Spatial Attention Shifts in Addition and Subtraction Arithmetic: Evidence of Eye Movement

Abstract

Recently, it has been proposed that solving addition and subtraction problems can evoke horizontal shifts of spatial attention. However, prior to this study, it remained unclear whether orienting shifts of spatial attention relied on actual arithmetic processes (i.e., the activated magnitude) or the semantic spatial association of the operator. In this study, spatial–arithmetic associations were explored through three experiments using an eye tracker, which attempted to investigate the mechanism of those associations. Experiment 1 replicated spatial–arithmetic associations in addition and subtraction problems. Experiments 2 and 3 selected zero as the operand to investigate whether these arithmetic problems could induce shifts of spatial attention. Experiment 2 indicated that addition and subtraction problems (zero as the second operand, i.e., 2 + 0) do not induce shifts of spatial attention. Experiment 3 showed that addition and subtraction arithmetic (zero as the first operand, i.e., 0 + 2) do facilitate rightward and leftward eye movement, respectively. This indicates that the operator alone does not induce horizontal eye movement. However, our findings support the idea that solving addition and subtraction problems is associated with horizontal shifts of spatial attention.

Keywords

spatial attention shifts mental arithmetic spatial–arithmetic associations eye movement

Introduction

Several studies have indicated that there is a clear association between numbers and space; that is to say, numbers are spatially mapped (Dehaene, Bossini, & Giraux, 1993; Fischer, Castel, Dodd, & Pratt, 2003; Fischer & Knops, 2014). Recent studies have indicated that the understanding of number–space associations can also be applied to elementary arithmetic (Fischer & Shaki, 2014a, 2014b; McCrink, Dehaene, & Dehaene-Lambertz, 2007). In one seminal study, McCrink et al. (2007) showed that participants tended to overestimate the results of addition problems and underestimate the results of subtraction problems (operational momentum effect), a phenomenon that has been confirmed through observation in other research (Jiang, Cooper, & Alibali, 2014; Knops, Viarouge, & Dehaene, 2009).

Recently, it has been reported that addition can induce rightward spatial bias, whereas subtraction can cause leftward spatial attention shifts (Li et al., 2018; Liu, Cai, Verguts, & Chen, 2017; Masson & Pesenti, 2014; Mathieu, Gourjon, Couderc, Thevenot, & Prado, 2016). Masson and Pesenti (2014) used the target detection paradigm to investigate whether addition and subtraction arithmetic can induce spatial attention shifts. They found that single-digit subtraction facilitated the detection of leftward targets, while multidigit addition caused rightward spatial attention shifts. However, their study was limited insofar as they required that participants only detect targets at 450 ms after solving addition and subtraction arithmetic. To identify the particular stages in which spatial–arithmetic association manifested, further studies began to focus on the time course of spatial attention shifts induced by addition and subtraction arithmetic (Li et al., 2018; Liu et al., 2017; Masson, Letesson, & Pesenti, 2017; Zhu, Luo, You, & Wang, 2018). Researchers found that rightward targets were more quickly detected than targets in the left visual field when solving addition problems. Conversely, subtraction problems facilitated the detection of leftward targets. Notably, this spatial–arithmetic association was robust at 300 ms after solving addition and subtraction problems. This result indicates that spatial–arithmetic associations could occur in earlier stages.

Evidence of the association between elementary arithmetic and space also stems from eye movement. A study using functional magnetic resonance imaging (fMRI) found that addition induced activation patterns in the parietal cortex, which overlapped with those activated by rightward eye movement (Knops, Thirion, Hubbard, Michel, & Dehaene, 2009). Hartmann, Mast, and Fischer (2015) required participants to solve arithmetic problems presented in auditory form. They found that compared with subtraction problems, gaze position shifted more upward while calculating addition problems. Masson et al. (2017) found that gaze position shifted rightward while solving addition problems. Furthermore, Zhu et al. (2018) observed that addition and subtraction problems could facilitate rightward and leftward eye movement, respectively. Binda, Morrone, and Bremmer (2012) found that participants underestimated the results of mental additions and subtractions before the execution of saccadic eye movements. They suggested that there existed a link between the preparation of action and the representation of abstract quantities.

Although many studies have provided evidences for spatial–arithmetic associations, the mechanism of spatial–arithmetic association was still a hot issue. Current theories give different explanations for spatial–arithmetic associations. Knops, Thirion, et al. (2009), for instance, posited that adding and subtracting numbers was similar to moving to the right or left of what they called the mental number line (MNL). The MNL is spatially oriented and the hypothesis suggests that numbers are internally mapped along the line. Some studies have demonstrated that solving addition and subtraction problems was associated with horizontal shifts of attention (Klein, Huber, Nuerk, & Moeller, 2014; Knops, Dehaene, Berteletti, & Zorzi, 2014; Masson et al., 2017; Mathieu et al., 2016). The neural computational model provides a potential explanation for the shifts of spatial attention in addition and subtraction arithmetic (Chen & Verguts, 2010). The basis function networks for spatial transformations correspond to multimodal parietal areas (such as the lateral intraparietal area or the ventral intraparietal area), which play a key role in numerical arithmetic, saccadic, and attentional control.

