Acta Psychologica 142 (2013) 308–313
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Acta Psychologica journal homepage: www.elsevier.com/ locate/actpsy
Your neighbors deﬁne your value: A study of spatial bias in number comparison Samuel Shaki a,⁎, Martin H. Fischer b a b
Ariel University Center, Israel University of Potsdam, Germany
a r t i c l e
i n f o
Article history: Received 29 July 2012 Received in revised form 6 January 2013 Accepted 10 January 2013 Available online 16 February 2013 PsycINFO classiﬁcation: 2300 Human Experimental Psychology 2340 Cognitive Processes
a b s t r a c t Several chronometric biases in numerical cognition have informed our understanding of a mental number line (MNL). Complementing this approach, we investigated spatial performance in a magnitude comparison task. Participants located the larger or smaller number of a pair on a horizontal line representing the interval from 0 to 10. Experiments 1 and 2 used only number pairs one unit apart and found that digits were localized farther to the right with “select larger” instructions than with “select smaller” instructions. However, when numerical distance was varied (Experiment 3), digits were localized away from numerically near neighbors. This repulsion effect reveals context-speciﬁc distortions in number representation not previously noticed with chronometric measures. © 2013 Elsevier B.V. All rights reserved.
Keywords: Magnitude comparison Mental number line Numerical cognition Spatial bias
1. Introduction When we compare two numbers, say 8 and 3, we seem to instantly decide which is larger. This ability reﬂects our longstanding experience with Arabic digit symbols and is a hallmark of numeracy. Nevertheless, the speed and accuracy of such decisions vary with number magnitude and reﬂect systematic biases in numerical cognition. These biases reveal, in turn, the conceptual representation of numbers and are of great interest to cognitive scientists. Complementing this chronometric approach, the present paper looks at biases in spatial behavior during numerical cognition to obtain further insights into the cognitive representation of numbers. Several biases in numerical cognition were ﬁrst established decades ago with chronometric methods. For example, small digits (e.g., 1 or 2) are classiﬁed faster with the left hand and larger digits (e.g., 8 or 9) with the right hand, indicating a spatial–numerical association of response codes (SNARC effect; Dehaene, Bossini, & Giraux, 1993). Also, magnitude comparison is faster and more accurate for the pair 8–3 (a number distance of 5) than for the pair 8–7 (a number distance of 1; the distance effect; Moyer & Landauer, 1967). Moreover, for ﬁxed numerical distances, as in the pairs 8–7 and 2–3, the comparison is easier for the smaller magnitude pair—the size effect ⁎ Corresponding author at: Ariel University Center, Ariel, 44837, Israel. Tel.: +972 9 7929092. E-mail address: [email protected]
(S. Shaki). 0001-6918/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.actpsy.2013.01.004
(for a review, see Zbrodoff & Logan, 2005). Together, such observations were used to infer a mental number line (MNL) with speciﬁc metrical properties. Speciﬁcally, the SNARC effect has been used to assign a spatial left–right orientation to the MNL (for a review, see Wood, Nuerk, Willmes, & Fischer, 2008). Moreover, numbers are supposedly represented in order of magnitude to accommodate the distance effect in terms of harder discriminability of similar magnitudes. Finally, in order to accommodate the size effect, it is either assumed that numbers are represented with logarithmic compression (e.g., Brysbaert, 1995) or with increased variability as their magnitudes increase (e.g., Whalen, Gallistel, & Gelman, 1999); either assumption would explain harder separability of larger compared to smaller number concepts. Only recently have cognitive scientists utilized spatial behavior as an additional source of information about the cognitive representation of number concepts: numerical tasks were modiﬁed to require goal-directed pointing responses instead of button presses, and the kinematics or endpoints of the resulting motor responses were analyzed. The spatial characteristics of these motor responses often converged with evidence from chronometric measures. Consider the three chronometric effects mentioned above: ﬁrst, a SNARC-like effect occurs in a string bisection task; endpoints are biased to the left of stimulus center when the string is made of small digits and to the right of center when the string is made of large digits (Fischer, 2001). Similarly, movement times in a pointing task are contaminated by the SNARC effect (Fischer, 2003; Ishihara et al., 2006). Secondly,
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there is a systematic increase in curvature of pointing movements during magnitude classiﬁcation with reduced numerical distance from the reference value; this is an instantiation of the distance effect in spatial behavior (Song & Nakayama, 2008). The present paper extends this spatial approach to investigate how number size is mapped onto locations in the magnitude classiﬁcation task. Speciﬁcally, we explore the precision of spatial responses when participants localize the larger or smaller digit of a pair. 2. Experiment 1 The main purpose of the ﬁrst experiment was to determine how spatial responses are affected by number magnitudes and task instructions in a magnitude comparison task. Participants located the position of either the larger or the smaller number of a pair on a visually presented horizontal line that represented the interval from 0 to 10. Number distance in a pair was held constant at one, and in addition to localization accuracy the response speed was recorded to detect possible trade-offs. 2.1. Method 2.1.1. Participants Twelve Israeli students from Ariel University Center (mean age 22.8 years; 4 male; 2 left-handed) participated in one 20-min. session for course credit. All participants reported normal or corrected-tonormal vision. 