Numbers and space Linking Numbers to Space

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Linking Numbers to Space: From the Mental Number Line towards a Hybrid .... modulate spatial attention according to their position in the working memory ...
Running head: Numbers and space

Linking Numbers to Space: From the Mental Number Line towards a Hybrid Account.

Jean-Philippe van Dijck1, Véronique Ginsburg!, Luisa Girelli" and Wim Gevers2

#Ghent University, Belgium 2

Centre de Recherche Neurosciences & Cognition, Université Libre de Bruxelles, Belgium "Dipartimento di Psicologia, Università degli studi di Milano-Bicocca, Italy

Corresponding author: Jean-Philippe van Dijck Faculty of Psychology and Educational Sciences Department of Experimental Psychology Henri Dunantlaan 2 B-9000 Gent Phone: +32 9 264 63 98 E-mail: [email protected]

Abstract Several psychophysical and neuropsychological investigations suggest that the processing of number and spatial information is strongly associated. A popular account argues that this association has its origin in the underlying mental representation of numbers taking the form of a horizontally oriented mental number line, which is isomorphic to the representation of physical lines. Recently however, several alternative explanations have been put forward. We describe those theories and argue that no current account is on itself able to explain the full range of observations. To do this, a hybrid account is proposed which takes into account the underlying representation but emphasizes the processing mechanisms required by the task at hand. KEYWORDS: numbers, space, attention, working memory SNARC, neglect

Linking Numbers to Space: From the Mental Number Line towards a Hybrid Account. Because of its relevance in daily life and its contribution to the expansion of our technological human societies, mathematics and the way in which the human mind deals with such abstract information triggered the curiosity of many cognitive neuroscientists. Numbers form the cornerstone of this intellectual achievement. It is therefore not surprising that much research was devoted to understand how our brain is able to represent and process this type of information. Both introspection and more formal research approaches converge on the idea that the processing of numbers and space are highly related, both at a functional and at an anatomical level. The first scientific articles, illustrating the link between numbers and space, date back to the 19th century. Sir Francis Galton published two papers in which he reported people who experienced vivid spatial images when processing numbers, that he termed “natural lines of thought” (1880a, Galton, 1880b). Later, large-scaled screening studies showed that about 15% of normal adults report such vivid visuo-spatial experiences when processing numbers (Seron et al., 1992). Although these observations support the current conception, the fact that part of the population reports such an explicit experience is, in itself, not sufficient to conclude that common or similar processing mechanisms underlie both of them. To illustrate, a certain part of the population (e.g. 2%, Simner et al., 2006) reports that they always ‘see’ Arabic numbers in a specific colour, a condition called colour-number synaesthesia. Not surprisingly, colour-number synaesthesia is never used as evidence for the existence of a close link between the processing of numbers and the processing of colour. Nevertheless, numerous other investigations using different research techniques in both healthy participants and in neurologically impaired patients have repeatedly demonstrated the close link between the processing of numbers and the processing of space (for overviews see Gevers and Lammertyn, 2005, Fias and Fischer, 2005, Hubbard et al., 2005, de Hevia et al., 2008). Although recently several alternative explanations have been formulated, the most popular account for the interaction between numbers and space argues that this association has its origin in

the underlying mental representation of numbers taking the form of a horizontally oriented mental number line (Dehaene et al., 1993). However, as we tried to illustrate on the basis of Galton’s study, it is an easy pitfall to describe empirical observations in line with current popular theoretical accounts. In the attempt to avoid this pitfall, we start with a descriptive overview of three key behavioural observations – i.e., the SNARC effect, the number interval bisection bias and the asymmetry of the distance effect in neglect – that are currently used to illustrate the association between numbers and space. In doing so, we first describe the empirical observations as much as possible free of imposed theoretical interpretations. In a second step, those theories that provide a unified framework for these observations are described. By using this approach, it will become clear that none of these accounts is able to fully explain the available evidence by itself. To go beyond this impasse, a hybrid account is proposed in which emphasis is put on both the mental representation and the processing mechanisms that operate on it.