Moreover, some studies showed that the operator could elicit spatial representation. Pinhas, Shaki, and Fischer (2014) observed that a quicker response was evident when the plus sign was associated with the right response key and similarly quick when the minus sign was matched with the left response key. This study suggests that spatial bias related to elementary arithmetic might be due to a semantic association between operation sign and space. Based on this finding, it is apparent that people tend to map the plus operator onto the right side and the minus operator onto the left side. This hypothesis has been confirmed by further research (Masson & Pesenti, 2014; Mathieu et al., 2018). Mathieu et al. (2018) used fMRI to test whether the neural mechanisms for space contributed to the processing of operation signs. They found that the simple perception of a “+” sign could activate several brain regions that supported in the orientation of spatial attention. In addition, Masson and Pesenti (2014) found that arithmetic problems involving zero could induce spatial bias. When the second operand was zero, it resulted in no eye movement on the MNL. In this study, it was observed that spatial bias was induced by a zero problem, which supports a semantic association hypothesis. However, other research found no evidence for operation-dependent spatial bias when the second operand was zero (Mathieu et al., 2016; Pinhas & Fischer, 2008). Hence, it remains unclear whether the operator could induce spatial–arithmetic associations.

Further studies on spatial–arithmetic associations include McCrink and Wynn (2009) account of operational momentum, which they named the accepting more or less heuristic. Their theory posited that individuals tended to associate plus with more and minus with less in their daily lives. This mapping association gradually aligns to a spatial dimension (right and left). Based on this heuristic, if zero plus (or minus) a digit, the magnitude will be more (or less). A greater magnitude will map to the rightward space and a lesser magnitude will relate to the left space. The magnitude of solutions may, thus, influence spatial association. This heuristic also fits with the semantic association of the operator. In the case of one number plus (or minus) zero, the magnitude will not become more (or less). Through this idea, we can investigate whether the operator influences spatial bias. Therefore, we chose the zero as the operand to explore the nature of spatial–arithmetic associations.

From the earlier studies, the mechanism of spatial–arithmetic association continues to be unknown. To clarify this issue, it is necessary to explore whether spatial–arithmetic associations are rooted in movements along the MNL or if there are semantic associations with the operator. Second, some research used arithmetic problems involving zero to investigate the role of operator (Masson & Pesenti, 2014). However, this research concentrated on arithmetic problems wherein zero was the second operand. Whether arithmetic problems in which zero is the first operand can induce similar effect remains a knowledge gap. Hence, to further examine whether the operator was crucial for spatial–arithmetic associations, we would compare the results between arithmetic problems wherein zero was the second operand and arithmetic problems in which zero is the first operand.

To address these limitations, three experiments have been conducted in this study. Experiment 1 aimed to examine whether elementary addition and subtraction problems could trigger horizontal eye movement. Experiment 2 comprised two parts, aiming to investigate the mechanism of spatial–arithmetic associations. Both experiments were designed to investigate whether addition and subtraction problems with zero as the second operand (e.g., 2 + 0) could induce spatial attention shifts. Both experiments, thus, used zero as the second operand, but in Experiment 2A, the first operand was a positive number, while in Experiment 2B, the first operand was a negative number. Experiment 2B, therefore, extended the consideration further: Whether the negative sign of the first operand (e.g., −2 + 0) could induce spatial attention shifts. Experiment 3 further examined whether the operator was related with spatial–arithmetic associations when zero as the first operand (e.g., 0 + 2).

Experiment 1

We first explored if solving addition problems would facilitate rightward eye movement and if solving subtraction problems would facilitate leftward eye movement.

Method

Participants

A total of 26 undergraduate students (10 males, all right-handed) with an age range of 18 to 25 years (M = 20.35, SD = 2.04) were recruited. All participants were naïve to the purpose of this experiment and had normal or corrected-to-normal vision. Before the experiment, all participants signed written informed consent as required by the ethics committee at Shaanxi Normal University.

Material and Apparatus

There were 40 arithmetic problems, which comprised Arabic numerals from 1 to 99. The solutions to these addition arithmetic problems were the same as the subtraction problems (such as 1 + 2 = 3 and 5 − 2 = 3). We used an eye movement tracker, Eyelink II, with a refresh rate of 500 Hz to record participants’ eye movement. Participants sat at a distance of 60 cm from the screen, and a chin and forehead stabilizer were used to prevent head movement.