2.1.2. Apparatus and stimuli Fig. 1 shows a sample trial sequence. Visual stimuli consisted of digit pairs, instructions, and a response line. All stimuli were presented on a 19" monitor with 1280× 1024 pixel resolution. Six pairs of Arabic digits with ﬁxed numerical distance (1–2; 2–3; 3–4; 6–7; 7–8; 8–9) deﬁned the stimulus set, thus yielding equally many larger and smaller responses to the four digits 2, 3, 7, and 8. All digits were shown in black 38 point Times New Roman font, separated by 36 mm, near the bottom of an otherwise white screen. Each digit pair was shown equally often in both spatial orders (e.g., 1–2 and 2–1). One-word instructions were used for each trial to indicate the location of the response. The Hebrew word “LARGER” indicated that participants should point to the location corresponding to the numerically Contact start
0 LARGER 3 4
10 LARGER 3 4
Fig. 1. Example of trial sequence, with the vertical arrow representing the mouse cursor controlled by the subject (not to scale).
larger of the two digits, and the Hebrew word “SMALLER” indicated that participants should point to the location corresponding to the numerically smaller of the two digits in the currently presented pair. These words were shown in uppercase Times New Roman font (38 point, bold), and appeared at the center of the lower third of the screen, 100 pixels above the digit pair. Participants responded with a mouse by clicking with the arrow tip of their standard mouse cursor on a horizontal black line (sized 10 × 400 pixels) that was ﬂanked by the digits 0 and 10 (Times New Roman with 30 point font size). This implies that the “true” locations of digits 1, 2, 3 etc. were at 40, 80, 120 pixels etc. along the line. The line was vertically centered on the screen but its horizontal position varied pseudo-randomly across trials between center (440 pixels), left (400 pixels) and right (480 pixels) screen positions to rule out screen-speciﬁc number–space associations. Event sequencing, randomization of trials and instructions, and recording of response locations and reaction times (RTs) were under the control of SuperLab® software (Cedrus, version 4.0.6). The experiment consisted of 216 trials, derived from the complete crossing of the factors digit pair (6), presentation order of the digit pair (2), task instruction (2), horizontal line position (3), and replication (3). The order of trials was fully randomized for each participant. 2.1.3. Procedure Participants were tested individually in a dimly lit room, seated approximately 50 cm from the center of the monitor. A trial sequence began with a start box (40 × 54 pixels) at the center of the lower third of the screen. This start box disappeared when the participant clicked on it with the mouse cursor, and 250 ms later the horizontal line, the instructional word, and the pair of digits appeared simultaneously (see Fig. 1). Participants were instructed to use the mouse cursor to accurately point to where the correct answer would be located on the line. Trials were response-terminated and the next trial began 500 ms later. Response time (RT) from onset of the digit pair to the mouse click was recorded to the nearest millisecond and the spatial (X and Y) coordinates of the mouse click were recorded to the nearest pixel. 2.2. Results We removed outliers in terms of response position (within the ﬁrst and last 10 pixels of the presented lines) and response speed (outside 950 and 4100 ms), leaving 96% of the data for analysis. Separate repeated measures analyses of variance (ANOVAs) evaluated the effects of digit magnitude (2, 3, 7, 8), instruction (select larger, select smaller) and spatial order (ascending, descending) on response position and response speed. 2.2.1. Response position There was a signiﬁcant main effect of digit magnitude, F(3, 33)= 892.17, pb .001, indicating that digits were localized farther to the right with increasing magnitude. Average positions for digits 2, 3, 7, and 8 were 68, 100, 278, and 311 pixels, respectively. All pair-wise comparisons were reliable, pb .001. These localizations were linear, Position= 41.7×Magnitude—19.3, R2 =0.998. This result signals task compliance of our participants and replicates previous ﬁndings (e.g., Siegler & Opfer, 2003). There was also a signiﬁcant main effect of instruction, F(1, 11)=10.31, pb .01, indicating that the same digits were localized farther on the right with “select larger” instructions (193 pixels on average), and farther on the left with “select smaller” instructions (186 pixels on average), respectively. All other effects were non-reliable, Fb 1. We also subtracted the “true” location from the selected location of each response, which results in negative scores for left-biased responses and positive scores for right-biased responses. Analyzing these scores conﬁrmed the reliable bias induced by task instruction,
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Spatial Bias (pixels) -25
3.1. Method 5 2
Target Digit Select Larger
Select Smaller 8
Fig. 2. Spatial bias (selected location minus “true” location on the line) in Experiment 1.
