The mental representation of ordinal sequences is spatially organized

Transcription

1 W. Gevers et al. / Cognition 87 (2003) B87 B95 B87 Cognition 87 (2003) B87 B95 Brief article The mental representation of ordinal sequences is spatially organized Wim Gevers*, Bert Reynvoet, Wim Fias Department of Experimental Psychology, Ghent University, H. Dunantlaan 2, B-9000 Ghent, Belgium Received 10 July 2002; accepted 13 November 2002 Abstract In the domain of numbers the existence of spatial components in the representation of numerical magnitude has been convincingly demonstrated by an association between number magnitude and response preference with faster left- than right-hand responses for small numbers and faster rightthan left-hand responses for large numbers (Dehaene, S., Bossini, S., & Giraux, P. (1993) The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, ). Because numbers convey not only real or integer meaning but also ordinal meaning, the question of whether non-numerical ordinal information is spatially coded naturally follows. While previous research failed to show an association between ordinal position and spatial response preference, we present two experiments involving months (Experiment 1) and letters (Experiment 2) in which spatial coding is demonstrated. Furthermore, the response-side effect was obtained with two different stimulus-response mappings. The association occurred both when ordinal information was relevant and when it was irrelevant to the task, showing that the spatial component of the ordinal representation can be automatically activated. q 2003 Elsevier Science B.V. All rights reserved. Keywords: Spatial representation; Ordinal sequences 1. Introduction Although considerable advances have been made in the understanding of the internal representation and processing of ordinal information (see Leth-Steensen & Marley, 2000, for a recent review), one aspect of it, the association between order and space, has barely been investigated. This is surprising because numbers, which convey ordinal meaning * Corresponding author. Tel.: ; fax: addresses: wim.gevers@rug.ac.be (W. Gevers), bert.reynvoet@rug.ac.be (B. Reynvoet), wim.fias@rug.ac.be (W. Fias) /03/$ - see front matter q 2003 Elsevier Science B.V. All rights reserved. doi: /s (02)

2 B88 W. Gevers et al. / Cognition 87 (2003) B87 B95 (hierarchically implied by their real and integer meaning; Gallistel & Gelman, 2000), are generally accepted to be spatially coded. An explicit indication of an association between numbers and space was first demonstrated by a response-side effect in number comparison (Dehaene, Dupoux, & Mehler, 1990; Hinrichs, Yurko, & Hu, 1981). When participants had to indicate whether a number was smaller or larger than a predetermined reference number by pressing one of two response buttons, it was found that large responses were faster with the right hand than with the left hand. The reverse was true for small numbers. In a study investigating the mental processing of parity judgement, Dehaene, Bossini, and Giraux (1993) provided further evidence concerning the response-side effect (and called it SNARC effect for Spatial Numerical Association of Response Codes). In evaluating a number s parity status, shorter latencies were obtained for small numbers responded to with the left hand than with the right hand. Conversely, large numbers were responded to preferentially with the right hand. Because parity is a numerical property which alternates between consecutive numbers, this study extends the response-side effect by showing that the effect is not specific to the stimulus-response mapping of the comparison task. It also follows that the effect is not related to the use of a reference number for comparison. Importantly, the results also show that the numerical value is automatically activated, because in principle the parity judgement task needs no access to numerical magnitude information for correct task performance. Further support for the possibility of automatic activation of spatially coded numerical magnitudes comes from a series of experiments which showed that the SNARC effect was elicited, even when digits served merely as a background distracter with a non-numerical target superimposed in an orientation discrimination task (Fias, Lauwereyns, & Lammertyn, 2001). In the present paper we investigate whether a similar association exists between nonnumerical ordered sequences and spatial properties of the response. Dehaene et al. (1993; Experiment 4) already used letters in situations that were structurally analogous to the parity judgement task. In a first condition subjects had to classify the letters A F into the categories ACE or BDF. No SNARC effect was obtained. One could argue, however, that the categories along which the letters had to be classified were arbitrary. This may have induced an atypical way of performing the task. This assumption is supported by the fact that the pattern of response times (RTs) was compatible with a serial search strategy. In a second condition, letters were used in a consonant vowel classification task spanning almost the entire alphabet, but a significant SNARC effect was not obtained. However, there was a tendency toward a SNARC effect. Given the small number of subjects (n ¼ 10), one could argue that the experiment did not have enough statistical power. Most importantly, in both conditions the ordering of the stimuli was irrelevant to the task. One could imagine that the mental representation of order is spatially organized, but that, unlike numbers, this spatially organized representation is not accessed automatically. Taken together, there are a number of reasons why the conditions to test a possible spatial association of order and space were suboptimal. In the following experiments we investigate the possible spatial organization of two ordered sequences, namely months of the year (Experiment 1) and letters (Experiment 2). For each stimulus type, subjects performed both an order-relevant task (comparison with a