Evidence from the SNARC effect The interaction between numbers and space has a strong empirical foundation and has been demonstrated in a variety of experimental designs and contexts. One of the most convincing and robust demonstrations for this intrinsic link is the Spatial Numerical Associations of Response Codes (SNARC) effect. This effect was originally reported by Dehaene et al. (1993, 1990). It was observed that participants’ left handed responses were faster when judging (relatively) small numbers while right-handed responses were faster when judging (relatively) large numbers. === INSERT FIGURE 1 HERE===

Automatic nature of the SNARC effect Following this seminal report, this phenomenon was of inspiration for several researchers who tested the effect with different tasks and in several experimental settings. The SNARC effect

was observed in tasks that do not require the explicit processing of numerical magnitude, like parity judgment (e.g. is the presented number odd or even; Dehaene et al., 1993), phoneme monitoring (e.g. does the name of the presented number contain the phoneme /e/; Fias et al., 1996), or even when numerical stimuli were completely task irrelevant (e.g. is a superimposed triangle pointing up or downward?; Fias et al., 2001, Lammertyn et al., 2002). A related observation was made in tasks where participants were required to detect a lateralized target preceded by a centrally displayed number. Even though this number was completely task irrelevant, left-sided targets were detected faster when preceded by a small number whereas right-sided targets were detected faster when preceded by a large number (Fischer et al., 2003, see Ruiz Fernández et al., 2011 for similar observations in free vision). A similar congruency was observed when participants had to judge the parity status (e.g. say ‘Ti’ if odd, ‘To’ if even) or the magnitude of a centrally presented number immediately followed (but not preceded) by a lateralized prime (Kramer et al., 2011, Stoianov et al., 2008).

Flexibility of the SNARC effect Critical to the present review, is the observation that the relative, rather than the absolute magnitude of a number drives the SNARC effect. That is, for example, the digits 4 and 5 elicited faster left than right responses when the digits ranged from 4 to 9, but elicited faster right than left responses when digits ranged from 0 to 5 (Dehaene et al., 1993, Fias et al., 1996). Recently, Ben Nathan et al. (2009, see also Ren et al., 2011) refined the range effect in a magnitude comparison task by dynamically changing the standard reference from trial to trial. They found that the SNARC effect was driven by the relative instead of the absolute magnitude as, for example, “7” was associated with “left” when the referent was “8” but with “right” when the referent was “6”. Bächtold et al. (1998, see also Vuilleumier et al., 2004) extended this idea to mental imagery by showing that inducing participants to imagine the numbers on a clock face, is sufficient to observe a reversed spatial

congruency effect (i.e. the smaller hours were associated with right, and the larger ones with left). Moreover, such a reversed congruency effect is observed even when no explicit reference to mental imagery occurs. For instance, Shaki and Fischer (2008) reported that Russian-Hebrew bilinguals showed a normal SNARC effect after reading a Russian text (which is read from left to right), but the same subjects showed a reversed SNARC effect after reading a Hebrew text (which is read from right to left). A further index of the flexible nature of the SNARC effect is provided by Lindemann and colleagues (2008). In this study, participants were asked to memorize the digits 3,4 and 5 in ascending, descending or random order while performing a SNARC task on the numbers 1,2, 8 and 9. The SNARC effect was modulated by means of the direction in which the other numbers had to be memorised. That is, a regular SNARC effect was observed only after memorizing ascending or random number sequences but disappeared after processing descending sequences (Lindemann et al., 2008 for further evidence that brief exposure to non-canonical presentation may alter the SNARC effect, see Fischer et al., 2010).