F(1, 11) = 10.36, p b .01. This novel result is depicted in Fig. 2 and indicates a task-speciﬁc bias in the representation of digit magnitude. 2.2.2. Response speed There was a signiﬁcant main effect of digit magnitude, F(3, 33) = 8.32, p b .001, indicating that smaller digits were localized faster than larger digits, with average localization times of 1821, 1909, 2131, and 2002 ms for digits 2, 3, 7, and 8, respectively. This result constitutes a size effect. Localizations for digit 7 were reliably slower than all other localizations, p b = .05, and also those for 2 compared to 8, p = .01. There was also a signiﬁcant main effect of instruction, F(1, 11) = 30.48, p b .01, indicating that the same digits were localized faster with “select larger” instructions (1919 ms) than with “select smaller” instructions (2013 ms), respectively. However, there was no reliable interaction, F(3, 33) = 1.43, p > .25, indicating the absence of a semantic congruity effect. The marginal effect of order, F(1, 11) = 4.43, p b .06, reﬂected 26 ms slower decisions with descending compared to ascending order of digit pairs. There were no further reliable interactions, all p values > .32. 2.3. Discussion The ﬁrst experiment documented the usefulness of spatial responses in magnitude comparisons to investigate biases in the cognitive representation of numbers. Speciﬁcally, we discovered a task-dependent distortion in the magnitude-to-location mapping, with different perceived magnitudes when selecting the larger or the smaller of two values. This effect generalized across all four magnitudes investigated and was not contaminated by a speed–accuracy trade-off. Does our result indicate that the meaning of numbers depends on their use? Before being in a position to answer this important question, we wished to address a limitation of the ﬁrst experiment. Specifically, we were concerned about the lack of a baseline condition of single digit localization, from which to decide whether the spatial bias is equally strong under both selection instructions. Experiment 2 addressed this issue by recording the localization of single digits. 3. Experiment 2 The main purpose of Experiment 2 was to replicate the novel ﬁnding of a task-speciﬁc spatial bias in the magnitude comparison task, and to include a single digit localization baseline that would allow us to determine whether this bias reﬂected equal contributions from both task instructions. In an attempt to add further value to this replication, we blocked the task instruction to assess whether this strategic factor would modulate the results (as was found in Shaki, Leth-Steensen, & Petrusic, 2006).
