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.     

Size congruity effects with two-digit numbers: Expanding the number line?

The size congruity effect is the interesting result that comparisons of the sizes of the physical formats in which numerals appear are affected by the numerical magnitudes of the respective numerals. We demonstrated that separating the physical and the numerical attributes in space leaves the effect unchanged. We then applied the spatially separated version to two-digit numerals and showed the effect to be comparable to that obtained with single numerals. We showed further that the effect is sensitive to the relative salience of the numeric and physical dimensions, to the extent that when the latter is the more salient dimension, a reverse effect obtains by which physical size interferes with number comparison. The results can be explained by a relative speed of processing account, but they are also compatible with an attention account that does not appeal to the notion of automaticity.     

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.     

Individual differences in children's mathematical competence are related to the intentional but not automatic processing of Arabic numerals

In recent years, there has been an increasing focus on the role played by basic numerical magnitude processing in the typical and atypical development of mathematical skills. In this context, tasks measuring both the intentional and automatic processing of numerical magnitude have been employed to characterize how children’s representation and processing of numerical magnitude changes over developmental time. To date, however, there has been little effort to differentiate between different measures of ‘number sense’. The aim of the present study was to examine the relationship between automatic and intentional measures of magnitude processing as well as their relationships to individual differences in children’s mathematical achievement. A group of 119 children in 1st and 2nd grade were tested on the physical size congruity paradigm (automatic processing) as well as the number comparison paradigm to measure the ratio effect (intentional processing). The results reveal that measures of intentional and automatic processing are uncorrelated with one another, suggesting that these tasks tap into different levels of numerical magnitude processing in children. Furthermore, while children’s performance on the number comparison paradigm was found to correlate with their mathematical achievement scores, no such correlations could be obtained for any of the measures typically derived from the physical size congruity task. These findings therefore suggest that different tasks measuring ‘number sense’ tap into different levels of numerical magnitude representation that may be unrelated to one another and have differential predictive power for individual differences in mathematical achievement.     

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.     

The development of automaticity in accessing number magnitude

This study traces developmental changes in automatic and intentional processing of Arabic numerals using a numerical-Stroop paradigm. In Study 1, university students compared the numerical or physical size of Arabic numerals varying along both dimensions. In Study 2, first graders (mean age = 6 years 6 months), third graders (mean age = 8 years 4 months), and fifth graders (mean age = 10 years 3 months) were tested to examine developmental changes in numerical and physical comparisons. In the numerical comparison task, a size congruity effect was found at all ages (i.e., relative to a neutral control, congruent physical sizes facilitated, and incongruent sizes interfered with, the numerical comparison). The pattern of facilitation and interference, however, was modulated by age. In the physical comparison task, the incongruity between physical and numerical size affected only older children and adults. These findings strongly suggest that the automatization in number processing is achieved gradually as numerical skills progress.     

Numerical and physical magnitudes are mapped into time

In two experiments we investigated mapping of numerical and physical magnitudes with temporal order. Pairs of digits were presented sequentially for a size comparison task. An advantage for numbers presented in ascending order was found when participants were comparing the numbers' physical and numerical magnitudes. The effect was more robust for comparisons of physical size, as it was found using both select larger and select smaller instructions, while for numerical comparisons it was found only for select larger instructions. Varying both the digits' numerical and physical sizes resulted in a size congruity effect, indicating automatic processing of the irrelevant magnitude dimension. Temporal order and the congruency between numerical and physical magnitudes affected comparisons in an additive manner, thus suggesting that they affect different stages of the comparison process.     

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.     

