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Heed the signs: Operation signs have spatial associations a
b
c
Michal Pinhas , Samuel Shaki & Martin H. Fischer a
Department of Psychology, Ben-Gurion University of the Negev, BeerSheva, Israel b
Department of Behavioral Sciences, Ariel University, Ariel, Israel
c
Department of Psychology, University of Potsdam, Potsdam, Germany Accepted author version posted online: 18 Feb 2014.Published online: 11 Mar 2014.
To cite this article: Michal Pinhas, Samuel Shaki & Martin H. Fischer (2014) Heed the signs: Operation signs have spatial associations, The Quarterly Journal of Experimental Psychology, 67:8, 1527-1540, DOI: 10.1080/17470218.2014.892516 To link to this article: http://dx.doi.org/10.1080/17470218.2014.892516
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THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2014 Vol. 67, No. 8, 1527–1540, http://dx.doi.org/10.1080/17470218.2014.892516
Heed the signs: Operation signs have spatial associations Michal Pinhas1, Samuel Shaki2, and Martin H. Fischer3 1
Department of Psychology, Ben-Gurion University of the Negev, Beer-Sheva, Israel Department of Behavioral Sciences, Ariel University, Ariel, Israel 3 Department of Psychology, University of Potsdam, Potsdam, Germany
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Mental arithmetic shows systematic spatial biases. The association between numbers and space is well documented, but it is unknown whether arithmetic operation signs also have spatial associations and whether or not they contribute to spatial biases found in arithmetic. Adult participants classified plus and minus signs with left and right button presses under two counterbalanced response rules. Results from two experiments showed that spatially congruent responses (i.e., right-side responses for the plus sign and left-side responses for the minus sign) were responded to faster than spatially incongruent ones (i.e., left-side responses for the plus sign and right-side responses for the minus sign). We also report correlations between this novel operation sign spatial association (OSSA) effect and other spatial biases in number processing. In a control experiment with no explicit processing requirements for the operation signs there were no sign-related spatial biases. Overall, the results suggest that (a) arithmetic operation signs can evoke spatial associations (OSSA), (b) experience with arithmetic operations probably underlies the OSSA, and (c) the OSSA only partially contributes to spatial biases in arithmetic. Keywords: Mental arithmetic; Mental number line; Operational momentum; Pointing; Spatial– numerical association of response codes effect.
Single numbers usually evoke spatial associations so that smaller numbers are associated with left space and larger numbers with right space (spatial– numerical association of response codes, SNARC, effect; Dehaene, Bossini, & Giraux, 1993). This effect is very pervasive, at least in left-to-right readers (Dehaene et al., 1993; Shaki, Fischer, & Petrusic, 2009) and obtains in a number of different tasks (for reviews see Fias & Fischer, 2005; Hubbard, Piazza, Pinel, & Dehaene, 2005; for a meta-analysis, see Wood, Nuerk, Willmes, & Fischer, 2008). Together with other performance signatures of numerical cognition (i.e., the distance
and size effects, Moyer & Landauer, 1967), the spatial association of numbers has been taken to suggest that numbers are cognitively represented as a spatially ordered continuum of magnitudes, often referred to as the “mental number line”. Recently, systematic biases with associated spatial errors were also found in mental arithmetic, thus extending the SNARC effect into the domain of everyday numerical cognition. First, McCrink, Dehaene, and Dehaene-Lambertz (2007) used dot pattern arithmetic to document a bias towards larger numerosities when evaluating addition outcomes and a bias towards smaller numerosities
Correspondence should be addressed to Michal Pinhas, Department of Psychology, Ben-Gurion University of the Negev, BeerSheva, 84105, Israel. E-mail: [email protected] We thank Gavin Revie and Clare Kirtley for their help with data collection. M.P. was supported by a Visiting Researcher’s Grant from the UK’s Experimental Psychology Society. M.H.F. was supported by the British Academy [grant number SG 46947] while working at the University of Dundee. © 2014 The Experimental Psychology Society
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when evaluating subtraction outcomes. This effect was termed “operational momentum” (OM) in accord with the metaphoric notion of calculations as movements along a mental number line (Lakoff & Nunez, 2000; see also McCrink & Wynn, 2009). Subsequently, Pinhas and Fischer (2008) used a spatial pointing task with singledigit problems to document a right bias after adding and left bias after subtracting, thereby extending OM to calculations with Arabic digits and into the spatial domain (hence, spatial OM or SOM). Similar spatial biases were also found with an extended numerical range in both symbolic and nonsymbolic arithmetic (Knops, Viarouge, & Dehaene, 2009). Using a neuroscientific approach, Knops, Thirion, Hubbard, Michel, and Dehaene (2009) identified common brain parietal structures that are involved in addition and subtraction as well as in right- and leftward-directed saccadic eye movements. After training a saccade classifier algorithm to infer the left or right direction of eye movements from brain activation measured in the posterior parietal cortex, Knops and colleagues tested whether this classifier would generalize to mental calculation. Without further training, functional magnetic resonance imaging (fMRI) images of brain activation during addition were classified 61% of the time as rightward saccades, across symbolic and nonsymbolic notations. However, the classification of subtraction images as leftward saccades was below chance level (49.1%). These results associate the cognitive representations of arithmetic operations to space but also raise questions about the underlying mechanism. Recently, Chen and Verguts (2012) modelled the OM and SOM effects based on an alternative account for OM, originally suggested by McCrink et al. (2007). Specifically, the authors assumed that over- and underestimation of additions and subtractions occur when these operations are performed on a compressed mental representation of magnitudes (e.g., logarithmic scaling of the metal number line). For example, adding two numbers n1 + n2 that are represented in a compressed manner [e.g., as log(n1) and log(n2)] will usually lead to larger values than the real outcome [e.g., as log
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(n1) + log(n2) equals log(n1) × log(n2)]. The model implements radial basis functions to simulate flawed “decompression” of numerical representations during parietal arithmetic computations as the source of OM. A closer inspection of the behavioural signatures of spatial–numerical biases and their interrelationship can help us to understand the mechanisms involved in SOM. At first glance, arithmetic operations can be spatially biased due to either the spatial associations of the individual digits of the equation (the operands) or the anticipated small or large result (e.g., from subtraction or addition operations), but also due to the operation sign (plus or minus) itself. Recent eye tracking studies show indeed that observers focus more on the operation signs than on the digits (Schneider, Maruyama, Dehaene, & Sigman, 2012), confirming a special attentional status of these signs. However, there has been no study of the possible effect of looking at a plus or minus sign on spatial biases in an observer. The current set of experiments addressed this specific shortcoming.
EXPERIMENT 1 Participants in our first study performed a new operation sign classification task, which tested whether the plus and minus signs (for addition and subtraction operations, respectively) evoke spatial associations. We would expect that the minus sign should be associated with left space and the plus sign with right space because subtractions typically lead to smaller numbers and additions to larger numbers (analogously to SNARC). If such an operation sign spatial association (or OSSA) effect would be obtained, then it might explain a part of the previously documented SOM. To explore the relationship between SOM, SNARC, and OSSA, we also tested participants in two further tasks. The typical parity judgement task was used to measure participants’ SNARC. The strength of SOM was measured using a calculation task in which participants pointed to number locations (1–9) on a visually presented number line,
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after computing the targets from addition or subtraction problems (as in Pinhas & Fischer, 2008). We tested the prediction that those participants who show OSSA should also show spatial biases in both other tasks. We hypothesized that if the OSSA is a consequence of moving along the mental number line to the left when subtracting and to the right when adding then OSSA and SOM should be correlated. A second prediction was that if OSSA reflects a more general experience of activating on average smaller results after subtraction versus larger results after addition then OSSA should correlate with the SNARC. Last, if the spatial associations evoked by the operands contribute to the SOM then it should be correlated with the SNARC.
Method Participants Sixty-two students (mean age 19.98 years; 10 male; 5 left-handers; 5 non-native English speakers) from the University of Dundee, Scotland, UK, participated in the experiment for course credit or payment. All had normal or corrected-to-normal vision. Apparatus and stimuli Stimuli appeared on an ELO 20′′ touch screen with 1024 × 768-pixel resolution. The software for all three tasks was programmed in E-Prime (Schneider, Eschman, & Zuccolotto, 2002). For the operation sign classification task we used “+” and “−” as stimuli to measure OSSA. Signs
were presented in black on a grey background (60-point boldface Courier New font). Each trial started with the presentation of a blank screen for 750 ms, followed by the presentation of the target sign until response. Each sign was randomly repeated 14 times per response rule (either “+” left key; or “+” right key), resulting in 28 trials per rule and 56 trials for this task in total. The assessment of SOM was designed after Pinhas and Fischer (2008). A typical display sequence (Figure 1) began with a green start box (40 × 40 pixels, 10 × 10 mm) at the bottom centre of the grey screen. All other stimuli were black. A horizontal line (20 × 400 pixels, 5 × 100 mm) flanked by 0 on the left and 10 on the right (Courier New 30 point font) appeared at fixed height on the screen (y coordinate = 350 pixels, 87.5 mm above the start box) but its left edge varied pseudorandomly across trials between centre (312 pixels), left (232 pixels), and right (392 pixels) positions. Arithmetic problems appeared inside a rectangle (166 × 75 pixels, 42 × 19 mm) with an operation sign (+ or −; 5 pixels wide, 20 pixels long; 1.25 × 5 mm) between the two operands. To evaluate the SOM effect while controlling for second operand size, we included problems between 1 and 9 that were equally often the result of an addition or a subtraction of 0, 1, or 2 as the second operand (20 target problems total). Problems with results 0, 5, and 10 were not accepted because the edges/midpoint of the line were much easier to attain in pilot tests than other target locations, presumably reflecting
Figure 1. Illustration of the experimental procedure in the SOM task (not drawn to scale). THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2014, 67 (8)
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extraneous processing strategies. Also, problems in which the operation was predictable from the magnitude of the first operand (e.g., 1 + X; 9 − X ) were not accepted. Additional fillers (16 problems) were used to increase the variability in pointing locations. See Table A1 (Appendix) for the full list of problems used in the task. Each problem appeared nine times, resulting in 324 trials. Problems were randomly presented in six successive blocks (54 trials per block). Finally, to assess SNARC we used the digits 1, 2, 8, and 9. Numbers were presented in black on a grey background (60-point boldface Courier New font). Each trial started with the presentation of a blank screen for 250 ms, followed by a central fixation cross for 500 ms, followed by the presentation of the target until participants made a parity decision. Each number was randomly repeated 14 times per each response rule (either even–left or even–right), resulting in 56 trials per rule, and 112 trials for this task in total. Procedure Participants were seated in front of the centre of the computer touch screen at a distance of approximately 50 cm. The keyboard was also centred in their midsagittal plane. In the operation sign classification task, participants decided whether a presented symbol was “+” or “−”. In the parity judgements task, participants decided whether each presented number was odd or even. In both tasks, left- and right-hand responses were recorded by pressing the “A” and “6” keys, respectively, on the numbers pad of a standard QWERTY keyboard. In both tasks, responding was made according to two different response-rules. Participants were asked to respond as quickly and as accurately as possible. An error beep was played in cases where an incorrect response was made. In the calculation task, participants touched the start box with the right index finger to trigger the display of a problem and the line with flankers. The problem disappeared after 200 ms. Participants were instructed to accurately point to where the result of their calculation would be located on the line. All touch coordinates (in pixels, relative to the start of the line) were recorded and were used
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later on to evaluate the presence of a spatial OM in the target problems. An error beep was played whenever pointing time (i.e., the time from onset of the problem to the touch on the presented number line) exceeded 800 ms, to induce fast responding. Order of tasks and response rules (for the parity judgements and operation sign classification tasks) were counterbalanced between and within participants, respectively.
