Expanding on the mental number line: Zero is perceived as the "smallest"

The representation of 0 in healthy adults was studied with the physical comparison task. Automatic processing of numbers, as indicated by the size congruity effect, was used for detecting the basic numerical representations stored in long-term memory. The size congruity effect usually increases with numerical distance between the physically compared numbers. This increase was attenuated for comparisons to 0 or 1 (but not to 2) when they were perceived as the smallest number in the set. Furthermore, the size congruity effect was enlarged in these cases. These results indicate an end effect in automatic processing of numbers and suggest that 0, or 1 in the absence of 0, is perceived as the smallest entity on the mental number line. The implications of these findings are discussed with regard to models of number representation. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Are Arabic and verbal numbers processed in different ways?

Four experiments were conducted in order to examine effects of notation--Arabic and verbal numbers--on relevant and irrelevant numerical processing. In Experiment 1, notation interacted with the numerical distance effect, and irrelevant physical size affected numerical processing (i.e., size congruity effect) for both notations but to a lesser degree for verbal numbers. In contrast, size congruity had no effect when verbal numbers were the irrelevant dimension. In Experiments 2 and 3, different parameters that could possibly affect the results, such as discriminability and variability (Experiment 2) and the block design (Experiment 3), were controlled. The results replicated the effects obtained in Experiment 1. In Experiment 4, in which physical size was made more difficult to process, size congruity for irrelevant verbal numbers was observed. The present results imply that notation affects numerical processing and that Arabic and verbal numbers are represented separately, and thus it is suggested that current models of numerical processing should have separate comparison mechanisms for verbal and Arabic numbers.     

A memory-based account of automatic numerosity processing

We investigated the mechanisms responsible for the automatic processing of the numerosities represented by digits in the size congruity effect (Henik & Tzelgov, 1982). The algorithmic model assumes that relational comparisons of digit magnitudes (e.g., larger than {8,2}) create this effect. If so, congruity effects ought to require two digits. Memory-based models assume that associations between individual digits and the attributes "small" and "large" create this effect. If so, congruity effects ought only to require one digit. Contrary to the algorithmic model and consistent with memory-based models, congruity effects were just as large when subjects judged the relative physical sizes of small digits paired with letters as when they judged the relative physical sizes of two digits. This finding suggests that size congruity effects can be produced without comparison algorithms.     

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.     

Paying attention to attention: evidence for an attentional contribution to the size congruity effect

Understanding the mechanisms supporting our comprehension of magnitude information represents a key goal in cognitive psychology. A major phenomenon employed in the pursuit of this goal has been the physical size congruity effect—namely, the observation that comparing the relative numerical sizes of two numbers is influenced by their relative physical sizes. The standard account of the physical size congruity effect attributes it to the automatic influence of the comparison of irrelevant physical magnitudes on numerical judgments. Here we develop an alternative account of this effect on the basis of the operation of attention in the typical size congruity display and the temporal dynamics of number comparison. We also provide a test of a number of predictions derived from this alternative account by combining a physical size congruity manipulation with a manipulation designed to alter the operation of attention within the typical size congruity display (i.e., a manipulation of the relative onsets of the digits). This test provides evidence consistent with an attentional contribution to the size congruity effect. Implications for our understanding of magnitude and the interactions between attention and magnitude are discussed.     

The psychophysics of numerical comparison: a reexamination of apparently incompatible data

Reaction-time studies of numerical comparison have used essentially two paradigms: classification, in which a target number must be labelled "larger" or "smaller" in comparison to a fixed standard, and selection, in which the larger (or smaller) number of a pair must be picked out. In previous studies, classification has yielded only a distance effect in RTs, whereas selection has also revealed magnitude (or minimum) and congruity effects. We used two experiments with two-digit number comparisons to find the reason for this discrepancy. In Experiment 1, we used a variant of the classification task with the standard changing on each trial. RTs increased along with the standard for "smaller" responses and decreased along with the standard for "larger" responses, in a manner reminiscent of magnitude and congruity effects. In Experiment 2, we again used classification, but the fixed standard 75 was not at the center of the range of target numbers (20, 21, ... 99). Close to the standard, RTs were faster for "larger" than for "smaller" responses, again a congruity effect. Our data show that magnitude and congruity effects can be obtained with two-digit numbers in classification as well as in selection tasks. A single equation, which implies that numbers are compared with respect to reference points at both ends of the continuum, describes the results from both tasks.     

The effect of orientation on number word processing

Besner and Coltheart [Besner, D., & Coltheart, M. (1979). Ideographic and alphabetic processing in skilled reading of English. Neuropsychologia, 17, 467-472] found a size congruity effect for Arabic numbers but not for number words. They proposed that Arabic numbers and number words are processed in different ways. However, in their study orientation of the stimuli and notation were confounded. In the present study, it is found that orientation of number words affects numerical processing. Orientation modulates both the size congruity effect and the distance effect; horizontal presentation produces similar results to those produced by Arabic numbers whereas vertical orientation produces different results. Accordingly, it is proposed that our cognitive system is endowed with two different mechanisms for numerical processing; one relies on a visual-spatial code and the other on a verbal code.     

