Psychological Research DOI 10.1007/s00426-015-0646-0
ORIGINAL ARTICLE
Measuring spatial–numerical associations: evidence for a purely conceptual link Martin H. Fischer • Samuel Shaki
Received: 24 October 2014 / Accepted: 17 January 2015 Ó Springer-Verlag Berlin Heidelberg 2015
Abstract Previous work on spatial–numerical association (SNAs) included either spatially distributed stimuli or responses. This raises the possibility that the inferred spatial nature of number concepts was a methodological artifact. We present results from a novel task that involves two categories (spatially oriented objects and number magnitudes) and dissociates spatial classification from number classification. The results reveal SNAs without inferential limitations of previous work and point to a working memory mechanism that transfers spatial coding across categories.
Introduction Our thinking about numbers is characterized by systematic performance signatures; perhaps most prominent are the distance effect (poorer number discrimination for more similar magnitudes), the size effect (poorer number comprehension for larger magnitudes), and spatial–numerical associations (SNAs, such as faster left/right responses for relatively small/large numbers, respectively). These findings led to the widely accepted inductive inference that number knowledge constitutes a ‘‘mental number line’’ representation that is spatially orientated, so that small numbers are represented to the left of larger numbers in
M. H. Fischer (&) Division of Cognitive Sciences, University of Potsdam, Karl-Liebknecht-Straße 24-25, House 14, 14476 Potsdam OT Golm, Germany e-mail: [email protected] S. Shaki Ariel University, Ariel, Israel
some mental space (e.g., Dehaene, 2012; Wood, Nuerk, Willmes & Fischer, 2008). How conceptual is this link between numbers and space? Is space truly part of our number knowledge, or is it merely the result of a contamination of cognition with spatial task demands? On the one hand, to assess the spatial nature of number concepts, we do need tasks which include some conceptual spatial feature, the processing efficiency of which can be measured. On the other hand, it is possible that the observed association is then merely a result of extraneous spatial task demands (e.g., Proctor & Cho, 2006). Previous research into SNAs can be classified according to four main approaches that result from crossing spatial stimulation with spatial responding. For example, measuring SNAs with non-lateralized stimuli but lateralized responses was the most popular approach (e.g., Dehaene, Bossini & Giraux, 1993 with left and right keys; Loetscher, Schwarz, Schubiger & Brugger, 2008, with directional head turns; for review, see Fischer & Shaki, 2014). Some studies documented SNAs with lateralized stimuli and a non-lateralized response (e.g., Fischer, Castel, Dodd & Pratt, 2003; with lateralized probe detection; Ranzini et al., 2014, with drifting gratings). Approaches where both stimuli and responses were lateralized (e.g., Mapelli, Rusconi & Umilta`, 2003; Keus & Schwarz, 2005) are less diagnostic for the current issue of assessing the spatial nature of number concepts. However, the few studies where neither stimuli nor responses were lateralized are important here. We discuss these now because our own work builds on their approach. Zorzi, Priftis, and Umilta (2002) asked hemi-neglect patients to verbally report the midpoint of a verbally stated number interval; they exhibited a bias toward larger numbers, consistent with the spatial nature of number
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Psychological Research
knowledge. Lachmair, Dudschig, de la Vega & Kaup, (2014) presented digits followed by nouns at fixation; participants performed speeded lexical decisions with a gonogo instruction. Small digits were associated with downassociated words such as ‘‘mouse’’, and large digits with up-associated words such as ‘‘roof’’, replicating previous work on vertical as opposed to horizontal SNAs (e.g., Schwarz & Keus, 2004; Ito & Hatta, 2004; Shaki & Fischer, 2012). Finally, Gevers et al., (2010) reported SNAs when participants used verbal responses to classify centrally presented digits as odd or even. These three studies have eliminated physical spatial features from both the stimuli and the responses, thus showing that SNAs are not a result of peripheral spatial coding (e.g., Proctor & Cho, 2006) but probably exist at the conceptual level. However, we cannot be sure that SNAs were present at the conceptual level regardless of spatial task demands because participants in all studies still had to use spatial features to solve the task. The instruction to neglect patients was ‘‘to state the midpoint number without making calculations’’ (Zorzi et al., 2002, 138); word stimuli in all