Results First, we report effects in each task separately. Then we calculate the correlations between the bias measures derived from each task. Operation sign classification task—OSSA effect Mean reaction times (RTs) of correct responses (96.98% of the data) were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms (2 trials). A 3 × 2 × 2 repeated measures analysis of variance (ANOVA) evaluated the effects of task order (first, second, third) as a between-participants variable and operation sign (+, −) and hand (left, right) as within-participants variables. Both hand and operation sign main effects were not significant (ps . .11). Importantly, the analysis revealed a significant interaction between hand and operation sign, F(1, 59) = 4.11, MSE = 1861, p , .05, η2p = .07, demonstrating significantly faster responses for the “+” sign with the right hand (17 ms), F(1, 59) = 6.66, MSE = 1353, p = .01, η2p = .10, and a smaller, nonsignificant, advantage for the “−” sign with the left hand (5 ms), F , 1, η2p = .01. Next, we evaluated whether spatially congruent trials (“+” right; “−” left) significantly differed from spatially incongruent trials (“−” right; “+” left). As can be seen in Figure 2, congruent trials (378 ms) were indeed responded to faster than incongruent ones (389 ms), t(61) = 2.09, SD = 42.69, p , .05. Furthermore, the ANOVA did not reveal a significant main effect for task order, F(2, 59) = 1.38, MSE = 8598, ns, η2p = .05, and task order also did not interact with the other factors: Task Order × Operation Sign, F(2, 59) = 1.64, MSE = 1105, ns, η2p = .05; for Task
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resulting set of individual regression weights was tested against zero to assess the SNARC effect. Mean RTs for the digits 1, 2, 8, and 9 were 523, 525, 513, and 538 ms, respectively. The mean nonstandardized regression slope was −8.72 ms/digit (range: −30.11 to 18.65 ms/digit), and it differed significantly from zero, t(61) = −6.39, SD = 10.73, p , .01, indicating a SNARC effect.
Figure 2. Operation sign spatial association (OSSA) effect, showing mean reaction times (RTs) as a function of responding hand and operation sign. Vertical bars denote standard errors.
Order × Hand and Task Order × Hand × Operation Sign interactions, Fs , 1. Calculation task—SOM effect Data were trimmed by successively eliminating trials with movement times (i.e., the time from the release of the start position to the next contact on the touch screen) outside 200–800 ms and lift-off coordinates outwith the start box or where the target number line was clearly missed, leaving 96.18% of data for statistical analysis. SOM was tested by evaluating the effect of operation (addition, subtraction) on horizontal landing coordinates of the target problems, using a t test for dependent samples. Participants pointed significantly further right following addition (181 pixels) than following subtraction (174 pixels), t(61) = 5.49, SD = 10.31, p , .01, indicating a SOM effect. Parity judgement task—SNARC effect Mean RTs of correct responses (94.93% of the data) were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms, leaving 94.74% of the data for statistical analysis. Next, RT difference of right- minus left-hand response was calculated for each digit and regressed on digit magnitude to determine the strength and direction of the SNARC effect for each participant. The
Does OSSA predict SOM or SNARC? In order to test our specific prediction about the influence of OSSA on SOM and SNARC, we computed individual scores for OSSA and SOM to be correlated with SNARC. Individual OSSA scores were computed by subtracting the mean RT of spatially congruent trials from that of spatially incongruent trials. Positive scores indicate an OSSA. Individual SOM scores were computed by subtracting the mean horizontal landing coordinate of subtractions from additions. Positive scores indicate SOM. Individual nonstandardized regression slopes were used as SNARC scores. Negative scores indicate SNARC. We conducted several correlational analyses in order to obtain an in-depth view concerning the potential relations between the three spatial bias measures. First, we calculated the Pearson correlations between the spatial bias measures for the full sample of participants (N = 62). This analysis did not yield any significant correlations: r(OSSA, SOM) = .02, p = .85, r (OSSA, SNARC) = −.18, p = .17, and r(SOM, SNARC) = −.16, p = .20. Next, we calculated Pearson correlations for a subsample of participants with individual scores within +2 standard deviations from the averaged spatial bias score per task (N = 54). This analysis allowed us to capture the potential relations between the “averaged spatial bias measures” across tasks because extreme bias scores were now excluded. One significant correlation was obtained for this subsample: SOM negatively correlated with SNARC, r(SOM, SNARC) = −.27, p = .04, such that the larger the right bias in pointing to addition results, the larger the spatial numerical associations evoked by single digits (instantiated by a more negative individual slope). This correlation is visualized in Figure 3. The other two correlations
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Figure 3. Spatial operational momentum (SOM) scores are negatively correlated with spatial–numerical association of response codes (SNARC) for participants with scores within +2 standard deviations from the averaged spatial bias score per task (N = 54).
were not significant: r(OSSA, SOM) = .01, p = .96, r(OSSA, SNARC) = −.22, p = .10. Last, we tested whether there were differences between the Spearman rank order correlations obtained for the individuals who exhibited an OSSA (indicated by a positive OSSA score; N = 38) and those obtained for the individuals who did not produce such bias (indicated by a negative OSSA score; N = 24).
Two noteworthy significant correlations were obtained for the individuals who produced an OSSA: First, the larger the OSSA, the stronger the SOM (Spearman’s rho = .34, p = .04). This novel pattern is shown in Figure 4a. Second, SOM negatively correlated with SNARC (Spearman’s rho = −.39, p = .02), similar to the Pearson correlation found with the larger sample
Figure 4. Participants who produced an operation sign spatial association (OSSA; N = 38). Panel A: OSSA scores are positively correlated with spatial operational momentum (SOM). Panel B: SOM scores are negatively correlated with spatial–numerical association of response codes (SNARC).
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of participants, which excluded outliers (see Figure 4b). Again, no reliable correlation emerged between OSSA and SNARC (Spearman’s rho = −.14, p = .39). In contrast, no significant Spearman correlations were found for the group of participants with negative OSSA scores (all ps . .67). Nevertheless, it should be pointed out that this latter group of participants still produced a significant SOM effect, t(23) = 5.62, SD = 8.42, p , .01 (indicated by a more rightward pointing after additions than after subtractions: 184 vs. 174 pixels, respectively), as well as a significant SNARC effect, t(23) = −2.53, SD = 11.34, p , .02 (mean slope = −5.86).
Discussion The primary purpose of the first experiment was to investigate whether there is an association between operation signs and space (OSSA). We discovered that spatially congruent trials were responded to faster than spatially incongruent trials, suggesting that there is a link between minus signs and lefthand responses and between plus signs and righthand responses. This newly discovered association was equally strong regardless of whether the sign classification had been performed first, second, or third in the order of tasks. This suggests that the spatial interpretation of a “+” or “−” symbol is spontaneous and is not dependent on the presence of an arithmetic context. However, OSSA was not statistically reliable for the minus sign in our first experiment. Thus, it was premature to discuss possible origins of OSSA and to interpret the correlational pattern between OSSA and other tasks. We conducted two replication attempts that assessed the generality and symmetry of OSSA.
EXPERIMENT 2 The main purpose of Experiment 2 was to determine whether OSSA would be present in other tasks. Previous work (e.g., Schneider et al., 2012) suggests that operation signs receive considerable attention during mental arithmetic. We therefore
modified the attentional SNARC paradigm introduced by Fischer and colleagues (Fischer, Castel, Dodd, & Pratt, 2003) where adults detected targets in their left/right hemifield faster after seeing a small/large digit at fixation. Instead of digits we presented operation signs, and we manipulated the stimulus onset asynchrony (SOA, i.e., the time between onset of fixated symbol and target onset) in order to determine the time course of lateralized attention shifts. Although task order had not interacted with OSSA in Experiment 1, we controlled possible contextual influences more carefully in the present experiment by assessing OSSA without presenting any additional numerical tasks prior to testing. This means that all participants were free to interpret the symbols “+” and “−” as they wished. The third experiment (see below) implemented the opposite context manipulation by presenting various numerical tasks prior to the assessment of OSSA. We tested two separate conditions: In one block only a plus sign was presented because this resembles the trial structure of many other reaction time experiments and would thus document an attentional bias in the absence of a polarity structure in the stimulus set (cf. Proctor & Cho, 2006). However, to be able to interpret any resulting right-side advantage unambiguously as OSSA we needed a control condition with at least one other sign, which was implemented in a second block. Following the rationale of Experiment 1, we expected to see faster detection times for rightlateralized targets following a plus sign at fixation, and perhaps also faster detection times for leftlateralized targets following a minus sign at fixation. Given that equal signs direct attention to the results of equations (which are normally stated on the right side), we expected either similar detection times for targets on both sides of an equal sign or a gradually developing right-side advantage following an equal sign at fixation.
