Toward a model framework of generalized parallel componential processing of multi-symbol numbers.

© 2014 American Psychological Association 0278-7393/15/$ 12.00 http://dx.doi.org/10.1037/xlm0000043

Journal of Experimental Psychology: Learning, Memory, and Cognition 2015, Vol. 41, No. 3, 732-745

Toward a Model Framework of Generalized Parallel Componential Processing of Multi-Symbol Numbers Stefan Huber

Sonja Cornelsen

Knowledge Media Research Center, Tubingen, Germany, and University of Tubingen

University of Tubingen

Korbinian Moeller

Hans-Christoph Nuerk

Knowledge Media Research Center, Tubingen, Germany, and University of Tubingen

University of Tubingen and Knowledge Media Research Center, Tubingen, Germany

In this article, we propose and evaluate a new model framework of parallel componential multi-symbol number processing, generalizing the idea of parallel componential processing of multi-digit numbers to the case of negative numbers by considering the polarity signs similar to single digits. In a first step, we evaluated this account by defining and investigating a sign-decade compatibility effect for the compar ison of positive and negative numbers, which extends the unit-decade compatibility effect in 2-digit number processing. Then, we evaluated whether the model is capable of accounting for previous findings in negative number processing. In a magnitude comparison task, in which participants had to single out the larger of 2 integers, we observed a reliable sign-decade compatibility effect with prolonged reaction times for incompatible (e.g., —97 vs. +53; in which the number with the larger decade digit has the smaller, i.e., negative polarity sign) as compared with sign-decade compatible number pairs (e.g., —53 vs. +97). Moreover, an analysis of participants’ eye fixation behavior corroborated our model of parallel componential processing of multi-symbol numbers. These results are discussed in light of concurrent theoretical notions about negative number processing. On the basis of the present results, we propose a generalized integrated model framework of parallel componential multi-symbol processing. Keywords: numerical cognition, negative numbers, eye tracking, sign-decade compatibility, parallel processing

Fischer, 2011). For this article, this controversy is important, because the types of multi-digit numbers mentioned before, but also negative numbers, may be conceptualized as multi-symbol numbers, which are considered to constitute a generalized number category, including different kinds of multi-digit numbers and negative numbers. The current eye-tracking study set out to pursue the idea that negative number processing (conceptualized as multi symbol number processing) can be framed along the same princi ples established for multi-digit numbers in the last decade. More specifically, we aimed at evaluating whether the processing of negative numbers can be explained by the model of parallel componential processing of single digits/symbols (Moeller, Fi scher, Nuerk, & Willmes, 2009; Nuerk & Willmes, 2005).

There is accumulating evidence that single-digit numbers are represented along a mental number line, with small numbers to the left and larger numbers to the right—at least in Western left-toright reading societies (e.g., Gobel. Shaki, & Fischer, 2011, for a review). However, the representation and processing of multi-digit numbers such as integers, unit, and decimal fraction is still under debate (for a review, see Nuerk, Moeller, Klein, Willmes, &

This article was published Online First July 28, 2014. Stefan Huber, Knowledge Media Research Center, IWM-KMRC, Tubin gen, Germany, and Department of Psychology, University of Tubingen; Sonja Cornelsen, Center for Neurology, Division of Neuropsychology, Hertie-Institue for Clinical Brain Research, University of Tubingen; Kor binian Moeller, Knowledge Media Research Center, IWM-KMRC. Tubin gen, Germany, and Department of Psychology, University of Tubingen; Hans-Christoph Nuerk, Department of Psychology, University of Tubin gen, and Knowledge Media Research Center, IWM-KMRC, Tubingen, Germany. Stefan Huber and Sonja Cornelsen contributed equally to the study and should be considered shared first authors. We are grateful to Klaus Willmes for proofreading of the manuscript and for his helpful comments. Correspondence concerning this article should be addressed to Stefan Huber, Knowledge Media Research Center, Schleichstrasse 6, 72076 Tubingen, Germany. E-mail: [email protected]

Theoretical Accounts on Negative Number Processing In general, three different accounts for negative number pro cessing can be distinguished (see Krajcsi & Igacs, 2010, for an overview of studies supporting either the holistic or the compo nential model): (a) The holistic (e.g., Fischer, 2003), (b) the features (Varma & Schwartz, 2011), and (c) the componential model (e.g., Ganor-Stern, Pinhas, Kallai, & Tzelgov, 2010; GanorStem & Tzelgov, 2008; Tzelgov, Ganor-Stern, & MaymonSchreiber, 2009). 732

PROCESSING OF MULTI-SYMBOL NUMBERS

The holistic model assumes that negative numbers are repre sented as an integrated entity mapped onto an analogue magnitude representation (i.e., holistic representation). Thus, negative num bers simply extend the analogue representation of positive num bers (Fischer, 2003). Usually, a regular distance effect (Moyer & Landauer, 1967) is interpreted to corroborate the notion of a holistic representation. The distance effect indicates faster re sponse times for comparing more distant rather than closer num bers (e.g., 1 vs. 8 faster than 4 vs. 5). This effect is usually observed for natural numbers, and, hence, its presence in positive versus negative number comparison supports the idea of negative numbers being represented on the same analogue magnitude rep resentation as natural numbers (Shaki & Petrusic, 2005; but see Ganor-Stern & Tzelgov, 2008; Tzelgov et al., 2009, for different findings with a larger set of positive and negative numbers). Moreover, participants seem to form a holistic representation of negative numbers when having to hold both the polarity sign and the numbers in mind (Ganor-Stern et al., 2010). The features model (or magnitude-symbol model) by Varma and Schwartz (2011) is a mathematical model describing the compar ison of single-digit positive and negative numbers. Complying with the holistic model, it suggests that the mental number line extends to negative numbers. However, diverging from the holistic model, the features model assumes componential processing of the polarity sign and the magnitude of digits. In the features model, natural numbers and negative numbers are represented as vectors of features Sj (i.e., a vector of numbers): [^0 s2 s 3 s* s 5 s6 si ss ■%]Features encode either polarity signs (one feature) or magnitudes of digits (nine features, one for each digit). The feature for the polarity signs can be either 0 (positive number) or 1 (negative number). The features of the digits are interpreted as a magnitude representation. For instance, the number “9” is represented by the vector [09~“ 8~“ 7-a 6-a 5~“ 4-a 3-a 2-a l - “] with a G (0,1], Thus, the features of 9 are s0 = 0, S] = 9““, = 8““, and so on. Moreover, the model assumes that the negative number line is a reflection of the positive number line. To account for slower reaction times (RTs) for negative numbers (e.g., Ganor-Stern & Tzelgov, 2008), the model involves an additional parameter that leads to compressed feature values for negative numbers. The difference between two representations is calculated by adding the differences between feature values. The smaller the difference is, the more similar numbers are, resulting in prolonged RTs. Varma and Schwartz (2011) showed that the model can even account for the reversed distance effect, which is found when comparing a negative with a positive number (e.g., Krajcsi & Igacs, 2010). For instance, features for —1 and 2 are more different than features for - 1 and 7 (please see Varma & Schwartz, 2011, for more details on the feature coding). Therefore, —1 and 2 are compared with each other faster than - 1 and 7, although the analogue distance of —1 and 7 is larger than the analogue distance of —1 and 2. Such a reverse distance effect is hard to reconcile with the idea of a holistic representation of negative numbers. Similar to the features model, the componential model assumes a separate representation of the components of negative num bers—the polarity sign and digits (e.g., Ganor-Stern & Tzelgov, 2008). However, the componential model does not assume a reflection of the mental number line for negative numbers. Evi dence for the componential model was deduced from the absence of a reliable distance effect when comparing a positive and a

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negative number (Fischer & Rottmann, 2005; Tzelgov et al., 2009), indicating that participants exclusively relied on the polarity signs when comparing numbers with different polarity and did not retrieve the number magnitude information. Although the compo nential model can account for this absence of a distance effect, the componential model cannot explain generally slower RTs for negative number pairs than for positive number pairs per se (e.g., Ganor-Stern & Tzelgov, 2008). However, Krajcsi and Igacs (2010) proposed a mirroring mechanism accounting for this effect by assuming that when comparing negative number pairs, participants ignore the minus sign in the first place and compare the natural numbers only before they invert their response in the second place (see also the very similar rule-based model of Blair, RosenbergLee, Tsang, Schwartz, & Menon, 2012). Accordingly, it is the inversion of responses, which requires additional reaction times resulting in prolonged response times compared with the compar ison of positive numbers.

