Elise Klein

The Influence of Implicit Hand-Based Representations on Mental Arithmetic

Recently, a strong functional relationship between finger counting and number processing has been suggested. It has been argued that bodily experiences such as finger counting may influence the structure of the basic mental representations of numbers even in adults. However, to date it remains unclear whether the structure of finger counting systems also influences educated adults’ performance in mental arithmetic. In the present study, we pursued this question by examining finger-based sub-base-five effects in an addition production task. With the standard effect of a carry operation (i.e., base-10 crossing) being replicated, we observed an additional sub-base-five effect such that crossing a sub-base-five boundary led to a relative response time increase. For the case of mental arithmetic sub-base-five effects have previously been reported only in children. However, it remains unclear whether finger-based numerical effects in mental arithmetic reflect an important but transitory step in the development of arithmetical skills. The current findings suggest that even in adults embodied representations such as finger counting patterns modulate arithmetic performance. Thus, they support the general idea that even seemingly abstract cognition in adults may at least partly be rooted in our bodily experiences.

Processing pathways in mental arithmetic--evidence from probabilistic fiber tracking.

Numerical cognition is a case of multi-modular and distributed cerebral processing. So far neither the anatomo-functional connections between the cortex areas involved nor their integration into established frameworks such as the differentiation between dorsal and ventral processing streams have been specified. The current study addressed this issue combining a re-analysis of previously published fMRI data with probabilistic fiber tracking data from an independent sample. We aimed at differentiating neural correlates and connectivity for relatively easy and more difficult addition problems in healthy adults and their association with either rather verbally mediated fact retrieval or magnitude manipulations, respectively. The present data suggest that magnitude- and fact retrieval-related processing seem to be subserved by two largely separate networks, both of them comprising dorsal and ventral connections. Importantly, these networks not only differ in localization of activation but also in the connections between the cortical areas involved. However, it has to be noted that even though seemingly distinct anatomically, these networks operate as a functionally integrated circuit for mental calculation as revealed by a parametric analysis of brain activation.

Considering structural connectivity in the triple code model of numerical cognition: differential connectivity for magnitude processing and arithmetic facts

The current study provides a generalizable account of the anatomo-functional associations as well as the connectivity of representational codes underlying numerical processing as suggested by the triple code model (TCM) of numerical cognition. By evaluating the neural networks subserving numerical cognition in two specific and substantially different numerical tasks with regard to both grey matter localizations as well as white matter tracts we (1) considered the possibility of additional memory-related cortex areas crucial for arithmetic fact retrieval (e.g., the hippocampus); (2) specified the functional involvement of prefrontal areas in number magnitude processing, and, finally; (3) identified the connections between these anatomo-functional instantiations of the representations involved in number magnitude processing and arithmetic fact retrieval employing probabilistic fiber tracking. The resulting amendments to the TCM are summarized in a schematic update, and ideas concerning the possible functional interplay between number magnitude processing and arithmetic fact retrieval are discussed.

The Exact vs. Approximate Distinction in Numerical Cognition May Not Be Exact, but Only Approximate: How Different Processes Work Together in Multi-Digit Addition.

Two types of calculation processes have been distinguished in the literature: approximate processes are supposed to rely heavily on the non-verbal quantity system, whereas exact processes are assumed to crucially involve the verbal system. These two calculation processes were commonly distinguished by manipulation of two factors in addition problems: the identity of the target and the distance of the distractor. However, in all previous studies, these two factors were not manipulated independently. In this fMRI study, we could disentangle the two factors by using a different (two-digit) number stimulus set. Both behavioral and neurofunctional data suggest that the cognitive processes involved could be best explained by the (independent) factors target and distractor distance. Based on these data we suggest that the exact/approximate distinction does not seem to be as generally valid as previously assumed. We conclude that this study may be a starting point for a closer examination of the experimental, procedural and strategic conditions of when the exact/approximate distinction is valid and when it is not.

