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.     

SNARC for numerosities is modulated by comparative instruction (and resembles some non-numerical effects)

The spatial–numerical association of response codes (SNARC) effect is observed for both numerical (Arabic digits) and non-numerical stimuli (size, duration, height). However, in a context of comparative judgment, Arabic numbers are mapped onto space differently from sizes and heights: SNARC for Arabic digits is formed consistently in a certain cultural reading direction, whereas SNARC for sizes and heights is additionally modulated by comparative instruction (it reverses when participants choose larger magnitudes). In the present study, we test whether the spatial characteristic of magnitude processing revealed in a context of comparison is determined by a presence or lack of numerical content of the processed information, or it depends on specific directional experience (e.g., left-to-right ordering) associated with the processed magnitude format. We examine the SNARC effect with the pairwise comparison design, by using non-symbolic numerical stimuli (objects’ collections), for which the left-to-right spatial structure is not as exceedingly overlearned as for Arabic numbers. We asked participants from two reading cultures (left-to-right vs. mixed reading culture) to compare numerosities of two sets, choosing either a larger or smaller one. SNARC emerged in both groups. Additionally, it was modulated by comparative instruction: It appeared in a left-to-right direction when participants selected a smaller set, but it tended to reverse when participants selected a larger set. We conclude that spatial processing of numerosities is dissociated from spatial processing of Arabic numbers, at least in a context of comparative judgment. This dissociation could reflect differences in spatial ordering experience specific to a certain numerical input.     

Isolating the effects of symbolic distance,and semantic congruity in comparative judgments: An additive-factors analysis

The time needed to compare two symbols increases as the cognitive distance between them on the relevant dimension increases (symbolic distance effect). Furthermore, when subjects are told to choose either the larger or the smaller of two stimuli, the response time is shorter if the instruction is congruent with the overall size of the stimuli (semantic congruity effect). Three experiments were conducted to determine the locus of these effects in terms of a sequence of processing stages. The developmental aspects of these effects were also evaluated, as the subjects were from kindergarten, first grade, third grade, fifth grade, and college. By varying the visual quality of the stimulus in each experiment, it was determined that the distance effect resides in a comparison stage, whereas the congruity effect is an encoding phenomenon. Both distance and congruity effects were present at all grade levels, but they decreased in magnitude as grade increased. The results were interpreted relative to recent models of comparative judgments.     

Sequential dependencies and regression in psychophysical judgments

A tendency for judgments of stimulus magnitude to be biased in the direction of the value of the immediately preceding stimulus is found in magnitude estimations of loudness. This produces a bias in the empirical psychophysical function that results in underestimation of the exponent of the unbiased function presumed to relate number and stimulus intensity, N = aSⁿ. The biased judgment can be represented as a power product of focal and preceding stimulus intensity, N_ij= aS ^m S_j ^b. A bias-free estimate of the correct exponent, n, can be obtained from the relation n = m + b.     

Representation of numerical and non-numerical order in children

The representation of numerical and non-numerical ordered sequences was investigated in children from preschool to grade 3. The child's conception of how sequence items map onto a spatial scale was tested using the Number-to-Position task (Siegler & Opfer, 2003) and new variants of the task designed to probe the representation of the alphabet (i.e., letter sequence) and the calendar year (i.e., month sequence). The representation of non-numerical order showed the same developmental pattern previously observed for numerical representation, with a logarithmic mapping in the youngest children and a shift to linear mapping in older children. Although the individual ability to position non-numerical items was related to the child's knowledge of the sequence, a significant amount of unique variance was explained by her type of number-line representation. These results suggest that the child's conception of numerical order is generalized to non-numerical sequences and that the concept of linearity is acquired in the numerical domain first and progressively extended to all ordinal sequences.     