Procedure and Design

The sequence of an example trial is detailed as follows (see Figure 1). At the beginning of the experiment, a fixation dot was presented randomly in nine different locations on the screen. Participants were required to track the fixation dot to ensure that their eye movement was calibrated to the tracker. After finishing calibration and validation, a fixation dot was presented on the center of the screen for 500 ms. Thereafter, the first operand, operator, and second operand (Courier New 40-point font) appeared in the center of the screen successively for a duration of 400 ms each. Having seen the arithmetic problem, participants were asked to report the answer aloud. Then, a fixation dot with two lateral boxes was presented in the screen for a variable delay (150 ms, 300 ms, or 450 ms). Thereafter, the cues (presented in the shape of a star, 3.5° of visual angle) appeared for 1 second either to the right of the fixation dot or the left. Once the cues disappeared, participants were instructed to move their eyes to the leftward box or rightward box according to the location of the cues. Participants were required to practice the exercise in 10 trial runs before the official experiment began and there were 160 trials. The experiment lasted about 45 minutes.

Figure 1.

An example of one trial showing the sequence of events and timing. The calibration, fixation, first operand, operator, and second operand were presented successively at the center of the screen. Then participants were asked to report the answer of the arithmetic problems as accurately as possible. After a variable delay (150 ms/300 ms/450 ms), participants moved their eyes to the corresponding box (left or right) according to the cues (the shape of a star).

Results

Trials in which there were errors in either the mental arithmetic task or the eye movement task were excluded from further analysis (6.03%). Furthermore, trials were excluded from further analysis if they met the following conditions: (a) in the eye movement task, trials in which the response times (RTs) were more than three standard deviations away from the mean; and (b) in the eye movement task, trials in which the vertical amplitude of the saccade were more than 3°.

Oral Report: RTs

We analyzed the RT and accuracy in varied delay conditions, respectively. A total of 0.53% of incorrect and extreme trials were excluded from the analysis in the 150 ms delay. A pairwise t test indicated no significant difference between mean RTs for addition and subtraction problems (addition: 449 ± 66 ms; subtraction: 459 ± 85 ms), t(25) = −0.75, p > .05, Cohen’s d = .147. In the 300 ms delay, 0.34% of error and extreme trials were excluded from the analysis. There was no significant difference between mean RTs for addition and subtraction problems (addition: 440 ± 66 ms; subtraction: 451 ± 81 ms), t(25) = −0.822, p > .05, Cohen’s d = .161. In the 450 ms delay, 0.56% of inaccurate trials were excluded from the analysis. Results indicated that there was no significant difference between mean RTs for addition and subtraction arithmetic problems (addition: 459 ± 71 ms; subtraction: 465 ± 79 ms), t(25) = −0.351, p > .05, Cohen’s d = .069.

Saccadic Latency

The mean saccadic latency of different delay conditions is presented in Table 1.

Table 1.

Mean Saccadic Latency (and SD) as a Function of Arithmetic, Eye Movement Orientation, and Delay in Experiment 1.

Delay	Addition			Subtraction
Delay	150 ms	300 ms	450 ms	150 ms	300 ms	450 ms
Left	291 (90)	278 (72)	306 (65)	277 (89)	246 (60)	289 (75)
Right	276 (88)	251 (63)	292 (78)	287 (65)	273 (80)	278 (67)

A 2 × 2 × 3 repeated measures analysis of variance (ANOVA) was first conducted with operation (addition or subtraction), eye movement orientation (left or right), and varied delay (150 ms, 300 ms, or 450 ms) as within-subject factors. Results indicate that there was a main effect of delay, F(2, 50) = 3.975, p < .05, $η_{p}^{2}$ = 0.14 (see Figure 2). The mean saccadic latency was fastest in the 300 ms delay (150 ms condition: 283 ms; 300 ms condition: 262 ms; 450 ms condition: 291 ms). The interaction of Operation × Eye movement orientation was significant, F(1, 25) = 21.223, p < .01, $η_{p}^{2}$ = 0.50. The interaction of Operation × Eye movement orientation × Varied delay was also significant, F(2, 50) = 3.548, p < .05, $η_{p}^{2}$ = 0.12.

Figure 2.

The mean saccadic latency in different delay conditions.

Further simple effect analysis was carried out in different delay conditions to reveal the time course of spatial–arithmetic association. In the 150 ms delay condition, the results showed that the main effect of operation was not significant, F(1, 25) = 0.061, p > .05, $η_{p}^{2}$ = 0.002. The main effect of eye movement orientation was also not significant, F(1, 25) = 0.139, p > .05, $η_{p}^{2}$ = 0.006. The interaction of Operation × Eye movement orientation, however, was significant, F(1, 25) = 5.015, p < .05, $η_{p}^{2}$ = 0.167. Further pairwise t tests indicated that there was a marginally significant difference between eye movement to the left side and right side after solving addition problems, t(25) =2.016, p = .055, Cohen’s d = .395. No other significant differences were observed, p > . 05.

