Calibrating the mental number line

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.     

The SNARC effect does not imply a mental number line

In this study, we directly contrast two approaches that have been proposed to explain the SNARC effect. The traditional direct mapping account suggests that a direct association exists between the position of a number on the mental number line and the location of the response. On the other hand, accounts are considered that propose an intermediate step in which numbers are categorized as either small or large between the number magnitude and the response representations. In a magnitude comparison task, we departed from the usual bimanual left/right response dimension and instead introduced the unimanual close/far dimension. A spatial-numerical association was observed: small numbers were associated with a close response, while large numbers were associated with a far response, regardless of the movement direction (left/right). We discuss why these results cannot be explained by assuming a direct mapping from the representation of numbers on a mental number line to response locations and discuss how the results can be explained by the alternative accounts.     

Concepts are not represented by conscious imagery

According to theories of grounded cognition, conceptual representation and perception share processing mechanisms. We investigated whether this overlap is due to conscious perceptual imagery. Participants filled out questionnaires to assess the vividness of their imagery (Questionnaire on Mental Imagery) and the extent to which their imagery was object oriented and spatially oriented (Object-Spatial Imagery Questionnaire), and they performed a mental rotation task. One week later, they performed a verbal property verification task. In this task, involvement of modality-specific systems is evidenced by the modality-switch effect, the finding that performance on a target trial (e.g., apple—green) is better after a same-modality trial (e.g., diamond—sparkle) than after a different-modality trial (e.g., airplane—noisy). Results showed a modality-switch effect, but there was no systematic relation between imagery scores and modality switch. We conclude that conscious mental imagery is not fundamental to conceptual representation.     

Visuospatial priming of the mental number line

It has been argued that numbers are spatially organized along a "mental number line" that facilitates left-hand responses to small numbers, and right-hand responses to large numbers. We hypothesized that whenever the representations of visual and numerical space are concurrently activated, interactions can occur between them, before response selection. A spatial prime is processed faster than a numerical target, and consistent with our hypothesis, we found that such a spatial prime affects non-spatial, verbal responses more when the prime follows a numerical target (backward priming) then when it precedes it (forward priming). This finding emerged both in a number-comparison and a parity judgment task, and cannot be ascribed to a "Spatial-Numerical Association of Response Codes" (SNARC). Contrary to some earlier claims, we therefore conclude that visuospatial-numerical interactions do occur, even before response selection.     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

Cleaner fish <Emphasis Type="Italic">Labroides dimidiatus</Emphasis> discriminate numbers but fail a mental number line test

Several species of primates, including humans, possess a spontaneous spatial mental arrangement (i.e. mental number line MNL) of increasing numbers or continuous quantities from left to right. This cognitive process has recently been documented in domestic chicken in a spatial–numerical task, opening the possibility that MNL is a cognitive capacity that has been conserved across vertebrate taxa. In this scenario, fish might possess the MNL as well. Here we investigated whether cleaner fish Labroides dimidiatus show evidence for MNL in two experiments. In Experiment I, we tested fish’s abilities in number discrimination, presenting simultaneously either small (2 vs 5) or large (5 vs 8) continuous quantities where one quantity was systematically rewarded. Experiment II used a protocol of an MNL task similar to the study on chickens. We trained cleaners with a target number (i.e. 5 elements), then we presented them with an identical pair of panels depicting either 2 elements or 8 elements, and we recorded their spontaneous choice for the left or right panel on each presentation. Cleaner fish showed high abilities in discriminating small and large numbers in Experiment I. Importantly, cleaners achieved this discrimination using numerical cues instead of non-numerical cues such as the cumulative surface area, density, and overall space. In contrast, cleaners did not allocate continuous quantities to space in Experiment II. Our findings suggest that cleaner fish possess numbering skills but they do not have an MNL. While similar studies on animals from various clades are needed to trace the evolution of MNL within vertebrates, our results suggest that this cognitive process might not be a capacity conserved across all vertebrate taxa.     

A mental number line in human newborns

Humans represent numbers on a mental number line with smaller numbers on the left and larger numbers on the right side. A left‐to‐right oriented spatial–numerical association, (SNA), has been demonstrated in animals and infants. However, the possibility that SNA is learnt by early exposure to caregivers’ directional biases is still open. We conducted two experiments: in Experiment 1, we tested whether SNA is present at birth and in Experiment 2, we studied whether it depends on the relative rather than the absolute magnitude of numerousness. Fifty‐five‐hour‐old newborns, once habituated to a number (12), spontaneously associated a smaller number (4) with the left and a larger number (36) with the right side (Experiment 1). SNA in neonates is not absolute but relative. The same number (12) was associated with the left side rather than the right side whenever the previously experienced number was larger (36) rather than smaller (4) (Experiment 2). Control on continuous physical variables showed that the effect is specific of discrete magnitudes. These results constitute strong evidence that in our species SNA originates from pre‐linguistic and biological precursors in the brain.     

A synesthetic walk on the mental number line: the size effect

Are small and large numbers represented similarly or differently on the mental number line? The size effect was used to argue that numbers are represented differently. However, recently it has been argued that the size effect is due to the comparison task and is not derived from the mental number line per se. Namely, it is due to the way that the mental number line is mapped onto the task-relevant output component. Here synesthesia was used to disentangle these two alternatives. In two naming experiments a digit-color synesthete showed that the congruity effect was modulated by number size. These results support the existence of a mental number line with a vaguer numerical representation as numbers increase in size. In addition, the results show that in digit-color synesthesia, colors can evoke numerical representation automatically.     

