Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions

Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Koch, N. N., Huber, J. F., Lohmann, J., Cipora, K., Butz, M. V., & Nuerk, H.-C. (2023). Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions. Collabra: Psychology, 9(1). https://doi.org/10.1525/collabra.67908 Cognitive Psychology Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions 1 2 1 3 1,2 Nadine N. Koch , Julia F. Huber , Johannes Lohmann , Krzysztof Cipora , Martin V. Butz , 2,4 Hans-Christoph Nuerk 1 2 Department of Computer Science, University of Tuebingen, Tuebingen, Germany, Department of Psychology, University of Tuebingen, Tuebingen, 3 4 Germany, Department of Mathematics Education, Loughborough University, Loughborough, UK, LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany Keywords: mental number representation, number processing, numerical cognition, SNARC effect, spatialization, spatial-numerical associations https://doi.org/10.1525/collabra.67908 Collabra: Psychology Vol. 9, Issue 1, 2023 The Spatial-Numerical Association of Response Codes (SNARC) effect – i.e., faster responses to small numbers with the left compared to the right side and to large numbers with the right compared to the left side – suggests that numbers are associated with space. However, it remains unclear whether the SNARC effect evolves from a number’s magnitude or the ordinal position of a number in working memory. One problem is that, in different paradigms, the task demands influence the role of ordinality and magnitude. While single-task setups in which participants judge the parity of a displayed number indicate the importance of magnitude for the SNARC effect, evidence for ordinal influences usually comes from experiments where ordinal sequences have to be memorized or setups in which participants possess pre-existing knowledge of the ordinality of stimuli. Therefore, in this preregistered study, we employed a SNARC task without secondary ordinal sequence memorization. We dissociate ordinal and magnitude accounts by carefully manipulating experimental stimulus sets. The results indicate that even though the magnitude model better accounts for the observed data, the ordinal position seems to matter as well. Hence, numbers are associated with space in both a magnitude- and an order-respective manner, yielding a mixture of both compatibility effects. Moreover, a multiple coding framework may most accurately explain the roots of the SNARC effect. the writing direction (Dehaene et al., 1993; Restle, 1970). 1. Introduction In strong opposition to the mental number line account, Magnitude information seems to be mentally associated Casasanto and Pitt (2019) have argued that it may not be with space, as suggested by the Spatial-Numerical Associ- the numerical magnitude but the ordinal position of num- ation of Response Codes (SNARC) effect: responses to rel- bers that is spatially mapped. In most experiments, the atively small numbers are faster with the left compared to two alternative explanations cannot be tested against each the right side, and responses to relatively large numbers are other because the ordinality typically coincides with mag- faster with the right compared to the left side (Dehaene et nitude. Prpic et al. (2021) argued against the ordinal ac- al., 1993). The mental number line (Dehaene et al., 1993; count by referring to studies that show spatial-numerical Restle, 1970) and the working memory account (van Dijck associations in setups where no immediate stimulus order et al., 2011; van Dijck & Fias, 2011) suggest different men- can be established. Meanwhile, studies showing a spatial tal representations that induce this effect. mapping of non-numerical ordinal sequences, such as let- ters of the alphabet or months (Gevers & Lammertyn, 1.1. The Mental Number Line Account and its 2005), support the ordinal account. This opened the discus- Challenges sion on whether the numerical magnitude or ordinality is spatially mapped (Gevers & Lammertyn, 2005). Dehaene et al. (1993) proposed a visuospatial mental number line to account for the SNARC effect. This line is assumed to represent number magnitude horizontally, ei- ther from left to right or from right to left, depending on a Corresponding author: hc.nuerk@uni-tuebingen.de Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions mental representation of numbers. The mental number line 1.2. The Working Memory Account account claims that numbers are represented according to Further researchers described the importance of working their magnitude. In contrast, in our interpretation, the memory for temporarily and selectively mapping number working memory account assumes that only the presented and ordinality representations onto space. These observa- numbers are represented in working memory in order of tions laid the grounds for the working memory account (van their magnitude. Dijck et al., 2011; van Dijck & Fias, 2011), which suggests that a spatial coding of numbers is formed during task ex- 1.3. More Than one Explanation for the SNARC ecution by storing task-relevant numbers in working mem- Several authors (e.g., Huber et al., 2016) showed that ory. Thereby the ordinal position of a number in working both magnitude (as postulated by the mental number line) memory is associated with space: the beginning of a se- and ordinal position (in line with the working memory ac- quence is associated with the left and the end with the count) are important for the SNARC effect. A model by right (van Dijck & Fias, 2011). This association is created Prpic et al. (2016), which is motivated by spatial compat- by binding number items to active spatial templates, which ibility effects when processing musical note lengths, sug- are created from experience. Newer versions of the working gests that depending on the nature of the task, either mag- memory account further assume that the most common nitude or ordinality determines the SNARC effect: direct numbers and their typical relations are saved in long-term tasks (tasks requiring the processing of the magnitude/or- memory (Abrahamse et al., 2016). One such long-term dinal position of a number, i.e., magnitude classification) memory representation is the canonical number set repre- evoke an ordinality-dependent SNARC effect and indirect senting the numbers one to nine according to their magni- ones (tasks that do not require processing the magnitude/ tude, as this is the order in which those numbers typically the ordinal position of the number, i.e., parity judgment) occur (Abrahamse et al., 2016). A number item can prompt a magnitude-dependent SNARC effect. Similarly, Schroeder this representation. Abrahamse et al. (2016) explain the et al. (2017) proposed a multiple coding framework, which range dependence of the SNARC effect (i.e., an early obser- suggests that different mechanisms provoke spatial asso- vation that the SNARC effect seems to adapt to the range of ciations simultaneously (see also Abrahamse et al., 2016; numbers being used in a stimulus set, and the same num- Huber et al., 2016). More precisely, the framework assumes ber can be characterized by a left/right side advantage de- that simultaneous activation of the number representa- pending on other numbers present in the stimulus set; De- tions on a mental number line and in working memory is haene et al., 1993) by assuming that when only a subset possible. Although this model requires further validation of the canonical number set is perceived, this representa- (see also Cipora et al., 2020) and asks for the formalization tion “is ‘pruned’ to match the actually used items in the of a computational model, the basic idea that multiple cod- experiment” (Abrahamse et al., 2016, p. 6). The working ing mechanisms may be responsible for the SNARC effect memory account does not further specify which numbers is supported by the data of neglect patients (van Dijck & are loaded into working memory while conducting the task. Doricchi, 2019). As Toomarian and Hubbard (2018) argued, Thus, there are two possible interpretations: first, only the the spatial coding of magnitude may be related to inborn perceived numbers are activated in working memory in or- spatial biases, while the mapping of the ordinality could be der of their magnitude. In this case, numbers within the related to cultural factors. To sum up, different lines of ar- range of the presented numbers that are not presented gument support the mental number line account and the would not be activated in working memory (see the right working memory account, and both magnitude and ordinal- column of Figure 1 for examples of the numbers activated ity seem to be at play. What remains unclear is their relative in working memory). However, this wording could also be role in the most basic parity judgment task typically used to interpreted such that the mental representation includes measure the SNARC effect. all numbers within a particular range (see Figure 1 left col- umn) . We focus on the first interpretation as the predic- 1.4. The Current Study tions of the mental number line, and the working memory account can only be differentiated when assuming that only Even though several studies compared the mental num- numbers are activated that are perceived during the task. ber line and working memory accounts (Ginsburg et al., We are aware that some extensions to the original working 2014; Huber et al., 2016; Lindemann et al., 2008; van Dijck memory account allow for a much wider range of predic- & Fias, 2011), all of the cited studies used an ordinal sec- tions, which are harder to falsify. While we do not focus on ondary task (i.e., learning a sequence) or were based on these extensions in detail here in the introduction, we will some pre-existing knowledge structures. The pre-existing examine under which conditions they might also account knowledge structures used as an anchor for the ordinality for our data in the discussion section. representation were musical notation (Prpic et al., 2016) or In sum, the mental number line and the working mem- phone dial (Mingolo et al., 2021). These studies only ob- ory account differ concerning their assumption about the 1 We thank J.