Internal number magnitude representation is not holistic,either

Over the last years, evidence has accumulated that the magnitude of two-digit numbers is not only represented as one holistic entity, but also decomposed for tens and units. Recently, Zhang and Wang (2005) suggested such separate processing may be due to the presence of external representations of numbers, whereas holistic processing became more likely when one of the to-be-compared numbers was already internalised. The latter conclusion essentially rested on unit-based null effects. However, Nuerk and Willmes (2005) argued that unfavourable stimulus selection may provoke such null effects and misleading conclusions. Therefore, we tested the conclusion of Zhang and Wang for internal standards with a modified stimulus set. We observed reliable unit-based effects in all conditions contradicting the holistic model. Thus, decomposed representations of tens and units can also be demonstrated for internal standards. We conclude that decomposed magnitude processing of multidigit numbers does not rely on external representations. Rather, even when two-digit numbers are internalised, the magnitudes of tens and units seem to be (also) represented separately.     

Two-digit number processing: holistic,decomposed or hybrid? A computational modelling approach

Currently, there are three competing theoretical accounts concerning the nature of two-digit number magnitude representation: a holistic, a strictly decomposed, and a hybrid model. Observation of the unit-decade compatibility effect (Nuerk et al. in Cognition 82:B25–B33, 2001) challenged the view of two-digit number magnitude to be represented as one integrated entity. However, at the moment there is no study distinguishing between the decomposed and the hybrid model. The present study addressed this issue using a computational modelling approach. Three network models complying with the constraints of all three theoretical models were programmed and trained on two-digit number comparison. Models were compared as to how well they accounted for empirical effects in the most parsimonious way. Generally, this evaluation indicated that the empirical data were simulated best by the strictly decomposed model. Implications of these results for our understanding of the nature of human number magnitude representation are discussed.     

Semantic processing of crowded stimuli?

Effects of semantic processing of crowded characters were investigated using numbers as stimuli. In an identification task, typical spacing effects in crowding were replicated. Using the same stimuli in a magnitude comparison task, a smaller effect of spacing was observed as well as an effect of response congruency. These effects were replicated in a second experiment with varying stimulus-onset asynchronies. In addition, decreasing performance with increasing onset-asynchrony (so-called type-B masking) for incongruent flankers indicates semantic processing of target and flankers. The data show that semantic processing takes place even in crowded stimuli. This argues strongly against common accounts of crowding in terms of early stimulus-driven impairments of processing.     

Decade breaks in the mental number line? Putting the tens and units back in different bins.

Most models of number recognition agree that among other number representations there is a central semantic magnitude representation which may be conceptualized as a logarithmically compressed mental number line. Whether or not this number line is decomposed into different representations for tens and units is, however, controversial. We investigated this issue in German participants in a magnitude comparison (selection) task in which the larger of two visually presented Arabic two-digit numbers had to be selected. Most importantly, we varied unit-decade-compatibility: a number pair was defined as compatible if the decade magnitude comparison and the unit magnitude comparison of the two numbers would lead to the same response (e.g. 52 and 67) and as incompatible if this was not the case (e.g. 47 and 62). While controlling for overall numerical distance, size and other variables, we consistently found compatibility effects. A control experiment showed that this compatibility effect was not due to perceptual presentation characteristics. We conclude that the idea of one single number line representation that does not additionally assume different magnitude representations for tens and units is not sufficient to account for the data. Finally, we discuss why decade effects were not found in other experimental settings.     

Facet theory applied to the construction and validation of the Aachen Aphasia Test

The linguistic performance of 120 aphasic patients of the four standard syndromes assessed by the Aachen Aphasia Test (AAT) is analyzed by a nonmetric (ordinal) multidimensional scaling procedure (Smallest Space Analysis, SSA1). The linguistic structure of the test items is characterized within the framework of L. Guttman's facet theory. Three systematic components (facets) are discerned: linguistic modality, unit, and regularity. Properties of the facets as well as their relations are assessed and tested empirically by analyzing the interrelations among different items or sets of items. The spatial configurations obtained by the scaling procedure fit only partially the expectations derived from the facet-theory model. The modality facet was found to have a strong overriding influence on the aphasic test performance. The facets unit and regularity were only found for the most rigorously designed subtests, Written Language and Comprehension. The results suggest the introduction of a new combined facet linguistic complexity which reflects the dependency of the facets regularity and unit.     

On the development of the mental number line: more, less, or never holistic with increasing age?

Magnitude is assumed to be represented along a holistic mental number line in adults. However, the authors recently observed a unit-decade compatibility effect for 2-digit numbers that is inconsistent with this "holisticness" assumption (H.-C. Nuerk, U. Weger, & K. Willmes, 2001). This study used the compatibility effect to examine whether the mental number line representation of magnitude changes toward greater or less holisticness in children from Grades 2-5. The results indicate that decades and units of 2-digit numbers are processed separately rather than holistically from Grade 2 on. However, this separate processing seems to develop from a more sequential (left-to-right) to a more parallel processing mode. Moreover, children may use different strategies depending on task demands. The results are interpreted in the framework of Siegler's overlapping waves model.     

The universal SNARC effect: the association between number magnitude and space is amodal

It is thought that number magnitude is represented in an abstract and amodal way on a left-to-right oriented mental number line. Major evidence for this idea has been provided by the SNARC effect (Dehaene, Bossini, & Giraux, 1993): responses to relatively larger numbers are faster for the right hand, those to smaller numbers for the left hand, even when number magnitude is irrelevant. The SNARC effect has been used to index automatic access to a central semantic and amodal magnitude representation. However, this assumption of modality independence has never been tested and it remains uncertain if the SNARC effect exists in other modalities in a similar way as in the visual modality. We have examined this question by systematically varying modality/notation (auditory number word, visual Arabic numeral, visual number word, visual dice pattern) in a within-participant design. The SNARC effect was found consistently for all modality/notation conditions, including auditory presentation. The size of the SNARC effect in the auditory condition did not differ from the SNARC effect in any visual condition. We conclude that the SNARC effect is indeed a general index of a central semantic and amodal number magnitude representation.