A Model of Generalized Parallel Componential Processing of Multi-Symbol Numbers Varma and Schwartz (2011) suggested that the componential model cannot account for a reversed distance effect when com paring a positive and a negative number because participants should rely only on the polarity signs, and hence no distance effect should be observed (Ganor-Stern et al., 2010; Tzelgov et al., 2009). However, it is important to note that this argument only holds for a sequential componential model (Poltrock & Schwartz, 1984). Such a sequential processing model assumes that multidigit number comparison starts at the leftmost digit pair and continues in a sequential digit-by-digit fashion to the right until the decisive component is found. However, a parallel componential processing model (e.g., Moeller et al., 2009; Nuerk & Willmes, 2005) supposes that the constituting digits of a number are pro cessed in parallel. In case of single-digit negative numbers, the parallel componential model would suggest that both the polarity sign and the digit should be compared in parallel. Hence, a com patibility effect as in two-digit number comparison should be present. The unit-decade compatibility effect for two-digit numbers de scribes the finding that two-digit numbers are compared with each other faster when separate comparisons of tens, and units lead to the same response bias (i.e., compatible trials, e.g., +42 and +57, 4 < 5 and 2 < 7) compared with the case when the comparisons lead to opposing response biases (i.e., incompatible trials, e.g., +37 and +52, 3 < 5 but 7 > 2; e.g., Nuerk, Weger, & Willmes, 2001). When comparing a positive and a negative num ber, comparing the polarity signs is sufficient to decide which number is larger (e.g., - 1 vs. + 6 with sgn(—1) < sgn(+6) or, in short, - < +). However, like for two-digit numbers, the compar ison of the units might interfere with the comparison of polarity signs. Therefore, similar to two-digit numbers, when comparing a positive and a negative number, response biases of polarity signs and digits can be either compatible (e.g., —2 vs. +7, — < + and 2 < 7) or incompatible (e.g., —7 vs. +2, — < + but 7 > 2). As this effect would be similar to the unit-decade compatibility effect, it is referred to as sign compatibility effect in the remainder of the present article. Comparable to the unit-decade compatibility effect for two-digit numbers, such a generalized model of parallel com-

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ponential processing would suggest a larger interference effect for number pairs with large unit distances (e.g., —9 vs. 1) than for number pairs with small unit distances (e.g., - 2 vs. 1). Therefore, participants should need more time to compare number pairs with large unit distances than with small unit distances. However, number pairs with large unit distances also have larger absolute distances (10 for —9 vs. 1 compared with 3 for —2 vs. 1), resulting in a reversed distance effect. In sum, our generalized model of parallel componential processing can even account for the reversed distance effect for heterogeneous pairs, as found by Varma and Schwartz (2011). Taken together, the proposed generalized model of parallel componential processing offers an integrative account to concep tualize the processing of multi-symbol numbers. In the present study, we aimed at evaluating this new model.

The Present Study To appraise the validity of our generalized model of parallel componential multi-symbol number processing, we used a number comparison task in which participants have to indicate the larger of two two-digit integer numbers. Two-digit numbers are well suited to identify parallel componential processing, because they allow for identifying not only the separate processing of polarity signs but also the separate processing of tens and units. Moreover, we use heterogeneous number pairs consisting of a negative and a positive number (e.g., —19 vs. +11) and homogeneous numbers pairs (e.g., +11 vs. +19 or —11 vs. —19). Furthermore, we presented heterogeneous and homogeneous number pairs in a blocked condition and in a mixed condition to ensure that findings were not due to differences in the presentation mode (e.g., Shaki & Petrusic, 2005). Importantly, our main focus was on distinguishing between different models and not on differences between blocking conditions. Therefore, all effects for the differentiation between models presented below should be present in both conditions. To be able to differentiate between different models, we manipulated the compatibility of polarity signs and digits. For heterogeneous number pairs, we manipulated the compatibility of polarity signs and tens (sign-decade compatibility) and the compatibility of po larity signs and units (sign-unit compatibility). For homogeneous number pairs, we manipulated the compatibility of tens and units (unit-decade compatibility). Like for two-digit numbers, the presence of compatibility effects provides evidence for a componential representation of the multi symbol numbers. Hence, the holistic model would not predict a compatibility effect in negative numbers (e.g., Nuerk et al., 2001). Originally, the features model of Varma and Schwartz (2011) only describes the processing of single-digit integer numbers. However, when extending the model to the case of two-digit numbers, it would predict a reversed sign-decade compatibility effect due to the reflection of the mental number line for negative numbers. For instance, the model predicts slower RTs for the comparison of the sign-decade compatible number pair —14 and + 92 (— < + and 1 < 9) than for the comparison of the sign-decade incompatible number pair + 14 and —92 (+ > —, but 1 < 9), because —1 and + 9 have more features in common than + 1 and —9 (please see Varma & Schwartz, 2011, for more details on the feature coding). Moreover, the feature model makes no clear predictions for the unit-decade compatibility effect, because it is just a model

for single-digit numbers. The presence of a regular sign-decade and a sign-unit compatibility effect would corroborate the parallel componential model (cf. Moeller et al., 2009). Obviously, the parallel componential model would predict a unit-decade compat ibility effect for the comparison of two-digit numbers (Moeller et al., 2009). Contrarily, assuming a sequential componential model, neither a sign-decade compatibility effect nor a unit-decade com patibility effect would be expected. When processing negative numbers with different polarity signs sequentially, processing starts at the leftmost component of the number (i.e., the polarity sign) and then stops there, because this component is already decisive for the case of heterogeneous pairs. Thus, number mag nitude will not be processed and thus should not interfere with the comparison of polarity signs. Similarly, when comparing positive or negative numbers pairs, only tens would be compared, implying that a unit-decade compatibility effect should not be observed. Interestingly, all of the different models would assume that negative numbers are processed more slowly than positive num bers. According to the holistic model, comparisons of negative numbers should be slower because they are less frequent in daily life (e.g., Krajcsi & Igacs, 2010). Frequency of occurrence of a number in everyday life has previously been suggested to account for the problem size effect in natural number comparison (Dehaene & Mehler, 1992). Moreover, simulating magnitudes comparison on Arabic numbers by a computational model, Verguts, Fias, and Stevens (2005) also found that the frequency of occurrence of a number influenced comparison speed. Smaller numbers occur more often in everyday life (as indicated by a Google survey; see also Dehaene & Mehler, 1992) and, thus, in the training of the computational model, they were compared faster than larger num bers, indicating a problem size effect (e.g., Brysbaert, 1995). Hence, the frequency of occurrence of a number can indeed account for RT differences in magnitude comparison. As a conse quence, negative numbers may be reacted to more slowly because they occur less frequently in daily life. The features model sug gests that slower reaction times are due to the compression of the mental number line for negative numbers (Varma & Schwartz, 2011). The componential models can explain slower reaction times by the mirror mechanism (Krajcsi & Igacs, 2010). The sign-compatibility effect provides only indirect evidence concerning the processing of negative numbers. A more direct way to investigate the processing of negative numbers is recording participants’ eye movements, while they are engaged in a number magnitude comparison task involving negative numbers. Evaluat ing participants’ eye movement behavior has been shown to be valid methodology for differentiating between processing models of multi-digit number processing, because fixation location is a reliable indicator of that part of a stimulus that is actually pro cessed (e.g., Rayner, 1998). For two-digit numbers, Moeller et al. (2009) observed that participants’ eye fixations in a number com parison task indicated parallel componential processing of tens and units. However, for multi-digit numbers, recent (eye-tracking) data suggest that both sequential and parallel processing strategies are combined (for three-digit numbers, see Mann, Moeller, Pixner, Kaufmann, & Nuerk, 2012; for up to six-digit numbers, see Meyerhoff, Moeller, Debus, & Nuerk, 2012). Eye-tracking methodology has already been applied to differ entiate between the holistic, the sequential, and the parallel com ponential model in two-digit number comparison (Moeller et al.,


