According to Walsh’s (2003) theory of magnitude, the processing of size, space, time, and quantity is mediated by shared neural mechanisms within the parietal lobe. An overlapping set of areas also takes part in the visuomotor computations involved in reaching and grasping. According to Walsh’s model, it is not incidental that the two domains, magnitude processing and object-directed actions, are mediated by a shared neural network. After all, the processing of magnitude across different domains, such as time and spatial location, is required in order to effectively interact with objects in the environment. Several fMRI studies have provided supporting evidence, showing increased activation in the intraparietal sulcus (IPS) when processing magnitude (Cohen Kadosh et al., 2005) and when performing object-directed actions (Frey, Vinton, Norlund, & Grafton, 2005). Movement preparation was also found to be associated with adjacent cortical areas (i.e., the inferior and superior parietal lobule; Astafiev et al., 2003).

A different line of studies has looked more closely at the relationship between symbolic magnitude and action. For example, Glover, Rosenbaum, Graham, and Dixon (2004) found that small-size-hinting words (e.g., “grape”) resulted in smaller grip apertures than did large-size-hinting words (e.g., “apple”) when grasping a neutral object (Glover et al., 2004). Such semantic effects of size on grip aperture were found only at the initial stages of the movement, but not during the second half of the movement, when the hand approached the target object. This pattern was interpreted by the authors as supporting Glover and Dixon’s (2001) “planning-control” model. According to the planning control model, the initial programming of the movement is affected by various factors, such as the semantic and contextual properties of the visual scene. The influence of these factors is attenuated during the later stages of the movement, during which movement is based on visual cues such as the position of the hand and the object to be grasped (Glover & Dixon, 2001).

Lindemann, Abolafia, Girardi, and Bekkering (2007) provided the first evidence that numerical magnitude can affect the planning and execution of actions. In their study, visual presentation of numerically smaller digits led to faster response times (RTs) when reaching for small objects (using a precision grip), whereas numerically larger digits led to faster RTs when reaching for large objects (using a power grip; see also Moretto & Pellegrino, 2008). Numerical magnitude also affected the maximum aperture between the fingers during memory-guided grasping, with larger numbers leading to larger apertures than did smaller numbers (Lindemann et al., 2007). Recently, Chiou, Wu, Tzeng, Hung, and Chang (2012) have extended these findings by showing that the fingers’ aperture during memory-guided grasping is affected not only by absolute numerical value, but also by the relative value of the other numbers included in the task set (e.g., the numeral 5 led to a larger grip aperture when it was associated with the number 2 than when it was associated with 8). Numerical magnitude has also been found to influence estimations of the ability to effectively grasp an object between the fingers, with larger numbers leading to underestimation of grasping ability, as compared with smaller numbers (Badets, Andres, Di Luca, & Pesenti, 2007).

As we have recounted, the influence of numerical magnitude on action is limited to the initial part of the movement; it does not affect action at later stages. Fischer and Miller (2008) showed that numerical information affects the RT to initiate simple buttonpress responses but has only a weak effect on the grip force of the final response. These authors suggested that magnitude affects the planning of the movement rather than actual performance (see also more generic evidence on the role of semantic features in movement planning; e.g., Bub & Masson, 2010; Bub, Masson, & Lin, 2013). More direct evidence on the effect of numerical magnitude on movement planning was provided by Andres, Ostry, Nicol, and Paus (2008), who examined fingers trajectories during grasp. Their results showed that during the initial stages of the movement trajectory (i.e., the first half of the movement, at around 10 %–50 % of the movement trajectory), the fingers opened at a larger distance for objects with a numerically larger digit embedded on their surface than for the same objects entailing a numerically smaller digits. No effect of numerical magnitude was found during late portions of the movement (Andres et al., 2008).

Although it has been proposed that the effects of numerical magnitude on perception (Goldfarb & Tzelgov, 2005) and on action planning (Moretto & Pellegrino, 2008) are automatic in nature (but see Pansky & Algom, 1999, 2002), no evidence exists to support the idea that numerical magnitude exerts an automatic influence on visuomotor control. In the Andres et al. (2008) study, for example, a numerical property was the relevant attribute for responding, since participants were asked to place objects on the basis of the parity of the digit shown on the object.

Although magnitude is not strictly needed for a parity decision (see Ben Nathan, Shaki, Salti, & Algom, 2009; Dehaene, Bossini, & Giraux, 1993), some participants might nonetheless base their response on dividing the number by 2. Be this as it may, the participants in the Andres et al. (2008) study still focused their attention and action on a numerical feature of the presented digit (see also Andres et al., 2008; Berch, Foley, Hill, & Ryan, 1999).