-P. van Dijck for clarifying this. Collabra: Psychology 2 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Figure 1. The four number sets (each participant saw only one set) and their mental representation according to the mental number line and the working memory account. The fitting by the regression line differs considerably for the two accounts. For the stimulus set [1, 2, 3, 8], 8 is ordinally the fourth element in the set, so in an ordinal sequence fit- ting, it would be one unit to the right of number 3. However, in a mental number line fitting, the 8 should be five units to the right of number 3. Analog arguments hold for the other stimulus sets. served the ordinality SNARC in tasks in which the ordinal As a result, an advantage of the ordinality model would position of the stimulus was task-relevant (i.e., direct thus lend strong support to the working memory account. tasks), that is, when a sequence of items had to be tem- In contrast, an advantage of the magnitude model would porarily memorized. One could argue that the nature of favor the mental number line model. As mentioned above, the ordinal secondary task or the memory recall triggered there are extensions of the original working memory model the effect of the ordinal position of a number/object in a that might also incorporate result patterns, which are pre- sequence found in such experiments. Further, studies us- dicted by the magnitude model, making it almost empir- ing pre-existing knowledge structures (e.g., Mingolo et al., ically indistinguishable from the mental number line ac- 2021; Prpic et al., 2016) deviated from the typical SNARC count. We will address those in the discussion. setup. For those studies, it remains unclear whether ordi- 2. Methods nality or magnitude plays a decisive role in the spatial map- ping of numbers in the most basic SNARC setup with num- The ethics committee for psychological research at the bers and a typical numerical task. University of Tuebingen (Nürk_2020_0623_192) has ap- Therefore, the current study aimed to probe the mental proved the study’s protocol. The study was preregistered number line and the ordinal working memory accounts us- (https://osf.io/h7vpm). ing a parity judgment (i.e., indirect) task without a sec- ondary task (i.e., without learning a sequence explicitly and 2.1. Participants without referring to an existing ordinality-related knowl- edge structure). Further, we did not present the number An a priori power analysis revealed that at least 265 set used in the parity judgment task to the participants data sets must be collected to detect differences in SNARC in advance, avoiding the priming of a particular number slopes of Cohen’s d = 0.2 (power = .9, in paired two-sided representation. We used carefully selected stimulus sets to t-tests), which we considered the smallest effect size of in- dissociate predictions from ordinal and magnitude num- terest. Due to the online setting, we expected a dropout ber representations, always including two even and two odd rate of 20% (incomplete or unusable data sets). Accordingly, numbers with three consecutive numbers and a fourth one we aimed for at least 320 participants. In total, 465 partici- maximally distant from the single-digit range (see Figure 1 pants took part in the online study. Participants signed in- for a more detailed description of stimuli that enable this formed consent and could participate in a lottery of eleven dissociation). Amazon vouchers or receive course credit for their par- ticipation. After exclusions (see below), a sample size of 2 All effect sizes were calculated using the jmv R-package. Collabra: Psychology 3 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions N = 423 remained (47 to 61 participants per condition, appeared, stating that participation in the experiment was for the exact distribution of participants, see supplemen- only possible using a PC, laptop, or netbook. If neither of tary material S1 ). The participants reported an average age the three devices was chosen or the device rendered like of 25.90 years (SD = 9.60 years, range = 18 - 83 years); a smartphone, the experiment ended with a corresponding 290 participants reported being female, 130 reported being note. male, and three defined themselves as diverse. In total, 375 Next, the participant was instructed to classify the parity participants indicated being right-handed, 39 left-handed, of a number by either pressing the “d” or “k” key depending and nine ambidextrous. The sample comprised 366 Ger- on the parity-to-key-mapping. During the practice trials man, 11 English, and 48 other native speakers (language re- (see Figure 2), each number of the number set was pre- ported less than ten times). All except three participants sented twice in random order. The participants received reported languages reading from left to right (two Arabic, feedback for late (‘too late’), correct (‘correct’), and wrong one Pashto), and two participants did not state an identifi- (‘wrong’) responses. Participants that got less than 75% of able mother tongue. the practice trials correct had to repeat the trials. We chose 75% because 50% of correct trials were reached by chance. 2.2. Material and Apparatus Hence, 75% accuracy means that 50% of the trials were re- sponded to correctly, while the other 50% were correct by The main experiment was conducted via Pavlovia (2020) chance (25% correct, 25% incorrect). using jsPsych (de Leeuw, 2015). We additionally used SoSci After the training, the instruction was repeated, and the Survey (Leiner, 2019) to assign a participant number and to experimental trials followed. The experimental trials were save e-mail addresses for the lottery. analogous to the practice trials. However, feedback was only Texts (in the experimental trials) were presented in given for no or late responses, stating that none of the re- white in font size 22 px, numbers, fixation cross, and feed- sponse keys was pressed, and the parity-to-key mapping back (for the practice trials) in font size 72 px on a black was presented again. Each number of a number set was pre- background. We used four number sets ([1, 2, 3, *8*], [2, sented 30 times in pseudorandomized order. Thereby no 3, 4, *9*], [*1*, 6, 7, 8], [*2*, 7, 8, 9]) including one critical more than three numbers of the same parity were shown number (between asterisk). For the critical number, the dis- in a row, and a number was never repeated in consecutive tance toward the previous (following) number differs re- trials. The same practice and experimental procedure were garding the magnitude of a number and its ordinal position then repeated with the reversed parity-to-key mapping. in the sequence (see Figure 1). The study continued by collecting demographic data (cf. supplementary materials S3) and running an arithmetic 2.3. Design task consisting of 40 equations with basic arithmetic oper- ations and a time limit of two minutes (cf. supplementary Each participant was assigned to one of the four number materials S4, the analysis of this task goes beyond the scope sets and one of two parity-to-key mapping orders (either of the current research question). Then, two final ques- first “d” for odd numbers and “k” for even numbers and tions: “How noisy was your environment?” (“silent” to “ex- then “k” for odd numbers and “d” for even numbers, or vice tremely noisy”) and “If you were the experimenter, would versa) by order of participation. Within one participant, you use the data?” (“Yes.”, “Not all of them.”, “No.”) were each combination of each number of the number set and presented. The participant was then forwarded to SoSci parity-to-key mappings was repeated 30 times resulting in Survey for the lottery. 240 trials. This between-subject design prevented transfer effects between number sets. 2.5. Data Analyses 2.4. Procedure The data exclusion was derived from the one used by Cipora et al. (2019). Our analysis script can be found in After a welcome page on SoSci Survey, the participant the supplementary materials S5 and the data in S6. First, was forwarded to Pavlovia, where the main experiment was data sets from participants who reported an age below 18 executed in full screen (the experimental code can be found (0.43%, this exclusion criterium was not preregistered) and in the supplementary material S2). The participants were participants who indicated that they performed the task in asked to choose a conducting language (German or Eng- a very noisy or extremely noisy environment (1.94%) were lish), whether they wanted to look at the experiment or excluded. Afterward, incorrectly responded trials (4.74%; participate and which device they used (PC, laptop, net- prepared data set), and trials with reaction times below book, smartphone, e-reader, or other). Beneath, a red note 250 ms (0.59%; as in Cipora et al., 2019; Ginsburg et al., 3 The supplementary materials can be accessed on OSF (https://osf.io/u9wer/). 4 As preregistered, we did not exclude the data of right-to-left readers. However, excluding those participants did not substantially change the results: the magnitude model described significantly more variance and the deviance between prediction and actual data is signifi- cantly smaller for the ordinality model. Collabra: Psychology 4 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Figure 2. The procedure of one practice trial. For experimental trials, there was only feedback when no response key was pressed. The feedback was presented for 2000 ms in this case. 2014; van Dijck & Fias, 2011 this excludes accidental re- stance, for the number set 1, 2, 3, 8, we fitted a regression actions and ensures the comparability of the results) were line to the dRTs of the numbers 1, 2, and 3. Subsequently, discarded as anticipations. Participants with no correct re- we compared the dRT value for the number 8 with the dRT sponse in one of the conditions left were excluded (0.22%). of this number being predicted by either the magnitude Then sequentially, all reaction times beyond three SD below model (i.e., 8) or by the ordinality model (i.e., 4). The de- or above the individual mean reaction time were removed pendent measure was the deviance between the actual dRT (3.27%). To ensure an accurate estimate for mean reaction and predicted values for both magnitude and ordinality time in each experimental cell, data sets resulting in less models. This analysis was added as we wanted to look at than 70% (as in Cipora et al., 2019) of valid trials left for how the regression models behave when being estimated any number (of a number set) in each parity-to-key-map- solely based on consecutive numbers and not being affected ping were excluded (6.18%). by the critical number (which constitutes 25% of the data) The dRTs (mean reaction time of the right minus mean and which of