Simple addition and multiplication: No comparison

Some models of memory for arithmetic facts (e.g., 5+2=7, 6×7=42) assume that only the max-left order is stored in memory (e.g., 5+2=7 is stored but not 2+5=7). These models further assume an initial comparison of the two operands so that either operand order (5+2 or 2+5) can be mapped to the common internal representation. We sought evidence of number comparison in simple addition and multiplication by manipulating size congruity. In number comparison tasks, performance costs occur when the physical and numerical size of numerals are incongruent (8 3) relative to when they are congruent (8 3). Sixty-four volunteers completed a number comparison task, an addition task, and a multiplication task with both size congruent and size incongruent stimuli. The comparison task demonstrated that our stimuli were capable of producing robust size congruity and split effects. In the addition and multiplication task, however, we were unable to detect any of the RT signatures of comparison or reordering processes despite ample statistical power: Specifically, there was no evidence of size congruity, split, or order effects in either the addition or multiplication data. We conclude that our participants did not routinely engage a comparison operation and did not consistently reorder the operands to a preferred orientation.     

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 Numerical Stroop Effect in Primary School Children: A Comparison of Low,Normal, and High Achievers

Sixty-six primary school children were selected, of which 21 scored low on a standardized math achievement test, 23 were normal, and 22 high achievers. In a numerical Stroop experiment, children were asked to make numerical and physical size comparisons on digit pairs. The effects of congruity and numerical distance were determined. All children exhibited congruity and distance effects in the numerical comparison. In the physical comparison, children of all performance groups showed Stroop effects when the numerical distance between the digits was large but failed to show them when the distance was small. Numerical distance effects depended on the congruity condition, with a typical effect of distance in the congruent, and a reversed distance effect in the incongruent condition. Our results are hard to reconcile with theories that suggest that deficits in the automaticity of numerical processing can be related to differential math achievement levels. Immaturity in the precision of mappings between numbers and their numerical magnitudes might be better suited to explain the Stroop effects in children. However, as the results for the high achievers demonstrate, in addition to numerical processing capacity per se, domain-general functions might play a crucial role in Stroop performance, too.     

Approaching stimuli bias attention in numerical space

Increasing evidence suggests that common mechanisms underlie the direction of attention in physical space and numerical space, along the mental number line. The small leftward bias (pseudoneglect) found on paper-and-pencil line bisection is also observed when participants 'bisect' number pairs, estimating (without calculating) the number midway between two others. Here we investigated the effect of stimulus motion on attention in numerical space. A two-frame apparent motion paradigm manipulating stimulus size was used to produce the impression that pairs of numbers were approaching (size increase from first to second frame), receding (size decrease), or not moving (no size change). The magnitude of pseudoneglect increased for approaching numbers, even when the final stimulus size was held constant. This result is consistent with previous findings that pseudoneglect in numerical space (as in physical space) increases as stimuli are brought closer to the participant. It also suggests that the perception of stimulus motion modulates attention over the mental number line and provides further support for a connection between the neural representations of physical space and number.     

Interference Effects in a Numerical Stroop Paradigm in 9- to 12-year-old Children with ADHD-C

Attention deficit hyperactivity disorder (ADHD) is characterized by deficient self-regulation, poor attentional control, and poor response inhibition. To date, however, the extent to which these deficits affect basic interference control remains a matter of controversy. Secondly, ADHD has been reported to be associated with arithmetic deficits. It remains unclear whether such deficits are a secondary consequence of the above-mentioned characteristics of ADHD or whether basic numerical magnitude representations are also affected. In the present study we attempted to investigate these issues using a basic numerical interference paradigm.

Nine- to twelve-year-old children with ADHD-C (attention-deficit hyperactivity-disorder combined type) and control children without ADHD (each n = 16) were presented with two digits of possibly different physical sizes (e.g., 3 7). This numerical Stroop task requires subjects to make a magnitude classification concerning either the physical or the numerical stimulus dimension. The irrelevant dimension can be congruent (same response), incongruent (different response), or neutral (no response association).

Children with ADHD-C performed worse than control children in most analyses. The most important finding was a significant interaction of congruity effects with group in the numerical comparison task. Children with ADHD-C tended to show larger congruity and interference effects than controls, and these were not attributable to a speed-accuracy trade-off.

The results might reflect differential processing speeds, or a different degree of automatic activation of physical and numerical magnitudes in children with and without ADHD-C. Alternative explanations, such as insufficient inhibition of selective (domain-specific) attention are also discussed.     