The study by Moretto and Pellegrino (2008) is of further interest. The authors asked their participants to attend to a perceptual dimension of the digit (its color) and found that irrelevant numerical magnitude affected affordances for the action. Nevertheless, this study did not entail actual visuomotor control and was limited to motor responses. The two should be carefully distinguished, given the evidence, from patients and neurologically intact participants alike, that the mechanisms mediating object-directed visuomotor control can be dissociated from those mediating the mere execution of motor responses (Goodale, 2011; Ganel, Chajut, & Algom, 2008; Ganel, Freud, Chajut, & Algom, 2012; Ganel, Tanzer, & Goodale, 2008; Milner & Goodale, 2008). In the present study, our participants performed visually guided grasping movements of simple objects, while numerical magnitude served as the task-irrelevant dimension. If numerical magnitude still affects grip aperture with respect to the target (nonnumerical) attribute, the automatic influence of numerical magnitude on visually guided grasping would be supported.

We refer to an automatic process as one that occurs without intention, including cases in which the specific (automatic) process is not part of the behavioral goal, and can even hurt the explicit goal for action (Bargh, 1989; Ganor-Stern, Tzelgov, & Ellenbogen, 2007; Tzelgov, 1997). Espousing this definition of automaticity, we examined whether magnitude information influences grasping performance even when magnitude processing is irrelevant to the task at hand.

In the present study, the participants were not asked to attend to any numerical aspect of the stimulus. The stimuli were those used by Andres et al. (2008)—that is, numbers embedded in graspable objects in view. However, in the present study the numbers appeared printed in colors, and the experimental task required an action (grasping) based on the color of the digit. The participants were asked to place each object in a location defined by the digit’s color. The critical question was this: Would the numerical magnitude of the digit—irrelevant to the color classification task at hand—influence the color-induced grasping trajectories?

Method

Participants

A group of 17 right-handed students from Ben-Gurion University gave their consent to take part in the experiment (five males, 12 females; mean age 24.2 years, SD = 2.15). All had normal or corrected-to-normal vision. The participants received course credit for their participation in the experiment.

Stimuli and procedure

The overall design was a modified version of the task used by Andres et al. (2008), but in our study, participants were asked to base their grasping movements on color rather than on the digit’s parity. The experiment was conducted in a lighted room; the participants sat in front of a black tabletop while the tips of their right hand’s index finger and thumb rested on a small starting button fixed at the center of the tabletop.

Participants wore a set of LCD glasses (Translucent Technologies, Toronto, ON) with liquid-crystal shutter lenses that were used to control stimulus exposure time. The stimuli consisted of two sizes of white wooden blocks (50/60 mm in length, 17 mm high and 30 mm wide) with one of four digits embedded on each object (1, 2, 8, and 9, each one 11 mm high and 6 mm wide). The digits were colored red or blue. The experiment consisted of one experimental block in which every stimulus was presented nine times, resulting in 144 trials overall. Each trial began with the opening of the glasses for 2,000 ms. Participants were instructed to grasp the object in front of them by using their thumb and index finger. They were also instructed to place each object in one of two areas on the tabletop (10 cm closer or 10 cm farther from the initial position of the object on the vertical plain), on the basis of the color of the digit. The associations of color and location were counterbalanced across participants.

Kinematic recordings

An Optotrak Certus device (Northern Digital, Waterloo, ON) was used to track participants’ hand and finger grasping trajectories. Two infrared light-emitting diodes were placed on the right hand thumb and index finger (located on top of the center of the distal phalanges), and another diode was on the wrist of the right hand. We collected data on the location of each diode and the aperture between the index finger and thumb throughout the entire grasping movement trajectory (at a 200-Hz sampling rate). Movement onset was determined as the point in time at which the aperture between index finger and thumb increased by more than 0.1 mm for at least ten successive frames (50 ms). Movement offset was determined as the point in time at which the aperture between index finger and thumb changed by no more than 0.1 mm for at least ten successive frames (50 ms), but only after reaching the maximum grip aperture between fingers.

Trials were excluded from the analysis if (1) the participant made an error by placing the object in the wrong place after grasping it, (2) the infrared diodes placed on the participant’s fingers were not visible to the camera while moving toward the object, and (3) if the participant failed to grasp the object properly and dropped it while trying to lift it off the table. This resulted in 4 % of the trials being excluded.

Results

In order to test the effect of numerical magnitude on grip aperture throughout the movement trajectory, we used a method similar to that applied in previous studies (Andres et al., 2008; Ganel et al., 2012). We divided each trial into 11 equal movement segments (0 % to 100 %). Figure 1 shows the average effect of digit magnitude on grip aperture over time. The results depicted in the figure show an effect of numerical magnitude in earlier but not in later stages of the movement, closely replicating previous findings (see Andres et al., 2008; Glover et al., 2004).