these models is better in predicting the dRT reaction time of the left hand) for each participant and each for the critical number. number were calculated according to Fias et al. (1996). Two 3. Results linear regressions were fitted for each participant predict- ing dRTs either by the ordinal positions of a number (ordi- We used a significance level of .05, two-sided t-tests, and nality model) or by the magnitude of a number (magnitude corrected the four reported p-values with the Bonferroni- model). We tested both slopes against zero. For the linear Holm method. At first, the results from the model compari- regressions comparison (model comparison), logit transfor- son and afterward, the comparison of the deviance from re- mations were applied to the R²-values of the linear regres- gressions are reported. sions from each participant to approximate a normal dis- tribution. A paired-samples t-test was used to compare the 3.1. Model Comparison transformed R²-values of the models. Additionally, we conducted an alternative analysis. We The slopes of the magnitude model (M = -5.55, slope fitted a linear regression to the dRTs of the three consec- t(422) = -11.56, p < .001, 95%-CI = [-6.49, -4.61]), as slope utive numbers in a number set (the critical number was well as the slopes of the ordinality model (M = -13.33, slope excluded) for each participant. We then calculated the ab- t(422) = -12.41, p < .001, 95%-CI = [-15.45, -11.22]), dif- slope solute difference between the measured dRT and the pre- fered significantly from zero, showing influences of magni- dicted value from each model for the critical number for tude and ordinality, respectively. each participant (deviance) and compared them. For in- 5 Further model comparisons are described in supplementary materials S7 and S8. S7 reports results from the analyses of different number sets and number ranges separately. S8 describes results from a comparison of two linear mixed models considering ordinality and magnitude. The Akaike Information Criterion (AICc) was smaller for the magnitude model. In accordance with the reported model comparison, this indicates that the magnitude model described the data more accurately. Collabra: Psychology 5 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Compared to the ordinality model, the magnitude model Meanwhile, comparing the explained variance supports showed significantly greater R -values (M = -0.67/.42 the mapping of numbers according to their magnitude and mag [mean of the transformed/untransformed R -values], hence the mental number line account. However, it has also M = -0.96/.40 [mean of the transformed/untransformed been proposed that not only the perceived numbers them- ord R -values], M = 0.29/.02 [difference of the transformed/ selves but also the whole included range of numbers as dif untransformed R -values], t(422) = 3.33, p < .001, overlearned and stored in working memory (i.e., 1, 2, 3, 4, 95%-CI = [0.12, 0.46], Cohen’s d = 0.16) and hence de- 5, 6,…) may partially constitute the working memory con- dif scribed significantly more variance of the data. Figure 3 tent (Abrahamse et al., 2016). This poses a fundamental shows a visualization of the comparisons in the top panel. problem for distinguishing this extended working memory account from the magnitude account because the ordinal 3.2. Comparison of the Deviance metric of the overlearned absolute sequence in long-term memory and the linear magnitude metric of the mental The predicted dRTs for the ordinal position (ordinality number line account are identical. Seeing that our deviance model) and the magnitude (magnitude model) of the crit- comparisons of the critical fourth number favor the ordi- ical number were calculated for each participant. The ab- nality model without an intermediate, never shown num- solute distance between the prediction and the measured ber, the results indicate that only the perceived numbers dRT for the critical number was significantly smaller for appear to be ordered in working memory. These seemingly the ordinality model compared to the magnitude model contradicting results thus all hint towards a multiple coding (M = 106.84, M = 51.43, M = 55.41, t(422) = 15.57, mag ord dif framework-oriented explanation. p < .001, 95%-CI = [48.41, 62.40], Cohen’s d = 0.76). dif Hence, the deviance between the prediction for the critical 4.2. Multiple-Coding Frameworks of the SNARC number and the measured dRT in the magnitude model was about twice as large as the corresponding deviance of Even though the magnitude model describes the current the ordinality model. A visualization of the comparisons is data more accurately, the advantage is associated with a shown in Figure 3, bottom panel. very small effect size (d = 0.17), which is, in fact, smaller than the smallest effect of interest we defined in our a priori 4. Discussion power analysis. At the same time, the results also imply that both the mental number line and the working memory The exact mapping of numbers to space has been as- account play important roles (e.g., the deviance comparison sumed to depend on their magnitude (mental number line with Cohen’s d = 0.76 and the measured dRT lies between account) or their ordinal position (as implied by working the predictions of both models). We propose to further con- memory account and other theoretical accounting of the sider an integrative model, which may have the potential to primary/sole role of ordinality). The current study com- expand on previous multiple-coding-framework-related ex- pared the two assumptions while avoiding secondary tasks planations (e.g., Abrahamse et al., 2016; Huber et al., 2016; and memory recall of overlearned ordinal sequences. The Prpic et al., 2016; Schroeder et al., 2017). In particular, the results show that the magnitude model explained signifi- referenced literature seems to imply that working memory cantly more variance than the ordinality model. However, may be loaded with different task-relevant item-to-space when calculating the deviance between the measured dRT associations. for the critical fourth number and the predicted dRT via re- In our case, it seems that a mental number line activa- gression of the three consecutive numbers, the ordinality tion induced a linear magnitude-to-space association of the model yielded significantly smaller deviance. These results relevant number range. appear contradictory. Meanwhile, an ordinal encoding – whereby the order is, in our case, also influenced by a canonical magnitude 4.1. A Possible Explanation for This – induced a linear order-to-space association. Such a si- Contradiction multaneous activation could account for the deviances be- tween the model’s predictions and the measured data, as One possible source for the apparent inconsistency is the well as for the fact that both models fit the data well. In- deviance comparison. Because the regression line parame- deed, such a model could furthermore account for the dis- ters are estimated with a limited amount of data, the in- cussed differences in magnitude and ordinal influences in tercept and slope estimations are inevitably error-prone. other experiments as well – the stronger the focus on a While the error in the intercept affects all predicted po- particular magnitude or sequential ordering, the stronger sitions equally – and thus both deviances – errors in the may be its influence on the selection of (in-)compatible re- slope lead to linearly increasing errors with a distance of sponse codes. Hence, as Casasanto and Pitt (2019) stated, the critical fourth number from the other three. This dis- the culturally influenced mapping of the ordinality plays tance is one for the ordinality model but five for the magni- an important role in spatial mappings in such an integra- tude model – thus, the error in the slope adds up five times tive model. Those ordinal mappings can explain the associ- in the prediction of the critical number using the magni- ation between numerically unrelated concepts such as mu- tude model. Since the error is not known, the results de- sical pitches (Rusconi et al., 2006) or emotional valence (de mand further studies and should be interpreted with cau- la Vega et al., 2012) and space. tion at this point. Collabra: Psychology 6 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Figure 3. The four plots show mean dRTs in ms depending on the numbers’ magnitude or the ordinal position of the numbers when ordered according to their magnitude. The lines indicate the averaged regression lines over all participants’ regression lines in a particular number set. For calculating the positions of the circles, triangles, and squares, first, the dRTs of each participant were averaged. Then the mean over all participants for each number per number set was calculated. The plots in a) show the linear regression lines of all four digits of a number set. The plots in b) show the linear regression lines of the three consecutive points. Additionally, as described by Prpic et al. (2021), the mag- 4.3. Conclusion nitude also seems to have an influence. The interaction of The presented statistical model comparison indicates those mappings can explain the results of studies inves- that the mental number line account yields more accurate tigating the SNARC effect in right-to-left reading partic- regressions. Meanwhile, despite the described issues and ipants, where the SNARC effect is not consistently found the fact that a linearity assumption may be a slight over- (Fischer et al., 2009; Shaki et al., 2009; Shaki & Gevers, simplification, a linear extrapolation to the critical number 2011; Zebian, 2005; Zohar-Shai et al., 2017). Here the map- favors the (pure) working memory account. The results are ping of the potentially inborn magnitude seems to be in in line with a multiple-coding framework account, which contrast with the culturally learned mapping of ordinality. assumes that multiple task-relevant entity encodings play crucial roles dependent on the task setup and the directness Collabra: Psychology 7 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions with which the setup triggers particular item-space associ- Contributions ations. However, the data are also in line with an extended Contributed to conception and design: NNK, JFH, JL, KC, working memory account, where not only ordinal se- MVB, H-CN quences are actively kept in active working memory but Contributed to acquisition of data: NNK, JFH, JL also the ordinality of natural overlearned numbers in