2009). Meyerhoff et al. (2012) observed that participants process up to four characters of a number string in parallel. Nevertheless, although participants are able to process up to four digits in parallel, their fixation behavior differs for different item types (compatible vs. incompatible number pairs when comparing twodigit numbers; Moeller et al., 2009), and even the experiment context (Huber, Mann, Nuerk, & Moeller, 2014). In the study of Huber et al. (201), we manipulated the number of within-decade filler items (e.g., 23_27) in three different conditions and found that this led to a modulation of participants’ fixation behavior even for the identical and critical between-decade conditions upon which the compatibility effect was computed (e.g., 37_84). Par ticipants fixated tens more often than units, when there was only a small number of within-decade filler items (i.e., 25%), and units more often than tens, when there was a large number of withindecade filler items (i.e., 75%). Thus, although participants pro cessed both digits in parallel (as indicated by a regular unit-decade compatibility effect), they nevertheless adapted their fixation be havior to stimulus set characteristics. Moreover, we found that relative fixations on tens and units were indicative of the size unit-decade compatibility effect with a larger unit-decade compat ibility effect in the condition, with 75% fillers and more fixations on the units than in the condition with 25% fillers and more fixations on the tens. Thus, fixation data are important when examining what is causing the change of the unit-decade compat ibility effect. Furthermore, this study also implicates that fixation patterns are not stable even when comparing the same type of stimuli (i.e., two-digit numbers). Hence, different processing strat egies should be identifiable, considering eye fixation data in line with our suggestion that fixation patterns should differ when participants apply a sequential or a parallel processing strategy. A holistic processing of negative numbers would be indicated by most fixations on the tens, because they are located in the middle of the digit string and thereby should be the preferential position when processing the whole number at once, and not only its constituent components. Similar to the optimal viewing position in word recognition (e.g., O’Rregan & Jacobs, 1992), the center of a digit string seems to designate the default landing position of the eyes (see Meyerhoff et al., 2012, who found most fixations on the second digit when the first or the second digit of four-digit num bers were decisive). This pattern should be present for both het erogeneous and homogeneous number pairs. A similar fixation pattern should be present according to the features model. Regard ing the sequential componential model, we hypothesize that most fixations should be on the relevant symbol (i.e., polarity sign for heterogeneous numbers or tens/units for homogeneous numbers). However, the parallel componential model makes very similar predictions as the holistic and the feature model, because for processing numbers in parallel, the best fixation position would be the middle of the number (i.e., tens). Moreover, in accordance with the parallel componential model, we would expect that sign-decade incompatible and unit-decade incompatible number pairs elicit more fixations than sign-decade and unit-decade compatible number pairs like in two-digit number comparison (cf. Moeller et al., 2009). Such a difference between compatible and incompatible number pairs should not be present according to the holistic and the sequential componential model. The feature model again makes no predictions, whether compati bility effects are also observable in the eye-tracking pattern.

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For the holistic model, we did not expect any difference in the eye fixation behavior between positive and negative numbers, because positive as well as negative numbers should be processed similarly. However, in accordance with the features model, we would expect more fixations on negative numbers than positive numbers because of the compression of the mental number line for negative numbers, which makes the comparison of negative num bers more difficult. Moreover, according to the componential models, we would also assume more fixations for negative than for positive numbers because of the mirror mechanism. Taken together, the eye-tracking data are of particular im portance for differentiating between the sequential and the holistic processing model, which cannot be discerned by RT data. The holistic as well as the sequential componential model make the very same predictions for RT data. However, the predictions of the two models differ for the to-be-expected eye fixation pattern. The holistic model predicts that participants should fixate the center of the number string most often (i.e., the tens; comparable to the optimal viewing position in word reading, e.g., Rayner, 2009, for a review), whereas the sequen tial componential model suggests that fixation location should depend on number pair type, with most fixations on polarity signs for heterogeneous number pairs and most fixations on tens for homogeneous number pairs. Our hypotheses for different models are summarized in Table 1.

Method Participants Twenty-four native German participants (14 women) were tested in single sessions (mean age = 25.71 years, age range = 20-37 years, SD = 4.24 years). All participants were right-handed and reported normal or corrected-to-normal vision. They were all naive with regard to the aim of the study and were reimbursed for their participation.

Apparatus Eye movements were recorded by an EyeLink 1000 Tower Mount (SR-Research, Osgoode, Ontario, Canada). This device attains a spatial resolution of less than 0.5° of visual angle, when using a 9-point calibration cycle at the beginning of the experiment and a drift correction before each trial. Responses had to be given via the vertically aligned buttons (T and A) of a Microsoft Gamepad. All stimuli were presented on a 21-in. monitor driven at a resolution of 1024 X 768 pixels and a refresh rate of 120 Hz. A constant viewing distance of 80 cm was ensured by the use of a chin rest.

Stimuli and Design We created 60 homogeneous positive (e.g., +31 vs. +57), 60 homogeneous negative (e.g., - 3 2 vs. -6 5 ), and 120 heteroge neous (e.g., -3 1 vs. +95) two-digit number pairs. For the homo geneous number pairs, we manipulated the unit-decade compati bility, such that half of the number pairs were unit-decade compatible (e.g., +41 vs. +84; 4 < 8 and 1 < 4), whereas the other half was unit-decade incompatible (+34 and + 91; 3 < 9 ,


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Table 1 Hypotheses Tested in the Present Study and the Predictions o f the Different Models Model Componential Hypothesis Reaction times Sign-decade compatibility Unit-decade compatibility Difference positive vs. negative Fixation pattern Most fixations on tens Sign-decade compatibility Unit-decade compatibility Difference positive vs. negative Note.

Holistic

Features

Sequential

Parallel

no no yes

reversed — yes

no no yes

yes yes yes

yes no no no

yes

no no no yes

yes yes yes yes

—

— yes

A dash indicates that a hypothesis cannot be derived.

but 4 > 1). Furthermore, half of the heterogeneous number pairs (i.e., 60 number pairs) were sign-decade compatible and the other half sign-decade incompatible, which additionally were subdivided in sign-unit compatible and incompatible numbers pairs (see Table 2 for examples). To strengthen the importance of the unit digits, we added 80 within-decade filler items (i.e., 20 positive, 20 neg ative, and 40 heterogeneous). In sum, there were 320 stimuli (240 critical items and 80 fillers items). Multiples of tens (e.g., +70) and tie numbers (e.g., +66) were excluded. Decade distance as well as the unit distance ranged from 2 to 8. Unit distance, decade distance, and overall absolute distance' were matched across the respective stimulus sets. Problem sizes of number pairs were matched as far as possible (see Table 3). All number pairs were presented in Arabic notation using 36point boldface Courier New font in white against a black back ground. A stimulus number (i.e., the polarity sign and the two digits including spaces) extended to a visual angle of up to 3.8° horizontally (with about 1.0° for a single digit) and 1.4° vertically. The fixation mark was presented above the two numbers (x = 512, y = 192), which were shown above each other. To prevent the evocation of columnwise comparisons of the digits, the two num bers of a pair were jittered by one position to either the left or right (x/y coordinates upper number: 498/300 or 526/300, respectively; lower number: 498/576 or 526/576, respectively). The polarity sign was shown for positive and negative number pairs. They were presented in separate blocks and intermixed in two conditions. The order of the conditions was counterbalanced in a within-participant design. Moreover, item order was randomized within blocks for each participant individually. Furthermore, the upper and lower number was larger in 50% of the trials.