Nature of the spatial code driving the SNARC effect Another line of research deals with the nature of the spatial codes underlying the SNARC effect, by means of methodological manipulations that controlled for on-line number-space associations. Departing from the standard bimanual left/right response paradigm, Santens and Gevers (2008) required participants to perform a magnitude comparison task adopting unimanual responses in the close/far dimension (response locations were placed physically close or far from a starting position). They observed that ‘small’ numbers were associated with ‘close’ responses whereas ‘large’ numbers were associated with ‘far’ responses, regardless of the lateralized movement direction. Following this observation, Gevers et al. (2010) further investigated the possibility that the SNARC effect mainly reflects an association between number magnitude and spatial concepts associated with the responses such as ‘close’/‘far’ and ‘left’/‘right’. In their design, conceptual labels and physical

response locations were directly pitted against one another. The verbal labels ‘left’ and ‘right’ were presented either in their canonical position (left right) or not (right left) on the screen. In a series of four experiments, it was shown that the verbal-spatial coding of magnitude was dominant over the visuo-spatial one: participants associated numbers more strongly with the verbal labels ‘left’ and ‘right’ regardless of their physical position on the screen.

Ordinal information and the SNARC effect Numbers convey not only magnitude meaning but also ordinal meaning. Several studies showed that ordinal information is also associated with lateralized responses. Gevers et al. (2003) firstly reported that the SNARC effect holds when either letters or names of the months are used as stimuli. Later research demonstrated that lateralized responses to newly learned ordinal sequences could also lead to a SNARC effect. In two separate studies, participants either had to extensively learn new arbitrary sequences of words (Previtali et al., 2010) or of arbitrary visual elements (Van Opstal et al., 2009). Regardless of the type of stimuli, elements early in the sequence were responded to faster with the left hand side whereas elements later in the sequence were responded to faster with the right hand side, even when order was irrelevant to the task (Previtali et al., 2010). Critically, ordinal meaning does not need to be internalized on a long-term basis to be associated with a lateralized response. Recently, van Dijck and Fias (2011) directly investigated the role of working memory in associating ordinal position to lateralized response. In their study, participants were asked to memorize a sequence of (centrally presented) numbers. Subsequently, using a go-no go procedure, participants had to make a parity judgment, but only if the presented number belonged to the memorized sequence. Regardless of their magnitude (i.e. no SNARC effect was observed), numbers presented at the beginning of the memorized sequence were responded to faster with the left hand side whereas items presented at the end of the sequence were responded to faster with the right hand side. The same observation was made when words (e.g. fruits and vegetables) instead of numbers were used,

again strongly suggesting that it is the ordinal position in the sequence and not the cardinal meaning of the stimuli that interacts with the response side. More recent findings demonstrate that the association between the ordinal position in the sequence and space can be observed without lateralized responses. Similar to the design introduced by Fischer et al. (2003), van Dijck et al. (under review) asked participants to memorize a sequence of centrally presented digits in the order of presentation, and found that when centrally presented as a cue, these memorized numbers can modulate spatial attention according to their position in the working memory sequence, irrespective of their magnitude. That is, after retrieving these numbers from memory, subjects were faster to detect left sided dots (vocally or with a central key press) when the number was from the beginning of the sequence, and faster to detect right sided dots when the number was from the end of the sequence.

Evidence from the number interval bisection task Besides investigating the behaviour of neurologically intact participants, more insights in normal cognitive functioning can be obtained by studying how selective brain damage disturbs these functions. In this respect, valuable signatures of the association between numbers and space are provided by neuropsychological studies on neglect patients. Patients suffering from neglect after (mainly) right brain damage struggle with difficulties to report, respond and orient to stimuli in the left side of space (e.g. Driver and Mattingley, 1998). Neglect not only manifests itself in perception. Since the seminal study of Bisiach and Luzzatti (1978), it is known that neglect can also affect the contralesional side of mental representations, like familiar places or objects (Grossi et al., 1989). This observation led to the hypothesis that if numbers are mapped in the representational space along a left-to-right orientation, the left sided attentional deficit specific to neglect should also affect the processing of small numbers (Zorzi et al., 2002). To test this idea, neglect patients with intact arithmetic skills were recruited and orally presented with two numbers that identified the extremes of