long- Contributed to analysis and interpretation of data: NNK, term memory can influence the resulting pattern (cf. also JFH, JH, KC, MVB, H-CN Abrahamse et al., 2016; Huber et al., 2016; Prpic et al., Drafted and/or revised the article: NNK, JFH, JH, KC, 2016; Schroeder et al., 2017). Unfortunately, these two MVB, H-CN models are empirically indistinguishable when we are sat- Approved the submitted version for publication: NNK, isfied with a verbal phrasing of an in-principle influence JFH, JH, KC, MVB, H-CN of both codes/both ordinalities. Therefore, a computational framework is needed specifying how any entity-space en- Funding coding in working memory may influence behavioral deci- sion-making for any given model assumption, be it ordinal- This work was funded by the Deutsche Forschungsge- ity or magnitude. meinschaft (DFG – German Research Foundation) within In particular, model implementations of the extended the Research Unit FOR2718: Modal and Amodal Cognition working memory account and the multiple coding frame- [grant number FOR 2718; project numbers BU 1335/12-1 works would be necessary. An extended working memory and NU 265/5-1]. Martin Butz is a member of the Machine model implementation would need to specify how long- Learning Cluster of Excellence, EXC number 2064/1 – Pro- term ordinal sequences in long-term memory and short- ject number 390727645. Additionally, Hans-Christoph term ordinal sequences currently active in working memory Nuerk studies spatial-numerical associations and the may influence stimulus processing and response selection. SNARC effect in the DFG project. NU 265/8-1. We acknowl- On the other hand, a multiple coding framework implemen- edge support by Open Access Publishing Fund of University tation would need to define how magnitude and ordinality of Tuebingen. both influence stimulus processing and response selection. These model implementations should be able to generate Competing Interests exact predictions about how strong the influence of each component will be under which experimental setup. If such The authors declare that no competing interests exist. influences are properly implemented, then the two mod- els could be quantitatively tested against each other, poten- Data Accessibility Statement tially enabling the exclusion of one of the proposed expla- nations. The experimental code, stimuli, participant data, data The presented experimental data provide boundary con- analysis scripts, additional analyses, and more supplemen- tary material can be accessed on OSF (https://osf.io/ ditions for the future when such models are developed and u9wer/). quantitatively tested against empirical data. We hope this work will facilitate the development and refining of the Statement working memory account so that it is more specific on which numbers are activated in the typical SNARC setup. The ethical agreement was given by the Commission for Ethics in Psychological Research of the University of Tue- bingen (Nürk_2020_0623_192). Submitted: November 20, 2022 PST, Accepted: December 19, 2022 PST This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License (CCBY-4.0). View this license’s legal deed at http://creativecommons.org/licenses/by/4.0 and legal code at http://creativecom- mons.org/licenses/by/4.0/legalcode for more information. Collabra: Psychology 8 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions References Abrahamse, E., van Dijck, J.-P., & Fias, W. (2016). How Lindemann, O., Abolafia, J. M., Pratt, J., & Bekkering, H. does working memory enable number-induced spatial (2008). Coding strategies in number space: Memory biases? Frontiers in Psychology, 7. https://doi.org/10.3 requirements influence spatial–numerical 389/fpsyg.2016.00977 associations. Quarterly Journal of Experimental Psychology, 61(4), 515–524. https://doi.org/10.1080/1 Casasanto, D., & Pitt, B. (2019). The faulty magnitude detector: Why SNARC‐like tasks cannot support a generalized magnitude system. Cognitive Science, Mingolo, S., Prpic, V., Bilotta, E., Fantoni, C., Agostini, 43(10). https://doi.org/10.1111/cogs.12794 T., & Murgia, M. (2021). Snarcing with a phone: The role of order in spatial-numerical associations is Cipora, K., He, Y., & Nuerk, H.-C. (2020). The revealed by context and task demands. Journal of spatial–numerical association of response codes Experimental Psychology: Human Perception and effect and math skills: why related? Annals of the New Performance, 47(10), 1365–1377. https://doi.org/10.10 York Academy of Sciences, 1477(1), 5–19. https://doi.o 37/xhp0000947 rg/10.1111/nyas.14355 Pavlovia. (2020). http://www.pavlovia.org Cipora, K., Soltanlou, M., Reips, U.-D., & Nuerk, H.-C. (2019). The SNARC and MARC effects measured Prpic, V., Fumarola, A., De Tommaso, M., Luccio, R., online: Large-scale assessment methods in flexible Murgia, M., & Agostini, T. (2016). Separate cognitive effects. Behavior Research Methods, 51(4), mechanisms for magnitude and order processing in 1676–1692. https://doi.org/10.3758/s13428-019-0121 the spatial-numerical association of response codes 3-5 (SNARC) effect: The strange case of musical note values. Journal of Experimental Psychology: Human de la Vega, I., de Filippis, M., Lachmair, M., Dudschig, Perception and Performance, 42(8), 1241–1251. http C., & Kaup, B. (2012). Emotional valence and s://doi.org/10.1037/xhp0000217 physical space: Limits of interaction. Journal of Experimental Psychology: Human Perception and Prpic, V., Mingolo, S., Agostini, T., & Murgia, M. (2021). Performance, 38(2), 375–385. https://doi.org/10.1037/ Magnitude and order are both relevant in SNARC and a0024979 SNARC‐like effects: A commentary on Casasanto and Pitt (2019). Cognitive Science, 45(7). https://doi.org/1 de Leeuw, J. R. (2015). jsPsych: A JavaScript library for 0.1111/cogs.13006 creating behavioral experiments in a Web browser. Behavior Research Methods, 47(1), 1–12. https://doi.or Restle, F. (1970). Speed of adding and comparing g/10.3758/s13428-014-0458-y numbers. Journal of Experimental Psychology: General, 83(2, Pt.1), 274–278. https://doi.org/10.1037/h002857 Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: Rusconi, E., Kwan, B., Giordano, B. L., Umilta, C., & General, 122(3), 371–396. https://doi.org/10.1037/009 Butterworth, B. (2006). Spatial representation of 6-3445.122.3.371 pitch height: the SMARC effect. Cognition, 99(2), 113–129. https://doi.org/10.1016/j.cognition.2005.0 Fias, W., Brysbaert, M., Geypens, F., & d’Ydewalle, G. 1.004 (1996). The importance of magnitude information in numerical processing: Evidence from the SNARC Schroeder, P. A., Nuerk, H.-C., & Plewnia, C. (2017). effect. Mathematical Cognition, 2(1), 95–110. https://d Switching between multiple codes of SNARC-like oi.org/10.1080/135467996387552 associations: Two conceptual replication attempts with anodal tDCS in sham-controlled cross-over Fischer, M. H., Shaki, S., & Cruise, A. (2009). It takes design. Frontiers in Neuroscience, 11. https://doi.org/1 just one word to quash a SNARC. Experimental 0.3389/fnins.2017.00654 Psychology, 56(5), 361–366. https://doi.org/10.1027/1 618-3169.56.5.361 Shaki, S., Fischer, M. H., & Petrusic, W. M. (2009). Reading habits for both words and numbers Gevers, W., & Lammertyn, J. (2005). The hunt for contribute to the SNARC effect. Psychonomic Bulletin SNARC. Psychology Science, 47(1), 10–21. & Review, 16(2), 328–331. https://doi.org/10.3758/pb Ginsburg, V., van Dijck, J.-P., Previtali, P., Fias, W., & r.16.2.328 Gevers, W. (2014). The impact of verbal working Shaki, S., & Gevers, W. (2011). Cultural characteristics memory on number–space associations. Journal of dissociate magnitude and ordinal information Experimental Psychology: Learning, Memory, and processing. Journal of Cross-Cultural Psychology, Cognition, 40(4), 976–986. https://doi.org/10.1037/a0 42(4), 639–650. https://doi.org/10.1177/00220221114 Huber, S., Klein, E., Moeller, K., & Willmes, K. (2016). Toomarian, E. Y., & Hubbard, E. M. (2018). On the Spatial–numerical and ordinal positional genesis of spatial-numerical associations: associations coexist in parallel. Frontiers in Evolutionary and cultural factors co-construct the Psychology, 7. https://doi.org/10.3389/fpsyg.2016.004 mental number line. Neuroscience & Biobehavioral Reviews, 90, 184–199. https://doi.org/10.1016/j.neubi Leiner, D. J. (2019). SoSci Survey (Version 3.1.06). http orev.2018.04.010 s://www.soscisurvey.de Collabra: Psychology 9 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions van Dijck, J.-P., & Doricchi, F. (2019). Multiple left-to- Zebian, S. (2005). Linkages between number concepts, right spatial representations of number magnitudes? spatial thinking, and directionality of writing: The Evidence from left spatial neglect. Experimental Brain SNARC effect and the reverse SNARC effect in Research, 237(4), 1031–1043. https://doi.org/10.1007/ English and Arabic monoliterates, biliterates, and s00221-019-05483-5 illiterate Arabic speakers. Journal of Cognition and Culture, 5(1–2), 165–190. https://doi.org/10.1163/156 van Dijck, J.-P., & Fias, W. (2011). A working memory account for spatial–numerical associations. Cognition, 119(1), 114–119. https://doi.org/10.1016/ Zohar-Shai, B., Tzelgov, J., Karni, A., & Rubinsten, O. j.cognition.2010.12.013 (2017). It does exist! A left-to-right spatial–numerical association of response codes van Dijck, J.-P., Gevers, W., Lafosse, C., Doricchi, F., & (SNARC) effect among native Hebrew speakers. Fias, W. (2011). Non-spatial neglect for the mental Journal of Experimental Psychology: Human Perception number line. Neuropsychologia, 49(9), 2570–2583. htt and Performance, 43(4), 719–728. https://doi.org/10.1 ps://doi.org/10.1016/j.neuropsychologia.2011.05.005 037/xhp0000336 Collabra: Psychology 10 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Supplementary Materials Response to Reviewers Download: https://collabra.scholasticahq.com/article/67908-mental-number-representations-are-spatially-mapped- both-by-their-magnitudes-and-ordinal-positions/attachment/134837.docx?auth_token=rY3GKo6LntoJBILTEvZN Peer Review History Download: https://collabra.scholasticahq.com/article/67908-mental-number-representations-are-spatially-mapped- both-by-their-magnitudes-and-ordinal-positions/attachment/134838.docx?auth_token=rY3GKo6LntoJBILTEvZN Collabra: Psychology http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Collabra Psychology University of California Press

Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions

Download PDF
11 pages

Loading next page...