Method Participants Twenty students (mean age 20.55 years; 5 male; two left-handers; 2 non-native English speakers) from
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the University of Dundee, Scotland, UK, participated in the experiment for course credit or payment. All had normal or corrected-to-normal vision. Apparatus, stimuli, and task design All stimuli appeared on an ELO 20′′ touch screen with 1024 × 768-pixel resolution. E-Prime software (Schneider et al., 2002) controlled their presentation and recorded responses. The operation signs “+”, “−”, and “=” were coloured black (20point boldface Courier New font) and appeared at the centre of the screen on a grey background. The target was a black dot (100-point boldface Courier New font) presented 150 pixels left or right of the screen centre. Each trial began with the presentation of a blank screen for 500 ms, followed by a fixation sign randomly presented for 250/500/750 ms. Target trials then presented the target randomly on the left/right of fixation sign whereas catch trials continued without further event for another 1750 ms to detect erroneous responses. Two blocks were presented in counterbalanced order: one block in which only “+” was used and a second block in which “+”/“−”/“=” were randomly and equally often used as fixation signs. On the former block there were 72 target trials [12 repetitions × 2 (left/right) target sides × 3 SOA conditions = 72 trials] and 18 catch trials [6 repetitions × 3 SOA conditions = 18 trials], and in the latter block there were 216 target trials [12 repetitions × 3 fixation signs × 2 (left/right) target sides × 3 SOA conditions = 216 trials] and 54 catch trials (6 repetitions × 3 fixation signs × 3 SOA conditions = 18 trials), resulting in a total of 360 trials for this task. Procedure Participants were asked to look at the centre of the screen and to avoid any eye movements. They were instructed to respond as quickly as possible by pressing the space bar with the right index finger whenever they detected a target. An error beep was played following catch trial responses, and these were to be avoided. The session began with a short practice, and there was a short self-paced break after every 45 trials.
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Results The number of catch trial responses per participant was below 5.55%. Mean RTs of target trials were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms, leaving 99.8% of the data. A preliminary analysis revealed no effects of block order, all p values . .09, so we report the results from each condition block separately. Data from the block using only the “+” sign were analysed with a 2 × 3 repeated measures ANOVA with target side (left, right) and SOA (250, 500, 750) as within-participants variables. A main effect of target side, F(1, 19) = 9.41, MSE = 592, p , .01, η2p = .33, indicated faster responses when targets appeared to the right (315) than when they appeared to the left (328) of the plus sign. A main effect of SOA, F(2, 38) = 4.17, MSE = 760, p , .05, η2p = .18, reflected longer responses in the shortest than in both longer SOAs (332 vs. 317 and 316 ms for 250, 500, and 750 ms SOAs, respectively), F(1, 19) = 5.32, MSE = 1185, p , .05, η2p = .22. There was no significant Target Size × SOA interaction (F , 1). Next, we analysed the block in which “+”, “−”, and “ = ” were randomly used as fixation signs, using the same ANOVA design with the addition of fixation sign (+, −, =) as another within-participants variable. Responses were again faster for right than for left targets (318 and 327 ms, respectively), F(1, 19) = 7.42, MSE = 1079, p = .01, η2p = .28. A main effect of SOA, F(2, 38) = 4.75, MSE = 1140, p = .01, η2p = .20, indicated that targets were again detected more slowly in the shortest than in both longer SOA conditions (330 vs. 319 and 318 ms for 250, 500, and 750 ms SOAs, respectively), F(1, 19) = 6.34, MSE = 1706, p , .05, η2p = .25. No other effect was significant (all ps . .11).
Discussion The results of Experiment 2 are clear: Attention allocation was not systematically affected by the identity of the operation signs. This result might reflect the lack of processing of the operators, which were not task relevant. Alternatively, the lack of contextualization of the signs as arithmetic
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symbols may be responsible for the null result, thus perhaps indicating a more strategic origin of OSSA in Experiment 1; after all, participants saw no digits throughout this second experiment. The fact that right targets were detected slightly faster than left targets when following a plus sign was not specific for the plus sign and presumably reflects the use of the right hand to detect all targets (Simon, 1969). Similarly, the speed advantage of later over earlier SOAs merely reflects an unspecific foreperiod effect due to better target predictability at later SOAs (e.g., Niemi & Näätänen, 1981). Importantly, there was no indication of differential attention deployment following fixation on a plus compared to a minus sign (or indeed an equal sign). If the existence of OSSA could be further substantiated in other tasks, then this observation might possibly be in conflict with an attentional explanation of the OM effect as recently favoured by Knops, Zitzmann, and McCrink (2013). However, in the light of the present null result regarding OSSA, we must refrain from such interpretation now and instead report the results of a third experiment.
EXPERIMENT 3 We reasoned that perhaps the task irrelevance of operators in the detection task of Experiment 2 had worked against OSSA and therefore returned to the original operation classification task of Experiment 1, which demanded explicit processing of the symbols. We also ensured that participants were exposed to several numerical tasks prior to the operation classification task, thus inducing an arithmetic context to increase the likelihood of a strong and symmetrical OSSA. Moreover, we manipulated hand position of the participants such that half of them performed the task with their hands crossed and the other half with uncrossed hands. This manipulation allowed us to study whether the spatial information associated with the operation signs is referenced egocentrically to the observer’s body (or a part of it) or to some allocentric location. If OSSA is allocentrically coded then it should replicate across both hand
position conditions, thus also ruling out a hemispheric dominance account of OSSA (cf. Dehaene et al., 1993, Experiment 6; Wood, Nuerk, & Willmes, 2006).
Method Participants Thirty-four students (mean age 22.29 years; 3 male; two left-handers; 10 non-native English speakers) from the University of Dundee, Scotland, UK, participated in the experiment for course credit or payment. All had normal or corrected-to-normal vision. Participants were randomly assigned to one of two groups, which differed in their assigned hands position: uncrossed or crossed. Procedure Participants completed two other numerical tasks that are not reported here (a parity task and a number localization task; Pinhas, Shaki, & Fischer, 2014) before finally being tested with the operation sign classification task. The procedure of the operation classification task was identical to that reported in Experiment 1 except that half the participants performed the task with crossed hands such that their left hand was placed on the right response key, and their right hand was placed on the left response key. The other half of participants performed the task with uncrossed hands.
Results Data from one participant were excluded from the analysis due to high error rate (45.62%). Mean RTs of correct responses of the other 33 participants (95.3% of the data) were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms (11 trials). A 2 × 2 × 2 repeated measures ANOVA evaluated the effects of hand position (crossed, uncrossed) as a between-participants variable and operation sign (+, −) and response key (left, right) as within-participants variables. Importantly, the interaction between response key and operation sign was replicated, F(1, 31) = 4.62, MSE = 815, p , .05, η2p = .13. Planned comparisons demonstrated significantly faster left key responses for the
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revealed a significant advantage for the “+” sign with the right side (13 ms), F(1, 93) = 6.66, MSE = 1171, p = .01, η2p = .07, and a nonsignificant advantage for the “−” sign with left-side responses (9 ms), F(1, 93) = 3.05, MSE = 1049, p = .08, η2p = .03. Furthermore, spatially congruent (388 ms) trials were responded to faster than spatially incongruent trials (399 ms), t(94) = 2.81, MSE = 38.43, p , .01. All other effects in the ANOVA were not significant (ps . .08).
Discussion Figure 5. Operation sign spatial association (OSSA) effect, showing mean reaction times (RTs) as a function of response key and operation sign across hands position. Vertical bars denote standard errors.
“−” sign (12 ms), F(1, 31) = 4.77, MSE = 474, p , .05, η2p = .13, and a nonsignificant advantage for the “+” sign with the right key (10 ms), F (1, 31) = 1.75, MSE = 878, p = .19, η2p = .05 (see Figure 5). As in Experiment 1, we found that congruent trials (397 ms) were responded to faster than incongruent ones (408 ms), t(32) = 2.19, SD = 28.41, p , .05. There was no reliable three-way interaction between hand position, operation sign, and response key (F , 1), and no other effects in the analysis reached significance (all ps . .19). Combined analysis—OSSA effect Last, we conducted a combined analysis on the data reported from the operation classification task in both Experiments 1 (62 participants) and 3 (33 participants) to further evaluate the inconsistent OSSA asymmetry. Given that there was no effect of task order in Experiment 1 and no effect of hand position in Experiment 3, we ignored these variables in the combined analysis design. The data were analysed with a 2 × 2 × 2 repeated measures ANOVA with experiment (1, 3) as a between-participants variable and operation sign (+, −) and response side (left, right) as within-participants variables. As expected, the interaction between response side and operation sign was significant, F(1, 93) = 7.04, MSE = 1493, p , .01, η2p = .07. Planned comparisons
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Experiment 3 successfully replicated the OSSA using the operation sign classification task. We again found that spatially congruent trials were responded to faster than spatially incongruent trials, suggesting an association between left/right space and the minus/plus sign, respectively. Importantly, the statistical pattern was opposite to that found in Experiment 1: Here we found a significant association only for the minus sign, whereas there we had found a significant association only for the plus sign. Thus, even following a strong context of numerical tasks, the OSSA was again asymmetrical and did not affect both signs equally strongly. Last, a combined analysis on the data from both Experiments 1 and 3 revealed again an asymmetrical OSSA where only the plus sign showed a significant association. Together, the results of all three experiments allow us to attempt a more general discussion of our findings.