Procedure Participants were seated in a dimly lit room and instructed to indicate which one of two presented numbers was the larger one by pressing the upper (lower) button, when the upper (lower) number was larger. After a successful 9-point calibration, the experiment started. The fixation mark presented at the beginning of each trial was also used for drift correction. After a successful drift correc tion, the to-be-compared numbers appeared simultaneously and

remained on the screen until the participant responded. No feed back was given as to the correctness of the response, and the fixation mark for the next trial followed immediately. There was a short break after every 80 trials.

Analysis Training trials as well as filler trials were excluded from further analysis. Only correct responses were considered for RT analyses. A trimming procedure eliminated response latencies shorter than 200 ms and longer than 2,000 ms (cf. Nuerk et al., 2001) first. Afterward, response latencies deviating more than three standard deviations from an individual participant’s mean were excluded iteratively. Only items included in the RT analysis were considered for the analysis of the eye fixation data. In sum, this resulted in a loss of 4.68% of the data due to data trimming. Data were analyzed by conducting repeated measures analyses of variance (ANOVAs). In case the repeated measures ANOVA assumption of sphericity was violated, Greenhouse-Geisser (GG) correction was applied to adjust the degrees of freedom. For reasons of readability, the original degrees of freedom together with the GG coefficient are reported. To analyze different levels of main effects and significant interactions, we calculated Bonferroni-corrected pairwise comparisons. For the analysis of participants’ eye fixation behavior, rectan gular areas of interest (AOI; 28 pixels wide and 100 pixels high) were centered at each character (i.e., the polarity sign, decade, and unit digit). Fixations falling outside of these AOIs were discarded. We analyzed the number of fixations on the polarity sign as well as on the tens and unit digits. The number of fixations for each of these components was obtained by averaging the number of fixa tions from corresponding components of the lower and upper number. When collapsing across polarity signs, tens, and units, the mean number of fixations on the upper and lower symbol in the display was averaged. Thus, the number of fixations reported

1 We also conducted a control experiment matching analogue distance (see the Appendix and Table A l). Importantly, the main findings were replicated in the control experiment.


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Table 2 Examples fo r Sign-Decade and Sign-Unit Compatible and Incompatible Heterogeneous Number Pairs Number 1

Number 2

SD compatibility

SU compatibility

+38 +51 + 17 + 12

-1 4 -2 6 -6 3 -3 8

Comp. (+ > —and 3 > 1) Comp. (+ > - and 5 > 2) Incomp. (+ > - , but 1 < 6) Incomp. (+ > —, but 1 < 3)

Comp. (+ > - and 8 > 4) Incomp. (+ > - , but 1 < 6) Comp. (+ > - and 7 > 3) Incomp. (+ > - , but 2 < 8)

Note.

SD = sign-decade; SU = sign-unit; Comp. = compatible; Incomp. = incompatible.

represent the average number of fixations within an interest area, but not the sum of all fixations within an interest area.2 Results RT Sign-decade and sign-unit compatibility effects can only be analyzed for heterogeneous number pairs. Therefore, we con ducted two separate ANOVAs for heterogeneous number pairs as well as homogeneous number pairs. For heterogeneous number pairs, we ran a 2 X 2 X 2 repeated measures ANOVA, with sign-decade compatibility (compatible vs. incompatible), sign-unit compatibility (compatible vs. incompatible), and condition (blocked vs. mixed) as factors. Homogeneous number pairs were analyzed using a 2 X 2 X 2 repeated measures ANOVA, with polarity (positive vs. negative), unit-decade compatibility (com patible vs. incompatible), and condition (blocked vs. mixed) as factors. In Table 4, means and standard deviations of the respective conditions are given.Heterogeneous number pairs. The ANOVA for heteroge neous number pairs revealed a significant main effect of signdecade compatibility, F (l, 23) = 93.09, p < .001, rip = .80, indicating that participants compared sign-decade compatible number pairs faster than sign-decade incompatible number pairs (M = 621 ms vs. M = 660 ms). The main effect of sign-unit compatibility was not significant (F < 1). Moreover, we found a significant effect of condition, F( 1, 23) = 124.35, p < .001, -rip = .84, with RT being faster in the blocked than in the mixed condi tion (M = 550 ms vs. M = 731 ms).

Table 3 Distances and Problem Sizes fo r the Different Stimulus Groups ( + + , ---- , + —) and Compatible and Incompatible Items Distances Polarity

Compatibility

++

UD compatible UD incompatible UD compatible UD incompatible SD compatible SD incompatible

— + -

Absolute Analogue Tens Units 45.47 45.53 45.47 45.53 45.50 45.50

45.47 45.53 45.47 45.53 104.77 104.77

4.10 5.00 4.10 5.00 4.55 4.55

4.47 4.47 4.47 4.47 2.07 2.07

Problem size 52.33 52.33 52.33 52.33 52.38 52.38

Note. + + = comparison of two positive numbers;-----= comparison of two negative numbers; -I— = comparison of a positive and a negative number; UD = unit-decade; SD = sign-decade.

The interaction between sign-decade and sign-unit compatibility was significant, F (l, 23) = 4.53, p = .044, -rip = .17. The interaction indicated that the sign-decade compatibility effect was smaller for sign-unit compatible number pairs than sign-unit in compatible number pairs (M = 26 ms vs. M = 53 ms; see Figure 1A). This suggests that not only the magnitudes of the tens but also the magnitudes of the units seem to interfere with the comparison of the polarity signs. Furthermore, the significant interaction between sign-decade compatibility and condition, F (l, 23) = 8.96, p = .006, r|p = .28, indicated a smaller sign-decade compatibility effect in the blocked than in the mixed condition (M = 29 ms vs. M = 50 ms). Importantly, sign-decade compatibility effects were reliable in both conditions, both t(23) > 5.8, both p < .001. The interaction between sign-unit and condition as well as the three-way interac tion were not significant (both F < 1). Thus, RT data for heterogeneous number pairs clearly favor the parallel componential model, as we found significant regular signdecade compatibility effects under both conditions (see also Table 1). Hence, we obtained evidence that tens interfered with the comparison of the polarity signs. Moreover, the interaction be tween sign-decade and sign-unit compatibility suggests that also units interfered with the comparison of polarity signs, indicating that all components of the numbers were processed separately and in parallel. Homogeneous number pairs. For homogeneous number pairs, the ANOVA revealed a significant main effect of polarity, F (l, 23) = 22.35, p < .001, rip = .49, with positive numbers being compared faster than negative numbers (M = 780 ms vs. M = 822 ms). Moreover, there was a significant main effect of unit-decade compatibility, F (l, 23) = 58.52, p < .001, = .72, indicating faster RT for unit-decade compatible than incompatible number pairs (M = 787 ms vs. M = 816 ms; see Figure IB). As for heterogeneous number pairs, the main effect of condition was significant, F( 1, 23) = 89.49, p < .001, r\j = .80. Participants compared homogeneous number pairs faster in the blocked than in the mixed condition (M = 730 ms vs. M = 872 ms). Moreover, we observed a significant interaction between polar ity and condition, F (l, 23) = 13.92, p = .001, r|p = .38. The 2 Please note that it is possible that participants do not fixate all digits separately, but they may be able to (pre)process digits or the polarity sign parafoveally. Therefore, when one of these symbols (e.g„ the lower polar ity sign) was fixated once, while the corresponding one was not fixated at all, a mean number of fixations below 1 resulted. Thus, the mean number of fixations for the polarity sign, tens, and units was smaller than 1 when a participant did not fixate one of the symbols on a trial.


