, 2005; Nuerk et al., 2015).

Each pair of problem and outcome was only used once in the subject’s individual set. The averages of larger operands in the subtraction problems and the outcomes of addition, as well as the mean distances between the second operand and the outcome in both operations, were equalized at the group level.

The blocks were arranged so that addition and subtraction blocks appeared interchangeably. Consecutive blocks were separated by a pause that allowed subjects to rest as long as they wished. In every block, there were 3 instances of every combination of the following varied factors: four speed levels (2 levels of event duration × 2 path lengths), the direction of movement (leftward or rightward), and magnitude of the proposed outcome (correct, too high, or too low). Hereafter we refer to these outcome options as “c” (correct), “c + 2” (too high) and “c−2” (too low), in respect to actual absolute numerical values of the outcome option. For example, in the case of both 57−22 subtraction problem and the 23+12 addition problem c = 35, c + 2 = 37, and c−2 = 33. Note that this notation is based on numerical values, not distance and direction of motion, and therefore may be somewhat confusing to readers more familiar with the literature on representational momentum. Eight different sets of four blocks were randomly constructed, each of them in three different orders, which resulted in 24 individual, fully counterbalanced sets. The first test block was preceded by an instruction screen and 8 warm-up problems with feedback. No feedback was given during the test trials. The entire session lasted approximately 25–30 min.

The design described above allows to test the hypotheses presented in the introduction. According to the most prevalent hypothesis based on the common mechanisms for representational and operational momenta, we expected that increased velocity of moving numbers would also increase operational momentum. Thus, we expected increased decision times of correct decisions or/and increased acceptance (decreased accuracy) in the case of too-high outcomes in addition and too-low outcomes in subtraction. It is important to note that we mean relative effects of comparison between too-high and too-low options (c + 2 and c−2) within the same operation (addition or subtraction). This is because each operation may add a separate effect. Particularly addition may be generally easier than subtraction, so processing addition problems may be generally quicker and more accurate. Figure 2 illustrates expected distributions of RTs and accuracy.

Furthermore, the direction of motion has to be considered, with rightward motion making it harder to reject too-high outcome options and leftward motion leading to the acceptance of too-low outcome options. Since previous studies indicated that rightward cuing of attention, or distraction in the right visual hemifield, influences addition, and the factors operating in the left part of the visual field influence subtraction, we may primarily expect the effects of rightward-directed motion on addition and leftward-directed motion on subtraction. Thus, our main hypothesis is that fast motion to the right would prime the position on the number-line ahead the correct one in the direction of motion (i.e., too-high number) and thus increase overestimation bias in addition, making it harder to judge the too-high outcome option as incorrect. This may be evidenced either in terms of increased RTs or decreased accuracy, or both. Quick motion to the left would increase underestimation bias in subtraction, making it harder to decide on too-low outcome options. More generalized interaction is, however, also possible, with both directions influencing both operations (in the opposite way).

Such operationalization is grounded in many previous studies on numerical representations showing the effects of numerical distance and priming of a position on the mental number line (for review, see Zorzi et al., 2005; c.f. also van Opstal and Verguts, 2011, for the empirical and computational investigation of the priming and distance effects in the “same-different” task, which is particularly relevant to our experimental design, as it matches the decisions required from our participants). As noted previously, comparing the size of two numbers at a greater distance on the number line is easier and requires less processing time than comparing two close numbers. In turn, priming the position on the mental number line facilitates access to a corresponding number. Assuming this, we expect that the shift of attention focus, associated with representational momentum, will cause a corresponding shift of internal attention along the mental number line and prime the expected outcome ahead of the correct one in movement direction. It should be noted that most of the studies to date have used selection of the outcome option rather than a decision on its correctness. In such a procedure it is not clear what the effects of movement along the mental number line should be, but it cannot be ruled out that it should be the opposite effect (and this was usually obtained, e.g., Knops et al., 2009a, b). Finally, because only no-carry operations were used, the subjects’ task was relatively easy: adding/subtracting a unit digit is enough to judge the correctness of the outcome, and the majority of participants realized this during the experiment. For this reason, we expect more salient effects in the analysis of response times rather than in accuracy.