a numerical interval. The task required them to indicate, without calculation and with no explicit reference to spatial imagery, the midpoint of the interval. Neglect patients systematically overestimated the objective midpoint of the intervals (e.g. they indicated 7 as being the midpoint of the interval 1-9) as if they ignored the smallest numbers, and based their decision on the item-range entering their focus of attention. Interestingly, the bias increased as a function of the number interval length (with a cross over-effect for the smallest intervals), similarly to what observed in neglect patients’ bisection of physical lines (e.g. Marshall and Halligan, 1989). Moreover, Rossetti et al. (2004) showed that similarly to what observed in visuo-spatial tasks (Rossetti et al., 1998), their performance on the number bisection task benefit from prism adaptation. Also optokinetic stimulation, known to ameliorate perceptual neglect (for a review see Kerkhoff, 2003), exerts a positive influence on number interval bisection (Priftis et al., 2012). A bisection bias towards the smaller numbers was observed in a left brain damaged patient suffering from right-sided neglect (Pia et al., 2009). Similarly, schizophrenic patients, having difficulties in orienting attention to the right side of space, showed a number bisection bias towards the smaller numbers (Cavezian et al., 2007). Moreover, accordingly to the slight leftward bias generally shown by healthy participants when bisecting physical lines, (i.e. pseudo- neglect; for a review see Jewell and McCourt, 2000) a slight bias towards the smaller numbers was also observed when they bisected number intervals (Longo and Lourenco, 2010, Longo and Lourenco, 2007). However, defective attentional orienting towards the left side of physical space in neglect is not always associated with a number interval bisection bias towards the larger numbers. A double dissociation was reported between physical line bisection and interval bisection tasks (Doricchi et al., 2005). Specifically, neglect patients showing a strong bias in number bisection did not necessarily demonstrate a similar bias in line bisection, and vice versa. This dissociation was further corroborated by anatomical data, since the number bisection bias was found to be associated with damage to prefrontal areas, known to be involved in working memory, whereas the line bisection bias

was associated with more posterior damage. Accordingly, the size of the number interval bisection bias was associated with a reduction of working memory capacity, both in the verbal and in the spatial modality (Doricchi et al., 2009). Similarly in a single-case study, van Dijck et al. (2011) reported a patient showing right sided extra personal and representational neglect, as marked by a left bias in line bisection, but a number bisection bias that was oriented towards the larger numbers. Evaluation of the patient’s working memory resources confirmed a non-spatial origin of the number bisection bias revealing that the bias was associated with a reduced working memory capacity mainly affecting the first elements within verbal sequences (van Dijck et al., 2011). The observed dissociation between number interval and line bisection was confirmed in several subsequent studies entailing neglect (e.g. van Dijck et al., 2012, Rossetti et al., 2004), schizophrenic (Tian et al., 2011) and developmental dyscalculic (Ashkenazi and Henik, 2010) populations (see Rossetti et al., 2011 for an overview). Importantly, number interval bisection is a task that taps on a mental representation while physical line bisection does not. Since it has been shown that representational and extra personal neglect can doubly dissociate (e.g.Guariglia et al., 1993), the above reports can be an instantiation of this dissociation. This alternative interpretation, however, does not hold for the patient reported by van Dijck et al. (2011) who showed right-sided representational neglect together with a bias towards the larger numbers in a number interval bisection task. Given the limited generalizability of single case reports, larger group studies are needed on this debate (see for instance Aiello et al., 2012). === INSERT FIGURE 2 HERE===