 
/lp/university-of-california-press/mental-number-representations-are-spatially-mapped-both-by-their-586lmApZSm
Publisher
University of California Press
eISSN
2474-7394
DOI
10.1525/collabra.67908
Publisher site
See Article on Publisher Site

Abstract

Koch, N. N., Huber, J. F., Lohmann, J., Cipora, K., Butz, M. V., & Nuerk, H.-C. (2023). Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions. Collabra: Psychology, 9(1). https://doi.org/10.1525/collabra.67908 Cognitive Psychology Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions 1 2 1 3 1,2 Nadine N. Koch , Julia F. Huber , Johannes Lohmann , Krzysztof Cipora , Martin V. Butz , 2,4 Hans-Christoph Nuerk 1 2 Department of Computer Science, University of Tuebingen, Tuebingen, Germany, Department of Psychology, University of Tuebingen, Tuebingen, 3 4 Germany, Department of Mathematics Education, Loughborough University, Loughborough, UK, LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany Keywords: mental number representation, number processing, numerical cognition, SNARC effect, spatialization, spatial-numerical associations https://doi.org/10.1525/collabra.67908 Collabra: Psychology Vol. 9, Issue 1, 2023 The Spatial-Numerical Association of Response Codes (SNARC) effect – i.e., faster responses to small numbers with the left compared to the right side and to large numbers with the right compared to the left side – suggests that numbers are associated with space. However, it remains unclear whether the SNARC effect evolves from a number’s magnitude or the ordinal position of a number in working memory. One problem is that, in different paradigms, the task demands influence the role of ordinality and magnitude. While single-task setups in which participants judge the parity of a displayed number indicate the importance of magnitude for the SNARC effect, evidence for ordinal influences usually comes from experiments where ordinal sequences have to be memorized or setups in which participants possess pre-existing knowledge of the ordinality of stimuli. Therefore, in this preregistered study, we employed a SNARC task without secondary ordinal sequence memorization. We dissociate ordinal and magnitude accounts by carefully manipulating experimental stimulus sets. The results indicate that even though the magnitude model better accounts for the observed data, the ordinal position seems to matter as well. Hence, numbers are associated with space in both a magnitude- and an order-respective manner, yielding a mixture of both compatibility effects. Moreover, a multiple coding framework may most accurately explain the roots of the SNARC effect. the writing direction (Dehaene et al., 1993; Restle, 1970). 1. Introduction In strong opposition to the mental number line account, Magnitude information seems to be mentally associated Casasanto and Pitt (2019) have argued that it may not be with space, as suggested by the Spatial-Numerical Associ- the numerical magnitude but the ordinal position of num- ation of Response Codes (SNARC) effect: responses to rel- bers that is spatially mapped. In most experiments, the atively small numbers are faster with the left compared to two alternative explanations cannot be tested against each the right side, and responses to relatively large numbers are other because the ordinality typically coincides with mag- faster with the right compared to the left side (Dehaene et nitude. Prpic et al. (2021) argued against the ordinal ac- al., 1993). The mental number line (Dehaene et al., 1993; count by referring to studies that show spatial-numerical Restle, 1970) and the working memory account (van Dijck associations in setups where no immediate stimulus order et al., 2011; van Dijck & Fias, 2011) suggest different men- can be established. Meanwhile, studies showing a spatial tal representations that induce this effect. mapping of non-numerical ordinal sequences, such as let- ters of the alphabet or months (Gevers & Lammertyn, 1.1. The Mental Number Line Account and its 2005), support the ordinal account. This opened the discus- Challenges sion on whether the numerical magnitude or ordinality is spatially mapped (Gevers & Lammertyn, 2005). Dehaene et al. (1993) proposed a visuospatial mental number line to account for the SNARC effect. This line is assumed to represent number magnitude horizontally, ei- ther from left to right or from right to left, depending on a Corresponding author: hc.nuerk@uni-tuebingen.de Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions mental representation of numbers. The mental number line 1.2. The Working Memory Account account claims that numbers are represented according to Further researchers described the importance of working their magnitude. In contrast, in our interpretation, the memory for temporarily and selectively mapping number working memory account assumes that only the presented and ordinality representations onto space. These observa- numbers are represented in working memory in order of tions laid the grounds for the working memory account (van their magnitude. Dijck et al., 2011; van Dijck & Fias, 2011), which suggests that a spatial coding of numbers is formed during task ex- 1.3. More Than one Explanation for the SNARC ecution by storing task-relevant numbers in working mem- Several authors (e.g., Huber et al., 2016) showed that ory. Thereby the ordinal position of a number in working both magnitude (as postulated by the mental number line) memory is associated with space: the beginning of a se- and ordinal position (in line with the working memory ac- quence is associated with the left and the end with the count) are important for the SNARC effect. A model by right (van Dijck & Fias, 2011). This association is created Prpic et al. (2016), which is motivated by spatial compat- by binding number items to active spatial templates, which ibility effects when processing musical note lengths, sug- are created from experience. Newer versions of the working gests that depending on the nature of the task, either mag- memory account further assume that the most common nitude or ordinality determines the SNARC effect: direct numbers and their typical relations are saved in long-term tasks (tasks requiring the processing of the magnitude/or- memory (Abrahamse et al., 2016). One such long-term dinal position of a number, i.e., magnitude classification) memory representation is the canonical number set repre- evoke an ordinality-dependent SNARC effect and indirect senting the numbers one to nine according to their magni- ones (tasks that do not require processing the magnitude/ tude, as this is the order in which those numbers typically the ordinal position of the number, i.e., parity judgment) occur (Abrahamse et al., 2016). A number item can prompt a magnitude-dependent SNARC effect. Similarly, Schroeder this representation. Abrahamse et al. (2016) explain the et al. (2017) proposed a multiple coding framework, which range dependence of the SNARC effect (i.e., an early obser- suggests that different mechanisms provoke spatial asso- vation that the SNARC effect seems to adapt to the range of ciations simultaneously (see also Abrahamse et al., 2016; numbers being used in a stimulus set, and the same num- Huber et al., 2016). More precisely, the framework assumes ber can be characterized by a left/right side advantage de- that simultaneous activation of the number representa- pending on other numbers present in the stimulus set; De- tions on a mental number line and in working memory is haene et al., 1993) by assuming that when only a subset possible. Although this model requires further validation of the canonical number set is perceived, this representa- (see also Cipora et al., 2020) and asks for the formalization tion “is ‘pruned’ to match the actually used items in the of a computational model, the basic idea that multiple cod- experiment” (Abrahamse et al., 2016, p. 6). The working ing mechanisms may be responsible for the SNARC effect memory account does not further specify which numbers is supported by the data of neglect patients (van Dijck & are loaded into working memory while conducting the task. Doricchi, 2019). As Toomarian and Hubbard (2018) argued, Thus, there are two possible interpretations: first, only the the spatial coding of magnitude may be related to inborn perceived numbers are activated in working memory in or- spatial biases, while the mapping of the ordinality could be der of their magnitude. In this case, numbers within the related to cultural factors. To sum up, different lines of ar- range of the presented numbers that are not presented gument support the mental number line account and the would not be activated in working memory (see the right working memory account, and both magnitude and ordinal- column of Figure 1 for examples of the numbers activated ity seem to be at play. What remains unclear is their relative in working memory). However, this wording could also be role in the most basic parity judgment task typically used to interpreted such that the mental representation includes measure the SNARC effect. all numbers within a particular range (see Figure 1 left col- umn) . We focus on the first interpretation as the predic- 1.4. The Current Study tions of the mental number line, and the working memory account can only be differentiated when assuming that only Even though several studies compared the mental num- numbers are activated that are perceived during the task. ber line and working memory accounts (Ginsburg et al., We are aware that some extensions to the original working 2014; Huber et al., 2016; Lindemann et al., 2008; van Dijck memory account allow for a much wider range of predic- & Fias, 2011), all of the cited studies used an ordinal sec- tions, which are harder to falsify. While we do not focus on ondary task (i.e., learning a sequence) or were based on these extensions in detail here in the introduction, we will some pre-existing knowledge structures. The pre-existing examine under which conditions they might also account knowledge structures used as an anchor for the ordinality for our data in the discussion section. representation were musical notation (Prpic et al., 2016) or In sum, the mental number line and the working mem- phone dial (Mingolo et al., 2021). These studies only ob- ory account differ concerning their assumption about the 1 We thank J.