Semantic processing of Arabic,Kanji, and Kana numbers: Evidence from interference in physical and numerical size judgments

Two experiments were conducted with the following objectives (1) to replicate the finding of similar semantic representation of Arabic and written-word (Kanji and Kana) numbers with a direct numerical task, (2) to investigate the automatic semantic processing of Arabic and written-word numbers, and (3) to verify whether the assumption of a common semantic representation is valid in an indirect numerical task. Subjects were asked to judge which of two numbers (e.g, 6-8) was larger either in its numerical size (Experiment 1) or in its physical size (Experiment 2) using the three notations. Effects of two factors were analyzed: the congruity between numerical and physical size and the numerical distance. The effects of these factors were very similar across the three notations in Experiment 1, but were drastically different in Experiment 2. The results of Experiment 2 demonstrated the nonsemantic processing of Kana numbers, and suggest that there may be separate semantic representations for Arabic and Kanji numbers.     

The role of parity, physical size, and magnitude in numerical cognition: The SNARC effect revisited

People indicate the physical size or the parity status of small numbers faster by a left-hand key and those of larger numbers by a right-hand key. Because magnitude information is not required for successful performance in these tasks, the presence of a number-space association (the SNARC effect) has been taken to indicate the automatic activation of numerical magnitude in all tasks with numerals. In order to test this account, in a series of five experiments, we derived two consensual markers of automatic activation of irrelevant numerical magnitude, the size congruity effect (for judgments of physical size), and the Garner effect (for judgments of parity). Both markers were found independent of the SNARC effect. Consequently, we question the traditional explanation of the SNARC effect and offer an alternative account in terms of a highly overlearned stimulus-response loop.     

Mental comparison of size and magnitude: size congruity effects

Paivio (1975) found that the latency to choose the larger of two named objects does not depend on congruity between the object sizes and the sizes of the object names. Because size congruity does affect latencies for pictorially presented objects, Paivio interpreted this result as support for the dual coding hypothesis. However, Experiment 1 demonstrated that Paivio's results were an artifact of his experimental design. Size congruity does affect latencies to choose the larger of two named objects when object pairs are not repeated. When the same object pairs are used repeatedly, as in Paivio's experiment, the effect disappears. In this case the response is probably remembered, so that the objects need not be compared. To determine the processing stages affected by size congruity, both the distance between stimulus sizes and the size congruity were manipulated in Experiment 2. Three groups of subjects chose either the greater Arabic digit, the greater named digit, or the larger named object. Size congruity interacted with distance only for Arabic digits. For both Arabic digits and named digits, the interference caused by size incongruity was greater than the facilitation caused by size congruity, whereas for object names, the facilitation was greater than the interference. A model of the interaction between physical size comparisons and conceptual size comparisons is proposed to account for these results.     

The locus and nature of semantic congruity in symbolic comparison: Evidence from the Stroop effect

Pictures of animals with names of animals printed within the pictures were presented for comparative judgments of size based on either the pictures or the names. The picture-word compounds were compared faster with picture than with word as the relevant dimension. The comparisons of pictures were free of interference from the irrelevant names, but the comparisons of names suffered considerable Stroop interference from the irrelevant pictures. Large effects of semantic congruity characterized the comparisons of both pictures and words. Stroop congruity and semantic congruity did not interact even for comparison of words in which both were present, leading instead to additive effects. The results support theories that (1) place semantic congruity in the decision stage and (2) minimize the role of semantic processing as the basis of the semantic congruity effect.     

The size congruity effect: is bigger always more?

When comparing digits of different physical sizes, numerical and physical size interact. For example, in a numerical comparison task, people are faster to compare two digits when their numerical size (the relevant dimension) and physical size (the irrelevant dimension) are congruent than when they are incongruent. Two main accounts have been put forward to explain this size congruity effect. According to the shared representation account, both numerical and physical size are mapped onto a shared analog magnitude representation. In contrast, the shared decisions account assumes that numerical size and physical size are initially processed separately, but interact at the decision level. We implement the shared decisions account in a computational model with a dual route framework and show that this model can simulate the modulation of the size congruity effect by numerical and physical distance. Using other tasks than comparison, we show that the model can simulate novel findings that cannot be explained by the shared representation account.     

A common representation for semantic and physical properties: a cognitive-anatomical approach

This is the first report of a mutual interference between luminance and numerical value in magnitude judgments. Instead of manipulating the physical size of compared numbers, which is the traditional approach in size congruity studies, luminance levels were manipulated. The results yielded the classical congruity effect. Participants took more time to process numerically larger numbers when they were brighter than when they were darker, and more time to process a darker number when its numerical value was smaller than when it was larger. On the basis of neurophysiological studies of magnitude comparison and interference between semantic and physical information, it is proposed that the processing of semantic and physical magnitude information is carried out by a shared brain structure. It is suggested that this brain area, the left intraparietal sulcus, subserves various comparison processes by representing various quantities on an amodal magnitude scale.     