Evidence from the asymmetrical distance effect Additional studies on neglect patients have shown that the impact of the attentional deficit on number processing is not limited to bisection tasks. Vuilleumier et al. (2004) asked neglect patients to perform a series of magnitude comparison tasks in which numbers from 1 to 9 were to be compared

against different referents. For this purpose, numbers were visually presented in random order and participants were required to compare them against 5 or 7 in different blocks. Typically in such comparison tasks, reaction times to the targets increase as an inverse function of the distance between this number and the referent. This observation, also known as the distance effect (e.g. Moyer and Landauer, 1967), is symmetrical in healthy subjects, meaning that the increase in reaction times is similar for numbers either smaller or larger than the referent. Vuilleumier et al. (2004) showed that this is not the case for neglect patients as their distance effect was much more marked for small (associated to the left) than for large (associated to the right) numbers. Importantly this distance asymmetry in neglect patients was similar for both referents, suggesting that the asymmetry is driven by the relative, and not the absolute size of the numbers. === INSERT FIGURE 3 HERE=== Evidence from other tasks Beside what so far reviewed, there is abundant additional evidence in favor of the idea that numbers and space are closely related. For example, another demonstration comes from a study using the random digit generation task (Loetscher et al., 2008). In this study, subjects were asked to name, as randomly as possible, numbers between 1 and 30 in a sequence. Data were collected in two runs: a baseline condition in which the subjects had to look straight ahead and a condition in which left or right- sided head turns had to be made. When looking straight ahead, the participants generated a higher amount of small numbers (i.e. numbers smaller than 16) compared to larger ones (i.e. numbers larger than 15). This small number bias, however, changed when subjects moved their head during the generation process, i.e., small numbers were more frequently produced when the head was rotated leftwards while the small number bias decreased when rotating the head towards the right. Interestingly, when looking straight ahead, the direction in which the eye moves just before a number is said, betrays whether this will be a small or a large one (Loetscher et al., 2010).

Further evidence comes from a variant of the bisection task. When neurologically healthy participants bisect digit strings, their performance is biased towards the left when the line is composed of small numbers (e.g. 1 or 2) and towards the right when the line is composed of large numbers (e.g. 8 or 9; Fischer, 2001). Moreover, when the line to be bisected is flanked by irrelevant digits, participants systematically shift the subjective center toward the larger digit irrespective of its position (De Hevia et al., 2006, Ranzini and Girelli, 2012). More recently, a similar association has been observed when writing digits by hand. Analyses of the spatial properties of handwritten digits revealed that they were dislocated as a function of their magnitude with small numbers being written more leftwards relative to large numbers (Perrone et al., 2010). Both findings indicate that in healthy participants, the automatic spatial coding of numbers not only interacts with spatially defined response buttons (like in the SNARC tasks described above), but also with more spontaneous hand movements. Finally, beyond the mental representation of numbers itself, there is also evidence that spatial coding contributes to mental arithmetic. It has been shown that when participants are asked to point to number locations (1–9) on a visually presented number line after computing them from addition or subtraction problems, pointing was biased leftward after subtracting and rightward after adding (Pinhas and Fischer, 2008, McCrink et al., 2007). Moreover, by using multivariate classification techniques of brain imaging data, these spatial biases were associated with the brain circuitry involved in making left or right eye movements. This suggests that performing mental arithmetic operations, like the mental representation of numbers, co-opts neural circuitry and cognitive mechanisms associated with spatial coding (Knops et al., 2009).

Theories explaining the associations between numbers and space

The Mental Number Line A popular account for the empirical findings described above is that numbers are represented on a horizontal mental number line (MNL) with numbers oriented in ascending order from left-toright or from right-to-left (e.g. Dehaene et al., 1993). The orientation of the number line is thought to reflect a long-term association built up gradually across development according to the culturally dominant reading/writing direction: People educated in Western societies would have a long-term representation of numbers going from small-left to large-right whereas a reversed long-term association would be observed in societies reading from right-to-left. Processing of a number results in the automatic activation of the corresponding spatial location along the MNL. Activating such a location or shifting from one location to another on the MNL is supposed to involve spatial attentional processes (e.g. Fischer et al., 2003). Critically it is believed that this MNL is based on the same coordinate system used to represent physical space (Umilta et al., 2009).