Interference effects in a numerical Stroop paradigm in 9- to 12-year-old children with ADHD-C.

Attention deficit hyperactivity disorder (ADHD) is characterized by deficient self-regulation, poor attentional control, and poor response inhibition. To date, however, the extent to which these deficits affect basic interference control remains a matter of controversy. Secondly, ADHD has been reported to be associated with arithmetic deficits. It remains unclear whether such deficits are a secondary consequence of the above-mentioned characteristics of ADHD or whether basic numerical magnitude representations are also affected. In the present study we attempted to investigate these issues using a basic numerical interference paradigm. Nine- to twelve-year-old children with ADHD-C (attention-deficit hyperactivity-disorder combined type) and control children without ADHD (each n = 16) were presented with two digits of possibly different physical sizes (e.g., 3 7). This numerical Stroop task requires subjects to make a magnitude classification concerning either the physical or the numerical stimulus dimension. The irrelevant dimension can be congruent (same response), incongruent (different response), or neutral (no response association). Children with ADHD-C performed worse than control children in most analyses. The most important finding was a significant interaction of congruity effects with group in the numerical comparison task. Children with ADHD-C tended to show larger congruity and interference effects than controls, and these were not attributable to a speed-accuracy trade-off. The results might reflect differential processing speeds, or a different degree of automatic activation of physical and numerical magnitudes in children with and without ADHD-C. Alternative explanations, such as insufficient inhibition of selective (domain-specific) attention are also discussed.     

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).     

Dissociating between cardinal and ordinal and between the value and size magnitudes of coins

In mathematics, the ordinal (relative) magnitude of a numerical object conveys a separate meaning from its cardinal (absolute) magnitude, whereas its physical size bears no inherent relationship to its magnitude. In numerical cognition, the ordinal-cardinal distinction has been scarcely addressed, whereas the size-magnitude distinction has been studied extensively, with the surprising demonstration of an interaction between semantic magnitude and physical size (Besner & Coltheart, 1979). The present work used coins to study the intricate relations between these meanings. In two experiments, Israeli observers (Experiment 1) and American observers (Experiment 2) performed numerical and physical comparative judgments of coins. Consensual markers of magnitude activation (e.g., the size congruity effect and the distance effect) were obtained. The results of the two experiments converged on the same conclusions. Comparisons of value were governed by ordinal magnitude. Magnitude interfered with comparisons of size, but size did not affect value. The results provided a set of clear dissociations between cardinal and ordinal magnitude and between value and size of coins. They highlight the important role played by ordinal information in magnitude processing.     

Flexible mental processes in numerical size judgments: The case of hebrew letters that are used to convey numbers

In addition to its primary linguistic function, the Hebrew alphabet is sometimes used as a means of number notation (i.e., the system of gematria). Hebrew letters, Arabic numerals, Hebrew number names, and Hebrew letter names were used in a numerical size comparison task, in which two visually presented symbols were compared for numerical value while irrelevant variations in their physical size had to be ignored. A size congruity effect, indicated by faster responses when differences in physical and numerical size were consistent, was larger for Arabic numerals than for number names. The effect for Hebrew letters was similar to that for Arabic numerals and was stronger than that observed for letter names. These results suggest flexible processing of Hebrew letters, so that they function as ideographic symbols in an arithmetic context. A distance effect, indicated by an inverse relationship between reaction time and numerical distance, was found for all notations but was particularly strong for Hebrew letters.     

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.     

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.     

Effect of display size on visual attention

Attention plays an important role in the design of human-machine interfaces. However, current knowledge about attention is largely based on data obtained when using devices of moderate display size. With advancement in display technology comes the need for understanding attention behavior over a wider range of viewing sizes. The effect of display size on test participants' visual search performance was studied. The participants (N = 12) performed two types of visual search tasks, that is, parallel and serial search, under three display-size conditions (16 degrees, 32 degrees, and 60 degrees). Serial, but not parallel, search was affected by display size. In the serial task, mean reaction time for detecting a target increased with the display size.