go trials of Lachmair et al.’s study contained spatial features; and Gevers et al.’s participants responded to digit parity by saying ‘‘left’’ or ‘‘right’’, so spatially associated responses were still involved in each trial. As the authors themselves acknowledged, ‘‘hence there is verbal-spatial coding at the level of the responses’’ (Gevers et al., 2010, p. 182). Perhaps, the closest approximation to assessing SNAs without importing spatial biases is the study of Priftis et al., (2008) who recorded evoked brain potentials to spoken numbers in six left hemi-neglect patients and found delayed P3b onsets for small numbers, perhaps reflecting inefficient attention shifts into the neglected space. However, non-neglect control patients showed no such SNARC-congruent bias, thus challenging the generality of this finding. In summary, previous work has mostly utilized task-relevant spatial features in one or the other way when number processing was measured, even when both the stimuli and responses were nonlateralized. Here, we report a study with healthy adults that also presented all stimuli at the central fixation point and used a single response key in the mid-sagittal plane to ensure absence of physical spatial features while assessing the presence of SNAs. However, we went beyond previous approaches by also making conceptual spatial features task-irrelevant at the moment of SNA assessment. Specifically, we randomly presented stimuli from two categories (spatially oriented objects and number magnitudes) and imposed a conjunctive response rule where one of the two conditionals was about object orientation and the other conditional was about number magnitude (e.g., ‘‘respond if the object points to the left OR if the number is less than
123
5’’). With this task, we assured that participants’ decisions about numbers were not influenced by extraneous spatial features at the conceptual level; their decisions in those trials were only based on the non-spatial aspect of the response rule. Therefore, the measurement of SNAs was not contaminated by activation of spatial features at either the physical or conceptual level; this aspect of our task is the key innovation over previous work. A magnituderelated bias in participants’ performance permits the inference of a conceptual locus of the spatial–numerical bias.
Method Participants Twenty-seven Israeli adults (20–25 years old; 23 were right-handed, 2 were males) gave informed consent and were treated in accordance with the Ethical Guidelines of the Declaration of Helsinki. Stimuli Stimuli came from two categories: Four positive digits (1, 2, 8, and 9) and four object pictures (cartoon images of a duck or car, each facing either left or right) were shown in black on a white background. The digit’s size was approximately 2.5 9 1.5 cm, and the object’s size was approximately 2.5 9 4 cm. Procedure Participants performed a speeded go-nogo task that required pressing the space bar in 50 % of trials with their right hand in the mid-sagittal plane, and to withhold a response otherwise. They were instructed to respond to targets defined by a combination of number magnitude (‘‘smaller/larger than 5’’) and object orientation (‘‘leftfacing’’ or ‘‘right-facing’’). Thus, we presented only a single stimulus (either number or object) per trial but participants had to keep in mind a response rule involving both of these stimuli. Focusing on the absolute value of digits, we defined ‘‘smaller ? left’’ and ‘‘larger ? right’’ as congruent response instructions, and ‘‘smaller ? right’’ and ‘‘larger ? left’’ as incongruent response instructions. Each of those four counterbalanced blocks of 56 trials consisted of seven random repetitions of the eight stimuli and began with a self-paced display of the response instruction (e.g., ‘‘respond only to digits smaller than five or objects facing left’’). Each trial showed a randomly selected stimulus at fixation until response (i.e., go trials or commission errors) or until 2,000 ms had elapsed (i.e., no-
Psychological Research
go trials or omission errors). Response speed and accuracy were recorded, followed without feedback by the next trial. Data were collected in one 15-min session. Data analysis Data from two participants with over 25 % errors were discarded for lack of representativity. The remaining 25 participants’ error rate was too low for analysis (1.3 % commission and 1.4 % omission errors for number trials). We accepted correct reaction times to number stimuli from 300 to 1,500 ms (98.5 %) for analysis because we were interested in the speed of number comprehension in the context of left- vs. right-facing objects. We conducted a Number Magnitude (small, large) 9 Number-Object Relation (congruent, incongruent) analysis of variance and report all results with p \ 0.05.