GENERAL DISCUSSION The present study uncovered a novel association between operation signs and space that may play a role in arithmetic performance. Specifically, we found that minus signs can be associated with left space and plus signs with right space. These results were established (Experiment 1) and replicated (Experiment 3) in a classification task that required explicit processing and discrimination of the signs. A detection task with less explicit processing requirements for these signs (Experiment 2) did not reveal similar spatial biases, suggesting
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that the OSSA is not strong enough to evoke attentional biases, at least without contextual cueing. We can see at least two explanations for why OSSA might exist. First, it could be the consequence of arithmetic operations themselves—that is, moving one’s attention along the mental number line to the left when subtracting and to the right when adding. In this view, OSSA is a result of SOM. Secondly, OSSA might reflect the general experience of activating smaller magnitude results (on average) when subtracting versus larger magnitude results when adding. In this view, OSSA is an instance of the spatial association captured by SNARC. The correlation pattern from Experiment 1 suggests that the first explanation is more plausible: We found a correlation between OSSA and SOM but no such correlation between OSSA and SNARC. OSSA did not affect both signs equally strongly —this was true in both the first and the last experiment, which revealed opposite association patterns. This outcome does not seem to agree with a polarity correspondence interpretation of OSSA that predicts symmetrical spatial associations given an expected bipolar coding for both the operation signs and the response alternatives (Proctor & Cho, 2006; see also Santens & Gevers, 2008). We propose two possible explanations for the inconsistent asymmetry in the OSSA, which we consider not to be mutually exclusive. First, the minus sign is used for several different functions in the context of elementary algebra—namely, unary, binary and symmetric functions: It is a structural signifier in negative numbers, an operational signifier when subtracting in arithmetic and algebra, and an operational signifier for taking the opposite of a number, respectively (for discussion see Vlassis, 2008). These multiple uses within a mathematical context need to be distinguished and used in a flexible and appropriate manner in operations and equations, making it difficult to assign a specific spatial function to this sign. In support of this claim, Vlassis (2008) reported that eight graders had difficulties when solving equations with negatives, possibly reflecting just this multidimensionality of the minus sign. Similarly, the plus sign is a binary operator that indicates addition, and it also
serves as a unary operator that leaves its operand unchanged. This last notation is typically used in the context of contrasting positive numbers with negative ones. Second, the operation signs also have nonmathematical meanings: the minus sign as a hyphen that concatenates words and thus actually functions like a plus sign; the plus sign as a marker for location. Both signs are also frequently used for noting additional (plus) or fewer (minus) options in computer-based applications, as in the meaning of more and less, respectively. These extramathematical (and sometimes conceptually competing) uses of the signs might further dilute their spatial biases. Moreover, our finding of a rightward association evoked by the plus sign, with the lack of a corresponding leftward bias of the minus sign, reported both in Experiment 1 and in the combined analysis of the OSSA across experiments, is in line with the coactivation for additions and rightward saccades with no reliable coactivation of subtractions and leftward saccades (Knops, Thirion, et al., 2009). Accordingly, the multiple uses and interpretations of the plus and minus signs possibly resulted in asymmetric OSSA patterns and possibly led to the finding of a rather small, yet replicable, effect. Consistent with this view, recent theorizing about the cognitive representation of mathematical symbols emphasizes the importance of contextual (or “situated”) factors (cf. Fischer & Brugger, 2011; Hung, Hung, Tzeng, & Wu, 2008; Wasner, Moeller, Fischer, & Nuerk, 2013). Future research is needed to test whether presenting words instead of signs (e.g., “addition”, “subtraction”), or presenting the operation signs in the context of arithmetic operations, would result in stronger and symmetric OSSA. Concerning the relation between the OSSA and SOM, we found significant correlations between these two spatial biases among individuals who produced an OSSA, suggesting that spatial biases in arithmetic might be related to spatial associations evoked by the operation signs. Yet, we also found that individuals who did not produce an OSSA showed a SOM at the group level. Thus it seems that while OSSA partially contributes to the SOM in some individuals, other sources might
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play a role in deriving the effect, such as the spatial associations evoked by the operands. Future studies focusing on the interrelations between measures of numerical and arithmetic spatial biases and in individual differences obtained for these measures are needed to better understand the role of various contributors for the SOM. The lack of a differential bias from the two operation signs in Experiment 2 might be due to the lack of processing of the operators, which were not relevant for performing the detection task. However, we believe that it is more plausible that the lack of contextualization of the signs as arithmetic symbols is responsible for the null result. The fact that participants saw no digits throughout this second experiment suggests that OSSA might be strategy or context dependent. This idea provided the motivation for Experiment 3. In Experiment 3, the presence of OSSA with crossed hands suggests that OSSA is allocentrically coded because it replicated across both hand position conditions. This observation rules out a hemispheric dominance account of OSSA (cf. Dehaene et al., 1993, Experiment 6; Wood et al., 2006). In conclusion, spatial–numerical associations were almost exclusively obtained with single-digit tasks (Wood et al., 2008). Only recently have researchers extended this work to the level of arithmetic operations, thereby documenting spatial biases in addition and subtraction (SOM effect: Knops, Viarouge, et al., 2009; Pinhas & Fischer, 2008). The present study extends this work further by documenting, for the first time, that arithmetic operators can also induce spatial biases (OSSA effect) and that this effect contributes to the spatial biases in arithmetic. Original manuscript received 27 May 2013 Accepted revision received 6 November 2013 First published online 11 March 2014
REFERENCES Chen, Q. & Verguts, T. (2012). Spatial intuition in elementary arithmetic: A neurocomputational
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account. PloS one, 7(2), e31180. doi:10.1371/ journal.pone.0031180 Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122(3), 371–396. Fias, W. & Fischer, M. H. (2005). Spatial representation of numbers. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 43–54). New York: Psychology Press. Fischer,M.H.&Brugger,P.(2011).Whendigitshelpdigits: Spatial-numerical associations point to finger counting as prime example of embodied cognition. Frontiers in Psychology, 2, 260. doi:10.3389/fpsyg.2011.00260 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. Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6(6), 435–448. Hung, Y. H., Hung, D. L., Tzeng, O. J. L., & Wu, D. H. (2008). Flexible spatial mapping of different notations of numbers in Chinese readers. Cognition, 106 (3), 1441–1450. Knops, A., Thirion, B., Hubbard, E. M., Michel, V., & Dehaene, S. (2009). Recruitment of an area involved in eye movements during mental arithmetic. Science, 324(5934), 1583–1585. Knops, A., Viarouge, A., & Dehaene, S. (2009). Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect. Attention, Perception, & Psychophysics, 71(4), 803–821. Knops, A., Zitzmann, S., & McCrink, K. (2013). Examining the presence and determinants of operational momentum in childhood. Frontiers in Psychology, 4, 325. doi:10.3389/fpsyg.2013.00325 Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. McCrink, K., Dehaene, S., & Dehaene-Lambertz, G. (2007). Moving along the number line: Operational momentum in nonsymbolic arithmetic. Perception and Psychophysics, 69(8), 1324–1333. McCrink, K. & Wynn, K. (2009). Operational momentum in large-number addition and subtraction by 9-month-olds. Journal of Experimental Child Psychology, 103(4), 400–408. Moyer, R. S. & Landauer, T. K. (1967). Time required for judgment of numerical inequality. Nature, 215, 1519–1520.
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Niemi, P. & Näätänen, R. (1981). Foreperiod and simple reaction time. Psychological Bulletin, 89(1), 133–162. Pinhas, M. & Fischer, M. H. (2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109(3), 408–415. Pinhas, M., Shaki, S., & Fischer, M. H. (2014). Addition is where the big numbers are: Evidence for an inverse operational momentum effect. Manuscript in preparation. Proctor, R. W. & Cho, Y. S. (2006). Polarity correspondence: A general principle for performance of speeded binary classification tasks. Psychological Bulletin, 132 (3), 416–442. Santens, S. & Gevers, W. (2008). The SNARC effect does not imply a mental number line. Cognition, 108(1), 263–270. Schneider, E., Maruyama, M., Dehaene, S., & Sigman, M. (2012). Eye gaze reveals a fast, parallel extraction of the syntax of arithmetic formulas. Cognition, 125 (3), 475–490. Schneider, W., Eschman, A., & Zuccolotto, A. (2002). E-Prime user’s guide. Pittsburgh, PA: Psychology Software Tools, Inc.
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. Simon, J. R. (1969). Reaction toward the source of stimulation. Journal of Experimental Psychology, 81 (1), 1974–1976. Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21 (4), 555–570. Wasner, M., Moeller, K., Fischer, M. H., & Nuerk, H. C. (2014). How situated cognition influences memory retrieval: The case of finger counting habits. Cognitive Processing. doi:10.1007/s10339014-0599-z Wood, G., Nuerk, H. C., & Willmes, K. (2006). Crossed hands and the SNARC effect: A failure to replicate Dehaene, Bossini and Giraux (1993). Cortex, 42(8), 1069–1079. Wood, G., Willmes, K., Nuerk, H. C., & 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.
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APPENDIX Table A1. Problems used in the calculation task
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Problem type
Problem number
Operand 1
Operation
Operand 2
Result
Target problems
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 2 3 3 4 3 2 4 5 6 6 5 4 6 7 8 7 7 8 8
+ − + − + + + − − − + + + − − − + − + −
0 0 0 0 0 1 2 0 1 2 0 1 2 0 1 2 0 0 0 0
2 2 3 3 4 4 4 4 4 4 6 6 6 6 6 6 7 7 8 8
Filler problems
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
5 6 6 7 7 8 7 8 3 2 3 2 4 3 5 4
− − − − − − − − + + + + + + + +
4 5 4 5 4 5 3 4 3 4 4 5 4 5 4 5
1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9
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The Quarterly Journal of Experimental Psychology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/pqje20
Heed the signs: Operation signs have spatial associations a
b
c
Michal Pinhas , Samuel Shaki & Martin H. Fischer a
Department of Psychology, Ben-Gurion University of the Negev, BeerSheva, Israel b
Department of Behavioral Sciences, Ariel University, Ariel, Israel
c
Department of Psychology, University of Potsdam, Potsdam, Germany Accepted author version posted online: 18 Feb 2014.Published online: 11 Mar 2014.