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Table 4 Means and Standard Deviations (in Milliseconds) o f Reaction Times fo r the Different Stimulus Groups C + + , -------, + —) and Compatible and Incompatible Items in the Blocked and Mixed Condition Blocked Polarity ++ — + -

Compatibility UD compatible UD incompatible UD compatible UD incompatible SD compatible SD compatible SD incompatible SD incompatible

SU SU SU SU

compatible incompatible compatible incompatible

Mixed

M

SD

M

SD

699 743 722 755 541 530 557 572

93 98 97 89 60 56 59 74

827 850 898 915 713 699 749 763

140 133 144 143 95 98 116 97

Note. + + = comparison of two positive numbers;-----= comparison of two negative numbers; H— = comparison of a positive and a negative number; UD = unit-decade; SD = sign-decade; SU = sign-unit.

difference in RT between positive and negative numbers was smaller in the blocked condition than in the mixed condition (M = 17 ms vs. M = 68 ms). Post hoc tests revealed that the difference between positive and negative numbers was significant in the mixed condition, r(23) = 8.32, p < .001, but not in the blocked condition, f(23) = 1.27, p = .216. Finally, the significant interaction between condition and unitdecade compatibility indicated that the size of the unit-decade compatibility effect differed between the two conditions, F (l, 23) = 8.67, p = .007, rip = .27. The effect was larger in the blocked than in the mixed condition (M = 39 ms vs. M = 19 ms). Post hoc t tests indicated that the unit-decade compatibility effect was significant in both conditions (both ts > 3.5, ps < .01). The interaction between polarity and unit-decade compatibility as well as the three-way interaction were not significant (both F < 1). Taken together, we obtained further evidence corroborating the parallel componential model, because the unit-decade compatibil ity effect was present in both conditions. Interestingly, we ob served that participants compared negative numbers slower than positive numbers only in the mixed conditions. We suggested originally that negative numbers are processed slower than posi tive numbers conditions irrespective of a particular condition (see also Table 1). Yet, our finding suggests that negative numbers are not processed slower than positive numbers per se, but only under certain conditions.

Second, we investigated differences in number of fixations between positive and negative numbers. Therefore, we conducted a 2 X 2 X 3 X 2 repeated measures ANOVA, with polarity (positive vs. negative), unit-decade compatibility (compatible vs. incompatible), location of fixation (polarity sign, tens, and units), and condition (blocked vs. mixed) as independent variables. In Table 5, means and standard deviations of the respective condi tions are displayed. Heterogeneous number pairs. The main effect of signdecade compatibility reached significance, F (l, 23) = 12.47, p = .002, iqp = .35, indicating that sign-decade compatible number pairs were fixated less often than sign-decade incompatible num ber pairs (M = 0.31 vs. M = 0.33). Moreover, the main effect of location of fixation was significant, F(2, 46) = 57.48, p < .001, tip = .71, GG = .80. Tens were fixated most often, followed by polarity signs, whereas units were fixated least (polarity signs vs. tens vs. units: M = 0.25 vs. M = 0.60 vs. M = 0.10; pairwise comparisons: all p < .05). As with RT data, we found a significant main effect of condition, F (l, 23) = 105.45, p < .001, T|p = .82. There were fewer fixations in the blocked than in the mixed condition (M = 0.26 vs. M = 0.38). We did not observe a significant sign-unit compatibility effect (F < 1). Moreover, we found a significant interaction between signdecade compatibility and location of fixation, F(2, 46) = 4.77, p = .022, Tip = .17, GG = .76 (see Figure 2). Pairwise comparisons indicated that only sign-decade compatibility effects on the polar ity sign and on the units differed from each other reliably (p < .001, both other contrasts p > .05). The significant two-way interaction between location of fixation and condition indicated that fixations did not increase equally for

s ig n -u n it c o m p a tib le

B

s ig n -u n it in c o m p a tib le ■ u n it-d e c a d e c o m p a tib le

R T in m s

□ u n it-d e c a d e in c o m p a tib le

Eye-T racking Data Analyzing participants’ eye fixation behavior allowed us to investigate more closely how participants compared the different item types. As with RT data, participants’ eye fixation behavior was analyzed by running two ANOVAs. First, we conducted a 2 X 2 X 3 X 2 repeated measures ANOVA for heterogeneous number pairs, with sign-decade com patibility (compatible vs. incompatible), sign-unit compatibility (compatible vs. incompatible), location of fixation (polarity signs, tens, and units), and condition (blocked vs. mixed) as independent variables and number of fixations as the dependent variable.

p o s itiv e

n e g a tiv e

Figure 1. A: Mean reaction time (RT) of sign-decade and sign-unit compatible and incompatible number pairs for heterogeneous number pairs. B: Mean RT of unit-decade compatible and incompatible number pairs for positive and negative number pairs. Error bars depict 1 SEM.


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Table 5 Means and Standard Deviations (in Milliseconds) o f Number o f Fixations fo r the Different Stimulus Groups ( '+ + , ---- , + - ) and Compatible and Incompatible Items in the Blocked and Mixed Condition Blocked Polarity

Compatibility UD compatible

UD incompatible UD compatible

UD incompatible SD compatible

SU compatible

SD compatible

SU incompatible

SD incompatible

SU compatible

SD incompatible

SU incompatible

Mixed

Location of fixation

M

SD

M

Polarity signs Tens Units Polarity signs Tens Units Polarity signs Tens Units Polarity signs Tens Units Polarity signs Tens Units Polarity signs Tens Units Polarity signs Tens Units Polarity signs Tens Units

0.19 0.76 0.22 0.20 0.79 0.24 0.13 0.74 0.29 0.13 0.77 0.30 0.18 0.47 0.10 0.18 0.46 0.09 0.24 0.47 0.08 0.23 0.51 0.08

0.24 0.25 0.26 0.26 0.27 0.25 0.18 0.28 0.32 0.20 0.27 0.29 0.19 0.21 0.13 0.19 0.23 0.12 0.21 0.23 0.11 0.22 0.23 0.10

0.38 0.80 0.12 0.37 0.82 0.11 0.30 0.91 0.18 0.28 0.93 0.19 0.29 0.71 0.12 0.28 0.74 0.12 0.31 0.74 0.12 0.32 0.74 0.11

0.27 0.27 0.15 0.27 0.29 0.14 0.23 0.27 0.22 0.24 0.27 0.23 0.21 0.18 0.11 0.20 0.20 0.13 0.24 0.23 0.15 0.24 0.22 0.12

Note. + + - comparison of two positive numbers;-----= comparison of two negative numbers; H— = comparison of a positive and a negative number; UD = unit-decade; SD = sign-decade; SU = sign-unit.

all locations of fixations from blocked to mixed condition, F(2, 46) = 6.63, p = .003, pj; = .22. Differences between conditions on polarity signs, tens, and units were 0.09, 0.25, and 0.03, respec tively. Pairwise comparisons revealed that the difference between conditions was larger for tens than for units (p < .001), whereas other pairwise comparisons were not significant (both p > .05). Finally, the three-way interaction between sign-decade, signunit, and condition was significant, F (l, 23) = 6.85, p = .015, r|p = .23. We disentangled the three-way interaction by conducting two 2 X 2 ANOVAs separately for each condition. The two-way interaction between sign-decade and sign-unit compatibility ap proached significance in the blocked condition, F (l, 23) = 3.37, p = .079, T|p = .13, but was not significant in the mixed condition

polarity signs

tens

units

Figure 2. Average number of fixations (NFix) on each of the three locations separated for sign-decade compatible and incompatible items. Error bars depict 1 SEM.