According to the representational momentum-based hypothesis, the speed factor is expected to increase both OM effects: under/overestimation magnitude OM and direction-related OM. Thus, three separate interactions involving speed may be expected: (1) Speed × Operation × Outcome option and (2) Speed × Direction of motion × Outcome option, and (3) Speed × Direction of motion × Operation. However, because direction is known to interact with operation, more complex patterns (particularly an interaction of all four factors) are also possible. In these case, however, it should incorporate the effects listed above.

The task was relatively simple, so the overall accuracy was high. Because only no-carry problems were used, most of the subjects realized during the session that the problems could be solved by simply adding or subtracting the unit digits and comparing the result to the last digit of the outcome option. We trimmed not only false starts (< 250 ms) but also too long reactions (> 3 SD above individual mean), assuming that they most likely result form distraction and may therefore not be reliable. After trimming the reactions with excessive times, the accuracy ranged from 0.844 to 0.976 (mean = 0.913, SD = 0.046). Indeed, all subjects made some errors (minimum seven) in at least seven or more categories of trials (design cells), and although there were no apparent outliers, the participants differed in the proportion and distribution of errors. We also checked the shape of error distribution and sphericity of variances and decided that they were sufficient for analysis of the error rate with GLM statistics (with Greenhouse-Geisser correction, which was required only in case of the main effect of speed). We used Statistica v. 13 software (Tibco Software Inc., United States).

Four-way repeated-measures ANOVA: 2 (Operation: subtraction vs. addition) × 2 (Direction of motion: leftward vs. rightward) × 4 (Speed of motion: from 1-slowest, to 4-quickest) × 2 [Outcome option: too low (c−2), too high (c + 2)] revealed significant main effects of two factors: (1) Operation, with responses to addition problems being more accurate [Madd = 0.939, SE = 0.010, Msub = 0.902, SE = 0.011; F(1,23) = 14.24, G-G epsilon = 1, p < 0.001, eta2p = 0.382], and (2) Speed [speed listed in increasing order: M1 = 0.943, SE = 0.008, M2 = 0.931, SE = 0.011, M3 = 0.910, SE = 0.013, M4 = 0.898, SE = 0.013; F(3,23) = 6.92, G-G epsilon = 0.697, corrected p < 0.001, eta2p = 0.255]. Note, however, that the two slower speeds differ from the two higher speeds by the duration of the movement, where the lower speed in each pair is associated with a shorter path (distance). Thus, the speed effect can be seemingly reduced to the effect of the event duration, as the Bonferroni-corrected post hoc tests revealed significant or near-significant differences between shorter- and longer-lasting motion trials (p1vs3 = 0.007, p1vs4 = 0.003, p2vs3 = 0.067, and p2vs4 = 0.033), but not between trials of the same duration (both ps = 1). As stated in T. Hubbard’s (2015, 2017) reviews, duration of the movement does not matter for strength of RM; thus, this effect is neutral for RM-based explanation of OM. On the other hand, a shorter trial duration means a greater time stress, which in turn leads to a more automatic allocation of attention and less deliberate decisions (Schneider and Shiffrin, 1977). It should be noted, however, that although the speed differed depending on the distance in short-duration trials, this difference does not affected the accuracy of decisions.

