A general number-to-space mapping deficit in developmental dyscalculia.

Previous research on developmental dyscalculia (DD) suggested that deficits in the number line estimation task are related to a failure to represent number magnitude linearly. This conclusion was derived from the observation of logarithmically shaped estimation patterns. However, recent research questioned this idea of an isomorphic relationship between estimation patterns and number magnitude representation. In the present study, we evaluated an alternative hypothesis: impairments in the number line estimation task are due to a general deficit in mapping numbers onto space. Adults with DD and a matched control group had to learn linear and non-linear layouts of the number line via feedback. Afterwards, we assessed their performance how well they learnt the new number-space mappings. We found irrespective of the layouts worse performance of adults with DD. Additionally, in case of the linear layout, we observed that their performance did not differ from controls near reference points, but that differences between groups increased as the distance to reference point increased. We conclude that worse performance of adults with DD in the number line task might be due a deficit in mapping numbers onto space which can be partly overcome relying on reference points.

Operational momentum affects eye fixation behaviour.

The operational momentum effect (OM) indicates an association of mental addition with a rightward spatial bias, whereas subtraction is associated with a leftward bias. To evaluate the assumed attentional origin of the OM effect, we evaluated not only participants' relative estimation error in a task requiring them to locate addition and subtraction results on a given number line but also their eye-fixation behaviour. Furthermore, to investigate the situatedness of spatial-numerical associations, the orientation of the number line (left-to-right vs. right-to left) was manipulated. OM biases in participants' explicit number line estimations and more implicit eye-fixation behaviour are integrated into a two-process hypothesis of the OM effect suggesting a first rough spatial anticipation followed by an evaluation/correction process. This account not only is capable of accounting for the results observed for participants' relative estimation error but is also corroborated by the eye-fixation results. Importantly, the fact that all effects were found independent of the orientation of the number line indicates that spatial-numerical associations such as the OM effect may not be hard-wired associations of spatial and numerical representations but rather reflect influences of situatedness on numerical cognition.

Adaptive processing of fractions--evidence from eye-tracking.

Recent evidence indicated that fraction pair type determined whether a particular fraction is processed holistically, componentially or in a hybrid manner. Going beyond previous studies, we investigated how participants adapt their processing of fractions not only to fraction type, but also to experimental context. To examine adaptation in fraction processing, we recorded participants' eye-fixation behaviour in a fraction magnitude comparison task. Participants' eye fixation behaviour indicated componential processing of fraction pairs with common components for which the decision-relevant components are easy to identify. Importantly, we observed that fraction processing was adapted to experimental context: Evidence for componential processing was stronger, when experimental context allowed valid expectations about which components are decision-relevant. Taken together, we conclude that fraction processing is adaptive beyond the comparison of different fraction types, because participants continuously adjust to the experimental context in which fractions are processed.

Cognitive control in number magnitude processing: evidence from eye-tracking.

The unit-decade compatibility effect describes longer response times and higher error rates for incompatible (e.g., 37_52) than compatible (e.g., 42_57) number comparisons. Recent research indicated that the effect depends on the percentage of same-decade filler items. In the present study, we further examined this relationship by recording participants' eye-fixation behaviour. In four conditions, participants had to compare item sets with different filler item types (i.e., same-decade and same-unit filler items) and different numbers of same-decade filler items (i.e., 25, 50, and 75%). We found a weaker unit-decade compatibility effect with most fixations on tens in the condition with same-unit filler items. Moreover, the compatibility effect increased with the percentage of same-decade filler items which was accompanied by less fixations on tens and more fixations on units. Thus, our study provides first eye-tracking evidence for the influence of cognitive control in number processing.

Language influences on numerical development-Inversion effects on multi-digit number processing.

In early numerical development, children have to become familiar with the Arabic number system and its place-value structure. The present review summarizes and discusses evidence for language influences on the acquisition of the highly transparent structuring principles of digital-Arabic digits by means of its moderation through the transparency of the respective language's number word system. In particular, the so-called inversion property (i.e., 24 named as "four and twenty" instead of "twenty four") was found to influence number processing in children not only in verbal but also in non-verbal numerical tasks. Additionally, there is first evidence suggesting that inversion-related difficulties may influence numerical processing longitudinally. Generally, language-specific influences in children's numerical development are most pronounced for multi-digit numbers. Yet, there is currently only one study on three-digit number processing for German-speaking children. A direct comparison of additional new data from Italian-speaking children further corroborates the assumption that language impacts on cognitive (number) processing as inversion-related interference was found most pronounced for German-speaking children. In sum, we conclude that numerical development may not be language-specific but seems to be moderated by language.

One language, two number-word systems and many problems: numerical cognition in the Czech language.

Comparing numerical performance between different languages does not only mean comparing different number-word systems, but also implies a comparison of differences regarding culture or educational systems. The Czech language provides the remarkable opportunity to disentangle this confound as there exist two different number-word systems within the same language: for instance, "25" can be either coded in non-inverted order "dvadsetpät" [twenty-five] or in inverted order "pätadvadset" [five-and-twenty]. To investigate the influence of the number-word system on basic numerical processing within one culture, 7-year-old Czech-speaking children had to perform a transcoding task (i.e., writing Arabic numbers to dictation) in both number-word systems. The observed error pattern clearly indicated that the structure of the number-word system determined transcoding performance reliably: In the inverted number-word system about half of all errors were inversion-related. In contrast, hardly any inversion-related errors occurred in the non-inverted number-word system. We conclude that the development of numerical cognition does not only depend on cultural or educational differences, but is indeed related to the structure and transparency of a given number-word system.

Early place-value understanding as a precursor for later arithmetic performance--a longitudinal study on numerical development.

It is assumed that basic numerical competencies are important building blocks for more complex arithmetic skills. The current study aimed at evaluating this interrelation in a longitudinal approach. It was investigated whether first graders' performance in basic numerical tasks in general as well as specific processes involved (e.g., place-value understanding) reliably predicted performance in an addition task in third grade. The results indicated that early place-value understanding was a reliable predictor for specific aspects of arithmetic performance. Implications of the role of basic numerical competencies for the acquisition of complex arithmetic are discussed.

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.

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.

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.

Factors related to the magnitude of T2* MR signal changes during functional imaging.

Our aim was to determine whether age, sex, the degree of weakness, anticonvulsants, the histology of the underlying lesion(s), the presence of oedema or the distance of the lesion from the motor region have an impact on the blood oxygenation level-dependent (BOLD) signal strength and therefore on the validity of functional MRI (fMRI). We studied 98 patients with masses near the central region imaged for surgical planning at 1.5 tesla, employing a BOLD sequence during a motor task. We calculated percentage signal change in the primary motor cortex between rest and activation and carried out multiple linear regression to examine the impact of the above factors on signal strength. Using a stepwise analysis strategy, the distance of the lesion from the motor region had the strongest influence (r=0.653, P<0.001). The factor with largest uncorrelated additional impact on signal change was the presence of oedema. Both predictors together formed a highly significant multiple r=0.739 ( P<0.001). No other predictive factor was identified (all P>0.20). Disturbances of cerebral blood flow and metabolism induced by the tumour were presumed to be the causes of a decrease in signal in the adjacent cortex.

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.