In the past two decades, research has revealed a close link between number and space in elementary number processing (e.g., Dehaene et al., 1993; Fischer et al., 2003; Hubbard et al., 2005; Wood et al., 2008; Chen and Verguts, 2010). For example, in the seminal study of Dehaene et al. (1993), it was demonstrated that (in parity judgment) small numbers are associated with faster left-hand response and larger numbers with faster right-hand responses. This suggested a spatial-numerical association in horizontal space.

Spatial-numerical associations also exist in vertical space. In particular, there is a preference to associate small numbers with the lower part of space and larger numbers with the upper part (Schwarz and Keus, 2004; Gevers et al., 2006; Winter et al., 2015). For example, using eye movements, Schwarz and Keus (2004) found that responses to the lower space start earlier with smaller than with larger numbers; whereas responses to the upper space start earlier with larger than with smaller numbers. Evidence for a vertical spatial–numerical association also comes from the random number generation task, in which small numbers generation is associated with downward movement and large numbers generation with upward movement (Schwarz and Keus, 2004; Loetscher et al., 2010; Hartmann and Mast, 2012; Winter and Matlock, 2013). These studies suggest that there is a link between numbers and spatial bias in the vertical dimension. Hence, basic single-number processing has a well-established association with both the horizontal and the vertical dimension.

Recently, the focus of spatial–numerical associations has been extended from elementary number processing to mental arithmetic (McCrink et al., 2007; Knops et al., 2009b; Chen and Verguts, 2012; Fischer and Shaki, 2014; Masson and Pesenti, 2014). It has been reported that mental arithmetic causes spatial response biases (Knops et al., 2009a,b). Specifically, subjects prefer to select options on the right side after solving addition problems, and options on the left side after solving subtraction problems (Knops et al., 2009b; Marghetis et al., 2014). Moreover, performing addition or subtraction causes spatial shifts of attention rightward or leftward, respectively (Dormal et al., 2014; Masson and Pesenti, 2014; Mathieu et al., 2016; Liu et al., 2017; Masson et al., 2017). Using a target detection paradigm, Masson and Pesenti (2014) found that solving subtraction problems facilitates the detection of left-side targets (Experiment 1), whereas addition facilitates the detection of right-side targets (Experiment 2). This suggests that there is a close relationship between arithmetic problem solving and visuospatial attention orientation (Masson and Pesenti, 2014). These spatial shifts of covert attention in mental arithmetic are further confirmed and shown robustly at 300 ms after the arithmetic operation (Mathieu et al., 2016; Liu et al., 2017).

Such findings suggest a strong similarity between elementary number processing (e.g., Fischer et al., 2003) and mental arithmetic, and possibly similar cognitive mechanisms across these diverse tasks. However, to make this argument compelling, it needs to be demonstrated that a vertical (upper/lower) association also exists in mental arithmetic. Unfortunately, however, the investigation of spatial-arithmetic associations in the vertical dimension remains scarce. By analyzing spontaneous eye movements, it was found that during addition problems, the gaze is more directed upward than during subtraction (Hartmann et al., 2015; Masson et al., 2017). Furthermore, several studies indicate that vertical motions facilitate calculation in a congruent condition (i.e., solving addition problems when performing an upward motion or solving subtraction problems when performing a downward motion), but interfere with calculations in an incongruent condition (i.e., solving addition problems when performing a downward motion, or solving subtraction problems when performing an upward motion) (Lugli et al., 2013; Wiemers et al., 2014). These motion-arithmetic compatibility effects suggest that spatial-arithmetic associations may also occur in a vertical dimension.

If arithmetic operations are closely associated with space in vertical dimension, this raises the question of whether mental arithmetic can also induce a shift of attention to the upper or lower visual field. To address this question, in the present study we were the first to use symbolic magnitudes to investigate the spatial shifts of attention in both horizontal and vertical space in a within-subject design. This allowed us to investigate the strength of both horizontal and vertical associations, respectively, and compare them directly. If spatial-arithmetic associations have the same origins as those in single-number processing, the effect should at least be very robust in both dimensions. Thus, we adopted the target detection paradigm from our previous study (see Experiment 1 in Liu et al., 2017). A trial time course can be seen in Figure 1. Participants were instructed to perform two tasks: (1) solving an arithmetic problem and making a judgment on whether the proposed result (proposal) was correct or incorrect (mental arithmetic task); and (2) detecting whether the target (a white solid circle) was present or not (target detection task). The targets might appear horizontally (Figure 1A) or vertically (Figure 1B) arranged on bilateral sides of the fixation. RTs of detection response to targets were measured. In both horizontal and vertical conditions, the target detection task was preceded by a mental arithmetic task, and the magnitudes of the second operands and the proposals were matched for addition and subtraction. Furthermore, three variable delay (150, 300, and 450 ms) were introduced before target detection to investigate the time course in which spatial attention shifts were induced.