In the 300 ms delay condition, the results reveal that the main effect of the operation was not significant, F(1, 25) = 0.78, p > .05, $η_{p}^{2}$ = 0.03. The main effect of eye movement orientation was also not significant, F(1, 25) = 0.005, p > .05, $η_{p}^{2}$ = 0.01. The interaction of Operation × Eye movement orientation was, however, significant, F(1, 25) = 14.07, p < .01, $η_{p}^{2}$ = 0.36 (see Figure 3). Further pairwise t tests showed a significant difference between the leftward and rightward eye movement following addition, t(25) = 2.884, p < .01, Cohen’s d = .565. For subtraction, a significant difference between the leftward and rightward eye movement was found, t(25) = −2.946, p < .01, Cohen’s d = .577. Moreover, there was a significant difference between addition and subtraction for eye movement to the left side or right side, respectively, t(25) = 3.879, p < .01, Cohen’s d = .761 or t(25) = −2.105, p < .05, Cohen’s d = .413.

Figure 3.

The interaction between operation and orientation in the 300 ms delay. **p < .01.

In the 450 ms delay condition, the results indicate that the main effect of operation was not significant, F(1, 25) = 7.981, p > . 05, $η_{p}^{2}$ = .009. The main effect of eye movement orientation was not significant either, F(1, 25) =1.786, p > . 05, $η_{p}^{2}$ = 0.067. Finally, the interaction of Operation × Eye movement orientation was not significant, F(1, 25) = 0.056, p > . 05, $η_{p}^{2}$ = 0.002.

Experiment 2

Experiment 2 contains two experiments both of which aimed to investigate whether the semantic association between operator and space can initiate spatial attention shifts.

Experiment 2A

In this experiment, we investigated whether addition and subtraction problems with zero as the second operand (e.g., 2 + 0) could induce spatial attention shifts.

Method

Participants

A total of 26 undergraduate students (8 males, all right-handed) were recruited with an age range of 18 to 25 (M = 22.11, SD = 2.45). All participants were naïve to the purpose of this experiment and had normal or corrected-to-normal vision.

Material and Apparatus

There were 20 arithmetic problems, which comprised Arabic numerals from 0 to 9. In this experiment, we added a baseline task (e.g., 0 + 0 and 0 − 0) to explore whether the operator could cause a spatial attention shift.

Procedure and Design

The procedure was the same as that in Experiment 1. We used 2 (Operation: addition, subtraction) × 2 (Eye movement orientation: left, right) × 3 (Varied delay: 150 ms, 300 ms, 450 ms) as within-subject factors.

Results

We used the same exclusion criteria for data as in Experiment 1. Trials with errors in either mental arithmetic task or eye movement task were excluded from further analysis (5.36%). We first analyzed the data in the baseline task (0 + 0 or 0 − 0). The mean saccadic latency in different delay times is presented in Table 2.

Table 2.

Mean Saccadic Latency (and SD) as a Function of Arithmetic, Orientation, and Delay in the Baseline Task of Experiment 2A.

Delay	Addition			Subtraction
Delay	150 ms	300 ms	450 ms	150 ms	300 ms	450 ms
Left	289 (61)	284 (51)	291 (62)	284 (65)	279 (54)	281 (51)
Right	278 (58)	278 (51)	283 (52)	294 (60)	269 (54)	297 (58)

The Baseline Task

We conducted a repeated measures ANOVA on the mean saccadic latency with operation (addition or subtraction), eye movement orientation (left or right), and varied delay (150 ms, 300 ms, or 450 ms) as within-subject factors. Results revealed that the interaction among Operation × Eye movement orientation × Varied delay was not significant, F(2, 50) = 0.503, p > .05, $η_{p}^{2}$ = 0.02. The interaction between operation and varied delay was not significant either, F(1, 25) = 1.763, p > .05, $η_{p}^{2}$ = 0.07.

The Main Task

Following the aforementioned investigation, we analyzed the data in the main task (e.g., 2 + 0). The mean saccadic latency (and SD) as a function of operation, eye movement orientation, and variable delay is presented in Table 3.

Table 3.

Mean Saccadic Latency (and SD) as a Function of Arithmetic, Orientation, and Delay in the Main Task of Experiment 2A.

Delay	Addition			Subtraction
Delay	150 ms	300 ms	450 ms	150 ms	300 ms	450 ms
Left	296 (66)	285 (53)	301 (62)	300 (66)	270 (54)	292 (51)
Right	289 (55)	282 (53)	296 (54)	302 (55)	285 (55)	304 (54)

A 2 × 2 × 3 repeated measures ANOVA was conducted with operation (addition or subtraction), eye movement orientation (left or right) and varied delay (150 ms, 300 ms, or 450 ms) as within-subject factors. The main effect of the delay was marginally significant, F(2, 50) = 3.19, p = .05, $η_{p}^{2}$ = 0.11. This result shows that the mean saccadic latency was the fastest in the 300 ms delay condition (150 ms condition: 297 ms; 300 ms condition: 281 ms; 450 ms condition: 298 ms). We found no interaction neither between Operation × Eye movement orientation, F(1, 25) = 2.30, p > .05, $η_{p}^{2}$ = 0.08; nor in the three-way interaction among Operation × Eye movement orientation × Varied delay, F(2, 50) = 0.08, p > .05, $η_{p}^{2}$ = 0.03.