-P. van Dijck for clarifying this. Collabra: Psychology 2 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Figure 1. The four number sets (each participant saw only one set) and their mental representation according to the mental number line and the working memory account. The fitting by the regression line differs considerably for the two accounts. For the stimulus set [1, 2, 3, 8], 8 is ordinally the fourth element in the set, so in an ordinal sequence fit- ting, it would be one unit to the right of number 3. However, in a mental number line fitting, the 8 should be five units to the right of number 3. Analog arguments hold for the other stimulus sets. served the ordinality SNARC in tasks in which the ordinal As a result, an advantage of the ordinality model would position of the stimulus was task-relevant (i.e., direct thus lend strong support to the working memory account. tasks), that is, when a sequence of items had to be tem- In contrast, an advantage of the magnitude model would porarily memorized. One could argue that the nature of favor the mental number line model. As mentioned above, the ordinal secondary task or the memory recall triggered there are extensions of the original working memory model the effect of the ordinal position of a number/object in a that might also incorporate result patterns, which are pre- sequence found in such experiments. Further, studies us- dicted by the magnitude model, making it almost empir- ing pre-existing knowledge structures (e.g., Mingolo et al., ically indistinguishable from the mental number line ac- 2021; Prpic et al., 2016) deviated from the typical SNARC count. We will address those in the discussion. setup. For those studies, it remains unclear whether ordi- 2. Methods nality or magnitude plays a decisive role in the spatial map- ping of numbers in the most basic SNARC setup with num- The ethics committee for psychological research at the bers and a typical numerical task. University of Tuebingen (Nürk_2020_0623_192) has ap- Therefore, the current study aimed to probe the mental proved the study’s protocol. The study was preregistered number line and the ordinal working memory accounts us- (https://osf.io/h7vpm). ing a parity judgment (i.e., indirect) task without a sec- ondary task (i.e., without learning a sequence explicitly and 2.1. Participants without referring to an existing ordinality-related knowl- edge structure). Further, we did not present the number An a priori power analysis revealed that at least 265 set used in the parity judgment task to the participants data sets must be collected to detect differences in SNARC in advance, avoiding the priming of a particular number slopes of Cohen’s d = 0.2 (power = .9, in paired two-sided representation. We used carefully selected stimulus sets to t-tests), which we considered the smallest effect size of in- dissociate predictions from ordinal and magnitude num- terest. Due to the online setting, we expected a dropout ber representations, always including two even and two odd rate of 20% (incomplete or unusable data sets). Accordingly, numbers with three consecutive numbers and a fourth one we aimed for at least 320 participants. In total, 465 partici- maximally distant from the single-digit range (see Figure 1 pants took part in the online study. Participants signed in- for a more detailed description of stimuli that enable this formed consent and could participate in a lottery of eleven dissociation). Amazon vouchers or receive course credit for their par- ticipation. After exclusions (see below), a sample size of 2 All effect sizes were calculated using the jmv R-package. Collabra: Psychology 3 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions N = 423 remained (47 to 61 participants per condition, appeared, stating that participation in the experiment was for the exact distribution of participants, see supplemen- only possible using a PC, laptop, or netbook. If neither of tary material S1 ). The participants reported an average age the three devices was chosen or the device rendered like of 25.90 years (SD = 9.60 years, range = 18 - 83 years); a smartphone, the experiment ended with a corresponding 290 participants reported being female, 130 reported being note. male, and three defined themselves as diverse. In total, 375 Next, the participant was instructed to classify the parity participants indicated being right-handed, 39 left-handed, of a number by either pressing the “d” or “k” key depending and nine ambidextrous. The sample comprised 366 Ger- on the parity-to-key-mapping. During the practice trials man, 11 English, and 48 other native speakers (language re- (see Figure 2), each number of the number set was pre- ported less than ten times). All except three participants sented twice in random order. The participants received reported languages reading from left to right (two Arabic, feedback for late (‘too late’), correct (‘correct’), and wrong one Pashto), and two participants did not state an identifi- (‘wrong’) responses. Participants that got less than 75% of able mother tongue. the practice trials correct had to repeat the trials. We chose 75% because 50% of correct trials were reached by chance. 2.2. Material and Apparatus Hence, 75% accuracy means that 50% of the trials were re- sponded to correctly, while the other 50% were correct by The main experiment was conducted via Pavlovia (2020) chance (25% correct, 25% incorrect). using jsPsych (de Leeuw, 2015). We additionally used SoSci After the training, the instruction was repeated, and the Survey (Leiner, 2019) to assign a participant number and to experimental trials followed. The experimental trials were save e-mail addresses for the lottery. analogous to the practice trials. However, feedback was only Texts (in the experimental trials) were presented in given for no or late responses, stating that none of the re- white in font size 22 px, numbers, fixation cross, and feed- sponse keys was pressed, and the parity-to-key mapping back (for the practice trials) in font size 72 px on a black was presented again. Each number of a number set was pre- background. We used four number sets ([1, 2, 3, *8*], [2, sented 30 times in pseudorandomized order. Thereby no 3, 4, *9*], [*1*, 6, 7, 8], [*2*, 7, 8, 9]) including one critical more than three numbers of the same parity were shown number (between asterisk). For the critical number, the dis- in a row, and a number was never repeated in consecutive tance toward the previous (following) number differs re- trials. The same practice and experimental procedure were garding the magnitude of a number and its ordinal position then repeated with the reversed parity-to-key mapping. in the sequence (see Figure 1). The study continued by collecting demographic data (cf. supplementary materials S3) and running an arithmetic 2.3. Design task consisting of 40 equations with basic arithmetic oper- ations and a time limit of two minutes (cf. supplementary Each participant was assigned to one of the four number materials S4, the analysis of this task goes beyond the scope sets and one of two parity-to-key mapping orders (either of the current research question). Then, two final ques- first “d” for odd numbers and “k” for even numbers and tions: “How noisy was your environment?” (“silent” to “ex- then “k” for odd numbers and “d” for even numbers, or vice tremely noisy”) and “If you were the experimenter, would versa) by order of participation. Within one participant, you use the data?” (“Yes.”, “Not all of them.”, “No.”) were each combination of each number of the number set and presented. The participant was then forwarded to SoSci parity-to-key mappings was repeated 30 times resulting in Survey for the lottery. 240 trials. This between-subject design prevented transfer effects between number sets. 2.5. Data Analyses 2.4. Procedure The data exclusion was derived from the one used by Cipora et al. (2019). Our analysis script can be found in After a welcome page on SoSci Survey, the participant the supplementary materials S5 and the data in S6. First, was forwarded to Pavlovia, where the main experiment was data sets from participants who reported an age below 18 executed in full screen (the experimental code can be found (0.43%, this exclusion criterium was not preregistered) and in the supplementary material S2). The participants were participants who indicated that they performed the task in asked to choose a conducting language (German or Eng- a very noisy or extremely noisy environment (1.94%) were lish), whether they wanted to look at the experiment or excluded. Afterward, incorrectly responded trials (4.74%; participate and which device they used (PC, laptop, net- prepared data set), and trials with reaction times below book, smartphone, e-reader, or other). Beneath, a red note 250 ms (0.59%; as in Cipora et al., 2019; Ginsburg et al., 3 The supplementary materials can be accessed on OSF (https://osf.io/u9wer/). 4 As preregistered, we did not exclude the data of right-to-left readers. However, excluding those participants did not substantially change the results: the magnitude model described significantly more variance and the deviance between prediction and actual data is signifi- cantly smaller for the ordinality model. Collabra: Psychology 4 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Figure 2. The procedure of one practice trial. For experimental trials, there was only feedback when no response key was pressed. The feedback was presented for 2000 ms in this case. 2014; van Dijck & Fias, 2011 this excludes accidental re- stance, for the number set 1, 2, 3, 8, we fitted a regression actions and ensures the comparability of the results) were line to the dRTs of the numbers 1, 2, and 3. Subsequently, discarded as anticipations. Participants with no correct re- we compared the dRT value for the number 8 with the dRT sponse in one of the conditions left were excluded (0.22%). of this number being predicted by either the magnitude Then sequentially, all reaction times beyond three SD below model (i.e., 8) or by the ordinality model (i.e., 4). The de- or above the individual mean reaction time were removed pendent measure was the deviance between the actual dRT (3.27%). To ensure an accurate estimate for mean reaction and predicted values for both magnitude and ordinality time in each experimental cell, data sets resulting in less models. This analysis was added as we wanted to look at than 70% (as in Cipora et al., 2019) of valid trials left for how the regression models behave when being estimated any number (of a number set) in each parity-to-key-map- solely based on consecutive numbers and not being affected ping were excluded (6.18%). by the critical number (which constitutes 25% of the data) The dRTs (mean reaction time