3 W. Gevers et al. / Cognition 87 (2003) B87 B95 B89 fixed standard) and an order-irrelevant task (letter detection in Experiment 1 and consonant vowel classification in Experiment 2). The order-relevant task allows judging of the spatial coding of internally ordered information, and with the order-irrelevant task the automaticity of activation of the spatial codes can be evaluated. 2. Experiment 1: months 2.1. Method Twenty-five Dutch speaking participants (four left-handed, mean age: 21.4 years) completed both order-relevant and -irrelevant tasks. The range of stimuli consisted of eight months, ranging from January to April and from September to December, presented in Dutch ( januari, februari, maart, april, september, oktober, november and december ). During the order-relevant task, subjects were asked to judge the position of the months as coming before or after July. During the order-irrelevant task they were asked to judge whether the presented month ended with the letter R or not. Within each task subjects were tested twice: once with early months/months ending with R assigned to the left hand and once to the right hand. Order of task and response assignment were counterbalanced across subjects. In each trial # was presented as a fixation mark for 300 ms, immediately followed by the target month (width ranging from 2.98 to 5.98 depending on word length, height 0.88), until a response was given or 1200 ms elapsed. The screen then remained blank for 1500 ms after which a new trial was started. Every condition was preceded by a training list of 16 items and then each month was presented 20 times in random order with the restriction that the same month could never be presented in successive trials. The experiment was run on a PC-compatible computer Results and discussion Order-relevant task RTs more than three standard deviations from the mean were excluded from the analysis. As a result 2.63% of the data had to be removed. The average error rate over subjects was 4.06%. Mean correct RTs were 484, 513, 482, 479, 501, 497, 492 and 477 ms for the respective months. An ANOVA with Position (before or after reference), Distance from reference (1 4) and Side of response (left or right) as within-subjects factors revealed the following significant effects: a main effect of Distance (Fð3; 72Þ ¼13:06; MSE ¼ 975; P, 0:0001), a main effect of Side of response (Fð1; 24Þ ¼6:71; MSE ¼ 1552; P, 0:002) with faster right-hand responses (486 ms) than left-hand responses (496 ms), and an interaction between Distance and Position (Fð3; 72Þ ¼8:70; MSE ¼ 618; P, 0:0001) which is due to the fast responses to the short months March and April, and most importantly a Position Side of response interaction (Fð1; 24Þ ¼ 4:68; MSE ¼ 11304; P, 0:05), indicating a clear association between order and spatial co-ordinates of the response. To capture the nature of the Position Side of response interaction, a regression analy-

4 B90 W. Gevers et al. / Cognition 87 (2003) B87 B95 Fig. 1. Observed data and regression line representing RT differences between right-handed minus left-handed responses as a function of position in the sequence of months in (a) the order-relevant task and (b) the orderirrelevant task. sis was performed on the RT difference between right-hand minus left-hand responses (drt). From an association between order and space, a negative relation between position in the sequence and drt is predicted: if the beginning of the sequence is associated with left, responses will be faster with the left hand, resulting in a positive drt, whereas negative drts are expected towards the end of the sequence. For this purpose we used the regression analysis for repeated measures data recommended by Lorch and Myers (1990; Method 3) (for advantages of this method see Fias, Brysbaert, Geypens, & d Ydewalle, 1996). In a first step, for all participants the mean RT of the correct responses was computed for each number, separately for left and right responses. On the basis of these mean RTs, drts were computed by subtracting the RT for left-hand responses from the RT for right-hand responses. In a second step, for each individual participant, a regression analysis was computed with the absolute position of months in the complete sequence as the predictor variable (January to April: 1 4; September to December: 9 12). 1 In a third step, one-tailed t-tests were performed to test whether the regression weights of the group deviated significantly from zero. This resulted in the equation drt ¼ 26:17 2 5:60 (Position), with Position contributing significantly (tð24þ ¼ 22:30; SD ¼ 12:2; P, 0:05). This 1 Analyses were also performed with position coded relative to the used items of the sequence (values 1 8). Although intercept and regression weights were different, the patterns of results were the same for all four conditions in the present manuscript.