Experiment 2B

In this experiment, we chose a negative number as the first operand (e.g., −2 + 0) to explore whether the polarity of the sign could influence spatial–arithmetic associations.

Method

Participants

A total of 26 undergraduate students (8 males, all right-handed) were recruited with an age ranging from 18 to 25 (M = 22.11, SD = 2.45). All participants were naïve to the purpose of this experiment and had normal or corrected-to-normal vision.

Material and Apparatus

There were 18 arithmetic problems, which comprised Arabic numerals from −9 to 0. The solutions of the addition and subtraction problems were the same (−9 + 0 = −9 and −9 − 0 = −9).

Procedure and Design

Results

We used the same exclusion criteria for data as in Experiment 1. Here, 4.06% error and extreme trials were excluded from the analysis. The mean saccadic latency (and SD) as a function of arithmetic, eye movement orientation, and delay is presented in Table 4.

Table 4.

Mean Saccadic Latency (and SD) as a Function of Arithmetic, Eye Movement Orientation, and Delay in Experiment 2B.

Delay	Addition			Subtraction
Delay	150 ms	300 ms	450 ms	150 ms	300 ms	450 ms
Left	295 (51)	292 (56)	321 (73)	310 (62)	299 (51)	307 (65)
Right	311 (70)	294 (54)	314 (62)	317 (62)	277 (62)	316 (54)

First, a 2 (Operation: addition, subtraction) × 2 (Eye movement orientation: left, right) × 3 (Varied delay: 150 ms, 300 ms, 450 ms) repeated measures ANOVA was conducted. The main effect of the delay was marginally significant, F(2, 50) = 3.152, p = .051, $η_{p}^{2}$ = 0.112. This result shows that the mean saccadic latency was the fastest in the 300 ms delay condition (150 ms condition: 308 ms; 300 ms condition: 291 ms; 450 ms condition: 315 ms). We did not find evidence of a significant interaction of Operation × Eye movement orientation × Varied delay, F(2, 50) = 0.913, p > .05, $η_{p}^{2}$ = 0.04. The interaction of Operation × Eye movement orientation was also not significant, F(1, 25) = 0.289, p > .05, $η_{p}^{2}$ = 0.02.

Experiment 3

In this experiment, we investigated whether addition and subtraction problems with zero as the first operand (e.g., 0 + 2) could induce spatial attention shifts.

Method

Participants

A total of 26 undergraduate students (8 males, all right-handed) were recruited with an age ranging from 18 to 25 (M = 21.69, SD = 2.20). All participants were naïve to the purpose of this experiment and had normal or corrected-to-normal vision.

Procedure and Design

There were 20 arithmetic problems, which comprised Arabic numerals from 0 to 9. Each problem repeated 4 times and 40 trials in total. The procedure of this experiment was similar to Experiment 1. We used a 2 (Operation: addition, subtraction) × 2 (Eye movement orientation: left, right) × 3 (Varied delay: 150 ms, 300 ms, 450 ms) within-subject design.

Results

Oral Report: RT

We analyzed the RT respectively in the varied delay conditions. When the varied delay was 150 ms, 0.39% of error and extreme trials were excluded from the analysis. A pairwise t test indicated no significant difference between the mean RTs for addition and subtraction problems (addition: 551 ± 92 ms; subtraction: 559 ± 101 ms), t(25) = −1.692, p > .05, Cohen’s d = .182. In the 300 ms delay condition, 0.36% of error and extreme trials were excluded from the analysis. There was no significant difference between the mean RTs for addition and subtraction problems (addition: 527 ± 89 ms; subtraction: 525 ± 94 ms), t(25) = 0.278, p > .05, Cohen’s d = .022. In the 450 ms delay condition, 0.46% of error trials were excluded from the analysis. The results showed no significant difference between the mean RTs for addition and subtraction arithmetic problems (addition: 579 ± 113 ms; subtraction: 590 ± 104 ms), t(25) = −1.008, p > .05, Cohen’s d = .223.

Saccadic Latency

We used the same exclusion criteria for data as in Experiment 1. The mean saccadic latency of different delay conditions is presented in Table 5.

Table 5.

Mean Saccadic Latency (and SD) as a Function of Arithmetic, Eye Movement Orientation, and Delay in Experiment 3.

	Addition			Subtraction
Delay	150 ms	300 ms	450 ms	150 ms	300 ms	450 ms
Left	211 (61)	224 (75)	277 (68)	220 (76)	201 (61)	267 (56)
Right	224 (78)	204 (62)	267 (57)	219 (64)	232 (69)	279 (63)

First, a 2 (Operation: addition, subtraction) × 2 (Eye movement orientation: left, right) × 3 (Varied delay: 150 ms, 300 ms, 450 ms) repeated measures ANOVA was conducted. A main effect of delay was found, F(2, 50) = 6.255, p < .01, $η_{p}^{2}$ = 0.20. The mean saccadic latency was the fastest in the 300 ms delay condition (see Figure 4). The main effect of operation was not significant, F(1, 25) = 0.625, p > .05, $η_{p}^{2}$ = 0.024, while the interaction effect between operation and eye movement orientation was significant, F(1, 25) = 14.534, p < .01, $η_{p}^{2}$ = 0.368. The interaction of Operation × Eye movement orientation × Varied delay was also significant, F(2, 50) = 9.667, p < .01, $η_{p}^{2}$ = 0.279.