The mental time line: An analogue of the mental number line in the mapping of life events

A crucial aspect of the human mind is the ability to project the self along the time line to past and future. It has been argued that such self-projection is essential to re-experience past experiences and predict future events. In-depth analysis of a novel paradigm investigating mental time shows that the speed of this “self-projection” in time depends logarithmically on the temporal-distance between an imagined “location” on the time line that participants were asked to imagine and the location of another imagined event from the time line. This logarithmic pattern suggests that events in human cognition are spatially mapped along an imagery mental time line. We argue that the present time-line data are comparable to the spatial mapping of numbers along the mental number line and that such spatial maps are a fundamental basis for cognition.     

Bisecting the mental number line in near and far space

Much evidence suggests that common posterior parietal mechanisms underlie the orientation of attention in physical space and along the mental number line. For example, the small leftward bias (pseudoneglect) found in paper-and-pencil line bisection is also found when participants “bisect” number pairs, estimating (without calculating) the number midway between two others. For bisection of physical lines, pseudoneglect has been found to shift rightward as lines are moved from near space (immediately surrounding the body) to far space. We investigated whether the presentation of stimuli in near or far space also modulated spatial attention for the mental number line. Participants bisected physical lines or number pairs presented at four distances (60, 120, 180, 240 cm). Clear rightward shifts in bias were observed for both tasks. Furthermore, the rate at which this shift occurred in the two tasks, as measured by least-squares regression slopes, was significantly correlated across participants, suggesting that the transition from near to far distances induced a common modulation of lateral attention in physical and numerical space. These results demonstrate a tight coupling between number and physical space, and show that even such prototypically abstract concepts as number are modulated by our on-line interactions with the world.     

The revelation of the negative side of a mental number line depends on list context: Evidence from a sign priming paradigm

Two parity judgement experiments examined how the activation of spatial-numerical associations of a single, centrally presented digit, reflected by the Spatial-Numerical Association Response Codes (SNARC) effect, is modulated by a preceding + (plus) or ? (minus) prime. The centrally presented prime prior to a digit presentation presumably triggers its positive or negative attributes. When the plus- and minus-primed trials were blocked, the left-small/right-large SNARC effects occurred regardless of prime type. When the plus- and minus-primed trials were randomly intermixed, this left-small/right-large SNARC effect occurred for plus-primed digits, but was reversed for minus-primed digits. The implications of this finding for context-dependent SNARC effects are discussed.     

The mental number line in depth revealed by vection

To explore how numbers are represented in depth in our mental space, we asked participants to sequentially speak random numbers while they observed forward/backward vection. We found that participants tended to generate larger numbers when they perceived backward self-motion. The results suggest that numerical magnitudes were topographically mapped onto our mental space from front to rear in an ascending order.     

Exploring the mental number line: evidence from a dual-task paradigm

In a parity-judgment task smaller numbers are responded to faster with the left-hand key and vice versa for larger numbers (SNARC effect; Dehaene et al., in Journal of Experimental Psychology: General, 122, 371-396, 1993). We used the psychological refractory period paradigm involving a parity-judgment task and tone-discrimination task to address the question at which stage this effect arises. When the parity-judgment task is performed second, then we found equal SNARC effects for the short and the long SOA. According to the central bottleneck model, this indicates that the effect arises during the response-selection or execution stage. In Experiment 2 the parity-judgment task was performed first. The pattern of results indicates that the SNARC effect originates during the perceptual encoding or response-selection. Together, our results suggest that the SNARC effect originates while the response is selected.     

A colorful walk on the mental number line: Striving for the right direction

Verguts and Van Opstal [Verguts, T., & Van Opstal, F. (2008). A colorful walk, but is it on the mental number line? Reply to Cohen Kadosh, Tzelgov, and Henik, Cognition, 106, 558-563] cleverly explained the results of Cohen Kadosh, Tzelgov, and Henik [Cohen Kadosh, R., Tzelgov, J., & Henik, A. (2008). A synesthetic walk on the mental number line: The size effect, Cognition, 106, 548-557] as a result of different association strength between the size of a number and its color in synesthesia. Here we present three challenges to their alternative explanation, and support our original suggestion.     

On the mental representation of negative numbers: Context-dependent SNARC effects with comparative judgments

In one condition, positive and negative number pairs were compared in separate blocks of trials. In another condition, the positive and the negative number pairs were intermixed. In the intermixed condition, comparisons involving negative numbers were faster with the left hand than with the right, and comparisons were faster with the right hand than with the left hand with the positive numbers; that is, a spatial numerical association of response codes (SNARC) effect was obtained, in which the mental number line was extended leftward with the negative numbers. On the other hand, in the blocked condition, a reverse SNARC effect was obtained with the negative numbers; that is, negative number pairs have the same underlying spatial representation as the positive numbers in this context. Nongraded semantic congruity effects, obtained in both the blocked and the intermixed conditions, are consistent with the idea that magnitude information is extracted prior to the generation of discrete semantic codes.     

正负数混合呈现对负数SNARC效应的影响

采用数字大小判断任务,探讨正负数混合呈现对负数SNARC效应的影响。结果发现,负数单独呈现条件下,负数出现反转的SNARC效应;负数和无加号正数混合呈现,且只对负数作反应条件下,负数有反转SNARC效应;负数和有加号正数混合呈现,且只对负数作反应条件下,负数出现反转SNARC效应;负数和无加号正数混合呈现,并对正负数分别作反应的条件下,负数有反转SNARC效应出现,而正数出现SNARC效应。说明负数空间表征受其绝对值大小的影响,绝对值较小的负数(-1、-2)表征在心理数字线的左侧,绝对值较大的负数(-8、-9)表征在数字线的右侧,且不能延伸至心理数字线左侧。