Results The reliable effect of Number Magnitude, F(1, 24) = 6.95, g2p = 0.23, p \ 0.05, indicated the size effect (604 vs 631 ms, respectively). The significant main effect of congruency, F(1, 24) = 11.04, g2p = 0.32, p \ 0.05, demonstrated an association of small digits with left and large digits with right space. Responses to small numbers (1 or 2) were faster when the response-permitting pictures faced left instead of right (586 vs. 623 ms; t(24) = 2.77, Cohen’s d = -0.37, p \ 0.05), while responses to large numbers (8 or 9) were faster when response-permitting pictures faced right instead of left (616 vs. 647 ms; t(24) = 2.13, Cohen’s d = -0.29, p \ 0.05; see Fig. 1). To assess the persistence of the effect of object processing onto number processing, we compared the spatial congruency effect for those number trials that immediately followed on an object trial (lag zero) with those number
Fig. 1 Results of the non-spatial SNA assessment. Inserts illustrate the response-relevant direction of the objects (these were left- or right-facing ducks or cars)
trials that immediately followed on a first number trial (lag one). This sequential analysis with the new factor Lag (0 vs. 1) revealed a significant main effect of lag, F(1, 24) = 84.34, g2p = 0.78, p \ 0.05, with faster responses for numbers following a number trial compared to numbers following an object trial (589 vs 683 ms), consistent with the task switching literature (e.g., Monsell, 2003). The main effect of congruency was again significant, F(1, 24) = 11.66, g2p = 0.33, p \ 0.05, and, importantly, there were no other reliable effects or interactions. This shows that the congruency effect was not reliably weaker for the second compared to the first number trial in a sequence.
Discussion The field of numerical cognition is currently very interested in the widely reported associations between number magnitude and space. Although much research has been conducted to clarify the origin and cognitive mechanisms of this pervasive association, a crucial methodological limitation has been the potential contamination of cognitive representations with extraneous, task-specific spatial features. In the present study, we adopted a novel method to measure spatial–numerical associations (SNAs) when spatial features were absent from both the physical and the conceptual level during SNA measurement. We randomly presented stimuli from two categories (numbers and objects) and gave participants a conditional response rule with a spatial and a non-spatial component. The spatial component of the rule was only relevant for objects, while the non-spatial component of the response rule was only relevant for numbers. Importantly, with this novel method, numerical performance was exclusively based on nonspatial aspects of the given response rule. In this way, we were able to measure the processing efficiency of a conceptual spatial feature, while avoiding that any observed association merely reflects extraneous spatial task demands. We obtained a clear result: Numbers are indeed associated with space, even when both stimuli and responses are non-spatial and when there is also no spatial bias at the conceptual level during measurement of the critical association. More so than all previous work, this result justifies the inductive inference that SNAs are part of number knowledge itself. This result alleviates any concerns that the spatial associations of numbers that were so widely reported in previous work might be an artifact of the spatial task components. Our result thereby confirms the pervasive conceptual association between numbers and space (Dehaene et al., 1993; Wood et al., 2008; Fischer & Shaki, 2014) and rules out that SNAs are merely a reflection of response selection or of the coding of task-relevant spatial
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Psychological Research
parameters. Instead, this finding points to a working memory mechanism that transfers spatial coding across categories, in our case from spatially oriented objects to numbers. Working memory mechanisms are gaining much attention in the context of SNAs because they seem to account for such SNAs through a sequential ordering mechanism that is applied as part of a general problem solving strategy (e.g., van Dijck, Abrahamse, Majerus & Fias, 2013, Van Dijck, Abrahamse, Acar, Ketels & Fias, 2014). Our result has theoretical implications for the possible working memory mechanism underlying the SNAs. Specifically, Gevers et al., (2010, p. 182) recently postulated that ‘‘verbal-spatial coding is sufficient to obtain an association between numbers and space’’; our finding shows that this coding does not have to be task-relevant for number processing to yield a SNA. What seems to be necessary, though, is an active spatial coding dimension in working memory, even if this dimension is irrelevant for number classification (e.g., van Dijck et al., 2013, 2014; Proctor & Cho, 2006). The strength of this association did not depend on which rule component had been used in the immediately preceding trial. Further work will determine the persistence of this spatial referencing mechanism over time and might thereby inform our choice of educational strategies for the efficient acquisition of number knowledge. Acknowledgments MHF is funded by DFG grant FI 1915/2-1 ‘‘Manumerical cognition’’.