Figure 4.

The mean saccadic latency in different variable delay conditions in Experiment 3.

Second, a simple effect analysis was carried out in different delay conditions to reveal the time course of spatial–arithmetic association. In the 150 ms delay condition, the results indicate that the main effect of operation was not significant, F(1, 25) = 0.112, p > .05, $η_{p}^{2}$ = 0.004. The main effect of eye movement orientation was also not significant, F(1, 25) = 1.75, p > .05, $η_{p}^{2}$ = 0.065. Furthermore, the interaction of Operation × Eye movement orientation was not significant, F(1, 25) = 0.097, p > .05, $η_{p}^{2}$ = 0.107.

In the 300 ms delay condition, the results show that the main effect of operation was not significant, F(1, 25) = 0.593, p > .05, $η_{p}^{2}$ = 0.023. The main effect of eye movement orientation was also not significant, F(1, 25) = 1.443, p > .05, $η_{p}^{2}$ = 0.055. The interaction of Operation × Eye movement orientation was, however, significant, F(1, 25) = 31.997, p < .01, $η_{p}^{2}$ = 0.561 (see Figure 5). A further simple effect analysis indicated that there was a significant difference in mean saccadic latency between the left and right in the addition condition, F(1, 25) = 9.705, p < .01, $η_{p}^{2}$ = 0.28. Specifically, after solving addition problems, the mean saccadic latency was faster in the right side (M = 204, SD = 62) than the left side (M = 224, SD = 75). As for the subtraction condition, a significant difference between the mean saccadic latency for the left and right side was found, F(1, 25) = 24.609, p < .01, $η_{p}^{2}$ = 0.496. Specifically, after solving subtraction problems, the mean saccadic latency was faster on the left side (M = 201, SD = 61) than the right side (M = 232, SD = 69). There was also a significant difference between both operations for eye movement to the left, F(1, 25) = 20.379, p < .01, $η_{p}^{2}$ = 0.449. The mean saccadic latency to the left was faster after solving subtraction problems (M = 201, SD = 61) than after solving addition problems (M = 224, SD = 75). Similarly, there was a significant difference between both operations for eye movement to the right, F(1, 25) = 21.687, p < .01, $η_{p}^{2}$ = 0.465. More specifically, the mean saccadic latency to the right was faster after solving addition problems (M = 204, SD = 62) than after subtraction problems (M = 232, SD = 69). These results indicate that addition and subtraction arithmetic could induce rightward and leftward spatial bias, respectively, in the 300 ms delay condition.

Figure 5.

The mean saccadic latency as a function of operation (addition and subtraction) and orientation (left vs. right) with a 300 ms delay. **p < .01.

In the 450 ms delay condition, the main effect of operation was not significant, F(1, 25) = 0.033, p > .05, $η_{p}^{2}$ = 0.001. The main effect of eye movement orientation was also not significant, F(1, 25) = 0.139, p > .05, $η_{p}^{2}$ = 0.006. The interaction of Operation × Eye movement orientation was furthermore not significant, F(1, 25) = 3.279, p > .05, $η_{p}^{2}$ = 0.116.

Discussion

In this study, we investigated spatial–arithmetic associations through three experiments, each testing eye movement for different conditions of elementary arithmetic in relation to the semantics of the operator and operands and their values. Experiment 1 replicated spatial–arithmetic associations in addition and subtraction problems. Experiment 2 and Experiment 3 selected zero as the operand to investigate whether these arithmetic problems could induce shifts of spatial attention. The results indicated that solving addition problems facilitated rightward eye movement while solving subtraction problems accelerated leftward eye movement.

In agreement with previous studies, our findings demonstrate that addition and subtraction problems can facilitate horizontal shifts of spatial attention (Knops et al., 2014; Knops, Thirion, et al., 2009; Li et al., 2018; Liu et al., 2017; Masson et al., 2017; Mathieu et al., 2018). According to spatial attention shifts on the MNL hypothesis, when calculating arithmetic problems, people will shift spatial attention along the MNL to activate operands and outcomes. The spatial attention shifts in addition and subtraction arithmetic were consistent with the prediction of the neural network model, which indicated that the radial basis function networks for spatial transformations were recycled while solving addition and subtraction problems. The presence of spatial–arithmetic association resulted from interactions between different brain areas such as the lateral intraparietal area and the ventral intraparietal area, which play an important role in saccadic, attentional control, and numerical arithmetic (Chen & Verguts, 2010, 2012; Knops, Thirion, et al., 2009).