(F < 1). Although neither of the two-way interactions was signif icant in itself, this difference most likely caused the significant three-way interaction. No other interactions were significant (all F < 2.44, p > .05). In sum, we found the predicted fixation pattern over location of fixations with most fixations on tens. This pattern is in accordance with all models except the sequential componential model (see Table 1). However, the significant sign-decade compatibility effect was predicted only by the parallel componential model. Thus, results for fixation patterns of heterogeneous number pairs pro vided converging evidence for the parallel componential model. Homogeneous number pairs. The significant unit-decade compatibility effect, F (l, 23) = 12.44, p = .002, rip = .35, indicated fewer fixations for unit-decade compatible than incom patible trials (M = 0.42 vs. M = 0.43). Moreover, the ANOVA revealed a significant main effect of location of fixation, F(2, 46) = 66.52, p < .001, pj; = .74. Mean fixations on polarity signs, tens, and units were 0.25, 0.81, and 0.21, respectively. Pairwise comparison indicated that participants fixated tens more often than polarity signs and units (both p < .001), whereas fixations on polarity signs and units did not differ (p = 1.00). Finally, also the main effect of condition was significant, F (l, 23) = 28.16, p < .001, pp = .55. Participants fixated numbers more often in the mixed than in the blocked condition (M = 0.40 vs. M — 0.45). The main effect of polarity was not significant, F (l, 23) = 3.00, p = .097, i)j = .12. Moreover, polarity interacted significantly with location of fix ations, F(2, 46) = 5.15, p = .010, pj; = .18. Mean differences


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between negative and positive numbers on polarity signs, tens, and units were —0.07, 0.04, and 0.07, respectively. Values smaller than zero indicate more fixations for positive numbers, whereas values larger than zero indicate more fixations for negative numbers. Pairwise comparisons revealed that the differences on tens and units were larger than those on polarity signs (both p < .05), whereas the differences on tens did not differ significantly from the differences on units (p = 1.00). Comparable to the RT data, the interaction between polarity and condition was significant, F (l, 23) = 11.92, p = .002, rip = .34, with the difference between positive and negative numbers being smaller in the blocked condition than in the mixed condition (M = -0.01 vs. M = 0.03). Post hoc t tests indicated that the difference was only significant in the mixed condition, /(23) = 4.07, p < .001, but not in the blocked condition, t{23) = -0 .5 1 , p = .614. Finally, the interaction between location of fixation and condi tion was significant, F(2, 46) = 5.31, p = .008, T|p = .19. Mean differences between conditions for polarity signs, tens, and units were 0.17, 0.10, and —0.11, respectively. Pairwise comparisons indicated that only the difference for polarity signs was larger than the difference for units (p < .05). Furthermore, all other interac tions were not significant (Fs < 4.10, ps > .05). Taken together, we again found that participants fixated tens most, which is in line with all models apart from the sequential componential model (see Table 1). However, neither the unitdecade compatibility effect nor the difference between positive and negative numbers revealed a consistent pattern, because these effects differed between conditions. Thus, results for fixation pat terns of homogeneous number pairs are not conclusive regarding the different models.

Discussion In the present study, we set off to provide a first test of a generalized model of parallel componential multi-symbol number processing by evaluating how far it can account for empirical effects observed in a two-digit number magnitude comparison task involving positive and negative numbers. Therefore, the general ized parallel componential processing model is contrasted to three existing models of negative number processing: (a) the holistic model (e.g., Fischer, 2003), (b) the features model (Varma & Schwartz, 2011), and (c) the sequential componential model de rived from multi-digit number processing (Poltrock & Schwartz, 1984). RT and eye-tracking data provided converging and com pelling evidence corroborating the parallel componential process ing of negative numbers and thus the proposed model of general ized parallel componential processing of multi-symbol numbers. In particular, the observed regular sign-decade compatibility for het erogeneous number pairs as well as the unit-decade compatibility effect for homogeneous number pairs argue against holistic, fea tures, or sequential componential accounts. In the following, we first discuss the findings for the compatibility effects, followed by a discussion of the general advantage for the processing of positive numbers.

Sign-Decade and Unit-Decade Compatibility The holistic model (Fischer, 2003) cannot explain a sign-decade compatibility effect, because absolute as well as analogue dis

tances (see the Appendix and Table A l) were matched between sign-decade compatible and incompatible number pairs, and hence we should not have observed a sign-decade compatibility effect. Moreover, as previously shown for two-digit numbers, a holistic model cannot account for a unit-decade compatibility effect in homogeneous number pairs (e.g., Nuerk et al., 2001). Thus, our results clearly argue against a holistic model. The features model (Varma & Schwartz, 2011) is very similar to componential models as it suggests that polarity signs and digits are represented separately. Moreover, comparable to the parallel componential model, it assumes that both polarity signs and digits are processed in parallel. However, it comes to a different conclu sion regarding the sign-decade compatibility effect. According to the features model, we should have observed a reversed signdecade compatibility effect, because the model predicts slower RTs for the comparison of the sign-decade compatible number pair —14 and + 92 (— < + and 1 < 9) than for the comparison of sign-decade incompatible number pair + 14 and - 9 2 (+ > but 1 < 9), as —1 and + 9 have more features in common than + 1 and —9 (please see Varma & Schwartz, 2011, for more details on feature coding). Yet, as we found a regular sign-decade compati bility effect for the comparison of heterogeneous number pairs, this pattern of results cannot be accounted for by the features model. Furthermore, both the sign-decade as well as the unit-decade compatibility effect are hard to reconcile with the model of se quential componential processing. Because polarity signs are de cisive in heterogeneous number pairs, these should be processed exclusively when heterogeneous number pairs are processed se quentially. As a consequence, no sign-decade compatibility effect should have been observed. Thus, the observation of the signdecade compatibility effect rules out a strictly sequential process such as a sign shortcut explanation (e.g., Krajcsi & Igacs, 2010). The same line of argument applies to the case of the unit-decade compatibility effects. In contrast, the proposed generalized model of parallel compo nential processing of multi-symbol numbers can account for both the sign-decade as well as the unit-decade compatibility effect, as it suggests the comparison of the tens to interfere with the com parison of polarity signs, comparable to the case in which the comparison of units interferes with the comparison of tens in two-digit number comparison (e.g., Nuerk et al., 2001). Therefore, already the RT data corroborate the generalized model of multi symbol processing, which suggests that any multi-symbol number (multi-digit, but also negative numbers consisting of polarity signs and digits) is processed in a parallel and a componential fashion. This interpretation was further corroborated by our eye-tracking data. We hypothesized that all models except the sequential com ponential model predict participants to fixate the tens most. And indeed, we found such a fixation pattern, which is in line with the expectations derived from the holistic, the features, and the parallel componential model. However, comparable to RT data, the paral lel componential model is the only model, which can account for sign-decade and unit-decade compatibility effects also observed in participants’ eye fixation behavior, thus providing further evidence supporting the generalized model of parallel componential pro cessing. Taken together, both RT as well as eye-tracking data indicated regular sign-decade as well as unit-decade compatibility effects,


which are hard to reconcile with the holistic, the features, or the sequential componential model. In contrast, our findings are con sistent with the proposed model of generalized parallel componen tial processing of multi-symbol numbers.