Is three greater than five: The relation between physical and semantic size in comparison tasks

In this study, subjects were asked to judge which of two digits (e.g., 3 5) was larger either in physical or in numerical size. Reaction times were facilitated when the irrelevant dimension was congruent with the relevant dimension and were inhibited when the two were incongruent (size congruity effect). Although judgments based on physical size were faster, their speed was affected by the numerical distance between the members of the digit pair, indicating that numerical distance is automatically computed even when it is irrelevant to the comparative judgment being required by the task. This finding argues for parallel processing of physical and semantic information in this task.     

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.     

An effect size index for comparing two independent alpha coefficients

Since Cronbach proposed the α coefficient in 1951, researchers have contributed to the derivation of its sampling distribution and the testing of related statistical hypotheses. Yet, there has been no research on effect size index relevant to coefficient α to our knowledge. Considering the importance of effect size in understanding quantitative research findings, we therefore developed an effect size index Δ for the comparison of two independent αs with equal test length based on the asymptotic distribution of under the assumptions of normality and compound symmetry. Simulations indicated that the index was applicable when the sample size was at least 100. The robustness of the derived index when the required assumptions were violated was also explored. It is suggested that the index should be applicable in most cases of unequal test lengths and could be extended to non‐normally distributed component scores. Moreover, a small simulation was conducted to explore the behaviour of Δ with correlated errors, a frequently studied situation violating the assumption of compound symmetry. The proposed index was found to be robust unless a large number of highly correlated errors were present in the data.     

Automatic quantity processing in 5-year olds and adults

In this study adults performed numerical and physical size judgments on a symbolic (Arabic numerals) and non-symbolic (groups of dots) size congruity task. The outcomes would reveal whether a size congruity effect (SCE) can be obtained irrespective of notation. Subsequently, 5-year-old children performed a physical size judgment on both tasks. The outcomes will give a better insight in the ability of 5-year-olds to automatically process symbolic and non-symbolic numerosities. Adult performance on the symbolic and non-symbolic size congruity tasks revealed a SCE for numerical and physical size judgments, indicating that the non-symbolic size congruity task is a valid indicator for automatic processing of non-symbolic numerosities. Physical size judgments on both tasks by children revealed a SCE only for non-symbolic notation, indicating that the lack of a symbolic SCE is not related to the mathematical or cognitive abilities required for the task but instead to an immature association between the number symbol and its meaning.     

Typical and atypical development of basic numerical skills in elementary school

Deficits in basic numerical processing have been identified as a central and potentially causal problem in developmental dyscalculia; however, so far not much is known about the typical and atypical development of such skills. This study assessed basic number skills cross-sectionally in 262 typically developing and 51 dyscalculic children in Grades 2, 3, and 4. Findings indicate that the efficiency of number processing improves over time and that dyscalculic children are generally less efficient than children with typical development. For children with typical arithmetic development, robust effects of numerical distance, size congruity, and compatibility of ten and unit digits in two-digit numbers could be identified as early as the end of Grade 2. Only the distance effect for comparing symbolic representations of numerosities changed developmentally. Dyscalculic children did not show a size congruity effect but showed a more marked compatibility effect for two-digit numbers. We did not find strong evidence that dyscalculic children process numbers qualitatively differently from children with typical arithmetic development.     

Automaticity for numerical magnitude of two-digit Arabic numbers in children

This paper examines the automatic processing of the numerical magnitude of two-digit Arabic numbers using a Stroop-like task in school-aged children. Second, third, and fourth graders performed physical size judgments on pairs of two-digit numbers varying on both physical and numerical dimensions. To investigate the importance of synchrony between the speed of processing of the numerical magnitude and the physical dimensions on the size congruity effect (SCE), we used masked priming: numerical magnitude was subliminally primed in half of the trials, while neutral priming was used in the other half. The results indicate a SCE in physical judgments, providing the evidence of automatic access to the magnitude of two-digit numbers in children. This effect was modulated by the priming type, as a SCE only appeared when the numerical magnitude was primed. This suggests that young children needed a relative synchronization of numerical and physical dimensions to access the magnitude of two-digit numbers automatically.     

The development of internal representations of magnitude and their association with Arabic numerals.

Developmental aspects of number concepts were evaluated using participants from the beginning and end of first grade (6-7 years old), third and fifth grades (7-11 years old), and university (22 years old). Participants evaluated the numerical value or physical size of stimuli varying along both dimensions. The numerical distance effect appeared in all groups. In contrast, the size congruity effect started to appear only at the end of first grade. Based on our results, a model of internal representation of magnitude claiming that there are two different representations was propose. At the beginning of first grade children can automatically access only one of these representations and only from the end of first grade can they access both of these representations.