3.1.1. Participants Fourteen Israeli students from Ariel University Center (mean age 21.4 years; 5 male; 3 left-handed) participated in one 25-min session for course credit. All participants reported normal or corrected-tonormal vision. 3.1.2. Stimuli and design The experimental design was essentially the same as that in Experiment 1, except for the fact that instructions were constant over a block, instead of randomly varying from trial to trial. The order of “select smaller” and “select larger” blocks was counterbalanced between participants. The total number of trials was 144, derived from the complete crossing of the factors digit pair (6), presentation order of the digits (2), instruction (2), horizontal line position (3), and replication (2). Before the comparison task, all participants completed a simple pointing task, in which single digits (1–4; 6–9) appeared six times on the screen and participants had to indicate their locations on the horizontal line. 3.2. Results We removed outliers in terms of response position (within the ﬁrst and last 10 horizontal pixels of the presented lines, as well as those outside + or − 30 vertical pixels from the line) and response speed (slower than 6000 ms, to ensure comparable number of observations as in the ﬁrst study), leaving 94% of the data for analysis. 3.2.1. Localization task Separate repeated measures ANOVAs evaluated the effects of digit magnitude (1–4, 6–9) on response position and response speed. There was a reliable main effect of digit magnitude on response position, F(7, 91)= 641.05, p b .001. Average localizations for digits 1, 2, 3, 4, 6, 7, 8 and 9 were 34, 67, 94, 131, 237, 278, 312, and 346 pixels, respectively. All pair-wise comparisons were signiﬁcant, p b .01, indicating the typical linear performance pattern, Position = 40.98 × Magnitude— 17.47, R2 = 0.993 (e.g., Siegler & Opfer, 2003). There was also a reliable main effect of digit magnitude on response speed, F(7, 91) = 10.02, p b .001. Average localization speed for digits 1, 2, 3, 4, 6, 7, 8 and 9 was 1809, 2138, 2502, 2501, 2475, 2784, 2387, and 1875 ms, respectively. Pair-wise comparisons between 7 and both 1 and 2, as well as between 4 and 9 were reliable, p b .05, indicating an inverted U-shaped proﬁle with fastest localizations for the most extreme magnitudes at both ends of the line. 3.2.2. Magnitude comparison task Separate repeated measures ANOVAs evaluated the effects of instruction (select larger, select smaller), display order (ascending, descending) and digit magnitude (2, 3, 7, 8) on response position and response speed. There was a main effect of instruction, F(1, 13)= 6.09, pb .05, with more rightward responses for larger compared to smaller instructions (191 vs. 186 pixels, respectively). The main effect of order was not reliable, F(1, 13) b 1, but the main effect of digit was signiﬁcant, F(3, 39)= 591.74, p b .001. Digits 2, 3, 7, and 8 were localized at 68, 93, 280, and 313 pixels, respectively, with all pair-wise comparisons reliable, pb .001, and further evidence for a linear mapping of number to position, Position= 42.50 × Magnitude—24.13, R2 = 0.996 (e.g., Siegler & Opfer, 2003). There were no further reliable effects, all p values >.29. There were no reliable main effects of instruction or order, both F(1, 13) b 1. The signiﬁcant main effect of digit, F(3, 39) = 12.01, p b .001, was due to reliably faster localization of digit 2 compared to all other digits, all p values b .05 Average localization times for digits 2, 3, 7, and 8 were 2017, 2365, 2650, and 2341 ms, respectively. There were no further reliable effects, all p values >.24.
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3.2.3. Position comparison across tasks To determine spatial biases from the two number comparison instructions relative to number localization, we computed the localization difference (comparison task mean minus localization task mean), separately for each participant under each task instruction and for each digit. These difference scores were analyzed with a task (larger, smaller) × digit (2, 3, 7, 8) ANOVA. There was a reliable effect of task, F(1, 13) = 5.42, p b .05, due to positive (rightward) localization biases in the “select larger” conditions (on average + 2.9 pixels) and negative (leftward) localization biases in the “select smaller” conditions (on average −1.9 pixels). This result is depicted in Fig. 3. However, none of the difference scores for individual digits differed reliably from zero, all p values > .38, and the absolute deviations from zero were equally sized under “select smaller” instructions and under “select larger” instructions (5.2 and 5.6 pixels, respectively, p >.6). Neither the main effect of digit nor the interaction was reliable, both F valuesb 1. There were no reliable effects in the corresponding analysis of response speed, all F values> 1. 3.3. Discussion The results of Experiment 2 conﬁrmed our earlier ﬁnding of systematic instruction-dependent differences in the association between the magnitude of a digit and its position on the number line. The pattern of results (see Fig. 3) suggests that both instructions induced approximately equally-sized biases. This observation was not contaminated by trade-offs with response speed or by the change from randomized to blocked task instructions. The result implies that the meaning of a digit is affected by its use. Before we can interpret this apparent instruction bias, we need to consider a confound. Speciﬁcally, we cannot determine whether this bias reﬂects the instruction itself or the mere presence of a neighboring value: for a given instruction there was always a speciﬁc digit of similar magnitude present. For example, when locating the digit 3 under the instruction “select larger”, the other stimulus in the pair was always 2, while under the instruction “select smaller” its ﬁxed neighbor was always 4. The main motivation for Experiment 3 was to unconfound instruction and comparison magnitude, and to assess the effect of numerical distance on the spatial bias in magnitude comparison. 4. Experiment 3 In the last experiment, we compared the effects of numerical distances 1 and 6 on localization performance, in order to unconfound inﬂuences of instruction and neighboring value on spatial performance. If instructions alone are responsible for the previous results then we expect no differences between the two numerical distance conditions within each instruction. If, however, the neighboring value inﬂuences the localization of a target digit, then we expect one of two possible
Spatial Bias (pixels) -10
Target Digit Select Larger
Select Smaller 8
Fig. 3. Spatial performance in Experiment 2.