The Advantage for Positive Numbers Processing Apart from the hypotheses for the compatibility effects, we also had a very specific hypothesis for the difference between the comparisons of homogeneous positive and negative number pairs, expecting that positive number pairs should be processed faster than negative number pairs. This hypothesis is in line with all models, however, for different reasons. Interestingly, we found evidence for faster RTs of positive than negative number pairs only in the mixed condition, in which both heterogeneous and homogeneous number pairs were presented within one block. Importantly, these differential results between conditions cannot be explained by representational accounts, namely, the holistic and the features model, suggesting that negative numbers are repre sented differently than positive numbers. As representations should be quite stable, slower reaction times for negative number pairs should have been observed irrespective of blocking con straints. However, a strategy account like the mirror mechanism might explain the differences between conditions (Krajcsi & Igacs,

2010). The mirror mechanism is an algorithm proposed by Krajcsi and Igacs (2010), which participants may have applied to compare nega tive number pairs. According to the mirror mechanism, participants compare negative number pairs by ignoring the minus sign in a first step, resulting in a comparison of positive numbers, followed by an inversion of their response in a second step (i.e., choosing the smaller instead of the larger to indicate the larger number). In the blocked condition, participants did not have to switch between choosing the absolutely larger (to indicate the larger number for positive number pairs) and choosing the absolute smaller number (to indicate the larger number for negative number pairs). However, in the mixed presenta tion, they had to flexibly adapt their response strategy depending on the polarity signs (i.e., choosing the larger number for positive num bers or choosing the smaller number for negative numbers). Thus, one would expect longer RT for negative than for positive number pairs in the mixed presentation due to the inversion of response for negative number pairs, but not in the blocked presentation because only in the former, participants have to adapt their response strategy on a trialto-trial basis, whereas in the latter, they can stick to the same strategy throughout the block. This is exactly what we found: RT for positive and negative number pairs did not differ reliably in the blocked presentation, but did so in the mixed presentation condition. Thus, our data provide further evidence for the mirroring mechanism suggested by Krajcsi and Igacs (2010). Nevertheless, there is also another model, which can account for this finding: a hybrid model assuming both a componential and a holistic representation of negative numbers (cf. Ganor-Stern et al., 2010). For positive numbers, comparing the magnitude of the decision-relevant digit (e.g., tens) and the overall magnitude would result in the same response (e.g., +31 and +57: 3 > 5 and 31 > 57). For negative number pairs, however, this does not hold. Comparing the magnitude of the decision-relevant digit and the overall magnitude is necessarily incongruent (e.g., —32 and —65: 3 < 6, but —32 > —65). Therefore, similar to the response conflict

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evoked by incompatible decisions for sign-decade incompatible number pairs, there might also be a response conflict between the comparison of decision-relevant digits and the overall magnitude for negative numbers. Consequently, prolonged RT of negative num bers may also be conceptualized as another instance of a compat ibility effect (i.e., a representational compatibility effect: compat ibility between a holistic representation and a componential representation of negative numbers). To account for RT differ ences between blocked and mixed presentation, it has to be as sumed that participants adapt to the blocked condition by relying more on the holistic representation, and thus the representational compatibility effect would be reduced. For both accounts—the mirror mechanism as well as a hybrid model of negative number processing with the corresponding representational compatibility effect— some kind of adaption to the blocking constraints is assumed, which argues for influences of cognitive control in the processing of negative numbers (see Macizo & Herrera, 2011, for the case of compatibility effects in positive two-digit numbers; Huber et al„ 2014). This notion is discussed in the following.

Cognitive Control in Negative Number Processing Interestingly, apart from faster RT for homogeneous positive num ber pairs, we also found a larger sign-decade compatibility effect in the mixed condition. In the blocked condition, the item set comprised 100% heterogeneous number pairs, whereas the item set consisted of 50% homogeneous and 50% heterogeneous number pairs in the mixed condition. Consequently, in the blocked condition, only polar ity signs were decision relevant for heterogeneous number pairs. However, in the mixed condition, not only polarity signs but also tens were decision relevant, because 50% of all items were homogeneous number pairs, where tens are decisive for comparison. This resulted in an increase of the relevance of the tens in the comparison of hetero geneous number pairs, which was accompanied by more fixations on tens (for heterogeneous number pairs). Thus, tens interfered more with the comparison of polarity signs, increasing the sign-decade compatibility effect in the blocked condition. A similar effect has already been reported for two-digit number comparison (Huber et al., 2014; Macizo & Herrera, 2011) and threedigit number comparison (Huber, Moeller, Nuerk, & Willmes, 2013). Huber et al. (2014) manipulated the frequency of within-decade filler items in three stimulus sets, with 25%, 50%, and 75% within-decade filler items (e.g., 43 vs. 47). In line with the finding for negative numbers, they found that the more within-decade filler items a stim ulus set included, the larger was the unit-decade compatibility effect. This finding was accompanied by more fixations on units than on tens. Thus, they suggested that participants shifted their attention toward units in the item set with more within-decade filler items, because units were more relevant when there were more withindecade filler items. Moreover, Macizo and Herrera (2011) and Huber et al. (2014) interpreted this finding in terms of two-digit number processing being “under cognitive control.” Two prominent theories of cognitive control can explain the find ings of Macizo and Herrera (2011) and Huber et al. (2014) and, therefore, might also account for the larger sign-decade compatibility effect in the mixed condition: (a) the conflict monitoring theory (Botvinick, Braver, Barch, Carter, & Cohen, 2001; Botvinick, Cohen, & Carter, 2004) and (b) the adaption by binding theory (Verguts &

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HUBER. CORNELSEN. MOELLER. AND NUERK

Notebaert, 2008, 2009). Applying these theoretical frameworks to the case of two-digit positive numbers, Macizo and Herrera (2011, 2013) proposed a possible architecture of cognitive control in number pro cessing. They suggested a computational model involving (a) an input layer for the processing of tens and units, (b) a task demand layer to specify the demand to focus on either tens or units for the overall decision, (c) a response level for indicating the larger number, and (d) a conflict-monitoring system (see also Huber, Moeller, Nuerk, Ma cizo, et al., 2013, for a computational model). The model of Macizo and Herrera (2011, 2013) assumes that two-digit numbers are processed componentially and in parallel, which is in line with the proposed model of generalized parallel componential processing of multi-symbol numbers proposed in the present study. Interestingly, the model of Macizo and Herrera (2011, 2013) can be extended quite easily to be applicable to the generalized case of multi-symbol numbers. Most importantly, the model architec ture has to be adapted to incorporate symbols like polarity signs. To achieve this, one may include an additional representation for polarity signs in the input layer and an additional unit in the task layer, which specifies the demand to focus on polarity signs. This cognitive control network for generalized multi-symbol num bers should be able to explain our findings for heterogeneous number pairs: As with two-digit numbers, the degree of conflict depends on the activation of the two response nodes (i.e., top or bottom number being larger). The larger the activation of the response nodes, the larger is the conflict. When homogeneous and heterogeneous number pairs are presented in an intermingled manner, homogeneous number pairs would elicit more conflict than heterogeneous number pairs, because it takes longer for the response nodes to reach a fixed threshold, because comparing two polarity signs is faster than com paring digits. Hence, both response nodes will be activated, which results in a conflict. This conflict will then be resolved by increasing the activation of the demand nodes of tens (conflict-monitoring the ory) or strengthening the connection weights between the demand node of tens and the input layer (adaptation by binding theory). Consequently, the comparison of tens is speeded up, which, however, leads to larger interference of tens when comparing heterogeneous number pairs.