5 W. Gevers et al. / Cognition 87 (2003) B87 B95 B91 confirms that the Position Side of response interaction results from an association of the spatial components of the position in the ordered sequence and of the response (Fig. 1a). Regression analysis was also used to evaluate the nature of the distance effect with the effect of word length removed. For this purpose, the number of syllables and distance from the reference (1 4) were included as variables to predict RT. This resulted in the equation RT ¼ 476:4 1 10:53 (Number of syllables) (Distance). Both Number of syllables (tð24þ ¼ 4:89; SD ¼ 10:76; P, 0:0001) and Distance (tð24þ ¼ 23:93; SD ¼ 8:10; P, 0:001) contributed significantly. The distance effect shows that subjects solved the task by comparing the month to the reference and not by using a serial search strategy. In the latter case, distance is not predicted to affect comparison times for items positioned after the reference Order-irrelevant task Data-trimming removed 1.99% of the data. The average error rate was 4.13%. Mean correct RTs were 469, 488, 482, 461, 449, 458, 446 and 444 ms for the respective months. An ANOVA was performed with Position (beginning or end) and Side of response (left or right) as within-subjects factors and Order of tasks (first order-relevant or first orderirrelevant) as a between-subjects variable. Distance was not incorporated in the design because no reference was used in the order-irrelevant task. The Position factor perfectly coincided with the presence of the end-letter R (September December) or not (January April). For clarity we keep the same nomenclature as above. The between-subjects factor allows us to evaluate whether the possible Position Side of response interaction is a residue of the S-R mapping in the order-relevant task or not. There was a main effect of Position (Fð1; 23Þ ¼57:71; MSE ¼ 275; P, 0:0001) with faster responses to months positioned at the end (containing the final letter R; 449 ms) than in the beginning (not containing a final R; 475 ms) of the sequence. This probably reflects the general observation that target-present responses are faster than target-absent responses (e.g. Treisman & Gelade, 1980). The interaction between Position and Side of response (Fð1; 23Þ ¼16:61; MSE ¼ 639; P, 0:0005) was significant and was not modulated by Order of tasks (F, 1), from which it can be concluded that the association between spatial coding of Position and Side of response was not due to the persistence of the S-R mapping adopted in the order-relevant task. The regression of Position (coded 1 4 and 8 12) on drt (Fig. 1b) resulted in the equation drt ¼ 25:28 2 4:93 (Position). Position differed significantly from zero (tð1; 24Þ ¼ 24:21; SD ¼ 5:85; P, 0:0005). The regression slopes of Position in the order-irrelevant and the order-relevant task did not differ (dependent t-test: tð24þ ¼20:30). Because months ending in R were the last months of the year and vice versa, one could argue that participants solved the final letter identification task on the basis of ordinal position. If participants really employed this strategy, one would expect a distance effect in the orderirrelevant task. Because a regression analysis of distance on RT revealed no distance effect (Distance-slope ¼ 0:20; tð24þ ¼ 0:15), the strategy explanation can be rejected. 3. Experiment 2: letters Although the results of Experiment 1 are quite clear, one could argue that the conclusion

6 B92 W. Gevers et al. / Cognition 87 (2003) B87 B95 Fig. 2. Observed data and regression line representing RT differences between right-handed minus left-handed responses as a function of position in the alphabet in (a) the order-relevant task and (b) the order-irrelevant task. of spatial coding of ordinal sequences is not justified because it is common practice to numerically code the months of the year. Thus, in principle it is possible that the spatial coding derives from numerical recoding. While numerical recoding is indeed well-learned in the case of months, it is not a feasible option for other sequences. Assigning numbers to the letters of the alphabet, for instance, has recently been shown to require at least 2000 ms (Jou & Aldridge, 1999). To evaluate whether spatial effects can be obtained in the absence of numerical recoding we used letters instead of months in Experiment Method Twenty-four native Dutch speaking subjects (three left-handed, average age: 20.8 years) participated in the experiment. Participants performed an order-relevant comparison task (before or after O) and an order-irrelevant task (consonant vowel classification). We made sure that in the subset of letters used (E G I L R U W Y; width ranging from 0.68 to 4.68, height 3.48) distance from the reference was the same before and after the reference letter O and that consonants and vowels alternated successively. Therefore, the order-irrelevant task allows the investigation of the spatial coding of ordinal structure with a completely different stimulus-response mapping. It should be noted that in Dutch, unlike in English, Y is considered to be a vowel /i/. In all other respects, stimulus presentation and design were the same as in Experiment 1.