Semantic congruity and expectancy in symbolic judgments

These experiments assess the degree to which the semantic-congruity effect in comparative judgment can be explained by such expectancy effects as priming, perceptual "set," or strategies used in the task. The first experiment mixed a lexical-decision task with the comparative-judgment task and showed that neither automatic semantic priming nor deliberate preparation can account for the congruity effect. Experiments 2-4 assessed expectancy effects in a different way by presenting the instructions for comparative judgment either before or after the pair to be judged. These experiments included, among other things, a number of safeguards against artifacts in this paradigm. In these three experiments the congruity effect was obtained with both orders of stimuli and instructions, contrary to the prediction of an expectancy hypothesis. The results indicate that when stimuli are not degraded. The semantic-congruity effect depends largely on the relation between the stimuli and the instructions and only to a small degree, if at all, on expectancy.     

The malleable bicultural consumer: effects of cultural contexts on aesthetic judgments

• Grounded in the cognitive framework of processing fluency, this study proposes further support for the experiential perspective in aesthetics by positing that aesthetic response to the same object may be malleable, depending on how the symbolic properties of the object interact with different cultural contexts which either facilitate or debilitate the processing experience of the perceiver. The study employed an Internet experiment to test the hypotheses among 105 female Hispanic college‐aged students enrolled at a large midwestern university. The findings revealed that symbolic attributes of products interact with cultural contexts to affect aesthetic judgments of (Hispanic) consumers. Aesthetic judgments were more positive when evaluating culturally symbolic product attributes after exposure to congruent contextual cues that facilitate fluent processing. The study furnishes support for the impact of environment/context on consumer behavior and aesthetic judgment, thus establishing further support for the cognitive framework of conceptual fluency in explaining aesthetic response. The study also contributes to recent literature on “frame‐switching” among bicultural consumers by suggesting that these consumers navigate between competing cultural frames in response to visual primes, with resultant shifts in aesthetic judgments. Important marketing insights emerge from these findings.

Copyright © 2009 John Wiley & Sons, Ltd.     

Comparative judgments with numerical reference points

A model of subjective magnitude comparisons is explored, which assumes that subjects compare symbolic stimulus magnitudes with respect to a reference point. The reference point may be established implicitly by the question (e.g., “Which is larger?” vs “Which is smaller?”) or be presented explicitly (e.g., “Choose the stimulus closer to X.”). The model was tested in five experiments in which subjects judged which of two comparison digits was closer to (or further from) a reference digit. Regression analyses in three experiments revealed that reaction time depended on the ratio of the distances from the comparison items to the reference point. The other two experiments provided evidence that subjects can strategically vary the processes by which they compare stimuli to a reference point. The results indicated that subjects can perform various types of “analog arithmetic” using either the linear number scale or a nonlinear scale of subjective digit magnitude.     

The role of subtractions and comparisons in comparative judgments involving numerical reference points

Experiments in which subjects are asked to decide which of two digits is closer in magnitude to a third raise problems for many theories of linear orders. Holyoak (1978), for example, performed a number of these reference point experiments and concluded that they posed serious difficulties for a number of leading models. In their place, he offered the distance ratio model in which the ease of the decision in a reference point task is a function of the ratio of the distances between each stimulus and the reference point. In the present article, three experiments are presented that bear on the adequacy of Holyoak's position. In the first two studies, we present evidence that an important assumption of the distance ratio model is incorrect. In the third experiment, we compare the empirical adequacy of the distance ratio model with our own subtraction model. This model treats the reference point task as a concatenation of two subtractions and a simple digit comparison. This comparison operation is equivalent to the magnitude comparison required in standard linear order experiments. Overall, the subtraction model gives a somewhat better account than the distance ratio.     

The associations between space and order in numerical and non-numerical sequences

Spatial-numerical associations have been found across different studies, yet the basis for these associations remains debated. The current study employed an order judgment task to adjudicate between two competing accounts of such associations, namely the Mental Number Line (MNL) and Working Memory (WM) models. On this task, participants judged whether number pairs were in ascending or descending order. Whereas the MNL model predicts that ascending and descending orders should map onto opposite sides of space, the WM model predicts no such mapping. Moreover, we compared the spatial-order mapping for numerical and non-numerical sequences because the WM model predicts no difference in mapping. Across two experiments, we found consistent spatial mappings for numerical order along both horizontal and vertical axes, consistent with a MNL model. In contrast, we found no consistent mappings for letter sequences. These findings are discussed in the context of conflicting extant data related to these two models.     