Polarity/ conceptual coding This alternative view suggests that number-space associations are an instance of the many associations that can exist between categorical conceptual dimensions. In particular, Proctor et al. (2006) developed a polarity coding account in which the relation between numbers and space derives from a systematic association between the verbal concepts that are linked to stimulus and response properties (e.g. the size of the number being small or large, and the side of the response being left or right). Based on the assumption that such categorical conceptual dimensions have a specific polarity (e.g. “left” being negative and “right” being positive; and “small” being negative and “large” being positive), it is suggested that the congruency between polarities drives the association between numbers and space. Importantly, such conceptual coding is not limited to numbers as it applies to

other kinds of binary categories as well (e.g. good/ bad, odd/ even, yin/ yang, early/ late are also coded in terms of positive and negative polarities). In that way, this account can also explain the SNARC-like effects observed with other ordinal information, like days of the week or letters from the alphabet. According to a similar view, the conceptual coding account, numbers are considered to be first conceptually categorized as either ‘small’ or ‘large’ and then linked in an associative network to other dichotomous (output) dimensions, such as left-right, (Gevers et al., 2010, Gevers et al., 2006, Santens and Gevers, 2008). Unlike the MNL it is assumed that this network is verbal-spatial rather than visuo-spatial in nature.

Working memory In a final account it is argued that the ordinal position of information in working memory is spatially coded (Fias et al., 2011, van Dijck and Fias, 2011). The processing of numerical magnitude would use the same underlying architecture, resulting in an association between numbers and space. In itself, this working memory account does not specify whether the coding of ordinal position is visuo-spatial or verbal-spatial in nature. In contrast to the MNL account, it supposes that the association between numbers and space is not long-term but built up during task execution, probably as a strategy to facilitate task execution (for a similar strategic interpretation of the SNARC effect see Fischer, 2006, Fischer et al., 2010). The link between serial order coding in working memory and spatial attention (Van Dijck et al., under review) would associate the working memory account to theories which assume that working memory and attention are closely related and partly overlapping constructs (e.g. Cowan, 1995, Awh and Jonides, 2001).

Strengths and weaknesses of the described accounts

The Mental Number Line Because of the visuo-spatial nature of the representation, the MNL can well account for the observation of the SNARC effect: it reflects a congruency between the activated location on the MNL and the location of the response buttons. In addition, the close link to spatial attention explains why numbers can act as directional cues, so that the mere perception of them gives rise to lateralized shifts of spatial attention (e.g. Fischer et al., 2003) or, the other way around, that attentional modulations can influence the processing of numbers (e.g. Stoianov et al., 2008). Assuming such spatial attention as processing mechanism, the MNL can easily account for the number bisection bias and the distance asymmetry observed in neglect. The deficit in orienting attention towards the left affects the access to the left side of the MNL and thereby the processing of small numbers. Despite this explanatory strength, the MNL account cannot explain several of the observations described above. For example, it remains unclear why the association between numerical magnitude and space (as reflected in the SNARC effect) is mediated by verbal-conceptual processes (e.g. Gevers et al., 2010). Moreover, without making additional assumptions, the MNL account cannot explain why the severity of neglect and the size of the number bisection bias do not consistently correlate (e.g. Rossetti et al., 2011, van Dijck et al., 2012, Doricchi et al., 2009). Because the MNL account assumes long-term associations between numbers and space, it may hardly justify the flexible nature of the SNARC effect. To this aim, it requires the extra assumption that short-term associations can easily and rapidly overrule the existing long-term ones.