of the right minus mean and which of these models is better in predicting the dRT reaction time of the left hand) for each participant and each for the critical number. number were calculated according to Fias et al. (1996). Two 3. Results linear regressions were fitted for each participant predict- ing dRTs either by the ordinal positions of a number (ordi- We used a significance level of .05, two-sided t-tests, and nality model) or by the magnitude of a number (magnitude corrected the four reported p-values with the Bonferroni- model). We tested both slopes against zero. For the linear Holm method. At first, the results from the model compari- regressions comparison (model comparison), logit transfor- son and afterward, the comparison of the deviance from re- mations were applied to the R²-values of the linear regres- gressions are reported. sions from each participant to approximate a normal dis- tribution. A paired-samples t-test was used to compare the 3.1. Model Comparison transformed R²-values of the models. Additionally, we conducted an alternative analysis. We The slopes of the magnitude model (M = -5.55, slope fitted a linear regression to the dRTs of the three consec- t(422) = -11.56, p < .001, 95%-CI = [-6.49, -4.61]), as slope utive numbers in a number set (the critical number was well as the slopes of the ordinality model (M = -13.33, slope excluded) for each participant. We then calculated the ab- t(422) = -12.41, p < .001, 95%-CI = [-15.45, -11.22]), dif- slope solute difference between the measured dRT and the pre- fered significantly from zero, showing influences of magni- dicted value from each model for the critical number for tude and ordinality, respectively. each participant (deviance) and compared them. For in- 5 Further model comparisons are described in supplementary materials S7 and S8. S7 reports results from the analyses of different number sets and number ranges separately. S8 describes results from a comparison of two linear mixed models considering ordinality and magnitude. The Akaike Information Criterion (AICc) was smaller for the magnitude model. In accordance with the reported model comparison, this indicates that the magnitude model described the data more accurately. Collabra: Psychology 5 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Compared to the ordinality model, the magnitude model Meanwhile, comparing the explained variance supports showed significantly greater R -values (M = -0.67/.42 the mapping of numbers according to their magnitude and mag [mean of the transformed/untransformed R -values], hence the mental number line account. However, it has also M = -0.96/.40 [mean of the transformed/untransformed been proposed that not only the perceived numbers them- ord R -values], M = 0.29/.02 [difference of the transformed/ selves but also the whole included range of numbers as dif untransformed R -values], t(422) = 3.33, p < .001, overlearned and stored in working memory (i.e., 1, 2, 3, 4, 95%-CI = [0.12, 0.46], Cohen’s d = 0.16) and hence de- 5, 6,…) may partially constitute the working memory con- dif scribed significantly more variance of the data. Figure 3 tent (Abrahamse et al., 2016). This poses a fundamental shows a visualization of the comparisons in the top panel. problem for distinguishing this extended working memory account from the magnitude account because the ordinal 3.2. Comparison of the Deviance metric of the overlearned absolute sequence in long-term memory and the linear magnitude metric of the mental The predicted dRTs for the ordinal position (ordinality number line account are identical. Seeing that our deviance model) and the magnitude (magnitude model) of the crit- comparisons of the critical fourth number favor the ordi- ical number were calculated for each participant. The ab- nality model without an intermediate, never shown num- solute distance between the prediction and the measured ber, the results indicate that only the perceived numbers dRT for the critical number was significantly smaller for appear to be ordered in working memory. These seemingly the ordinality model compared to the magnitude model contradicting results thus all hint towards a multiple coding (M = 106.84, M = 51.43, M = 55.41, t(422) = 15.57, mag ord dif framework-oriented explanation. p < .001, 95%-CI = [48.41, 62.40], Cohen’s d = 0.76). dif Hence, the deviance between the prediction for the critical 4.2. Multiple-Coding Frameworks of the SNARC number and the measured dRT in the magnitude model was about twice as large as the corresponding deviance of Even though the magnitude model describes the current the ordinality model. A visualization of the comparisons is data more accurately, the advantage is associated with a shown in Figure 3, bottom panel. very small effect size (d = 0.17), which is, in fact, smaller than the smallest effect of interest we defined in our a priori 4. Discussion power analysis. At the same time, the results also imply that both the mental number line and the working memory The exact mapping of numbers to space has been as- account play important roles (e.g., the deviance comparison sumed to depend on their magnitude (mental number line with Cohen’s d = 0.76 and the measured dRT lies between account) or their ordinal position (as implied by working the predictions of both models). We propose to further con- memory account and other theoretical accounting of the sider an integrative model, which may have the potential to primary/sole role of ordinality). The current study com- expand on previous multiple-coding-framework-related ex- pared the two assumptions while avoiding secondary tasks planations (e.g., Abrahamse et al., 2016; Huber et al., 2016; and memory recall of overlearned ordinal sequences. The Prpic et al., 2016; Schroeder et al., 2017). In particular, the results show that the magnitude model explained signifi- referenced literature seems to imply that working memory cantly more variance than the ordinality model. However, may be loaded with different task-relevant item-to-space when calculating the deviance between the measured dRT associations. for the critical fourth number and the predicted dRT via re- In our case, it seems that a mental number line activa- gression of the three consecutive numbers, the ordinality tion induced a linear magnitude-to-space association of the model yielded significantly smaller deviance. These results relevant number range. appear contradictory. Meanwhile, an ordinal encoding – whereby the order is, in our case, also influenced by a canonical magnitude 4.1. A Possible Explanation for This – induced a linear order-to-space association. Such a si- Contradiction multaneous activation could account for the deviances be- tween the model’s predictions and the measured data, as One possible source for the apparent inconsistency is the well as for the fact that both models fit the data well. In- deviance comparison. Because the regression line parame- deed, such a model could furthermore account for the dis- ters are estimated with a limited amount of data, the in- cussed differences in magnitude and ordinal influences in tercept and slope estimations are inevitably error-prone. other experiments as well – the stronger the focus on a While the error in the intercept affects all predicted po- particular magnitude or sequential ordering, the stronger sitions equally – and thus both deviances – errors in the may be its influence on the selection of (in-)compatible re- slope lead to linearly increasing errors with a distance of sponse codes. Hence, as Casasanto and Pitt (2019) stated, the critical fourth number from the other three. This dis- the culturally influenced mapping of the ordinality plays tance is one for the ordinality model but five for the magni- an important role in spatial mappings in such an integra- tude model – thus, the error in the slope adds up five times tive model. Those ordinal mappings can explain the associ- in the prediction of the critical number using the magni- ation between numerically unrelated concepts such as mu- tude model. Since the error is not known, the results de- sical pitches (Rusconi et al., 2006) or emotional valence (de mand further studies and should be interpreted with cau- la Vega et al., 2012) and space. tion at this point. Collabra: Psychology 6 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Figure 3. The four plots show mean dRTs in ms depending on the numbers’ magnitude or the ordinal position of the numbers when ordered according to their magnitude. The lines indicate the averaged regression lines over all participants’ regression lines in a particular number set. For calculating the positions of the circles, triangles, and squares, first, the dRTs of each participant were averaged. Then the mean over all participants for each number per number set was calculated. The plots in a) show the linear regression lines of all four digits of a number set. The plots in b) show the linear regression lines of the three consecutive points. Additionally, as described by Prpic et al. (2021), the mag- 4.3. Conclusion nitude also seems to have an influence. The interaction of The presented statistical model comparison indicates those mappings can explain the results of studies inves- that the mental number line account yields more accurate tigating the SNARC effect in right-to-left reading partic- regressions. Meanwhile, despite the described issues and ipants, where the SNARC effect is not consistently found the fact that a linearity assumption may be a slight over- (Fischer et al., 2009; Shaki et al., 2009; Shaki & Gevers, simplification, a linear extrapolation to the critical number 2011; Zebian, 2005; Zohar-Shai et al., 2017). Here the map- favors the (pure) working memory account. The results are ping of the potentially inborn magnitude seems to be in in line with a multiple-coding framework account, which contrast with the culturally learned mapping of ordinality. assumes that multiple task-relevant entity encodings play crucial roles dependent on the task setup and the directness Collabra: Psychology 7 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions with which the setup triggers particular item-space associ- Contributions ations. However, the data are also in line with an extended Contributed to conception and design: NNK, JFH, JL, KC, working memory account, where not only ordinal se- MVB, H-CN quences are actively kept in active working memory but Contributed to acquisition of data: NNK, JFH, JL also the ordinality of natural overlearned numbers in long- Contributed to