outcomes: on the one hand, the localization bias might decrease with increasing numerical distance in a pair because the context number is less close and hence has a smaller effect on the target number. On the other hand, the localization bias might be augmented by the numerical distance in the number pair, such that larger number distances augment the “select larger” effect and vice versa. Response speed was expected to replicate the well-documented distance effect, with faster decisions for larger distances. 4.1. Method 4.1.1. Participants Twenty-three Israeli students from Ariel University Center (mean age 22.2 years; 6 male; 4 left-handed) participated in one 30-min session for course credit. All participants reported normal or correctedto-normal vision. 4.1.2. Stimuli and design The experimental design was similar to Experiment 2, except for the fact that more digit pairs were tested to assess the distance effect. Speciﬁcally, we used the six pairs from before with a number distance of one (1–2, 2–3, 3–4, 6–7, 7–8, 8–9), as well as three new pairs with a number distance of six (1–7, 2–8, 3–9). Thus, we were in a position to assess distance effects (i.e., number distance 1 vs. 6) separately under the “select smaller” instruction for target digits 1, 2, and 3, and under the “select larger” instruction for target digits 7, 8, and 9. We presented each digit pair equally often in ascending and descending order (e.g., 1–2 and 2–1) and used a randomized order of task instructions. We displaced left and right lines by 10 pixels from the center and removed the central line condition to reduce the number of trials to 216, derived from the complete crossing of the factors digit pair (9), presentation order (2), instruction (2), horizontal line position (2), and replication (3). As before, a localization block (identical to that of Experiment 2) preceded the comparison block for each participant. 4.2. Results As before, we removed outliers in terms of response position (within the ﬁrst and last 10 horizontal pixels of the presented lines, as well as those outside + or − 30 vertical pixels from the line) and response speed (slower than 6000 ms), leaving 96% of the data for analysis. Data from one participant were discarded completely because of too few observations in some conditions of the comparison task following this trimming procedure, leaving 22 data sets. 4.2.1. Localization task Separate repeated measures ANOVAs evaluated the effects of digit magnitude (1–4, 6–9) on response position and response speed. There was a reliable main effect of digit magnitude on response position, F(7, 147) = 2322.07, p b .001. Average localizations for digits 1, 2, 3, 4, 6, 7, 8 and 9 were 36, 71, 101, 139, 238, 279, 311, and 353 pixels, respectively. All pair-wise comparisons were signiﬁcant, p b .01, indicating the typical linear performance pattern, Position= 40.71 × Magnitude—12.52, R 2 = 0.996 (e.g., Siegler & Opfer, 2003). There was also a reliable main effect of digit magnitude on response speed, F(7, 147) = 10.02, p b .001. Average localization speed for digits 1, 2, 3, 4, 6, 7, 8 and 9 was 1495, 1897, 2124, 2283, 2371, 2340, 2064, and 1563 ms, respectively. Pair-wise comparisons indicated again the inverted U-shaped proﬁle with fastest localizations for the most extreme magnitudes at both ends of the line. 4.2.2. Magnitude comparison task Repeated measures ANOVAs evaluated the effects of instruction (select smaller, select larger), distance (1 vs. 6), display order (ascending,