References Dehaene, S. (2012). The number sense: how the mind creates mathematics (2nd ed.). Oxford: University Press. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, 371–396. Fischer, M. H., Castel, A. D., Dodd, M. D., & Pratt, J. (2003). Perceiving numbers causes spatial shifts of attention. Nature Neuroscience, 6(6), 555–556. Fischer, M. H., & Shaki, S. (2014). Spatial associations in numerical cognition: from single digits to arithmetic. Quarterly Journal of Experimental Psychology, 67(8), 1461–1483.
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Gevers, W., Santens, S., Dhooge, E., Chen, Q., Van den Bossche, L., Fias, W., & Verguts, T. (2010). Verbal-spatial and visuospatial coding of number–space interactions. Journal of Experimental Psychology: General, 139(1), 180–190. Ito, Y., & Hatta, T. (2004). Spatial structure of quantitative representation of numbers: evidence from the SNARC effect. Memory & Cognition, 32, 662–673. Keus, I. M., & Schwarz, W. (2005). Searching for the locus of the SNARC effect: evidence for a response-related origin. Memory and Cognition, 33, 681–695. Lachmair, M., Dudschig, C., de la Vega, I., & Kaup, B. (2014). Relating numeric cognition and language processing: do numbers and words share a common representational platform? Acta Psychologica, 148, 107–114. Loetscher, T., Schwarz, U., Schubiger, M., & Brugger, P. (2008). Head turns bias the brain’s internal random generator. Current Biology, 18(2), 60–62. Mapelli, D., Rusconi, E., & Umilta`, C. (2003). The SNARC effect: an instance of the Simon effect? Cognition, 88, B1–B10. Monsell, S. (2003). Task switching. Trends. Cognitive Science, 7(3), 134–140. Priftis, K., Piccione, F., Giorgi, F., Meneghello, F., Umilta, C., & Zorzi, M. (2008). Lost in number space after right parietal neglect: a neural signature of representational neglect. Cortex, 44, 449–453. Proctor, R. W., & Cho, Y. S. (2006). Polarity correspondence: a general principle for performance of speeded binary classification tasks. Psychological Bulletin, 132, 416–442. Ranzini, M., Lisi, M., Blini, E., Treccani, B., Priftis, K., & Zorzi, M. (2014). Larger, smaller, odd or even? Task-specific effects of optokinetic stimulation on the mental number space. Journal of Cognitive Psychology,. doi:10.1080/20445911.2014.941847. Schwarz, W., & Keuss, I. (2004). Moving the eyes along the mental number line: comparing SNARC effects with manual and saccadic responses. Perception and Psychophysics, 66, 651–664. Shaki, S., & Fischer, M. H. (2012). Multiple spatial mappings in numerical cognition. Journal of Experimental Psychology: Human Perception and Performance, 38(3), 804–809. Van Dijck, J.-P., Abrahamse, E. L., Acar, F., Ketels, B., & Fias, W. (2014). A working memory account of the interaction between numbers and spatial attention. The Quarterly Journal of Experimental Psychology, 67(8), 1500–1513. Van Dijck, J.-P., Abrahamse, E. L., Majerus, S., & Fias, W. (2013). Spatial attention drives serial order retrieval in verbal working memory. Psychological Science, 24(9), 1854–1859. Wood, G., Nuerk, H.-C., Willmes, K., & Fischer, M. H. (2008). On the cognitive link between space and number: a meta-analysis of the SNARC effect. Psychology Science Quarterly, 50(4), 489–525. Zorzi, M., Priftis, K., & Umilta, C. (2002). Brain damage: neglect disrupts the mental number line. Nature, 417(6885), 138–139.
ORIGINAL ARTICLE
Measuring spatial–numerical associations: evidence for a purely conceptual link Martin H. Fischer • Samuel Shaki
Received: 24 October 2014 / Accepted: 17 January 2015 Ó Springer-Verlag Berlin Heidelberg 2015
Abstract Previous work on spatial–numerical association (SNAs) included either spatially distributed stimuli or responses. This raises the possibility that the inferred spatial nature of number concepts was a methodological artifact. We present results from a novel task that involves two categories (spatially oriented objects and number magnitudes) and dissociates spatial classification from number classification. The results reveal SNAs without inferential limitations of previous work and point to a working memory mechanism that transfers spatial coding across categories.