analysis and interpretation of data: NNK, term memory can influence the resulting pattern (cf. also JFH, JH, KC, MVB, H-CN Abrahamse et al., 2016; Huber et al., 2016; Prpic et al., Drafted and/or revised the article: NNK, JFH, JH, KC, 2016; Schroeder et al., 2017). Unfortunately, these two MVB, H-CN models are empirically indistinguishable when we are sat- Approved the submitted version for publication: NNK, isfied with a verbal phrasing of an in-principle influence JFH, JH, KC, MVB, H-CN of both codes/both ordinalities. Therefore, a computational framework is needed specifying how any entity-space en- Funding coding in working memory may influence behavioral deci- sion-making for any given model assumption, be it ordinal- This work was funded by the Deutsche Forschungsge- ity or magnitude. meinschaft (DFG – German Research Foundation) within In particular, model implementations of the extended the Research Unit FOR2718: Modal and Amodal Cognition working memory account and the multiple coding frame- [grant number FOR 2718; project numbers BU 1335/12-1 works would be necessary. An extended working memory and NU 265/5-1]. Martin Butz is a member of the Machine model implementation would need to specify how long- Learning Cluster of Excellence, EXC number 2064/1 – Pro- term ordinal sequences in long-term memory and short- ject number 390727645. Additionally, Hans-Christoph term ordinal sequences currently active in working memory Nuerk studies spatial-numerical associations and the may influence stimulus processing and response selection. SNARC effect in the DFG project. NU 265/8-1. We acknowl- On the other hand, a multiple coding framework implemen- edge support by Open Access Publishing Fund of University tation would need to define how magnitude and ordinality of Tuebingen. both influence stimulus processing and response selection. These model implementations should be able to generate Competing Interests exact predictions about how strong the influence of each component will be under which experimental setup. If such The authors declare that no competing interests exist. influences are properly implemented, then the two mod- els could be quantitatively tested against each other, poten- Data Accessibility Statement tially enabling the exclusion of one of the proposed expla- nations. The experimental code, stimuli, participant data, data The presented experimental data provide boundary con- analysis scripts, additional analyses, and more supplemen- tary material can be accessed on OSF (https://osf.io/ ditions for the future when such models are developed and u9wer/). quantitatively tested against empirical data. We hope this work will facilitate the development and refining of the Statement working memory account so that it is more specific on which numbers are activated in the typical SNARC setup. The ethical agreement was given by the Commission for Ethics in Psychological Research of the University of Tue- bingen (Nürk_2020_0623_192). Submitted: November 20, 2022 PST, Accepted: December 19, 2022 PST This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License (CCBY-4.0). View this license’s legal deed at http://creativecommons.org/licenses/by/4.0 and legal code at http://creativecom- mons.org/licenses/by/4.0/legalcode for more information. Collabra: Psychology 8 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions References Abrahamse, E., van Dijck, J.-P., & Fias, W. (2016). How Lindemann, O., Abolafia, J. M., Pratt, J., & Bekkering, H. does working memory enable number-induced spatial (2008). Coding strategies in number space: Memory biases? Frontiers in Psychology, 7. https://doi.org/10.3 requirements influence spatial–numerical 389/fpsyg.2016.00977 associations. Quarterly Journal of Experimental Psychology, 61(4), 515–524. https://doi.org/10.1080/1 Casasanto, D., & Pitt, B. (2019). The faulty magnitude detector: Why SNARC‐like tasks cannot support a generalized magnitude system. Cognitive Science, Mingolo, S., Prpic, V., Bilotta, E., Fantoni, C., Agostini, 43(10). https://doi.org/10.1111/cogs.12794 T., & Murgia, M. (2021). Snarcing with a phone: The role of order in spatial-numerical associations is Cipora, K., He, Y., & Nuerk, H.-C. (2020). The revealed by context and task demands. Journal of spatial–numerical association of response codes Experimental Psychology: Human Perception and effect and math skills: why related? Annals of the New Performance, 47(10), 1365–1377. https://doi.org/10.10 York Academy of Sciences, 1477(1), 5–19. https://doi.o 37/xhp0000947 rg/10.1111/nyas.14355 Pavlovia. (2020). http://www.pavlovia.org Cipora, K., Soltanlou, M., Reips, U.-D., & Nuerk, H.-C. (2019). The SNARC and MARC effects measured Prpic, V., Fumarola, A., De Tommaso, M., Luccio, R., online: Large-scale assessment methods in flexible Murgia, M., & Agostini, T. (2016). Separate cognitive effects. Behavior Research Methods, 51(4), mechanisms for magnitude and order processing in 1676–1692. https://doi.org/10.3758/s13428-019-0121 the spatial-numerical association of response codes 3-5 (SNARC) effect: The strange case of musical note values. Journal of Experimental Psychology: Human de la Vega, I., de Filippis, M., Lachmair, M., Dudschig, Perception and Performance, 42(8), 1241–1251. http C., & Kaup, B. (2012). Emotional valence and s://doi.org/10.1037/xhp0000217 physical space: Limits of interaction. Journal of Experimental Psychology: Human Perception and Prpic, V., Mingolo, S., Agostini, T., & Murgia, M. (2021). Performance, 38(2), 375–385. https://doi.org/10.1037/ Magnitude and order are both relevant in SNARC and a0024979 SNARC‐like effects: A commentary on Casasanto and Pitt (2019). Cognitive Science, 45(7). https://doi.org/1 de Leeuw, J. R. (2015). jsPsych: A JavaScript library for 0.1111/cogs.13006 creating behavioral experiments in a Web browser. Behavior Research Methods, 47(1), 1–12. https://doi.or Restle, F. (1970). Speed of adding and comparing g/10.3758/s13428-014-0458-y numbers. Journal of Experimental Psychology: General, 83(2, Pt.1), 274–278. https://doi.org/10.1037/h002857 Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: Rusconi, E., Kwan, B., Giordano, B. L., Umilta, C., & General, 122(3), 371–396. https://doi.org/10.1037/009 Butterworth, B. (2006). Spatial representation of 6-3445.122.3.371 pitch height: the SMARC effect. Cognition, 99(2), 113–129. https://doi.org/10.1016/j.cognition.2005.0 Fias, W., Brysbaert, M., Geypens, F., & d’Ydewalle, G. 1.004 (1996). The importance of magnitude information in numerical processing: Evidence from the SNARC Schroeder, P. A., Nuerk, H.-C., & Plewnia, C. (2017). effect. Mathematical Cognition, 2(1), 95–110. https://d Switching between multiple codes of SNARC-like oi.org/10.1080/135467996387552 associations: Two conceptual replication attempts with anodal tDCS in sham-controlled cross-over Fischer, M. H., Shaki, S., & Cruise, A. (2009). It takes design. Frontiers in Neuroscience, 11. https://doi.org/1 just one word to quash a SNARC. Experimental 0.3389/fnins.2017.00654 Psychology, 56(5), 361–366. https://doi.org/10.1027/1 618-3169.56.5.361 Shaki, S., Fischer, M. H., & Petrusic, W. M. (2009). Reading habits for both words and numbers Gevers, W., & Lammertyn, J. (2005). The hunt for contribute to the SNARC effect. Psychonomic Bulletin SNARC. Psychology Science, 47(1), 10–21. & Review, 16(2), 328–331. https://doi.org/10.3758/pb Ginsburg, V., van Dijck, J.-P., Previtali, P., Fias, W., & r.16.2.328 Gevers, W. (2014). The impact of verbal working Shaki, S., & Gevers, W. (2011). Cultural characteristics memory on number–space associations. Journal of dissociate magnitude and ordinal information Experimental Psychology: Learning, Memory, and processing. Journal of Cross-Cultural Psychology, Cognition, 40(4), 976–986. https://doi.org/10.1037/a0 42(4), 639–650. https://doi.org/10.1177/00220221114 Huber, S., Klein, E., Moeller, K., & Willmes, K. (2016). Toomarian, E. Y., & Hubbard, E. M. (2018). On the Spatial–numerical and ordinal positional genesis of spatial-numerical associations: associations coexist in parallel. Frontiers in Evolutionary and cultural factors co-construct the Psychology, 7. https://doi.org/10.3389/fpsyg.2016.004 mental number line. Neuroscience & Biobehavioral Reviews, 90, 184–199. https://doi.org/10.1016/j.neubi Leiner, D. J. (2019). SoSci Survey (Version 3.1.06). http orev.2018.04.010 s://www.soscisurvey.de Collabra: Psychology 9 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions van Dijck, J.-P., & Doricchi, F. (2019). Multiple left-to- Zebian, S. (2005). Linkages between number concepts, right spatial representations of number magnitudes? spatial thinking, and directionality of writing: The Evidence from left spatial neglect. Experimental Brain SNARC effect and the reverse SNARC effect in Research, 237(4), 1031–1043. https://doi.org/10.1007/ English and Arabic monoliterates, biliterates, and s00221-019-05483-5 illiterate Arabic speakers. Journal of Cognition and Culture, 5(1–2), 165–190. https://doi.org/10.1163/156 van Dijck, J.-P., & Fias, W. (2011). A working memory account for spatial–numerical associations. Cognition, 119(1), 114–119. https://doi.org/10.1016/ Zohar-Shai, B., Tzelgov, J., Karni, A., & Rubinsten, O. j.cognition.2010.12.013 (2017). It does exist! A left-to-right spatial–numerical association of response codes van Dijck, J.-P., Gevers, W., Lafosse, C., Doricchi, F., & (SNARC) effect among native Hebrew speakers. Fias, W. (2011). Non-spatial neglect for the mental Journal of Experimental Psychology: Human Perception number line. Neuropsychologia, 49(9), 2570–2583. htt and Performance, 43(4), 719–728. https://doi.org/10.1 ps://doi.org/10.1016/j.neuropsychologia.2011.05.005 037/xhp0000336 Collabra: Psychology 10 Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions Supplementary Materials Response to Reviewers Download: https://collabra.scholasticahq.com/article/67908-mental-number-representations-are-spatially-mapped- both-by-their-magnitudes-and-ordinal-positions/attachment/134837.docx?auth_token=rY3GKo6LntoJBILTEvZN Peer Review History Download: https://collabra.scholasticahq.com/article/67908-mental-number-representations-are-spatially-mapped- both-by-their-magnitudes-and-ordinal-positions/attachment/134838.docx?auth_token=rY3GKo6LntoJBILTEvZN Collabra: Psychology

Journal

Collabra Psychology – University of California Press

Published: Jan 31, 2023

Keywords: mental number representation; number processing; numerical cognition; SNARC effect; spatialization; spatial-numerical associations

There are no references for this article.