To cite this article: Michal Pinhas, Samuel Shaki & Martin H. Fischer (2014) Heed the signs: Operation signs have spatial associations, The Quarterly Journal of Experimental Psychology, 67:8, 1527-1540, DOI: 10.1080/17470218.2014.892516 To link to this article: http://dx.doi.org/10.1080/17470218.2014.892516
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THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2014 Vol. 67, No. 8, 1527–1540, http://dx.doi.org/10.1080/17470218.2014.892516
Heed the signs: Operation signs have spatial associations Michal Pinhas1, Samuel Shaki2, and Martin H. Fischer3 1
Department of Psychology, Ben-Gurion University of the Negev, Beer-Sheva, Israel Department of Behavioral Sciences, Ariel University, Ariel, Israel 3 Department of Psychology, University of Potsdam, Potsdam, Germany
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Mental arithmetic shows systematic spatial biases. The association between numbers and space is well documented, but it is unknown whether arithmetic operation signs also have spatial associations and whether or not they contribute to spatial biases found in arithmetic. Adult participants classified plus and minus signs with left and right button presses under two counterbalanced response rules. Results from two experiments showed that spatially congruent responses (i.e., right-side responses for the plus sign and left-side responses for the minus sign) were responded to faster than spatially incongruent ones (i.e., left-side responses for the plus sign and right-side responses for the minus sign). We also report correlations between this novel operation sign spatial association (OSSA) effect and other spatial biases in number processing. In a control experiment with no explicit processing requirements for the operation signs there were no sign-related spatial biases. Overall, the results suggest that (a) arithmetic operation signs can evoke spatial associations (OSSA), (b) experience with arithmetic operations probably underlies the OSSA, and (c) the OSSA only partially contributes to spatial biases in arithmetic. Keywords: Mental arithmetic; Mental number line; Operational momentum; Pointing; Spatial– numerical association of response codes effect.
Single numbers usually evoke spatial associations so that smaller numbers are associated with left space and larger numbers with right space (spatial– numerical association of response codes, SNARC, effect; Dehaene, Bossini, & Giraux, 1993). This effect is very pervasive, at least in left-to-right readers (Dehaene et al., 1993; Shaki, Fischer, & Petrusic, 2009) and obtains in a number of different tasks (for reviews see Fias & Fischer, 2005; Hubbard, Piazza, Pinel, & Dehaene, 2005; for a meta-analysis, see Wood, Nuerk, Willmes, & Fischer, 2008). Together with other performance signatures of numerical cognition (i.e., the distance
and size effects, Moyer & Landauer, 1967), the spatial association of numbers has been taken to suggest that numbers are cognitively represented as a spatially ordered continuum of magnitudes, often referred to as the “mental number line”. Recently, systematic biases with associated spatial errors were also found in mental arithmetic, thus extending the SNARC effect into the domain of everyday numerical cognition. First, McCrink, Dehaene, and Dehaene-Lambertz (2007) used dot pattern arithmetic to document a bias towards larger numerosities when evaluating addition outcomes and a bias towards smaller numerosities
Correspondence should be addressed to Michal Pinhas, Department of Psychology, Ben-Gurion University of the Negev, BeerSheva, 84105, Israel. E-mail: [email protected] We thank Gavin Revie and Clare Kirtley for their help with data collection. M.P. was supported by a Visiting Researcher’s Grant from the UK’s Experimental Psychology Society. M.H.F. was supported by the British Academy [grant number SG 46947] while working at the University of Dundee. © 2014 The Experimental Psychology Society
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when evaluating subtraction outcomes. This effect was termed “operational momentum” (OM) in accord with the metaphoric notion of calculations as movements along a mental number line (Lakoff & Nunez, 2000; see also McCrink & Wynn, 2009). Subsequently, Pinhas and Fischer (2008) used a spatial pointing task with singledigit problems to document a right bias after adding and left bias after subtracting, thereby extending OM to calculations with Arabic digits and into the spatial domain (hence, spatial OM or SOM). Similar spatial biases were also found with an extended numerical range in both symbolic and nonsymbolic arithmetic (Knops, Viarouge, & Dehaene, 2009). Using a neuroscientific approach, Knops, Thirion, Hubbard, Michel, and Dehaene (2009) identified common brain parietal structures that are involved in addition and subtraction as well as in right- and leftward-directed saccadic eye movements. After training a saccade classifier algorithm to infer the left or right direction of eye movements from brain activation measured in the posterior parietal cortex, Knops and colleagues tested whether this classifier would generalize to mental calculation. Without further training, functional magnetic resonance imaging (fMRI) images of brain activation during addition were classified 61% of the time as rightward saccades, across symbolic and nonsymbolic notations. However, the classification of subtraction images as leftward saccades was below chance level (49.1%). These results associate the cognitive representations of arithmetic operations to space but also raise questions about the underlying mechanism. Recently, Chen and Verguts (2012) modelled the OM and SOM effects based on an alternative account for OM, originally suggested by McCrink et al. (2007). Specifically, the authors assumed that over- and underestimation of additions and subtractions occur when these operations are performed on a compressed mental representation of magnitudes (e.g., logarithmic scaling of the metal number line). For example, adding two numbers n1 + n2 that are represented in a compressed manner [e.g., as log(n1) and log(n2)] will usually lead to larger values than the real outcome [e.g., as log
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(n1) + log(n2) equals log(n1) × log(n2)]. The model implements radial basis functions to simulate flawed “decompression” of numerical representations during parietal arithmetic computations as the source of OM. A closer inspection of the behavioural signatures of spatial–numerical biases and their interrelationship can help us to understand the mechanisms involved in SOM. At first glance, arithmetic operations can be spatially biased due to either the spatial associations of the individual digits of the equation (the operands) or the anticipated small or large result (e.g., from subtraction or addition operations), but also due to the operation sign (plus or minus) itself. Recent eye tracking studies show indeed that observers focus more on the operation signs than on the digits (Schneider, Maruyama, Dehaene, & Sigman, 2012), confirming a special attentional status of these signs. However, there has been no study of the possible effect of looking at a plus or minus sign on spatial biases in an observer. The current set of experiments addressed this specific shortcoming.
EXPERIMENT 1 Participants in our first study performed a new operation sign classification task, which tested whether the plus and minus signs (for addition and subtraction operations, respectively) evoke spatial associations. We would expect that the minus sign should be associated with left space and the plus sign with right space because subtractions typically lead to smaller numbers and additions to larger numbers (analogously to SNARC). If such an operation sign spatial association (or OSSA) effect would be obtained, then it might explain a part of the previously documented SOM. To explore the relationship between SOM, SNARC, and OSSA, we also tested participants in two further tasks. The typical parity judgement task was used to measure participants’ SNARC. The strength of SOM was measured using a calculation task in which participants pointed to number locations (1–9) on a visually presented number line,
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after computing the targets from addition or subtraction problems (as in Pinhas & Fischer, 2008). We tested the prediction that those participants who show OSSA should also show spatial biases in both other tasks. We hypothesized that if the OSSA is a consequence of moving along the mental number line to the left when subtracting and to the right when adding then OSSA and SOM should be correlated. A second prediction was that if OSSA reflects a more general experience of activating on average smaller results after subtraction versus larger results after addition then OSSA should correlate with the SNARC. Last, if the spatial associations evoked by the operands contribute to the SOM then it should be correlated with the SNARC.
Method Participants Sixty-two students (mean age 19.98 years; 10 male; 5 left-handers; 5 non-native English speakers) from the University of Dundee, Scotland, UK, participated in the experiment for course credit or payment. All had normal or corrected-to-normal vision. Apparatus and stimuli Stimuli appeared on an ELO 20′′ touch screen with 1024 × 768-pixel resolution. The software for all three tasks was programmed in E-Prime (Schneider, Eschman, & Zuccolotto, 2002). For the operation sign classification task we used “+” and “−” as stimuli to measure OSSA. Signs
were presented in black on a grey background (60-point boldface Courier New font). Each trial started with the presentation of a blank screen for 750 ms, followed by the presentation of the target sign until response. Each sign was randomly repeated 14 times per response rule (either “+” left key; or “+” right key), resulting in 28 trials per rule and 56 trials for this task in total. The assessment of SOM was designed after Pinhas and Fischer (2008). A typical display sequence (Figure 1) began with a green start box (40 × 40 pixels, 10 × 10 mm) at the bottom centre of the grey screen. All other stimuli were black. A horizontal line (20 × 400 pixels, 5 × 100 mm) flanked by 0 on the left and 10 on the right (Courier New 30 point font) appeared at fixed height on the screen (y coordinate = 350 pixels, 87.5 mm above the start box) but its left edge varied pseudorandomly across trials between centre (312 pixels), left (232 pixels), and right (392 pixels) positions. Arithmetic problems appeared inside a rectangle (166 × 75 pixels, 42 × 19 mm) with an operation sign (+ or −; 5 pixels wide, 20 pixels long; 1.25 × 5 mm) between the two operands. To evaluate the SOM effect while controlling for second operand size, we included problems between 1 and 9 that were equally often the result of an addition or a subtraction of 0, 1, or 2 as the second operand (20 target problems total). Problems with results 0, 5, and 10 were not accepted because the edges/midpoint of the line were much easier to attain in pilot tests than other target locations, presumably reflecting
Figure 1. Illustration of the experimental procedure in the SOM task (not drawn to scale). THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2014, 67 (8)
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extraneous processing strategies. Also, problems in which the operation was predictable from the magnitude of the first operand (e.g., 1 + X; 9 − X ) were not accepted. Additional fillers (16 problems) were used to increase the variability in pointing locations. See Table A1 (Appendix) for the full list of problems used in the task. Each problem appeared nine times, resulting in 324 trials. Problems were randomly presented in six successive blocks (54 trials per block). Finally, to assess SNARC we used the digits 1, 2, 8, and 9. Numbers were presented in black on a grey background (60-point boldface Courier New font). Each trial started with the presentation of a blank screen for 250 ms, followed by a central fixation cross for 500 ms, followed by the presentation of the target until participants made a parity decision. Each number was randomly repeated 14 times per each response rule (either even–left or even–right), resulting in 56 trials per rule, and 112 trials for this task in total. Procedure Participants were seated in front of the centre of the computer touch screen at a distance of approximately 50 cm. The keyboard was also centred in their midsagittal plane. In the operation sign classification task, participants decided whether a presented symbol was “+” or “−”. In the parity judgements task, participants decided whether each presented number was odd or even. In both tasks, left- and right-hand responses were recorded by pressing the “A” and “6” keys, respectively, on the numbers pad of a standard QWERTY keyboard. In both tasks, responding was made according to two different response-rules. Participants were asked to respond as quickly and as accurately as possible. An error beep was played in cases where an incorrect response was made. In the calculation task, participants touched the start box with the right index finger to trigger the display of a problem and the line with flankers. The problem disappeared after 200 ms. Participants were instructed to accurately point to where the result of their calculation would be located on the line. All touch coordinates (in pixels, relative to the start of the line) were recorded and were used
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later on to evaluate the presence of a spatial OM in the target problems. An error beep was played whenever pointing time (i.e., the time from onset of the problem to the touch on the presented number line) exceeded 800 ms, to induce fast responding. Order of tasks and response rules (for the parity judgements and operation sign classification tasks) were counterbalanced between and within participants, respectively.