In Experiment 2, we did not find evidence of spatial–arithmetic association. The baseline task (e.g., 0 + 0) and main task (e.g., 2 + 0) did not facilitate horizontal eye movement. This suggests that a single operator (“+” or “−”) does not induce spatial bias. These results indicate that the semantic association between operator and space might not induce spatial–arithmetic mapping. Importantly, these results are inconsistent with previous studies. Pinhas et al. (2014), for instance, found that the “+” sign was associated with right space, while the “−” sign was related to the left space. However, in their study, the spatial association with the operation sign occurred only in the context of arithmetic: outside of an addition or subtraction problem, there was no operation sign-related spatial bias. In addition, Mathieu et al. (2018) observed that the spatial region was activated when perceiving “+” sign and other mathematical symbols (i.e., multiple sign) also could evoke spatial representations. From these studies, an operator classification task (i.e., classifying “+” and “−” with a right-hand or left-hand response) and an operator priming task (i.e., presenting the sign before calculating addition and subtraction) all emphasize the process of the operator. They found that operator processing was able to activate a spatial semantic association.

In Experiment 3, we observed that addition and subtraction problems, in which the first operand was zero (e.g., 0 + 2 or 0 − 2), could induce rightward and leftward eye movement, respectively. This result further supports the theory of spatial attention shifts in elementary arithmetic. In comparison, the operand and operator were the same in the addition condition (e.g., 2 + 0 in Experiment 2A, 0 + 2 in Experiment 3) and the subtraction condition (e.g., 2 − 0 in Experiment 2A, 0 − 2 in Experiment 3) in both Experiments 2 and 3. However, we found evidence of spatial–arithmetic association in Experiment 3 but not in Experiment 2. This different finding indicated that negative sign might cause spatial–arithmetic associations in Experiment 3. The solution of subtraction problems was different in Experiment 2A (i.e., 2 − 0 = 0) and Experiment 3 (0 − 2 = −2).

To further test whether a negative solution contributed to spatial–arithmetic associations and, thus, added subtraction problems where the solution was a negative value in Experiments 2 and 3 (e.g., 0 − 2 = −2). First, we did not find evidence of any horizontal eye movement in Experiment 2B (−2 + 0 or −2 − 0). This reveals that a single negative digit does not affect spatial–arithmetic associations. Second, many prior studies indicate that the spatial representation of negative digits is related with their absolute value, not with the polarity of the negative sign (Shaki & Petrusic, 2005; Zhang & You, 2012). This study further confirms that the negative sign does not facilitate spatial–arithmetic associations. Finally, previous research established that the most significant spatial bias was observed in the condition of a 450 ms delay when representing a single negative digit (Fischer, 2003; Kong, Zhao, You, & Zhang, 2012). In our study, however, although there were subtraction problems (e.g., 0 − 2 = −2) in which the solution was a negative value, in Experiment 3, the spatial bias was observed only in the condition of the 300 ms delay and not in the 450 ms delay. It is plausible that, in this period, spatial–arithmetic associations were irrelevant to negative solutions. Therefore, we can eliminate the influence of the operand value and arithmetic solutions.

Our findings reveal that addition and subtraction problems can facilitate horizontal eye movement, which is inconsistent with the results of the study by Hartmann et al. (2015). This inconsistency might be related to the mode of presentation. Hartmann et al. (2015) presented the stimulus in an auditory form, while visual presentation was used in this study. In addition, the operand and operator were presented successively in an auditory form in their experiment, which might strengthen the semantic processing of the operator; it would take up more cognitive resources to process the semantics of the operator but not the process of the calculation. Hence, auditory and visual presentation differences might influence spatial–arithmetic representations. Future studies should further investigate the mechanism of the difference between auditory and visual channels.

A possible limitation of this study was that we did not investigate whether the operand order could induce different spatial representations. In Experiments 2 and 3, we exchanged the location of the zero in the arithmetic problems (e.g., 2 + 0 or 0 + 2) but observed different results (Experiment 2: no evidence of spatial–arithmetic associations; Experiment 3: significant evidence of spatial–arithmetic associations). However, previous studies found that two independent representation systems exist between the a + b expression and the b + a expression (Shaki, Sery, & Fischer, 2014), while other studies indicate that they shared a common representation (Zhou, Zhao, Chen, & Zhou, 2012). Whether the order of the operands cause different spatial representation systems should be further explored in future research.

Conclusion

In summary, we investigated, for the first time, the association between arithmetic problems involving zero and space. We found that solving addition and subtraction problems can induce rightward and leftward eye movement, respectively. Our findings indicate that the presence of spatial–arithmetic associations is related to horizontal shifts of spatial attention.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the “major project of medicine science and technology of PLA (AWS17J012)” awarded to Xuqun You.

References

Binda

Morrone

M. C.

Bremmer

(2012). Saccadic compression of symbolic numerical magnitude. PLoS One, 7, e49587.

Chen

Verguts

(2010). Beyond the mental number line: A neural network model of number-space interactions. Cognitive Psychology, 60, 218–240.

Chen

Verguts

(2012). Spatial intuition in elementary arithmetic: A neurocomputational account. PLoS One, 7, e31180.