Only one interaction, Operation × Outcome option, appeared to be significant [F(1,23) = 5.24, G-G epsilon = 1, p < 0.035, eta2p = 0.179]. Bonferroni post hoc tests revealed that accuracy in addition is significantly higher than that in subtraction in the case of too-high outcome trials (p < 0.001). Even more importantly, for addition, the difference in accuracy between c−2 and c + 2 trials also approached significance (p = 0.065; Bonferroni corrected). Since this contrast was expected from our hypotheses and from previous OM studies, the planned comparison approach may be applied here, which in this case makes the contrast significant [F(1,23) = 4,385, p < 0.05]. No difference was found for c−2 and c + 2 outcome trials in subtraction [F(1,23) = 0.17, p = 0.74]. The results are visualized on Figure 3. This interaction seems to show some form of operational momentum, but it is not consistent with the representational momentum-based hypothesis. As stated previously, the RM-based hypothesis predicts that attentional movement along a mental number line overshoots the correct result, which should lead to hindering accuracy judgments of too-high outcome options in addition and too-low outcome options in subtraction. Participants in our experiment were better at recognizing too-high outcomes in addition as incorrect, while no difference was revealed in subtraction.

Although the required computation was easy, the task as a whole was relatively demanding on attentional resources, requiring from participants continuous tracking of the numbers moving on the screen, which, according to McCrink and Hubbard (2017), should also increase the bias, especially in the case of shorter duration trials. Unfortunately, direct comparison of our results to the previous studies is problematic, as most of them used a choice from a large set of outcome options rather than judging the correctness of a single proposed outcome. These two tasks pose different requirements on decision making processes. However, most importantly, none of the expected interactions involving the speed factor was found, which also does not support the RM-based explanation of the operational momentum effect.

Repeated-measures ANOVA 2 (Operation: subtraction vs. addition) × 2 (Direction of motion: leftward vs. rightward) × 4 ([Speed of motion (from 1 – the slowest, to 4 – the fastest)] × 2 (Outcome option: c−2 vs. c + 2) on response times revealed the main effects of Operation [F(1,23) = 31.71, p < 0.0001, eta2p = 0.580] and Outcome option [F(2,46) = 6.79, p < 0.016, eta2p = 0.228]. Addition was processed faster than subtraction, and too-low outcome options were processed quicker than too-high ones, which is generally consistent with previous findings in numerical cognition but not related to the hypotheses tested in our study. More importantly, there were also two significant interactions: Outcome option x Direction × Speed [F(3,69) = 3.36, p < 0.025, eta2p = 0.127], and the interaction of all four factors [F(3,69) = 2.89, p < 0.05, eta2p = 0.111].

The presence of these two interactions seems to confirm findings from previous studies, showing the interference of motion-induced attention with numerical processes, and extends them onto velocity-related aspects of motion. The interpretation of these interactions is not, however, straightforward. Assuming common mechanisms for the RM and OM effects, our hypotheses predicted increased RT for c−2 option in the trials with left-directed movement of increased speed and in subtraction and for c + 2 option in the trials with rightward movement of increased speed and in addition. However, since the increased speed led to higher RTs itself (likely because of increased attentional load), the above effects should be considered relative rather than absolute. The results presented in Figure 4, illustrating the Direction of motion × Outcome option × Speed interaction, do not seem to confirm the hypothesis about the effects of direction of motion. In the case of the left-directed movement, all RTs generally increase with speed, and the only differences between too-high and too-low outcome options concern differences between slowest speed levels, which seems not to be related to RM-like mechanisms. For c−2 type outcomes, the RTs decrease between the two slowest speeds and then increase constantly. Transversely, RTs increase most between the first two (slowest) levels of speed for c + 2 type outcomes. Moreover, the pattern for rightward motion results seems to overtly contradict the hypothesis, with RTs decreasing for too-high outcomes and increasing for too-low outcomes in relation to increasing speed.

In summary, we have found an operational momentum-like effect in the analysis of response accuracy (at least in addition). Participants were more efficient in recognizing too-high than too-low outcomes of addition as incorrect. This effect, although consistent with the previous studies, does not fit the representational momentum-based hypothesis. Moreover, in this analysis speed did not interact with the numerical factors, although the main effect of speed alone was significant (which seems to be related to general attentional load caused by quick motion).