Fig. 1
figure 1

Average differences in grip aperture between high and low numerical magnitudes along normalized time trajectories of the movement. Magnitude affected grip aperture over the first half of the movement trajectory but did not affect later stages of the trajectory, consistent with earlier findings (Andres et al., 2008). Note that at no point of the experiment were the participants asked to address any numerical aspect of the digits. The error bar denotes the confidence interval of the interaction between numerical magnitude and movement time for repeated measures designs (Jarmasz & Hollands, 2009)

Full size image

We entered the grip aperture data into a repeated measures 11 (movement) × 2 (size) × 2 (digit magnitude) analysis of variance. A significant interaction was found between movement and magnitude [F(10, 160) = 3.69, p < .05, η 2 = .19], indicating that magnitude processing had a differential effect on grip aperture at different parts of the movement. Also, we observed main effects of object (physical) size [F(1, 16) = 196, p < .001, η 2 = .92] and movement [F(10, 160) = 352.5, p < .001, η 2 = .96]. The main effect of numerical magnitude was marginally significant [F(1, 16) = 3.61, p = .076, η 2 = .18]. To complement the analysis, we used planned comparisons to test the significance of the magnitude effects during each segment of the movement. The effect of magnitude was significant at 20 % and 30 % of the movement [F(1, 16) = 11.2, p < .01, η 2 = .41; F(1, 16) = 8.9, p < .01, η 2 = .36, respectively], and approached significance at 40 % of the movement [F(1, 16) = 3.5, p < .01, η 2 = .18]. The effect was not significant at 10 % or 50 % of the movement [F(1, 16) = 1.99, p > .05, η 2 = .11; F(1, 16) = 1.16, p > .05, η 2 = .7, respectively]. No effects of magnitude were found at 60 %–90 % of the movement (all Fs < 1).

Previous findings had shown that the effect of magnitude was evident only in the first half of the movement (Andres et al., 2008; Glover et al., 2004). A planned comparison confirmed that the effect of magnitude was significant in the first half of the movement [10 %–50 %; F(1, 16) = 8.8, p < .01, η 2 = .35], but not significant in the second half [60 %–90 %; F(1, 16) < 1]. The difference between the effects of magnitude during the first and second parts of the movements was also significant [F(1, 16) = 6.4, p < .05, η 2 = .28].

To exclude the possibility that the final position of the object affected the kinematics of the reach-to-grasp movements, we compared the maximum grip apertures (MGAs) that preceded movements to the far and close locations (on the basis of the digit’s color). The average MGAs were 89.86 mm for the closer location and 89.90 mm for the farther location, and the difference in MGA between the two locations was not significant [t(16) = 0.18, p > .1].

Discussion

The main purpose of the present study was to examine whether visual presentation of a digit automatically affects motor control, so that a number’s symbolic magnitude would affect grip aperture during grasp. To test the hypothesis of automatic symbolic influence on action, we asked participants to grasp objects on the basis of the magnitude-irrelevant dimension of color. Nevertheless, the results revealed that magnitude information was processed even when it was irrelevant to the task at hand. In particular, numerical magnitude was found to affect grip aperture only during the initial stages of the grasp, with larger digits leading to larger apertures.

The pattern of data is similar to that found in a previous study using a similar design (Andres et al., 2008). Critically, however—unlike in the Andres et al. study, in which the participants were asked to base their grasping movement on a numerical aspect of the digit (parity)—our task required participants to attend to a nonnumerical, perceptual dimension of the digit (color) without any reference to a numeric property. Therefore, our results lend support to the idea of an automatic influence of numerical value on grasping preparation. Our findings are also consistent with the idea that the planning of visuomotor control and the processing of numeric magnitude are mediated by shared neural mechanisms. A set of overlapping neural network located mainly in the parietal lobe are assumed to mediate the processing of magnitude as well as the programming of visuomotor control (Göbel, Johansen-Berg, Behrens, & Rushworth, 2004; Simon, Mangin, Cohen, Le Bihan, & Dehaene, 2002). However, further behavioral and imaging studies will still be needed to establish the precise pattern of the relationship across the different modalities involved in magnitude processing and visuomotor control.

We note that, although the effect of magnitude in the early stages of movement was statistically robust, the effect size (0.7 mm) was smaller than those reported in previous studies. For example, Andres et al. (2008) reported an effect size of 1.7 mm, whereas Glover et al. (2004) reported an effect size of about 1.4 mm. Could the difference derive from the automatic nature of the responses in our study? The values in previous studies, by contrast, characterized nonautomatic responding. This explanation is consistent with dimensional-overlap models, which predict larger effects on motor responses as the (cognitive and neural) overlap between (numerical and nonnumerical) dimensions grows larger (see Fias, Lauwereyns, & Lammertyn, 2001; Lammertyn, Notebaert, Gevers, & Fias, 2007).

In conclusion, let us highlight the novel contribution of this study. A voluminous literature on numerical cognition has underscored the importance of numbers in all ways of (modern) life. Given their paramount role, it has been suggested that numerical magnitude is perceived in an automatic fashion, so that it affects performance even when it is irrelevant to the task at hand, and even when it hurts performance (Henik & Tzelgov, 1982; but see Algom, Dekel, & Pansky, 1996; Fitousi, Shaki, & Algom, 2009, for a somewhat different perspective). In this study, we moved beyond perception. We provide pioneering evidence that the automatic influence of numerical magnitude extends also to actions and visuomotor control. Our results and conclusions join in a larger body of current research that demonstrates a major role for the semantic features of objects in the planning of action with respect to those objects.