Results First, we report effects in each task separately. Then we calculate the correlations between the bias measures derived from each task. Operation sign classification task—OSSA effect Mean reaction times (RTs) of correct responses (96.98% of the data) were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms (2 trials). A 3 × 2 × 2 repeated measures analysis of variance (ANOVA) evaluated the effects of task order (first, second, third) as a between-participants variable and operation sign (+, −) and hand (left, right) as within-participants variables. Both hand and operation sign main effects were not significant (ps . .11). Importantly, the analysis revealed a significant interaction between hand and operation sign, F(1, 59) = 4.11, MSE = 1861, p , .05, η2p = .07, demonstrating significantly faster responses for the “+” sign with the right hand (17 ms), F(1, 59) = 6.66, MSE = 1353, p = .01, η2p = .10, and a smaller, nonsignificant, advantage for the “−” sign with the left hand (5 ms), F , 1, η2p = .01. Next, we evaluated whether spatially congruent trials (“+” right; “−” left) significantly differed from spatially incongruent trials (“−” right; “+” left). As can be seen in Figure 2, congruent trials (378 ms) were indeed responded to faster than incongruent ones (389 ms), t(61) = 2.09, SD = 42.69, p , .05. Furthermore, the ANOVA did not reveal a significant main effect for task order, F(2, 59) = 1.38, MSE = 8598, ns, η2p = .05, and task order also did not interact with the other factors: Task Order × Operation Sign, F(2, 59) = 1.64, MSE = 1105, ns, η2p = .05; for Task
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resulting set of individual regression weights was tested against zero to assess the SNARC effect. Mean RTs for the digits 1, 2, 8, and 9 were 523, 525, 513, and 538 ms, respectively. The mean nonstandardized regression slope was −8.72 ms/digit (range: −30.11 to 18.65 ms/digit), and it differed significantly from zero, t(61) = −6.39, SD = 10.73, p , .01, indicating a SNARC effect.
Figure 2. Operation sign spatial association (OSSA) effect, showing mean reaction times (RTs) as a function of responding hand and operation sign. Vertical bars denote standard errors.
Order × Hand and Task Order × Hand × Operation Sign interactions, Fs , 1. Calculation task—SOM effect Data were trimmed by successively eliminating trials with movement times (i.e., the time from the release of the start position to the next contact on the touch screen) outside 200–800 ms and lift-off coordinates outwith the start box or where the target number line was clearly missed, leaving 96.18% of data for statistical analysis. SOM was tested by evaluating the effect of operation (addition, subtraction) on horizontal landing coordinates of the target problems, using a t test for dependent samples. Participants pointed significantly further right following addition (181 pixels) than following subtraction (174 pixels), t(61) = 5.49, SD = 10.31, p , .01, indicating a SOM effect. Parity judgement task—SNARC effect Mean RTs of correct responses (94.93% of the data) were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms, leaving 94.74% of the data for statistical analysis. Next, RT difference of right- minus left-hand response was calculated for each digit and regressed on digit magnitude to determine the strength and direction of the SNARC effect for each participant. The
Does OSSA predict SOM or SNARC? In order to test our specific prediction about the influence of OSSA on SOM and SNARC, we computed individual scores for OSSA and SOM to be correlated with SNARC. Individual OSSA scores were computed by subtracting the mean RT of spatially congruent trials from that of spatially incongruent trials. Positive scores indicate an OSSA. Individual SOM scores were computed by subtracting the mean horizontal landing coordinate of subtractions from additions. Positive scores indicate SOM. Individual nonstandardized regression slopes were used as SNARC scores. Negative scores indicate SNARC. We conducted several correlational analyses in order to obtain an in-depth view concerning the potential relations between the three spatial bias measures. First, we calculated the Pearson correlations between the spatial bias measures for the full sample of participants (N = 62). This analysis did not yield any significant correlations: r(OSSA, SOM) = .02, p = .85, r (OSSA, SNARC) = −.18, p = .17, and r(SOM, SNARC) = −.16, p = .20. Next, we calculated Pearson correlations for a subsample of participants with individual scores within +2 standard deviations from the averaged spatial bias score per task (N = 54). This analysis allowed us to capture the potential relations between the “averaged spatial bias measures” across tasks because extreme bias scores were now excluded. One significant correlation was obtained for this subsample: SOM negatively correlated with SNARC, r(SOM, SNARC) = −.27, p = .04, such that the larger the right bias in pointing to addition results, the larger the spatial numerical associations evoked by single digits (instantiated by a more negative individual slope). This correlation is visualized in Figure 3. The other two correlations
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Figure 3. Spatial operational momentum (SOM) scores are negatively correlated with spatial–numerical association of response codes (SNARC) for participants with scores within +2 standard deviations from the averaged spatial bias score per task (N = 54).
were not significant: r(OSSA, SOM) = .01, p = .96, r(OSSA, SNARC) = −.22, p = .10. Last, we tested whether there were differences between the Spearman rank order correlations obtained for the individuals who exhibited an OSSA (indicated by a positive OSSA score; N = 38) and those obtained for the individuals who did not produce such bias (indicated by a negative OSSA score; N = 24).
Two noteworthy significant correlations were obtained for the individuals who produced an OSSA: First, the larger the OSSA, the stronger the SOM (Spearman’s rho = .34, p = .04). This novel pattern is shown in Figure 4a. Second, SOM negatively correlated with SNARC (Spearman’s rho = −.39, p = .02), similar to the Pearson correlation found with the larger sample
Figure 4. Participants who produced an operation sign spatial association (OSSA; N = 38). Panel A: OSSA scores are positively correlated with spatial operational momentum (SOM). Panel B: SOM scores are negatively correlated with spatial–numerical association of response codes (SNARC).
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of participants, which excluded outliers (see Figure 4b). Again, no reliable correlation emerged between OSSA and SNARC (Spearman’s rho = −.14, p = .39). In contrast, no significant Spearman correlations were found for the group of participants with negative OSSA scores (all ps . .67). Nevertheless, it should be pointed out that this latter group of participants still produced a significant SOM effect, t(23) = 5.62, SD = 8.42, p , .01 (indicated by a more rightward pointing after additions than after subtractions: 184 vs. 174 pixels, respectively), as well as a significant SNARC effect, t(23) = −2.53, SD = 11.34, p , .02 (mean slope = −5.86).
Discussion The primary purpose of the first experiment was to investigate whether there is an association between operation signs and space (OSSA). We discovered that spatially congruent trials were responded to faster than spatially incongruent trials, suggesting that there is a link between minus signs and lefthand responses and between plus signs and righthand responses. This newly discovered association was equally strong regardless of whether the sign classification had been performed first, second, or third in the order of tasks. This suggests that the spatial interpretation of a “+” or “−” symbol is spontaneous and is not dependent on the presence of an arithmetic context. However, OSSA was not statistically reliable for the minus sign in our first experiment. Thus, it was premature to discuss possible origins of OSSA and to interpret the correlational pattern between OSSA and other tasks. We conducted two replication attempts that assessed the generality and symmetry of OSSA.
EXPERIMENT 2 The main purpose of Experiment 2 was to determine whether OSSA would be present in other tasks. Previous work (e.g., Schneider et al., 2012) suggests that operation signs receive considerable attention during mental arithmetic. We therefore
modified the attentional SNARC paradigm introduced by Fischer and colleagues (Fischer, Castel, Dodd, & Pratt, 2003) where adults detected targets in their left/right hemifield faster after seeing a small/large digit at fixation. Instead of digits we presented operation signs, and we manipulated the stimulus onset asynchrony (SOA, i.e., the time between onset of fixated symbol and target onset) in order to determine the time course of lateralized attention shifts. Although task order had not interacted with OSSA in Experiment 1, we controlled possible contextual influences more carefully in the present experiment by assessing OSSA without presenting any additional numerical tasks prior to testing. This means that all participants were free to interpret the symbols “+” and “−” as they wished. The third experiment (see below) implemented the opposite context manipulation by presenting various numerical tasks prior to the assessment of OSSA. We tested two separate conditions: In one block only a plus sign was presented because this resembles the trial structure of many other reaction time experiments and would thus document an attentional bias in the absence of a polarity structure in the stimulus set (cf. Proctor & Cho, 2006). However, to be able to interpret any resulting right-side advantage unambiguously as OSSA we needed a control condition with at least one other sign, which was implemented in a second block. Following the rationale of Experiment 1, we expected to see faster detection times for rightlateralized targets following a plus sign at fixation, and perhaps also faster detection times for leftlateralized targets following a minus sign at fixation. Given that equal signs direct attention to the results of equations (which are normally stated on the right side), we expected either similar detection times for targets on both sides of an equal sign or a gradually developing right-side advantage following an equal sign at fixation.