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descending) and digit magnitude (3 levels) on response position and response speed. 22.214.171.124. Response position. Response position showed a signiﬁcant main effect of instruction, F(1, 21) =7083, p b .001, due to localizing overall more smaller target digits under “select smaller” than under “select larger” instructions (mean of 69 pixels for target digits 1, 2, and 3 vs. mean of 313 pixels, for target digits 7, 8 and 9, respectively). There was also a signiﬁcant main effect of digit, F(2, 42) =462.71, p> .001, reﬂecting task compliance. More interesting are the two reliable interactions we found. One was between instruction and distance, F(1, 21) = 13.72, p b .001. Under “select smaller” instructions, we observed overall more leftward responses with the smaller compared to the larger numerical distance (64 vs. 73 pixels, respectively). Thus, participants localized a small target digit (1, 2, or 3) more leftward in the presence of another small digit (2, 3, or 4) and less leftward in the presence of another larger digit (7, 8 or 9). In contrast, with the “select larger” instruction, we observed overall more rightward responses with the smaller compared to the larger numerical distance (316 vs. 309 pixels, respectively). Thus, participants localized the large target digits (7, 8 or 9) more rightward in the presence of another large digit (6, 7, or 8) and less rightward in the presence of another smaller digit (1, 2, or 3). The other reliable interaction was between instruction and digit magnitude, F(2, 42) = 4.24, p b .05. Under “select smaller” instructions, we observed an average shift of 30.8 pixels per digit, whereas under “select larger” instructions we observed an average shift of 36.9 pixels per digit. 126.96.36.199. Response speed. Response speed showed a reliable main effect of instruction, F(1, 21) = 5.36, p b .05, with faster decisions under “select smaller” instructions compared to “select larger” instructions (means of 2148 ms and 2251 ms, respectively). The only other reliable main effect was for digit magnitude, F(2, 42)= 462.71, p b .001, due to faster localization of the spatially more lateral number positions, with average localization times of 2163, 2249, and 2188 ms for the small, medium and large digits of each set, respectively. The interaction between digit magnitude and instruction, F(2, 42)= 42.93, p b .001, further clariﬁed this interpretation by showing that, under “select smaller” instructions, there was a selective advantage for digit 1 over 2 and 3 (means of 1826, 2166, and 2452 ms, respectively), while under “select larger” instructions, there was a selective advantage for digit 9 over 8 and 7 (means of 1924, 2332, and 2499 ms, respectively). The ﬁnal signiﬁcant result was a triple interaction of instruction, distance, and digit magnitude, F(2, 42) = 6.66, p b .01. This was due to a stronger distance effect in the midrange of target digits compared to the extreme target digits. 4.2.3. Position comparison across tasks As before, we determined spatial biases from the two number comparison instructions relative to number localization. We computed the localization difference (comparison task mean minus localization task mean), separately for each participant under each task instruction and for each digit. These difference scores were analyzed with a task instruction (larger, smaller) × numerical distance (1, 6) × digit magnitude (3 levels) ANOVA, where digit magnitudes were 1, 2, and 3 for the “select smaller” level of the Instruction factor, and 7, 8, and 9 for the “select larger” level of the Instruction factor. This analysis of position differences yielded no reliable main effects, all F values b 1. The only reliable result was a signiﬁcant interaction between instruction and distance, F(1, 21) = 13.99, p b .001, that is depicted in Fig. 4. Under “select smaller” instructions (black bars, digits 1–3), numerically near neighbors induced reliable left shifts of − 5.0 pixels away from the neighbor, t(21) = 3.45, p b . 01, while numerically far neighbors tended to induce right shifts of 3.6 pixels, t(21) = 1.07, p > .29. Under “select larger” instructions (white bars, digits 7–9), there were no reliable effects although numerically
Spatial Bias (pixels) -10
Neigbour Digit's Distance
Select Smaller Select Larger far
Fig. 4. Spatial performance in Experiment 3. The upper part (near neighbors) represents conditions with number distance 1. The lower part (far neighbors) represents conditions with number distance 6.