7 3.2. Results and discussion W. Gevers et al. / Cognition 87 (2003) B87 B95 B Order-relevant task Due to data-trimming 3.50% of the data was removed. The average error rate was 2.62%. Average RTs for the respective letters were 560, 619, 568, 653, 624, 619, 550 and 583 ms. The response latencies are far below those required to convert letters to numbers (Jou & Aldridge, 1999). Hence, numerical recoding can be rejected as an explanation for the results described below. An ANOVA was performed, including Position (before or after reference), Distance from reference (1 4) and Side of response (left or right) as repeated measures. There were main effects of Distance (Fð3; 69Þ ¼18:74; MSE ¼ 4341; P, 0:0001) and of Side of response (Fð1; 23Þ ¼12:59; MSE ¼ 2069; P, 0:005; RT left ¼ 605 ms; RT right ¼ 589 ms). There was a significant interaction between Position and Distance (Fð3; 69Þ ¼24:98; MSE ¼ 2745; P, 0:0001). Inspection of RTs shows that the interaction is not due to the absence of a distance effect at one or the other side of the reference. Thus, there is no evidence that participants used a serial search strategy. Most importantly, the Position Side of response interaction was significant (Fð1; 23Þ ¼ 10:95; MSE ¼ 13240; P, 0:005). The nature of this interaction was determined by regression of Position (coded according to the absolute letter position in the alphabet) on drt (Fig. 2a), which resulted in the equation drt ¼ 58:05 2 4:97 (Position). Position differed significantly from zero (tð1; 23Þ ¼23:49; SD ¼ 6:98; P, 0:001). Hence, the results provide clear evidence in support of the spatial coding of letter positions Order-irrelevant task Data-trimming removed 0.76% of the data. The average error rate was 2.04%. RTs for the respective letters were 488, 513, 483, 540, 497, 508, 484 and 541 ms. An ANOVA with Position (1: E G, 2: I L, 3: R U or4:w Y), Letter category (consonant or vowel) and Side of response (left or right) as within-subjects variables and Order of task as a between-subjects variable led to main effects of Position (Fð3; 66Þ ¼5:34; MSE ¼ 690; P, 0:005) and of Side of response (Fð1; 22Þ ¼15:04; MSE ¼ 1333; P, 0:001). Position interacted with Letter category (Fð3; 66Þ ¼ 59:81; MSE ¼ 956; P, 0:0001) and, most importantly, with Side of response (Fð3; 66Þ ¼3:55; MSE ¼ 2546; P, 0:01). Thus, even with a radically different response mapping as in the order-relevant task and under conditions where this ordinal information is irrelevant to the task, the spatial component of the ordinal representation manifests itself. Finally, the spatial coding is not due to perseveration of the order-relevant task, because there was no Order of task Position Side of response interaction (F, 1). The regression analyses support the conclusions of the ANOVA (Fig. 2b): drt ¼ 2:37 2 1:12 (Position). Position differed significantly from zero (tð1; 23Þ ¼22:42; SD ¼ 2:26; P, 0:05) which was not due solely to the letter R, because the slope still was reliable when calculated omitting the letter R from the analysis (tð1; 23Þ ¼21:95; SD ¼ 2:38; P, 0:05). It should be noted that the effect size is weaker than in the orderrelevant task (dependent t-test: tð23þ ¼ 22:39; SD ¼ 7:9; P, 0:05). The intercept was also different (dependent t-test: tð23þ ¼ 2:22; SD ¼ 122:80, P, 0:036). The more complex alternating Position to response mapping might be at the basis of these differ-