Polarity/ Conceptual coding The conceptual coding account, on the other hand, provides a parsimonious explanation for the conceptual nature of the SNARC effect, its automatic emergence, its flexibility and for the

observations of spatial coding for ordinal information. After all, the categorical concepts (e.g. small/large; early/late; close/far) or the polarity codes (negative/positive) associated to a specific number or ordinal position are determined spontaneously by contextual factors. For instance, in a magnitude related context, the number 5 can be either categorized as large (or positive polarity), when it occurs in the range from 0 to 5, or as small (negative polarity) when the range goes from 4 to 9. Similarly, in an ordinal context, the letter E can be categorized as late (positive polarity) within the A to E range but as early (negative polarity) when the range goes from E to I. However, it is unclear how this theory can be applied to the interval bisection and distance asymmetry effects or to the observations that number-space interactions are observed in situations where no lateralized responses were required (e.g. Fischer et al., 2003).

Working memory Finally, by assuming that ordinal information in working memory is spatially coded and that during task execution, numbers are spontaneously and strategically stored in their canonical order in working memory, the working memory account can well explain the SNARC effect. Along the same line, a working memory deficit for the initial or final elements of ordered sequences, including number intervals, would result in a mental bisection task, in a directional bias towards the end or the beginning of the list. Due to the involvement of working memory, this account can also explain the flexibility of the effects and why spatial coding is not limited to numbers. It is unclear, however, how working memory may account for the asymmetry in the distance effect observed in neglect patients. After all, in magnitude comparison (in which the asymmetry of the distance effect is observed) the need to store information in working memory is only minimal. In addition, the working memory account can only explain the beneficial effect of prism adaptation and optokinetic stimulation in neglect patients’ interval bisection (Rossetti et al., 2004, Priftis et al., 2012), assuming that this improvement would extend to any serial order in working memory.

Towards a hybrid account for spatial numerical associations In the previous section we described several theories that can account for most of the observed number-space interactions. Apparently none of them is able to explain the full range of observations. This suggests that the relation between numbers and space is not due to one single underlying processing mechanism or representation. This idea is well supported by a dual-task study where participants were asked to perform a parity judgement and magnitude comparison task while memorizing spatial and verbal information (van Dijck et al., 2009). A double dissociation was observed between the type of task and the type of working memory load: The SNARC effect in parity judgment disappeared when verbal working memory was taxed, and no SNARC effect in magnitude comparison was found when memorizing spatial information. These findings provide evidence against the view that all behavioural signatures of the association between numbers and space have their origin in a single underlying mental representation. Besides, they show that the nature of these associations and the type of working memory resources involved, depend on the task at hand. Elaborating upon this idea, van Dijck et al. (2012) investigated the functional relationship between the several tasks and effects typically used to illustrate the link between numbers and space. Therefore, a group of right-brain damaged patients (with and without neglect) and healthy participants performed physical line bisection, number interval bisection, parity judgment and magnitude comparison. After replicating the previously reported ANOVA patterns and the dissociations at the level of the individual subjects, the data were submitted to a principal component analyses (PCA) to unravel the “internal structure of number-space”. Further confirming the heterogeneous nature of the association between numbers and space, the results showed that a threecomponent solution provided the best fit of the data pattern (see Figure 4). The first component was loaded by the magnitude comparison SNARC effect and number interval bisection, the second component by physical line bisection and the distance asymmetry and the third component by the parity judgment SNARC effect and number interval bisection and the distance asymmetry. These