Method Participants Twenty students (mean age 20.55 years; 5 male; two left-handers; 2 non-native English speakers) from
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the University of Dundee, Scotland, UK, participated in the experiment for course credit or payment. All had normal or corrected-to-normal vision. Apparatus, stimuli, and task design All stimuli appeared on an ELO 20′′ touch screen with 1024 × 768-pixel resolution. E-Prime software (Schneider et al., 2002) controlled their presentation and recorded responses. The operation signs “+”, “−”, and “=” were coloured black (20point boldface Courier New font) and appeared at the centre of the screen on a grey background. The target was a black dot (100-point boldface Courier New font) presented 150 pixels left or right of the screen centre. Each trial began with the presentation of a blank screen for 500 ms, followed by a fixation sign randomly presented for 250/500/750 ms. Target trials then presented the target randomly on the left/right of fixation sign whereas catch trials continued without further event for another 1750 ms to detect erroneous responses. Two blocks were presented in counterbalanced order: one block in which only “+” was used and a second block in which “+”/“−”/“=” were randomly and equally often used as fixation signs. On the former block there were 72 target trials [12 repetitions × 2 (left/right) target sides × 3 SOA conditions = 72 trials] and 18 catch trials [6 repetitions × 3 SOA conditions = 18 trials], and in the latter block there were 216 target trials [12 repetitions × 3 fixation signs × 2 (left/right) target sides × 3 SOA conditions = 216 trials] and 54 catch trials (6 repetitions × 3 fixation signs × 3 SOA conditions = 18 trials), resulting in a total of 360 trials for this task. Procedure Participants were asked to look at the centre of the screen and to avoid any eye movements. They were instructed to respond as quickly as possible by pressing the space bar with the right index finger whenever they detected a target. An error beep was played following catch trial responses, and these were to be avoided. The session began with a short practice, and there was a short self-paced break after every 45 trials.
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Results The number of catch trial responses per participant was below 5.55%. Mean RTs of target trials were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms, leaving 99.8% of the data. A preliminary analysis revealed no effects of block order, all p values . .09, so we report the results from each condition block separately. Data from the block using only the “+” sign were analysed with a 2 × 3 repeated measures ANOVA with target side (left, right) and SOA (250, 500, 750) as within-participants variables. A main effect of target side, F(1, 19) = 9.41, MSE = 592, p , .01, η2p = .33, indicated faster responses when targets appeared to the right (315) than when they appeared to the left (328) of the plus sign. A main effect of SOA, F(2, 38) = 4.17, MSE = 760, p , .05, η2p = .18, reflected longer responses in the shortest than in both longer SOAs (332 vs. 317 and 316 ms for 250, 500, and 750 ms SOAs, respectively), F(1, 19) = 5.32, MSE = 1185, p , .05, η2p = .22. There was no significant Target Size × SOA interaction (F , 1). Next, we analysed the block in which “+”, “−”, and “ = ” were randomly used as fixation signs, using the same ANOVA design with the addition of fixation sign (+, −, =) as another within-participants variable. Responses were again faster for right than for left targets (318 and 327 ms, respectively), F(1, 19) = 7.42, MSE = 1079, p = .01, η2p = .28. A main effect of SOA, F(2, 38) = 4.75, MSE = 1140, p = .01, η2p = .20, indicated that targets were again detected more slowly in the shortest than in both longer SOA conditions (330 vs. 319 and 318 ms for 250, 500, and 750 ms SOAs, respectively), F(1, 19) = 6.34, MSE = 1706, p , .05, η2p = .25. No other effect was significant (all ps . .11).
Discussion The results of Experiment 2 are clear: Attention allocation was not systematically affected by the identity of the operation signs. This result might reflect the lack of processing of the operators, which were not task relevant. Alternatively, the lack of contextualization of the signs as arithmetic
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symbols may be responsible for the null result, thus perhaps indicating a more strategic origin of OSSA in Experiment 1; after all, participants saw no digits throughout this second experiment. The fact that right targets were detected slightly faster than left targets when following a plus sign was not specific for the plus sign and presumably reflects the use of the right hand to detect all targets (Simon, 1969). Similarly, the speed advantage of later over earlier SOAs merely reflects an unspecific foreperiod effect due to better target predictability at later SOAs (e.g., Niemi & Näätänen, 1981). Importantly, there was no indication of differential attention deployment following fixation on a plus compared to a minus sign (or indeed an equal sign). If the existence of OSSA could be further substantiated in other tasks, then this observation might possibly be in conflict with an attentional explanation of the OM effect as recently favoured by Knops, Zitzmann, and McCrink (2013). However, in the light of the present null result regarding OSSA, we must refrain from such interpretation now and instead report the results of a third experiment.
EXPERIMENT 3 We reasoned that perhaps the task irrelevance of operators in the detection task of Experiment 2 had worked against OSSA and therefore returned to the original operation classification task of Experiment 1, which demanded explicit processing of the symbols. We also ensured that participants were exposed to several numerical tasks prior to the operation classification task, thus inducing an arithmetic context to increase the likelihood of a strong and symmetrical OSSA. Moreover, we manipulated hand position of the participants such that half of them performed the task with their hands crossed and the other half with uncrossed hands. This manipulation allowed us to study whether the spatial information associated with the operation signs is referenced egocentrically to the observer’s body (or a part of it) or to some allocentric location. If OSSA is allocentrically coded then it should replicate across both hand
position conditions, thus also ruling out a hemispheric dominance account of OSSA (cf. Dehaene et al., 1993, Experiment 6; Wood, Nuerk, & Willmes, 2006).
Method Participants Thirty-four students (mean age 22.29 years; 3 male; two left-handers; 10 non-native English speakers) from the University of Dundee, Scotland, UK, participated in the experiment for course credit or payment. All had normal or corrected-to-normal vision. Participants were randomly assigned to one of two groups, which differed in their assigned hands position: uncrossed or crossed. Procedure Participants completed two other numerical tasks that are not reported here (a parity task and a number localization task; Pinhas, Shaki, & Fischer, 2014) before finally being tested with the operation sign classification task. The procedure of the operation classification task was identical to that reported in Experiment 1 except that half the participants performed the task with crossed hands such that their left hand was placed on the right response key, and their right hand was placed on the left response key. The other half of participants performed the task with uncrossed hands.
Results Data from one participant were excluded from the analysis due to high error rate (45.62%). Mean RTs of correct responses of the other 33 participants (95.3% of the data) were trimmed to exclude RTs shorter than 100 ms or longer than 2000 ms (11 trials). A 2 × 2 × 2 repeated measures ANOVA evaluated the effects of hand position (crossed, uncrossed) as a between-participants variable and operation sign (+, −) and response key (left, right) as within-participants variables. Importantly, the interaction between response key and operation sign was replicated, F(1, 31) = 4.62, MSE = 815, p , .05, η2p = .13. Planned comparisons demonstrated significantly faster left key responses for the
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revealed a significant advantage for the “+” sign with the right side (13 ms), F(1, 93) = 6.66, MSE = 1171, p = .01, η2p = .07, and a nonsignificant advantage for the “−” sign with left-side responses (9 ms), F(1, 93) = 3.05, MSE = 1049, p = .08, η2p = .03. Furthermore, spatially congruent (388 ms) trials were responded to faster than spatially incongruent trials (399 ms), t(94) = 2.81, MSE = 38.43, p , .01. All other effects in the ANOVA were not significant (ps . .08).
Discussion Figure 5. Operation sign spatial association (OSSA) effect, showing mean reaction times (RTs) as a function of response key and operation sign across hands position. Vertical bars denote standard errors.
“−” sign (12 ms), F(1, 31) = 4.77, MSE = 474, p , .05, η2p = .13, and a nonsignificant advantage for the “+” sign with the right key (10 ms), F (1, 31) = 1.75, MSE = 878, p = .19, η2p = .05 (see Figure 5). As in Experiment 1, we found that congruent trials (397 ms) were responded to faster than incongruent ones (408 ms), t(32) = 2.19, SD = 28.41, p , .05. There was no reliable three-way interaction between hand position, operation sign, and response key (F , 1), and no other effects in the analysis reached significance (all ps . .19). Combined analysis—OSSA effect Last, we conducted a combined analysis on the data reported from the operation classification task in both Experiments 1 (62 participants) and 3 (33 participants) to further evaluate the inconsistent OSSA asymmetry. Given that there was no effect of task order in Experiment 1 and no effect of hand position in Experiment 3, we ignored these variables in the combined analysis design. The data were analysed with a 2 × 2 × 2 repeated measures ANOVA with experiment (1, 3) as a between-participants variable and operation sign (+, −) and response side (left, right) as within-participants variables. As expected, the interaction between response side and operation sign was significant, F(1, 93) = 7.04, MSE = 1493, p , .01, η2p = .07. Planned comparisons
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Experiment 3 successfully replicated the OSSA using the operation sign classification task. We again found that spatially congruent trials were responded to faster than spatially incongruent trials, suggesting an association between left/right space and the minus/plus sign, respectively. Importantly, the statistical pattern was opposite to that found in Experiment 1: Here we found a significant association only for the minus sign, whereas there we had found a significant association only for the plus sign. Thus, even following a strong context of numerical tasks, the OSSA was again asymmetrical and did not affect both signs equally strongly. Last, a combined analysis on the data from both Experiments 1 and 3 revealed again an asymmetrical OSSA where only the plus sign showed a significant association. Together, the results of all three experiments allow us to attempt a more general discussion of our findings.