near neighbors tended to induce right shifts of 1.9 pixels while numerically far neighbors tended to induce left shifts of − 5.5 pixels (both p values > .11). The absolute deviations from zero were equally sized under “select smaller” instructions and under “select larger” instructions (7.8 and 8.6 pixels, respectively, p > .67). 4.3. Discussion The third experiment unconfounded the inﬂuences of instruction and neighboring number on spatial performance in magnitude classiﬁcation. The results clearly indicated a repulsion bias when there was a nearby distractor number of similar magnitude (i.e., small distance). This outcome shows that the results of the previous two experiments were not necessarily due to instructional bias; instead they suggest that participants tried to make their responses to the targets spatially distinct from the neighboring numbers. This is evidenced by the interaction between instruction and distance: Across all magnitudes, participants were more strongly biased when the distance to the neighbor was small, thereby inducing the repulsion effect; the direction of this bias was away to the left when selecting a smaller number and to the right when selecting a larger number. A potential confound is the fact that trials with larger numerical distances required responses into contralateral hemi-space, thereby introducing a biomechanical contribution to the results. However, the fact that the repulsion effect occurred at both ends of the line argues against this possibility. Furthermore, the data also argue against an attentional account of our results. According to such an account, trials with small numerical distance contain either two large or two small digits, thus biasing attention to the right or left side, respectively (Fischer et al., 2003). However, the results of Experiment 2 already showed that, even when only small digits are present, there is a right bias under the instruction “select larger” (see Fig. 3). 5. General discussion Three experiments studied the spatial behavior in a magnitude comparison task. Speciﬁcally, we measured how the task instructions and the numerical context affect participants' performance of locating numbers on a horizontal line. Experiment 1 (mixed instructions) and Experiment 2 (blocked instructions) established that the same digits were localized farther on the right with “select larger” instructions and farther on the left with “select smaller” instructions. Furthermore, Experiment 3 revealed a spatial repulsion from the other number when it was numerically close. We now turn to a contextualization and interpretation of our ﬁnding. When locating a number in a magnitude comparison task there is a spatial repulsion that originates from projecting the magnitudes of both the target and the distractor number onto space. This novel
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observation extends the recent trend to apply motor versions of numerical cognition paradigms (Fischer, 2001, 2003; Ishihara et al., 2006; Pinhas & Fischer, 2008; Song & Nakayama, 2008) by documenting a systematic spatial bias in magnitude comparison. The result also generalizes distractor effects that have previously been observed in simple pointing tasks (e.g., Fischer & Adam, 2001) to the domain of symbolic magnitudes. The effect indicates that participants tried to make their responses to the targets spatially distinct from the neighboring numbers. How can this novel result be interpreted? A perceptual analogy to our ﬁnding is the well-known Ebbinghaus (or Titchener) illusion that results when a target is embedded in distractors of different size: a disk surrounded by smaller disks appears larger, and v.v. the same disk surrounded by larger disks appears smaller. Finding a similar size–contrast illusion with symbolic digits indicates that magnitude is spontaneously computed for both target and distractor digits and suggests that the mentally represented value of a digit depends to some extent on the value of its neighbors. Is the spatial behavior bias found in our study related to the well-known semantic congruity effect in chronometric studies of comparative judgments (e.g., Banks, Fujii, & Kayra-Stuart, 1976; Shaki et al., 2006)? A typical semantic congruity effect in number comparison involves an interaction between the direction mentioned in the instruction (select smaller or larger number) and the relative size of the pair (small or large). In contrast to this, our “repulsion” effect depends only on the relative size of the neighbor. Nevertheless, it may be the case that the apparent inﬂuence of the selection instruction (see Experiments 1 and 2) can throw light on the mechanism of the semantic congruity effect that has so far always been documented in response speed. We found that with the instruction to choose the smaller of two small numbers (e.g. 1, 2) participants will locate the small number farther to the left, hence closer to the end of the continuum. Similarly, with the instruction to choose the larger of two large numbers (e.g. 8, 9) participants will locate the large number farther to the right, hence again closer to the end of the continuum. Instead, locating the larger of two small numbers or the smaller of two large numbers will yield a bias toward the center of the continuum. Transferring this process to the mental representation of numbers, when there is congruity between the instructional word and the pair size, participants will locate the target numbers closer to both ends