In the current study, we adopted a target detection paradigm to investigate the spatial-arithmetic association in both horizontal and vertical dimensions within subjects. The results indicated that solving addition problems facilitated target detection on the right side, while targets on the left side were detected faster after solving subtraction problems. The finding confirmed that mental arithmetic induces spatial shifts of attention in horizontal space. However, we did not find this spatial-arithmetic association effect in vertical space. These two aspects of our results are now discussed in more detail.

Our results of a horizontal spatial-arithmetic association were consistent with a series of earlier studies (e.g., Masson and Pesenti, 2014; Mathieu et al., 2016; Liu et al., 2017; Masson et al., 2017). Some studies have proposed that the spatial attention shifts in mental arithmetic are induced by the spatial nature of number representation and processing (e.g., Masson and Pesenti, 2014; Mathieu et al., 2016). However, in our study the second operands and the proposed results were matched in addition and subtraction. Thus, the spatial biases in mental arithmetic were unlikely to be caused directly by the spatial representation of the specific numbers, but instead must be due to the arithmetic operation (Liu et al., 2017).

In elementary number processing, small magnitudes are typically associated with left/lower space and larger magnitudes with right/upper space (e.g., Schwarz and Keus, 2004; Holmes and Lourenco, 2012; Winter et al., 2015). In spatial arithmetic, a strong case has been made that subtraction is associated with left, and addition with right (e.g., Masson and Pesenti, 2014; Mathieu et al., 2016; Liu et al., 2017). This finding suggests a strong similarity between elementary number and arithmetic processing, and suggests that also subtraction could be associated with lower space and addition with upper space. However, we found no such vertical effect for mental arithmetic, although the Bayes factor analysis indicated that more data were needed to make this conclusive. Given that also in elementary number processing the horizontal bias is stronger than the vertical one (Holmes and Lourenco, 2012), more powerful designs are needed to determine whether there is a (small) vertical bias in mental arithmetic or none at all.

Thus, what is the nature of spatial shifts of attention underlying arithmetic operations? An earlier neural computational model proposed that elementary arithmetic is based on a cognitive system for spatial transformations that is “recycled” for the purpose of elementary arithmetic (Chen and Verguts, 2012). Consistent with this model, the neural structures recruited during addition and subtraction could resemble those observed during rightward or leftward eye movement, respectively (Knops et al., 2009a). The brain substrates implicated in spatial orienting even activate when an arithmetic operator (e.g., addition sign “+”) is shown (Mathieu et al., 2017). In this account, because arithmetic only hijacks the horizontal-dimension spatial transformation system, the spatial compatibility effect does not extend to the vertical dimension.

Other variables that may play an important role in the dissociated spatial-arithmetic associations in horizontal and vertical dimensions are the reading and writing direction (e.g., Dehaene et al., 1993), or subjects’ learning environment more generally. It has been suggested that left-to-right reading is responsible for the horizontal spatial-arithmetic association (Winter et al., 2015). However, for Chinese speakers (subjects used in the current study) the vertical reading direction is rarely used, which might thus cause the null effect in our vertical dimension. Furthermore, it is also plausible that the dissociated effects in our study might be caused by the fact that for Chinese speakers’ learning environment, mental arithmetic is performed from left to right but rarely from top to down. In contrast, elementary number processing typically occurs in both directions. Consistently, existing studies (see review, Winter et al., 2015), show that a horizontal direction is used in education for both elementary number processing and mental arithmetic; but that the vertical direction is used exclusively in education of elementary number processing. Future research should test the vertical spatial-arithmetic association in a Western (i.e., non-Chinese) population.

In order to test the hypothesis that there is a similar system between elementary number processing and mental arithmetic we investigated the spatial-arithmetic associations in both horizontal and vertical dimensions. The dissociated effects in the current study cautions against overgeneralizations from elementary number processing to arithmetic based on number-space identity theories; and it points to the importance of considering individual subjects’ learning environment (Verguts and Chen, in press).

Conceived and supervised the experiments: QC and TV. Designed the experiments: QC and DL. Implemented the experiments and collected data: DL, ML, and ZL. Analyzed the results: DL. Wrote and revised the paper: DL, TV, ML, ZL, and QC.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.