GENERAL DISCUSSION The present study uncovered a novel association between operation signs and space that may play a role in arithmetic performance. Specifically, we found that minus signs can be associated with left space and plus signs with right space. These results were established (Experiment 1) and replicated (Experiment 3) in a classification task that required explicit processing and discrimination of the signs. A detection task with less explicit processing requirements for these signs (Experiment 2) did not reveal similar spatial biases, suggesting
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that the OSSA is not strong enough to evoke attentional biases, at least without contextual cueing. We can see at least two explanations for why OSSA might exist. First, it could be the consequence of arithmetic operations themselves—that is, moving one’s attention along the mental number line to the left when subtracting and to the right when adding. In this view, OSSA is a result of SOM. Secondly, OSSA might reflect the general experience of activating smaller magnitude results (on average) when subtracting versus larger magnitude results when adding. In this view, OSSA is an instance of the spatial association captured by SNARC. The correlation pattern from Experiment 1 suggests that the first explanation is more plausible: We found a correlation between OSSA and SOM but no such correlation between OSSA and SNARC. OSSA did not affect both signs equally strongly —this was true in both the first and the last experiment, which revealed opposite association patterns. This outcome does not seem to agree with a polarity correspondence interpretation of OSSA that predicts symmetrical spatial associations given an expected bipolar coding for both the operation signs and the response alternatives (Proctor & Cho, 2006; see also Santens & Gevers, 2008). We propose two possible explanations for the inconsistent asymmetry in the OSSA, which we consider not to be mutually exclusive. First, the minus sign is used for several different functions in the context of elementary algebra—namely, unary, binary and symmetric functions: It is a structural signifier in negative numbers, an operational signifier when subtracting in arithmetic and algebra, and an operational signifier for taking the opposite of a number, respectively (for discussion see Vlassis, 2008). These multiple uses within a mathematical context need to be distinguished and used in a flexible and appropriate manner in operations and equations, making it difficult to assign a specific spatial function to this sign. In support of this claim, Vlassis (2008) reported that eight graders had difficulties when solving equations with negatives, possibly reflecting just this multidimensionality of the minus sign. Similarly, the plus sign is a binary operator that indicates addition, and it also
serves as a unary operator that leaves its operand unchanged. This last notation is typically used in the context of contrasting positive numbers with negative ones. Second, the operation signs also have nonmathematical meanings: the minus sign as a hyphen that concatenates words and thus actually functions like a plus sign; the plus sign as a marker for location. Both signs are also frequently used for noting additional (plus) or fewer (minus) options in computer-based applications, as in the meaning of more and less, respectively. These extramathematical (and sometimes conceptually competing) uses of the signs might further dilute their spatial biases. Moreover, our finding of a rightward association evoked by the plus sign, with the lack of a corresponding leftward bias of the minus sign, reported both in Experiment 1 and in the combined analysis of the OSSA across experiments, is in line with the coactivation for additions and rightward saccades with no reliable coactivation of subtractions and leftward saccades (Knops, Thirion, et al., 2009). Accordingly, the multiple uses and interpretations of the plus and minus signs possibly resulted in asymmetric OSSA patterns and possibly led to the finding of a rather small, yet replicable, effect. Consistent with this view, recent theorizing about the cognitive representation of mathematical symbols emphasizes the importance of contextual (or “situated”) factors (cf. Fischer & Brugger, 2011; Hung, Hung, Tzeng, & Wu, 2008; Wasner, Moeller, Fischer, & Nuerk, 2013). Future research is needed to test whether presenting words instead of signs (e.g., “addition”, “subtraction”), or presenting the operation signs in the context of arithmetic operations, would result in stronger and symmetric OSSA. Concerning the relation between the OSSA and SOM, we found significant correlations between these two spatial biases among individuals who produced an OSSA, suggesting that spatial biases in arithmetic might be related to spatial associations evoked by the operation signs. Yet, we also found that individuals who did not produce an OSSA showed a SOM at the group level. Thus it seems that while OSSA partially contributes to the SOM in some individuals, other sources might
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play a role in deriving the effect, such as the spatial associations evoked by the operands. Future studies focusing on the interrelations between measures of numerical and arithmetic spatial biases and in individual differences obtained for these measures are needed to better understand the role of various contributors for the SOM. The lack of a differential bias from the two operation signs in Experiment 2 might be due to the lack of processing of the operators, which were not relevant for performing the detection task. However, we believe that it is more plausible that the lack of contextualization of the signs as arithmetic symbols is responsible for the null result. The fact that participants saw no digits throughout this second experiment suggests that OSSA might be strategy or context dependent. This idea provided the motivation for Experiment 3. In Experiment 3, the presence of OSSA with crossed hands suggests that OSSA is allocentrically coded because it replicated across both hand position conditions. This observation rules out a hemispheric dominance account of OSSA (cf. Dehaene et al., 1993, Experiment 6; Wood et al., 2006). In conclusion, spatial–numerical associations were almost exclusively obtained with single-digit tasks (Wood et al., 2008). Only recently have researchers extended this work to the level of arithmetic operations, thereby documenting spatial biases in addition and subtraction (SOM effect: Knops, Viarouge, et al., 2009; Pinhas & Fischer, 2008). The present study extends this work further by documenting, for the first time, that arithmetic operators can also induce spatial biases (OSSA effect) and that this effect contributes to the spatial biases in arithmetic. Original manuscript received 27 May 2013 Accepted revision received 6 November 2013 First published online 11 March 2014
REFERENCES Chen, Q. & Verguts, T. (2012). Spatial intuition in elementary arithmetic: A neurocomputational
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account. PloS one, 7(2), e31180. doi:10.1371/ journal.pone.0031180 Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122(3), 371–396. Fias, W. & Fischer, M. H. (2005). Spatial representation of numbers. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 43–54). New York: Psychology Press. Fischer,M.H.&Brugger,P.(2011).Whendigitshelpdigits: Spatial-numerical associations point to finger counting as prime example of embodied cognition. Frontiers in Psychology, 2, 260. doi:10.3389/fpsyg.2011.00260 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. Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6(6), 435–448. Hung, Y. H., Hung, D. L., Tzeng, O. J. L., & Wu, D. H. (2008). Flexible spatial mapping of different notations of numbers in Chinese readers. Cognition, 106 (3), 1441–1450. Knops, A., Thirion, B., Hubbard, E. M., Michel, V., & Dehaene, S. (2009). Recruitment of an area involved in eye movements during mental arithmetic. Science, 324(5934), 1583–1585. Knops, A., Viarouge, A., & Dehaene, S. (2009). Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect. Attention, Perception, & Psychophysics, 71(4), 803–821. Knops, A., Zitzmann, S., & McCrink, K. (2013). Examining the presence and determinants of operational momentum in childhood. Frontiers in Psychology, 4, 325. doi:10.3389/fpsyg.2013.00325 Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. McCrink, K., Dehaene, S., & Dehaene-Lambertz, G. (2007). Moving along the number line: Operational momentum in nonsymbolic arithmetic. Perception and Psychophysics, 69(8), 1324–1333. McCrink, K. & Wynn, K. (2009). Operational momentum in large-number addition and subtraction by 9-month-olds. Journal of Experimental Child Psychology, 103(4), 400–408. Moyer, R. S. & Landauer, T. K. (1967). Time required for judgment of numerical inequality. Nature, 215, 1519–1520.
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Downloaded by [University of Nebraska, Lincoln] at 17:13 13 October 2014
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Niemi, P. & Näätänen, R. (1981). Foreperiod and simple reaction time. Psychological Bulletin, 89(1), 133–162. Pinhas, M. & Fischer, M. H. (2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109(3), 408–415. Pinhas, M., Shaki, S., & Fischer, M. H. (2014). Addition is where the big numbers are: Evidence for an inverse operational momentum effect. Manuscript in preparation. Proctor, R. W. & Cho, Y. S. (2006). Polarity correspondence: A general principle for performance of speeded binary classification tasks. Psychological Bulletin, 132 (3), 416–442. Santens, S. & Gevers, W. (2008). The SNARC effect does not imply a mental number line. Cognition, 108(1), 263–270. Schneider, E., Maruyama, M., Dehaene, S., & Sigman, M. (2012). Eye gaze reveals a fast, parallel extraction of the syntax of arithmetic formulas. Cognition, 125 (3), 475–490. Schneider, W., Eschman, A., & Zuccolotto, A. (2002). E-Prime user’s guide. Pittsburgh, PA: Psychology Software Tools, Inc.
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. Simon, J. R. (1969). Reaction toward the source of stimulation. Journal of Experimental Psychology, 81 (1), 1974–1976. Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21 (4), 555–570. Wasner, M., Moeller, K., Fischer, M. H., & Nuerk, H. C. (2014). How situated cognition influences memory retrieval: The case of finger counting habits. Cognitive Processing. doi:10.1007/s10339014-0599-z Wood, G., Nuerk, H. C., & Willmes, K. (2006). Crossed hands and the SNARC effect: A failure to replicate Dehaene, Bossini and Giraux (1993). Cortex, 42(8), 1069–1079. Wood, G., Willmes, K., Nuerk, H. C., & 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.
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APPENDIX Table A1. Problems used in the calculation task
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Problem type
Problem number
Operand 1
Operation
Operand 2
Result
Target problems
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 2 3 3 4 3 2 4 5 6 6 5 4 6 7 8 7 7 8 8
+ − + − + + + − − − + + + − − − + − + −
0 0 0 0 0 1 2 0 1 2 0 1 2 0 1 2 0 0 0 0
2 2 3 3 4 4 4 4 4 4 6 6 6 6 6 6 7 7 8 8
Filler problems
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
5 6 6 7 7 8 7 8 3 2 3 2 4 3 5 4
− − − − − − − − + + + + + + + +
4 5 4 5 4 5 3 4 3 4 4 5 4 5 4 5
1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9
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