Extending the Parallel Componential Model to Negative Numbers As outlined above, the present study suggests that the model of parallel componential processing can be generalized to negative num bers. In Figure 3, we provide a first sketch of how such a hypothetical model might look like. We suggest that in line with the previous model of Huber, Moeller, Nuerk, Macizo et al. (2013), magnitude representations for each digit of a number should exist (see Figure 3, tens and units comparison network). These representations should follow a place-coding scheme, with a corresponding node for each digit and nodes of neighboring digits being activating to a lesser degree depending on their distance to the respective digit (Verguts et al., 2005). In Figure 3, the amount of activation of a respective node is depicted by its darkness with white nodes representing no activa tion, black nodes representing the highest activation, and the different gray values as the continuum in between. In addition to the model of Huber, Moeller, Nuerk, Macizo et al. (2013), we suggest that a comparison network for the comparison of polarity signs with nodes coding the presence of either a plus or a

minus sign should exist. Moreover, in line with a hybrid model for negative numbers, a holistic representation for positive and negative two-digit numbers should be added. These representations should also follow a place-coding scheme for the magnitude representation of numbers like the representation for single-digit numbers (cf. Verguts et al., 2005). In the comparison layer, we suggest that each component of a number should be compared with the respective component of the other number. For instance, as indicated in Figure 3, the tens digit of the first number (i.e., “3”) should be compared with the tens digit of the second number (i.e., “1”). Importantly, the comparison of each component would yield different decision biases. For instance, the comparison of the polarity signs of the numbers “- 3 8 ” and “+12” would indicate that the right number is larger than the left number, and hence the right node of the comparison layer should be activated more strongly than the left node. However, we suggest that these separate comparisons are then weighted by the task demand layer, which ensures that the outcome of the comparison of the polarity signs is more important than the result of the comparison of the tens digits. The task demand layer might be conceptualized as an intentional weighting mechanism (cf. Memelink & Hommel, 2013) putting more weight on the comparison of the decision-relevant components when comparing multi-symbol num bers: The more a node of the task demand layer is activated, the larger is the respective weight of the associated component, and thus its importance for the overall decision. In the response layer, we propose that separate decisions are integrated to form an overall decision of either the left or the right number being larger. In the example depicted in Figure 3, the right number (i.e., “+12”) is larger than the left number (i.e., “ - 3 8 ”), and therefore the right node of the response layer is activated more strongly than the left node. However, also the left node gets some activation due to the opposing decision biases when comparing polarity signs and tens. As in the model of Huber, Moeller, Nuerk, Macizo et al. (2013), we suggest that a cognitive control unit should detect the amount of conflict and thereby should modulate the connection weights between the task demand node of the tens and the input layer (adaptation by binding theory; Verguts & Notebaert, 2008, 2009). In Figure 3, we outlined a model for the comparison of up to two digits. Nevertheless, the model can be extended without any problems to three and even multi-digit numbers by simply adding additional comparison networks for respective digits as shown in the study of Huber, Moeller, Nuerk, and Willmes (2013). Moreover, the general idea of the model is that numbers are represented componentially with a potential additional holistic representation (as depicted in Figure 3, but see the alternative idea of a mirror mechanism of Krajcsi & Igacs, 2010). Hence, it suggests a similar magnitude representation for all signed numbers adhering to the place-value structure of the Arabic number system, also including decimal fractions in addition to multidigit numbers (Huber, Klein, Willmes, Nuerk, & Moeller, 2014). However, the most important extension to the previous model of Huber, Moeller, Nuerk, Macizo et al. (2013) is the comparison net work for the polarity signs. Symbols encoding magnitude information seem to be processed and represented very similarly to digital com ponents of multi-symbol numbers. Moreover, the findings of the present study suggest that when ever symbols or digits are presented within a character string and/or compared with another character string containing a symbol or digits, they are processed and compared in parallel. Only such parallel comparisons can account for the proposed compatibility


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Model of generalized parallel componential processing of multi-symbol numbers

Response Layer

Task Demand Layer

Comparison Layer

Input Layer

-38 vs. +12 Figure 3. Schematic illustration of the model architecture: Network A depicts the polarity sign comparison network, Networks B and C digit comparison networks for tens (B) and units (C), Network D the holistic comparison network for whole numbers, and Network E the cognitive control network. L = "left digit larger” node; R = “right digit larger” node.

effects (i.e., sign-decade, sign-unit, and unit-decade compatibility effects). Finally, Figure 3 depicts a network suggesting a hybrid representation of integer numbers. Nevertheless, as discussed above, the mirror mechanism might be a valid alternative expla nation for prolonged reaction times in the mixed condition of the present study. Therefore, further studies are needed to evaluate whether a hybrid representation of negative numbers or a mirror mechanism describes the comparison of negative numbers best.

Conclusion In the present study, we proposed and evaluated a generalized model of parallel componential processing of multi-symbol num bers by means of its applicability to the processing of negative numbers. RT and eye fixation data provide converging evidence for the validity of our model, suggesting that negative number processing can indeed be described in terms of parallel compo nential multi-symbol processing, in which the polarity sign is processed separately from, but in parallel to, the constituting digits of negative numbers. In contrast, the holistic, the features, and the sequential componential models cannot account for the present

data pattern, because they cannot explain sign-decade as well as unit-decade compatibility effects observed in the study. Thus, the present data provide convincing evidence for the concept of gen eralized parallel componential multi-symbol number processing by transferring recent evidence from multi-digit integers to negative numbers. Importantly, this also included recent evidence on influ ences of cognitive control on multi-digit number processing, which is also corroborated by the present data. Thereby, these results represent an important step toward a generalized model framework for the processing of a multi-symbol number, including two-digit, three-digit positive and negative numbers as well as decimal numbers. Importantly, this suggests that the processing of all of these number types relies on the same basic mechanism of parallel componential number processing without the need for separate mechanisms for separate number types (e.g.. mirroring mecha nisms) to be assumed.

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Appendix Control Experiment W hen matching the absolute distance of heterogeneous number pairs, the analogue distance is, by definition, larger for heteroge neous number pairs than homogeneous number pairs. If partici pants compare number pairs holistically, RT for heterogeneous number pairs will be shorter due to the distance effect. However, a similar pattern is expected when number pairs are processed componentially. Although the fixation pattern of the main exper iment argued against a holistic model, we conducted a control experiment with matched analogue distance to investigate its in fluences on potential differences. Additionally, the control exper iment served to exclude that our results are caused by the selection of stimuli rather than by the respective processing style.

Table A l

Method and Analysis

Note. + + = comparison of two positive numbers; — - = comparison of two negative numbers; H— = comparison of a positive and a negative number; UD = unit-decade; SD = sign-decade.

The method and participants were identical to the main exper iment. The crucial difference between experiments was the dis tance between a pair of numbers. In the analogue set o f the control experiment, the distance was matched with regard to the analogue numerical distance (e.g., 1+81 - 341 = 47 = 1—32 - 151), whereas the set used in the main studies was matched with regard to the absolute distance (e.g., 11+811 - 13411 = 47 = 11—321 — 1+7911; see also Table A l). Comparable to the set matched for absolute dis tance, a set of 320 number pairs matched for analogue distance was chosen. Because the sign-decade compatibility effect allowed us to differentiate between models, we only report the results for this effect for the RT data and the eye-tracking data. The same ANOVAs as in the main experiment were used to analyze RT and number of fixations.

Results and Discussion As in the main experiment, we found a significant main effect of sign-decade compatibility, F (l, 23) = 36.03, p < .001, tip = .61.

Distances and Problem Sizes fo r the Different Stimulus Groups ( + + , — , + —) and Compatible and Incompatible Items in the Control Experiment Distances Polarity

Compatibility

++

UD compatible UD incompatible UD compatible UD incompatible SD compatible SD incompatible

— + -

Absolute Analogue Tens Units 66.80 66.53 66.80 66.20 26.50 26.67

66.80 66.53 66.80 66.20 66.77 66.60

6.23 7.10 6.23 7.07 2.65 2.67

4.47 4.47 4.47 4.47 1.67 1.97

Problem size 53.20 53.37 53.23 53.30 33.38 33.30

Participants responded to sign-decade compatible pairs faster than sign-decade incompatible pairs, replicating the finding of the main experiment (M = 636 vs. M = 667). Moreover, the sign-decade compatibility effect was also present in the eye-tracking data, F (l, 23) = 11.10, p = .003, T)p = .33, with fewer fixations for sign-decade compatible than incompatible number pairs (M = 0.32 vs. M = 0.33). Thus, the sign-decade compatibility effect, suggesting parallel componential processing of negative numbers, did not diminish because o f the different matching procedure, substantiating the results of the main experiments and providing additional evidence for the parallel componential model of multi-symbol number pro cessing. Received December 21, 2013 Revision received May 22, 2014 Accepted May 23, 2014 ■

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