of the mental number line (thus generating an apparent “instruction effect”). In turn, this may facilitate participant's use of an “end item strategy” (Moyer & Dumais, 1978) which generates the well-known semantic congruity effect in response speed (e.g., Shaki et al., 2006). Support for such converging insight about the mental representation of magnitudes from temporal and spatial performance measures comes from our ﬁnding of faster localization times for more laterally positioned digits. Response speed was mainly analyzed to protect the interpretation of spatial biases from the possible contamination of speed–accuracy trade-offs. Aside from succeeding in this objective, we also found that spatially more lateralized digits 1 and 9 were projected onto space more easily than the more central digits, despite the longer physical distances from the start box that was clicked to initiate each trial. Other response speed results probably reﬂect a complex combination of faster attention allocation as well as slower response discrimination for adjacent magnitudes. A possible concern could be the use of Israeli participants in the present study because Israeli participants do not show a SNARC effect (Shaki et al., 2009). The present results might therefore not reﬂect number representation in the general population. However, we
believe that the visually available ﬂankers in our task impose a directionality that overrules mental conﬂicts or cultural biases which account for unusual SNARC in Israeli participants. This was shown by Fischer, Mills, and Shaki (2010, Experiment 2), where the visual position of small and large numbers within a text induced a SNARC effect in Israeli participants. However, more work is needed to establish a better understanding of the preconditions for spatial–numerical mappings (Fischer, 2012; Shaki & Fischer, 2012). 6. Conclusions In conclusion, we have documented a novel spatial bias in numerical comparison that suggests that the cognitive representation of number magnitude is distorted in relation to the value of simultaneously processed numbers. In other words, a number's neighbors deﬁne its value. This cognitive distortion of magnitude contributes to the ubiquitous size effect, thus demonstrating how spatial biases can reveal previously unknown aspects of number representation. References Banks, W. P., Fujii, M., & Kayra-Stuart, F. (1976). Semantic congruity effects in comparative judgments of magnitude of digits. Journal of Experimental Psychology. Human Perception and Performance, 2(3), 435–447. Brysbaert, M. (1995). Arabic number reading: On the nature of the numerical scale and the origin of phonological recoding. Journal of Experimental Psychology. General, 124, 434–452. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology. General, 122, 371–396. Fischer, M. H. (2001). Number processing induces spatial performance biases. Neurology, 57(5), 822–826. Fischer, M. H. (2003). Spatial representations in number processing — Evidence from a pointing task. Visual Cognition, 10, 493–508. Fischer, M. H. (2012). A hierarchical view of grounded, embodied and situated numerical cognition. Cognitive Processing, 13, S161–S164. Fischer, M. H., & Adam, J. J. (2001). Distractor effects in pointing: The role of spatial layout. Experimental Brain Research, 136(4), 507–513. Fischer, M. H., Castel, A. D., Dodd, M. D., & Pratt, J. (2003). Perceiving numbers causes spatial shifts of attention. Nature Neuroscience, 6(6), 555–556. Fischer, M. H., Mills, R. A., & Shaki, S. (2010). How to cook a SNARC: Number placement in text rapidly changes spatial–numerical associations. Brain and Cognition, 72, 333–336. Ishihara, M., Jacquin-Courtois, S., Flory, V., Salemme, R., Imanaka, K., & Rossetti, Y. (2006). Interaction between space and number representations during motor preparation in manual aiming. Neuropsychologia, 44, 1009–1016. Moyer, R. S., & Dumais, S. T. (1978). Mental comparison. In G. H. Bower (Ed.), The psychology of learning and motivation, Vol. 12. (pp. 117–155)New York: Academic Press. Moyer, R. S., & Landauer, T. K. (1967). Time required for judgments of numerical inequality. Nature, 215, 1519–1520. Pinhas, M., & Fischer, M. H. (2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109, 408–415. Shaki, S., & Fischer, M. H. (2012). Multiple spatial mappings in numerical cognition. Journal of Experimental Psychology. Human Perception and Performance, 38(3), 804–809. Shaki, S., Fischer, M. H., & Petrusic, W. M. (2009). Reading habits for both words and numbers contribute to the SNARC effect. Psychonomic Bulletin and Review, 16(2), 328–331. Shaki, S., Leth-Steensen, C., & Petrusic, W. M. (2006). Effects of instruction presentation mode in comparative judgments. Memory & Cognition, 34(1), 196–206. Siegler, R. S., & Opfer, J. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237–243. Song, J. H., & Nakayama, K. (2008). Numeric comparison in a visually-guided manual reaching task. Cognition, 106, 994–1003. Whalen, J., Gallistel, C., & Gelman, R. (1999). Nonverbal counting in humans: The psychophysics of number representation. Psychological Science, 10(2), 130–137. Wood, G., Nuerk, H. -C., Willmes, K., & Fischer, M. H. (2008). On the cognitive link between space and number: A meta-analysis of the SNARC effect. Psychology Science Quarterly, 50(4), 489–525. Zbrodoff, J. N., & Logan, G. D. (2005). What everyone ﬁnds: The problem size effect. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 331–347). New York: Psychology Press.