Analyses of reaction times revealed a more complex pattern of results. We obtained a number of statistically significant effects, including those involving the speed factor, which shows that our design was suitable for testing the hypotheses concerning the relation between external spatial attention and mental arithmetic. However, while each of these effects can be meaningfully interpreted individually, together they do not provide a coherent picture and only some of them confirm the hypothesis about the common mechanism of RM and OM, while others contradict it. In the case of addition, quick rightward motion quickened evaluation of c + 2 outcomes, while quick leftward motion enhanced processing c + 2 outcomes in subtraction. Only this second effect conforms the RM-based hypotheses. Thus, alternative hypotheses should also be considered.

After analyzing the results of accuracy judgments, it seems that none of the motion-related factors (speed and direction) interacted with number-related factors (operation and outcome option). There was the main effect of speed, which can be reduced to the duration of the event. Accuracy was improved in longer-lasting trials, which seems trivial, as subjects had more time to conduct the calculation (an analogous effect of speed has been demonstrated in some response time analyses). Importantly, as shown in Hubbard (2015, 2017) review, distance and duration are not factors that significantly influence representational momentum and therefore cannot be part of the common RM and OM mechanism. The only significant interaction did not include any motion-related factor and showed that too-high addition results are recognized better than too-low ones, with a weak, non-significant reverse trend in subtraction, results that were contrary to what was predicted, i.e., that representational momentum would move the expected result further on the number-line in the direction of operation and thus make it harder to recognize too-high outcomes in addition and too-low outcomes in subtraction.

Our results are more consistent with another version of the attentional explanation of OM. Operation may prime attention to the operation-consistent direction on the mental number line (larger numerical magnitudes in addition and smaller magnitudes in subtraction) and thus enhance operating in this part of the numerical space, at least making decisions about higher outcomes easier in addition. Subtraction outcomes are located on the “small number” or “left” part of the number-line, which primes attention by default because of either cultural (e.g., left-to-right script direction; cf. Azhar et al., 2020; Masson et al., 2020), or biological factors e.g., right hemisphere dominance in spatial attention, which results in left visual hemifield prevalence and has been well documented in the literature on attention (Corbetta and Shulman, 2002; see also Toomarian and Hubbard, 2018). If so, no strong cuing effect is to be expected for subtraction. Note also, that in most previous studies (e.g., Knops et al., 2009b), participants were asked to choose the result from a set of options. In this case, both the hypothesis proposed above and the hypothesis based on representational momentum lead to similar predictions. True/false decisions based on actually displayed numbers and expected outcomes were forced in our study, which consequently may explain the divergence of our results with previous works.

At least some results of RT analyses may also support the hypotheses diverging from that appealing to the representational momentum mechanism. The main effect of the outcome option factor revealed faster responses to lower (c−2) outcomes. This may be a manifestation of the size effect reported in research on number processing (c.f. Feigenson et al., 2004; Dehaene, 2011), or it may be the result of a default setting of the internal focus of attention on small numbers (on the left side of the number line), which does not contradict the previous explanation. However, the analyses of reaction times also revealed a more complicated picture. In addition to findings from previous studies, which showed relationships between RTs in arithmetic and spatial directions (left/right), outcome size, and operation, our study also revealed that these relationships are modified or even radically altered by the speed of motion. Indeed, although the prediction of interactions involving a speed factor was the most innovative part of our test of the common RM and OM mechanism hypothesis, the whole pattern of these interactions does not seem to be consistent with predictions derived from this hypothesis. Only the effect of the interaction of speed and direction of movement on the processing time of too-high and too-low outcome options in subtraction (enhanced processing of too-high outcomes in the fastest trials with leftward motion) seems to conform the RM-based hypothesis. Some early works on OM (e.g., McCrink et al., 2007) also showed this effect only for subtraction. However, many later works revealeded a stronger effect for addition (e.g., Masson et al., 2018). The inverted pattern of results in addition contradicts the common OM and RM mechanism hypothesis. So, assuming that the common mechanism hypothesis is insufficient, we discuss other alternative explanations.