results clearly refute a single mechanism account by showing that multiple factors are needed to capture the correlations between the tasks. Furthermore, the observed component structure fitted the dissociations previously described in the literature. This led to the suggestion that the three components reflect the involvement of spatial attention, verbal and spatial working memory and that the different effects draw differently upon these processing mechanisms. In an attempt to explain the component loadings, it was proposed that the parity judgment SNARC effect emerges by using verbal working memory while the magnitude comparison SNARC effect emerges by using visuospatial working memory. The number interval bisection bias and distance asymmetry would arise from the interplay of different processes. Both effects draw upon verbal working memory (probably for the recollection of the numbers defining the interval and the encoding the ordinal relations with regard the comparison referent, respectively) but whereas the distance asymmetry is further related to spatial attention, the number interval bisection recruits spatial working memory (probably to construct a spatial representation in mental imagery in searching for the numerical midpoint). === INSERT FIGURE 4 HERE=== In the PCA analyses, no direct measures of working memory capacity were included. This makes it difficult to determine exactly the functional details of these underlying mechanisms. Therefore we acknowledge that the current proposal should be consider a starting point for an alternative framework since a fully elaborated functional elaboration is still to be reached. The observations from the dual-task paradigm and the group-study, however, clearly indicate that the relation between numbers and space is more complex than so far proposed by each theory in isolation. Specifically, they indicate that the association between numbers and space results from the interplay of different processing mechanisms and that the degree of involvement of those mechanisms depends on the task at hand (Chen and Verguts, 2010; see also the chapter by Roggeman et al., this volume).

Conclusion In the present review we directly opposed the existing theories to explain the association between numbers and space. Although we acknowledge that not everyone will fully agree with our interpretation of the described theoretical account, we aimed at illustrating their individual strengths and weaknesses. In doing so, we concluded that none of the described theoretical accounts by itself can explain the full range of empirical findings and we proposed a hybrid account to overcome this impasse. Here it is argued that instead of one single underlying representation associated with attentional processes; at least two additional independent components (spatial and verbal working memory) and their interactions are characterizing the internal structure of the “number-space”. The cases of the SNARC effect, number bisection bias and the distance asymmetry have been worked out. How other tasks and effects providing additional evidence for the numbers-space association are situated with respect to the identified components is a matter of further investigation. We believe that future research should consider the heterogenic nature of these associations to come to an exhaustive understanding of the relation between numbers and space. In doing so, it is not only important to consider the content and format of the mental representations, but also the full range of (cognitive) processes that operate upon them and the way in which they interact.

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FIGURE CAPTIONS Figure 1. Schematic depiction of the SNARC effect. A. Participants are required to judge whether a number is odd or even, or whether it is smaller or larger than a reference number. In both conditions, left hand responses are faster for smaller numbers compared to larger numbers, whereas the opposite is observed for right hand responses. B. The results of a magnitude comparison task. The reaction time difference (dRT) between right hand and left hand responses. Values greater than zero indicate that the response is given faster with the left hand, whereas values under zero indicate that the response is given faster with the right. Adapted from Journal of Experimental Psychology: Human Perception and Performance, Vol. 36, no. 1, Gevers, W., Verguts, T., Reynvoet, B., Caessens, B., Fias, W., Pages 32-44, Copyright (2006), with permission from APA.

Figure 2. Schematic depiction of the number interval bisection task. A. Several pairs of numbers are orally presented and subjects are asked to verbally indicate the numerical midpoint between two different numbers. B. This figure shows the percentage of deviation from the numerical midpoint. The zero corresponds to a correct response whereas positive values indicate an overestimation of the midpoint and negative values indicate an underestimation of this midpoint. Reprinted from Frontiers in Human Neuroscience, Vol. 5:182 , Van Dijck, J.P. Gevers, W., Lafosse, C., Fias, W., (2012).

Figure 3. Depiction of the magnitude comparison distance effect in neglect patients. Participants have to judge whether numbers are smaller or larger than the reference number (5 or 7). Neglect patients show increasing RTs for those numbers immediately preceding the referent

(e.g., in the condition with number 5 as reference, neglect patients have particularly slower RTs for the number « 4 »). Reprinted from Cortex, Vol. 40, Vuilleumier, P., Ortigue S., Brugger, P., Pages 399-410, Copyright (2004), with permission from Elsevier.

Figure 4. Results of the Principal Component Analyses (PCA) as described by van Dijck et al. (2012). To unravel the « internal structure » of the association between numbers and space, a PCA was used. Instead of 1 single underlying component (spatial attention), this analysis revealed that at least 2 additional factors (verbal and spatial working memory) are involved.

FIGURES Figure 1


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Figure 2

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