Dissecting the symbolic distance effect: Comparison and priming effects in numerical and nonnumerical orders

When participants are asked to compare two stimuli, responses are slower for stimuli close to each other on the relevant dimension than for stimuli further apart. Previously, it has been proposed that this comparison distance effect originates from overlap in the representation of the stimuli. This idea is generally accepted in numerical cognition, where it is assumed that representational overlap of numbers on a mental number line accounts for the effect (e.g., Cohen Kadosh et al., 2005). In contrast, others have emphasized the role of response-related processes to explain the comparison distance effect (e.g., Banks, 1977). In the present study, numbers and letters are used to show that the comparison distance effect can be dissociated from a more direct behavioral signature of representational overlap, the priming distance effect. The implication is that a comparison distance effect does not imply representational overlap. An interpretation is given in terms of a recently proposed model of quantity comparison (Verguts, Fias, & Stevens, 2005).     

Backward-compatibility effects with irrelevant stimulus-response overlap: the case of the SNARC effect

In 3 dual-task experiments, the relationship between primary-task response (R1) and secondary-task response (R2) was varied. In general, R1-left responses were faster when followed by the word one, and right responses were faster when followed by the word two. This backward-compatibility (BWC) effect indicated (a) that activation of R2 was not delayed until R1 selection was completed, and (b) that activation of the vocal responses was accompanied by the automatic activation of magnitude codes, known to be associated with spatial left-right codes (spatial-numerical association of response codes [the SNARC effect]). These findings supported the hypotheses (a) that BWC effects persist even with irrelevant R1-R2 overlap, (b) that the SNARC effect is based on associations between magnitude and spatial representations underlying response processing, and (c) that the production and perception of magnitudes relies on common codes.     

The spatial representation of numerical and non-numerical ordered sequences: Insights from a random generation task

It is widely believed that numbers are spatially represented from left to right on the mental number line. Whether this spatial format of representation is specific to numbers or is shared by non-numerical ordered sequences remains controversial. When healthy participants are asked to randomly generate digits they show a systematic small-number bias that has been interpreted in terms of “pseudoneglect in number space”. Here we used a random generation task to compare numerical and non-numerical order. Participants performed the task at three different pacing rates and with three types of stimuli (numbers, letters, and months). In addition to a small-number bias for numbers, we observed a bias towards “early” items for letters and no bias for months. The spatial biases for numbers and letters were rate independent and similar in size, but they did not correlate across participants. Moreover, letter generation was qualified by a systematic forward direction along the sequence, suggesting that the ordinal dimension was more salient for letters than for numbers in a task that did not require its explicit processing. The dissociation between numerical and non-numerical orders is consistent with electrophysiological and neuroimaging studies and suggests that they rely on at least partially different mechanisms.     

Associations of non‐symbolic and symbolic numerical magnitude processing with mathematical competence: a meta‐analysis

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non‐symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta‐analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two‐level random‐effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non‐symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non‐symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school‐aged children and for adults.     

Biases in social comparative judgments: the role of nonmotivated factors in above-average and comparative-optimism effects

Biases in social comparative judgments, such as those illustrated by above-average and comparative-optimism effects, are often regarded as products of motivated reasoning (e.g., self-enhancement). These effects, however, can also be produced by information-processing limitations or aspects of judgment processes that are not necessarily biased by motivational factors. In this article, the authors briefly review motivational accounts of biased comparative judgments, introduce a 3-stage model for understanding how people make comparative judgments, and then describe how various nonmotivational factors can influence the 3 stages of the comparative judgment process. Finally, the authors discuss several unresolved issues highlighted by their analysis, such as the interrelation between motivated and nonmotivated sources of bias and the influence of nonmotivated sources of bias on behavior.     

