Korbinian Moeller

On the language specificity of basic number processing: Transcoding in a language with inversion and its relation to working memory capacity

Transcoding Arabic numbers from and into verbal number words is one of the most basic number processing tasks commonly used to index the verbal representation of numbers. The inversion property, which is an important feature of some number word systems (e.g., German einundzwanzig [one and twenty]), might represent a major difficulty in transcoding and a challenge to current transcoding models. The mastery of inversion, and of transcoding in general, might be related to nonnumerical factors such as working memory resources given that different elements and their sequence need to be memorized and manipulated. In this study, transcoding skills and different working memory components in Austrian (German-speaking) 7-year-olds were assessed. We observed that inversion poses a major problem in transcoding for German-speaking children. In addition, different components of working memory skills were differentially correlated with particular transcoding error types. We discuss how current transcoding models could account for these results and how they might need to be adapted to accommodate inversion properties and their relation to different working memory components.

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking.

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants’ eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.

Impairments of the mental number line for two-digit numbers in neglect

Humans represent numbers along a left-to-right oriented Mental Number Line (MNL). Neglect patients seem to neglect the left part of the MNL, namely the smaller numbers within a given numerical interval. However, until now all studies examining numerical representation have focussed on single-digit numbers or two-digit numbers smaller than 50. In this study, the full range of two-digit numbers was assessed in neglect patients and two control groups. Participants were presented with number triplets (e.g., 10_13_18) and asked whether or not the central number is also the arithmetical middle of the interval. The factors manipulated were decade crossing (e.g., 22_25_28 vs 25_28_31), distance to the arithmetical middle (e.g., 18_19_32 vs 18_24_32), and, most importantly, whether the central number was smaller or larger than the arithmetical middle (e.g., 11_12_19 vs 11_18_19). Neglect patients differed from controls in that they benefited less when the middle number was smaller than the arithmetical middle of the interval. Neglect patients thus seem to have particular problems when accessing the left side of numerical intervals, also when adjusted to two-digit numbers. Such an impaired magnitude representation in neglect seems to have detrimental effects on two-digit number processing as the helpful spatial metric of magnitude cannot be properly activated.

Whorf reloaded: Language effects on nonverbal number processing in first grade—A trilingual study

The unit-decade compatibility effect is interpreted to reflect processes of place value integration in two-digit number magnitude comparisons. The current study aimed at elucidating the influence of language properties on the compatibility effect of Arabic two-digit numbers in Austrian, Italian, and Czech first graders. The number word systems of the three countries differ with respect to their correspondence between name and place value systems; the German language is characterized by its inversion of the order of tens and units in number words as compared with digital notations, whereas Italian number words are generally not inverted and there are both forms for Czech number words. Interestingly, the German-speaking children showed the most pronounced compatibility effect with respect to both accuracy and speed. We interpret our results as evidence for a detrimental influence of an intransparent number word system place value processing. The data corroborate a weak Whorfian hypothesis in children, with even nonverbal Arabic number processing seeming to be influenced by linguistic properties in children.

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.

Decomposed but Parallel Processing of Two-Digit Numbers in 1st Graders

It has been suggested that decomposed processing of two-digit numbers develops from sequential (left-to-right) to parallel with age (Nuerk et al., 2004). However, task demands may have provoked sequential processing as a specific rather than a universal processing style. In the current study a standard unit-decade compatibility effect observed in two-digit number magnitude comparison indicated that first graders were already able to process the single digit magnitudes of tens and units separately and in parallel. Consequently, previous findings of sequential processing may be specific for stimulus characteristics in which such a processing style is useful. It is concluded that even first graders seem to be able to adapt their individual processing styles depending on stimulus properties. More generally, this suggests that the manner by which children process two-digit numbers is strategically adaptive rather than fixed at a particular developmental stage.

Eye fixation behaviour in the number bisection task: evidence for temporal specificity.

Together with magnitude representations, knowledge about multiplicativity and parity contributes to numerical problem solving. In the present study, we used eye tracking to document how and when multiplicativity and parity are recruited in the number bisection task. Fourteen healthy adults evaluated whether the central number of a triplet (e.g., 21_24_27) corresponds to the arithmetic integer mean of the interval defined by the two outer numbers. We observed multiplicativity to specifically affect gaze duration on numbers, indicating that the information of multiplicative relatedness is activated at early processing stages. In contrast, parity only affected total reading time, suggesting involvement in later processing stages. We conclude that different representational features of numbers are available and integrated at different processing stages within the same task and outline a processing model for these temporal dynamics of numerical cognition.

Oscillatory EEG correlates of an implicit activation of multiplication facts in the number bisection task.

Neuroimaging evidence points towards the left inferior parietal cortex to be crucial for the representation and retrieval of multiplication facts. However, to date studies allowing a functional interpretation of neuroimaging data are still scarce. In the current study we aimed at evaluating the functional involvement of left inferior parietal cortex areas in the implicit retrieval of multiplication fact knowledge in a number bisection task by examining event-related desynchronization (ERD) in the upper alpha band. Upper alpha ERD is generally agreed to be modulated by processes of memory retrieval. It was observed that upper alpha ERD decreased for multiplicative triplets (e.g. 3_6_9) but not for non-multiplicative (e.g. 2_5_8) triplets at left parietal electrodes but increased at left prefrontal electrodes. These results are interpreted to suggest that after multiplicativity has been recognized further magnitude evaluations in the left hemisphere may be abated by prefrontal processes of executive control.

The influence of math anxiety on symbolic and non-symbolic magnitude processing

Deficits in basic numerical abilities have been investigated repeatedly as potential risk factors of math anxiety. Previous research suggested that also a deficient approximate number system (ANS), which is discussed as being the foundation for later math abilities, underlies math anxiety. However, these studies examined this hypothesis by investigating ANS acuity using a symbolic number comparison task. Recent evidence questions the view that ANS acuity can be assessed using a symbolic number comparison task. To investigate whether there is an association between math anxiety and ANS acuity, we employed both a symbolic number comparison task and a non-symbolic dot comparison task, which is currently the standard task to assess ANS acuity. We replicated previous findings regarding the association between math anxiety and the symbolic distance effect for response times. High math anxious individuals showed a larger distance effect than less math anxious individuals. However, our results revealed no association between math anxiety and ANS acuity assessed using a non-symbolic dot comparison task. Thus, our results did not provide evidence for the hypothesis that a deficient ANS underlies math anxiety. Therefore, we propose that a deficient ANS does not constitute a risk factor for the development of math anxiety. Moreover, our results suggest that previous interpretations regarding the interaction of math anxiety and the symbolic distance effect have to be updated. We suggest that impaired number comparison processes in high math anxious individuals might account for the results rather than deficient ANS representations. Finally, impaired number comparison processes might constitute a risk factor for the development of math anxiety. Implications for current models regarding the origins of math anxiety are discussed.