Introduction Our thinking about numbers is characterized by systematic performance signatures; perhaps most prominent are the distance effect (poorer number discrimination for more similar magnitudes), the size effect (poorer number comprehension for larger magnitudes), and spatial–numerical associations (SNAs, such as faster left/right responses for relatively small/large numbers, respectively). These findings led to the widely accepted inductive inference that number knowledge constitutes a ‘‘mental number line’’ representation that is spatially orientated, so that small numbers are represented to the left of larger numbers in
M. H. Fischer (&) Division of Cognitive Sciences, University of Potsdam, Karl-Liebknecht-Straße 24-25, House 14, 14476 Potsdam OT Golm, Germany e-mail: [email protected] S. Shaki Ariel University, Ariel, Israel
some mental space (e.g., Dehaene, 2012; Wood, Nuerk, Willmes & Fischer, 2008). How conceptual is this link between numbers and space? Is space truly part of our number knowledge, or is it merely the result of a contamination of cognition with spatial task demands? On the one hand, to assess the spatial nature of number concepts, we do need tasks which include some conceptual spatial feature, the processing efficiency of which can be measured. On the other hand, it is possible that the observed association is then merely a result of extraneous spatial task demands (e.g., Proctor & Cho, 2006). Previous research into SNAs can be classified according to four main approaches that result from crossing spatial stimulation with spatial responding. For example, measuring SNAs with non-lateralized stimuli but lateralized responses was the most popular approach (e.g., Dehaene, Bossini & Giraux, 1993 with left and right keys; Loetscher, Schwarz, Schubiger & Brugger, 2008, with directional head turns; for review, see Fischer & Shaki, 2014). Some studies documented SNAs with lateralized stimuli and a non-lateralized response (e.g., Fischer, Castel, Dodd & Pratt, 2003; with lateralized probe detection; Ranzini et al., 2014, with drifting gratings). Approaches where both stimuli and responses were lateralized (e.g., Mapelli, Rusconi & Umilta`, 2003; Keus & Schwarz, 2005) are less diagnostic for the current issue of assessing the spatial nature of number concepts. However, the few studies where neither stimuli nor responses were lateralized are important here. We discuss these now because our own work builds on their approach. Zorzi, Priftis, and Umilta (2002) asked hemi-neglect patients to verbally report the midpoint of a verbally stated number interval; they exhibited a bias toward larger numbers, consistent with the spatial nature of number
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knowledge. Lachmair, Dudschig, de la Vega & Kaup, (2014) presented digits followed by nouns at fixation; participants performed speeded lexical decisions with a gonogo instruction. Small digits were associated with downassociated words such as ‘‘mouse’’, and large digits with up-associated words such as ‘‘roof’’, replicating previous work on vertical as opposed to horizontal SNAs (e.g., Schwarz & Keus, 2004; Ito & Hatta, 2004; Shaki & Fischer, 2012). Finally, Gevers et al., (2010) reported SNAs when participants used verbal responses to classify centrally presented digits as odd or even. These three studies have eliminated physical spatial features from both the stimuli and the responses, thus showing that SNAs are not a result of peripheral spatial coding (e.g., Proctor & Cho, 2006) but probably exist at the conceptual level. However, we cannot be sure that SNAs were present at the conceptual level regardless of spatial task demands because participants in all studies still had to use spatial features to solve the task. The instruction to neglect patients was ‘‘to state the midpoint number without making calculations’’ (Zorzi et al., 2002, 138); word stimuli in all go trials of Lachmair et al.’s study contained spatial features; and Gevers et al.’s participants responded to digit parity by saying ‘‘left’’ or ‘‘right’’, so spatially associated responses were still involved in each trial. As the authors themselves acknowledged, ‘‘hence there is verbal-spatial coding at the level of the responses’’ (Gevers et al., 2010, p. 182). Perhaps, the closest approximation to assessing SNAs without importing