Position effects in comparative judgments of serial order: List structure vs. differential strength

Two experiments are reported that test the hypothesis that the serial position effect in comparative judgment of ordinal position in arbitrary serial lists results from differential memory or associative strength among list items. The serial position effect in comparative judgment is typically a pattern in which pairs that contain a term from one of the two extremes of the list are processed faster and more accurately than pairs that contain no end terms. The experiments show that a new term added to either the end or the middle of a well-practiced fourterm series behaves almost immediately like the end or central term, respectively, of a well-practiced five-term series. Furthermore, when the added term is removed, the list reverts immediately to the position effect obtained in a four-term series. Theories that explain the position effect by differential build-up of item strength or of interitem associative strength over practice cannot explain these effects. We propose instead that learning of a serial list is accomplished by assigning list members to positions in a general-purpose linear order schema and that subjects can make these assignments rapidly and flexibly.     

The role of numerical and non-numerical cues in nonsymbolic number processing: Evidence from the line bisection task

In line bisection tasks, adults and children bisect towards the numerically larger of two nonsymbolic numerosities [de Hevia, M. D., & Spelke, E. S. (2009 de Hevia, M. D., & Spelke, E. S. (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003]. However, it is not clear whether this effect is driven by number itself or rather by visual cues such as subtended area [Gebuis, T., & Gevers, W. (2011 Gebuis, T., & Gevers, W. (2011). Numerosities and space: Indeed a cognitive illusion! A reply to de Hevia and Spelke (2009). Cognition, 121, 248–252. doi:10.1016/j.cognition.2010.09.008[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Numbers and space: Indeed a cognitive illusion! A reply to de Hevia and Spelke (2009 de Hevia, M. D., & Spelke, E. S. (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Cognition, 121, 248–252. doi:10.1016/j.cognition.2010.09.008]. Furthermore, this effect has only been demonstrated with flanking displays of two and nine items. Here, we report three studies that examined whether this “spatial bias” effect occurs across a range of absolute and ratio numerosity differences; in particular, we examined whether the bias would occur when both flankers were outside the subitizing range. Additionally, we manipulated the subtended area of the stimulus and the aggregate surface area to assess the influence of visual cues. We found that the spatial bias effect occurred for a range of flanking numerosities and for ratios of 3:5 and 5:6 when subtended area was not controlled (Experiment 1). However, when subtended area and aggregate surface area were held constant, the biasing effect was reversed such that participants bisected towards the flanker with fewer items (Experiment 2). Moreover, when flankers were identical, participants bisected towards the flanker with larger subtended area or larger aggregate surface area (Experiments 2 and 3). On the basis of these studies, we conclude that the spatial bias effect for nonsymbolic numerosities is primarily driven by visual cues.     

Directional verbal probabilities: inconsistencies between preferential judgments and numerical meanings

Verbal probability expressions (e.g., it is possible or doubtful) convey not only vague numerical meanings (i.e., probability) but also semantic functions, called directionality. We performed two experiments to examine whether preferential judgments are consistent with numerical meanings of verbal probabilities regardless of directionality. The results showed that because of the effects of directionality, perceived degrees of certainty for verbal probabilities differed between a binary choice and a numerical translation (Experiment 1), and decisions based on a verbal probability do not correspond to those based on a numerical translation for verbal probabilities (Experiment 2). These findings suggest that directionality of verbal probabilities is an independent feature from numerical meanings; hence numerical meanings of verbal probability alone remain insufficient to explain the effects of directionality on preferential judgments.     

Children's equivalence judgments: Crossmapping effects

Preschoolers made numerical comparisons between sets with varying degrees of shared surface similarity. When surface similarity was pitted against numerical equivalence (i.e., crossmapping), children made fewer number matches than when surface similarity was neutral (i.e, all sets contained the same objects). Only children who understood the number words for the target sets performed above chance in the crossmapping condition. These findings are consistent with previous research on children's non-numerical comparisons (e.g., [Rattermann, M. J., & Gentner, D. (1998). The effect of language on similarity: The use of relational labels improves young children's performance in a mapping task. In K. Holyoak, D. Gentner, & B. Kokinov (Eds.), Advances in analogy research: Integration of theory and data from cognitive, computational, and neural sciences (pp. 274–282). Sofia: New Bulgarian University; Smith, L. B. (1993). The concept of same. In H. W. Reese (Ed.), Advances in child development and behavior, Vol. 24 (pp. 215–252). New York: Academic Press]) and suggest that the same mechanisms may underlie numerical development.