Dehaene

Bossini

Giraux

(1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: Human Perception and Performance, 122, 25.

Fischer

(2003). Cognitive representation of negative numbers. Psychological Science, 14, 278–282.

Fischer

M. H.

Castel

A. D.

Dodd

M. D.

Pratt

(2003). Perceiving numbers causes spatial shifts of attention. Nature Neuroscience, 6, 555–556.

Fischer

M. H.

Knops

(2014). Attentional cueing in numerical cognition. Frontiers in Psychology, 5, 1381.

Fischer

M. H.

Shaki

(2014a). Spatial associations in numerical cognition—From single digits to arithmetic. The Quarterly Journal of Experimental Psychology, 67, 1461–1483.

Fischer

M. H.

Shaki

(2014b). Spatial biases in mental arithmetic. The Quarterly Journal of Experimental Psychology (Hove), 67, 1457–1460.

10.

Hartmann

Mast

F. W.

Fischer

M. H.

(2015). Spatial biases during mental arithmetic: Evidence from eye movements on a blank screen. Frontiers in Psychology, 6, 12.

11.

Jiang

M. J.

Cooper

J. L.

Alibali

M. W.

(2014). Spatial factors influence arithmetic performance: The case of the minus sign. The Quarterly Journal of Experimental Psychology (Hove), 67, 1626–1642.

12.

Klein

Huber

Nuerk

H. C.

Moeller

(2014). Operational momentum affects eye fixation behaviour. The Quarterly Journal of Experimental Psychology (Hove), 67, 1614–1625.

13.

Knops

Dehaene

Berteletti

Zorzi

(2014). Can approximate mental calculation account for operational momentum in addition and subtraction? The Quarterly Journal of Experimental Psychology (Hove), 67, 1541–1556.

14.

Knops

Thirion

Hubbard

E. M.

Michel

Dehaene

(2009). Recruitment of an area involved in eye movements during mental arithmetic. Science, 324, 1583–1585.

15.

Knops

Viarouge

Dehaene

(2009). Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect. Attention, Perception, Psychophysics, 71, 803–821.

16.

Kong

Zhao

You

Zhang

(2012). The attentional SNARC effect caused by low-level processing of negative numbers in auditory modality. Studies of Psychology and Behavior, 10, 12–17.

17.

Liu

Dong

Huang

Chen

(2018). Addition and subtraction but not multiplication and division cause shifts of spatial attention. Frontiers in Human Neuroscience, 12, 183.

18.

Liu

Cai

Verguts

Chen

(2017). The time course of spatial attention shifts in elementary arithmetic. Scientific Reports, 7, 921.

19.

Masson

Letesson

Pesenti

(2017). Time course of overt attentional shifts in mental arithmetic: Evidence from gaze metrics. Quarterly Journal of Experimental Psychology (Hove), 71, 1009–1019.

20.

Masson

Pesenti

(2014). Attentional bias induced by solving simple and complex addition and subtraction problems. The Quarterly Journal of Experimental Psychology (Hove), 67, 1514–1526.

21.

Mathieu

Epinat-Duclos

Sigovan

Breton

Cheylus

Fayol

, . . . Prado

(2018). What’s behind a “+” sign? Perceiving an arithmetic operator recruits brain circuits for spatial orienting. Cerebral Cortex, 28, 1673–1684.

22.

Mathieu

Gourjon

Couderc

Thevenot

Prado

(2016). Running the number line: Rapid shifts of attention in single-digit arithmetic. Cognition, 146, 229–239.

23.

McCrink

Dehaene

Dehaene-Lambertz

(2007). Moving along the number line: Operational momentum in nonsymbolic arithmetic. Perception & Psychophysics, 69, 1324–1333.

24.

McCrink

Wynn

(2009). Operational momentum in large-number addition and subtraction by 9-month-olds. Journal of Experimental Child Psychology, 103, 400–408.

25.

Pinhas

Fischer

M. H.

(2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109, 408–415.

26.

Pinhas

Shaki

Fischer

M. H.

(2014). Heed the signs: Operation signs have spatial associations. The Quarterly Journal of Experimental Psychology (Hove), 67, 1527–1540.

27.

Shaki

Petrusic

W. M.

(2005). On the mental representation of negative numbers: Context-dependent SNARC effects with comparative judgments. Psychonomic Bulletin & Review, 12, 931–937.

28.

Shaki

Sery

Fischer

M. H.

(2014). 1 + 2 is more than 2 + 1: Violations of commutativity and identity axioms in mental arithmetic. Journal of Cognitive Psychology, 27, 471–477.

29.

Zhang

Y. U.

You

(2012). Extending the mental number line—How do negative numbers contribute? Perception, 41, 1323–1335.

30.

Zhou

Zhao

Chen

Zhou

(2012). Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis. The Quarterly Journal of Experimental Psychology (Hove), 65, 661–674.

31.

Zhu

Luo

You

Wang

(2018). Spatial bias induced by simple addition and subtraction: From eye movement evidence. Perception, 47, 143–157.