spatial biases is the study of Priftis et al., (2008) who recorded evoked brain potentials to spoken numbers in six left hemi-neglect patients and found delayed P3b onsets for small numbers, perhaps reflecting inefficient attention shifts into the neglected space. However, non-neglect control patients showed no such SNARC-congruent bias, thus challenging the generality of this finding. In summary, previous work has mostly utilized task-relevant spatial features in one or the other way when number processing was measured, even when both the stimuli and responses were nonlateralized. Here, we report a study with healthy adults that also presented all stimuli at the central fixation point and used a single response key in the mid-sagittal plane to ensure absence of physical spatial features while assessing the presence of SNAs. However, we went beyond previous approaches by also making conceptual spatial features task-irrelevant at the moment of SNA assessment. Specifically, we randomly presented stimuli from two categories (spatially oriented objects and number magnitudes) and imposed a conjunctive response rule where one of the two conditionals was about object orientation and the other conditional was about number magnitude (e.g., ‘‘respond if the object points to the left OR if the number is less than
123
5’’). With this task, we assured that participants’ decisions about numbers were not influenced by extraneous spatial features at the conceptual level; their decisions in those trials were only based on the non-spatial aspect of the response rule. Therefore, the measurement of SNAs was not contaminated by activation of spatial features at either the physical or conceptual level; this aspect of our task is the key innovation over previous work. A magnituderelated bias in participants’ performance permits the inference of a conceptual locus of the spatial–numerical bias.
Method Participants Twenty-seven Israeli adults (20–25 years old; 23 were right-handed, 2 were males) gave informed consent and were treated in accordance with the Ethical Guidelines of the Declaration of Helsinki. Stimuli Stimuli came from two categories: Four positive digits (1, 2, 8, and 9) and four object pictures (cartoon images of a duck or car, each facing either left or right) were shown in black on a white background. The digit’s size was approximately 2.5 9 1.5 cm, and the object’s size was approximately 2.5 9 4 cm. Procedure Participants performed a speeded go-nogo task that required pressing the space bar in 50 % of trials with their right hand in the mid-sagittal plane, and to withhold a response otherwise. They were instructed to respond to targets defined by a combination of number magnitude (‘‘smaller/larger than 5’’) and object orientation (‘‘leftfacing’’ or ‘‘right-facing’’). Thus, we presented only a single stimulus (either number or object) per trial but participants had to keep in mind a response rule involving both of these stimuli. Focusing on the absolute value of digits, we defined ‘‘smaller ? left’’ and ‘‘larger ? right’’ as congruent response instructions, and ‘‘smaller ? right’’ and ‘‘larger ? left’’ as incongruent response instructions. Each of those four counterbalanced blocks of 56 trials consisted of seven random repetitions of the eight stimuli and began with a self-paced display of the response instruction (e.g., ‘‘respond only to digits smaller than five or objects facing left’’). Each trial showed a randomly selected stimulus at fixation until response (i.e., go trials or commission errors) or until 2,000 ms had elapsed (i.e., no-
Psychological Research
go trials or omission errors). Response speed and accuracy were recorded, followed without feedback by the next trial. Data were collected in one 15-min session. Data analysis Data from two participants with over 25 % errors were discarded for lack of representativity. The remaining 25 participants’ error rate was too low for analysis (1.3 % commission and 1.4 % omission errors for number trials). We accepted correct reaction times to number stimuli from 300 to 1,500 ms (98.5 %) for analysis because we were interested in the speed of number comprehension in the context of left- vs. right-facing objects. We conducted a Number Magnitude (small, large) 9 Number-Object Relation (congruent, incongruent) analysis of variance and report all results with p \ 0.05.