8 B94 W. Gevers et al. / Cognition 87 (2003) B87 B95 ences. This hypothesis is supported by the fact that equal effect sizes and intercepts were obtained in Experiment 1, where the mapping was the same in both tasks. The regression of Distance on RT did not produce a significant distance effect (Distance-slope ¼ 20:67; tð23þ ¼ 20:76). Hence, there was no implicit comparison involved in solving the task. 4. General discussion In line with the findings of Dehaene et al. (1990, 1993) in the domain of numbers, we obtained clear evidence that the mental representation of ordinal sequences is spatially coded. In Experiment 1 we found that months from the beginning of the year are responded to faster with the left hand than with the right hand, whereas the reverse pattern was obtained for months towards the end of the year. This was true not only in a task in which ordinal position had to be compared to a reference position, but also in a letterdetection task which requires only superficial processing of the month names and for which, consequently, the ordinal position was irrelevant. The latter result shows that the spatial coding is not limited to the explicit use of a reference and that the spatial components of the sequence representation can be activated automatically. Moreover, the orderirrelevant condition does not produce a distance effect, on the basis of which an automatically induced implicit comparison process can be ruled out as the underlying factor of the response-side effect. With letters instead of months, Experiment 2 confirmed and extended the conclusions obtained in Experiment 1: the same spatial response bias was observed. The mental representation of ordinal information is spatially defined and can affect performance automatically. These conclusions are strengthened by the fact that the spatial effects are not attributable to the translation of positions to numerical values, as may have been the case for months, because the RTs were much shorter than the 2000 ms reported by Jou and Aldridge (1999) to be required for this translation process. Moreover, the presence of the response-side effect in the order-irrelevant task, where the S-R mapping alternated between successive items, shows that the spatial coding is more precise than a crude beginning-left and end-right classification. Because a similar spatial coding has been established in the domain of numbers (Dehaene et al., 1993) and because numbers convey ordinal meaning, one could wonder whether non-numerical ordered sequences and numbers share the same processing mechanism or, alternatively, use different mechanisms with similar properties. While the present experiments do not allow these two possibilities to be disentangled, the performance of neurological patients might be informative. In particular, Dehaene and Cohen (1997) describe a double dissociation in two patients. Patient BOO performed normally in tasks involving quantitative aspects of number processing but was impaired in tasks involving ordered sequences like reciting the alphabet, musical notes, and so on. Patient MAR, however, was impaired in quantitative number processing but was intact in tasks (e.g. bisection) involving ordered series. Understanding of both number and ordinal sequence processing may benefit from charting the commonalities and differences established by systematically and directly

9 W. Gevers et al. / Cognition 87 (2003) B87 B95 B95 contrasting number to order. The research presented here was an effort in this respect and shows that not only numerical sequences but also ordinal sequences are associated with space. Hence, the association observed in numbers should not necessarily be attributed to their number-specific properties at integer or real level, as has been postulated before (Dehaene et al., 1993). Acknowledgements This research is supported by IUAP P5/04 and by grant D of the Flemish Fund for Scientific Research. Bert Reynvoet is a post-doc researcher supported by the Flemish Fund for Scientific Research. The authors wish to thank Sam Gilbert for checking the English. References Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33 (2), Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16 (3), Fias, W., Brysbaert, M., Geypens, F., & d Ydewalle, G. (1996). The importance of magnitude information in numerical processing: evidence from the SNARC effect. Mathematical Cognition, 2, Fias, W., Lauwereyns, J., & Lammertyn, J. (2001). Irrelevant digits affect feature-based attention depending on the overlap of neural circuits. Cognitive Brain Research, 12 (3), Gallistel, C. R., & Gelman, R. (2000). Non-verbal numerical cognition: from reals to integers. Trends in Cognitive Sciences, 4 (2), Hinrichs, J. V., Yurko, D. S., & Hu, J. M. (1981). Two-digit number comparison: use of place information. Journal of Experimental Psychology: Human Perception and Performance, 7, Jou, J., & Aldridge, J. W. (1999). Memory representation of alphabetic position and interval information. Journal of Experimental Psychology: Learning, Memory, and Cognition, 25 (3), Leth-Steensen, C., & Marley, A. A. J. (2000). A model of response time effects in symbolic comparison. Psychological Review, 107 (1), Lorch Jr., R. F., & Myers, J. L. (1990). Regression analyses of repeated measures data in cognition research. Journal of Experimental Psychology: Learning, Memory and Cognition, 16, Treisman, A. M., & Gelade, G. (1980). A feature-integration theory of attention. Cognitive Psychology, 12,