Results The reliable effect of Number Magnitude, F(1, 24) = 6.95, g2p = 0.23, p \ 0.05, indicated the size effect (604 vs 631 ms, respectively). The significant main effect of congruency, F(1, 24) = 11.04, g2p = 0.32, p \ 0.05, demonstrated an association of small digits with left and large digits with right space. Responses to small numbers (1 or 2) were faster when the response-permitting pictures faced left instead of right (586 vs. 623 ms; t(24) = 2.77, Cohen’s d = -0.37, p \ 0.05), while responses to large numbers (8 or 9) were faster when response-permitting pictures faced right instead of left (616 vs. 647 ms; t(24) = 2.13, Cohen’s d = -0.29, p \ 0.05; see Fig. 1). To assess the persistence of the effect of object processing onto number processing, we compared the spatial congruency effect for those number trials that immediately followed on an object trial (lag zero) with those number
Fig. 1 Results of the non-spatial SNA assessment. Inserts illustrate the response-relevant direction of the objects (these were left- or right-facing ducks or cars)
trials that immediately followed on a first number trial (lag one). This sequential analysis with the new factor Lag (0 vs. 1) revealed a significant main effect of lag, F(1, 24) = 84.34, g2p = 0.78, p \ 0.05, with faster responses for numbers following a number trial compared to numbers following an object trial (589 vs 683 ms), consistent with the task switching literature (e.g., Monsell, 2003). The main effect of congruency was again significant, F(1, 24) = 11.66, g2p = 0.33, p \ 0.05, and, importantly, there were no other reliable effects or interactions. This shows that the congruency effect was not reliably weaker for the second compared to the first number trial in a sequence.
Discussion The field of numerical cognition is currently very interested in the widely reported associations between number magnitude and space. Although much research has been conducted to clarify the origin and cognitive mechanisms of this pervasive association, a crucial methodological limitation has been the potential contamination of cognitive representations with extraneous, task-specific spatial features. In the present study, we adopted a novel method to measure spatial–numerical associations (SNAs) when spatial features were absent from both the physical and the conceptual level during SNA measurement. We randomly presented stimuli from two categories (numbers and objects) and gave participants a conditional response rule with a spatial and a non-spatial component. The spatial component of the rule was only relevant for objects, while the non-spatial component of the response rule was only relevant for numbers. Importantly, with this novel method, numerical performance was exclusively based on nonspatial aspects of the given response rule. In this way, we were able to measure the processing efficiency of a conceptual spatial feature, while avoiding that any observed association merely reflects extraneous spatial task demands. We obtained a clear result: Numbers are indeed associated with space, even when both stimuli and responses are non-spatial and when there is also no spatial bias at the conceptual level during measurement of the critical association. More so than all previous work, this result justifies the inductive inference that SNAs are part of number knowledge itself. This result alleviates any concerns that the spatial associations of numbers that were so widely reported in previous work might be an artifact of the spatial task components. Our result thereby confirms the pervasive conceptual association between numbers and space (Dehaene et al., 1993; Wood et al., 2008; Fischer & Shaki, 2014) and rules out that SNAs are merely a reflection of response selection or of the coding of task-relevant spatial
123
Psychological Research
parameters. Instead, this finding points to a working memory mechanism that transfers spatial coding across categories, in our case from spatially oriented objects to numbers. Working memory mechanisms are gaining much attention in the context of SNAs because they seem to account for such SNAs through a sequential ordering mechanism that is applied as part of a general problem solving strategy (e.g., van Dijck, Abrahamse, Majerus & Fias, 2013, Van Dijck, Abrahamse, Acar, Ketels & Fias, 2014). Our result has theoretical implications for the possible working memory mechanism underlying the SNAs. Specifically, Gevers et al., (2010, p. 182) recently postulated that ‘‘verbal-spatial coding is sufficient to obtain an association between numbers and space’’; our finding shows that this coding does not have to be task-relevant for number processing to yield a SNA. What seems to be necessary, though, is an active spatial coding dimension in working memory, even if this dimension is irrelevant for number classification (e.g., van Dijck et al., 2013, 2014; Proctor & Cho, 2006). The strength of this association did not depend on which rule component had been used in the immediately preceding trial. Further work will determine the persistence of this spatial referencing mechanism over time and might thereby inform our choice of educational strategies for the efficient acquisition of number knowledge. Acknowledgments MHF is funded by DFG grant FI 1915/2-1 ‘‘Manumerical cognition’’.
References Dehaene, S. (2012). The number sense: how the mind creates mathematics (2nd ed.). Oxford: University Press. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, 371–396. Fischer, M. H., Castel, A. D., Dodd, M. D., & Pratt, J. (2003). Perceiving numbers causes spatial shifts of attention. Nature Neuroscience, 6(6), 555–556. Fischer, M. H., & Shaki, S. (2014). Spatial associations in numerical cognition: from single digits to arithmetic. Quarterly Journal of Experimental Psychology, 67(8), 1461–1483.
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