To carry or not to carry — Is this the question? Disentangling the carry effect in multi-digit addition

Recent research has suggested addition performance to be determined by both the need for a carry operation and problem size. Nevertheless, it has remained debatable, how these two factors are interrelated. In the current study, this question was pursued by orthogonally manipulating carry and problem size in two-digit addition verification. As the two factors interacted reliably, our results indicate that the carry effect is moderated by number magnitude processing rather than representing a purely procedural, asemantic sequence of processing steps. Moreover, it was found that the carry effect may not be a purely categorical effect but may be driven by continuous characteristics of the sum of the unit digits as well. Since the correct result of a carry problem can only be derived by integrating and updating the magnitudes of tens and units within the place-value structure of the Arabic number system, the present study provides evidence for the idea that decomposed processing of tens and units also transfers to mental arithmetic.

A review on functional and structural brain connectivity in numerical cognition

Only recently has the complex anatomo-functional system underlying numerical cognition become accessible to evaluation in the living brain. We identified 27 studies investigating brain connectivity in numerical cognition. Despite considerable heterogeneity regarding methodological approaches, populations investigated, and assessment procedures implemented, the results provided largely converging evidence regarding the underlying brain connectivity involved in numerical cognition. Analyses of both functional/effective as well as structural connectivity have consistently corroborated the assumption that numerical cognition is subserved by a fronto-parietal network including (intra)parietal as well as (pre)frontal cortex sites. Evaluation of structural connectivity has indicated the involvement of fronto-parietal association fibers encompassing the superior longitudinal fasciculus dorsally and the external capsule/extreme capsule system ventrally. Additionally, commissural fibers seem to connect the bilateral intraparietal sulci when number magnitude information is processed. Finally, the identification of projection fibers such as the superior corona radiata indicates connections between cortex and basal ganglia as well as the thalamus in numerical cognition. Studies on functional/effective connectivity further indicated a specific role of the hippocampus. These specifications of brain connectivity augment the triple-code model of number processing and calculation with respect to how gray matter areas associated with specific number-related representations may work together.

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.

Bilateral Bi-Cephalic Tdcs with Two Active Electrodes of the Same Polarity Modulates Bilateral Cognitive Processes Differentially

Transcranial direct current stimulation (tDCS) is an innovative method to explore the causal structure-function relationship of brain areas. We investigated the specificity of bilateral bi-cephalic tDCS with two active electrodes of the same polarity (e.g., cathodal on both hemispheres) applied to intraparietal cortices bilaterally using a combined between- and within-task approach. Regarding between-task specificity, we observed that bilateral bi-cephalic tDCS affected a numerical (mental addition) but not a control task (colour word Stroop), indicating a specific influence of tDCS on numerical but not on domain general cognitive processes associated with the bilateral IPS. In particular, the numerical effect of distractor distance was more pronounced under cathodal than under anodal stimulation. Moreover, with respect to within-task specificity we only found the numerical distractor distance effect in mental addition to be modulated by direct current stimulation, whereas the effect of target identity was not affected. This implies a differential influence of bilateral bi-cephalic tDCS on the recruitment of different processing components within the same task (number magnitude processing vs. recognition of familiarity). In sum, this first successful application of bilateral bi-cephalic tDCS with two active electrodes of the same polarity in numerical cognition research corroborates the specific proposition of the Triple Code Model that number magnitude information is represented bilaterally in the intraparietal cortices.

Rehabilitation of arithmetic fact retrieval via extensive practice: A combined fMRI and behavioural case-study

The present study investigates the effects of a training of arithmetic fact retrieval in a patient suffering from particular difficulties with multiplication facts. Over a period of four weeks simple multiplication facts were trained extensively. The outcome of the training was assessed behaviourally and changes in cerebral activation patterns were investigated using fMRI. The training led to a change in calculation strategies: Prior to training, the patient used predominantly time-consuming back-up strategies, after training he relied increasingly on the direct retrieval of arithmetic facts from long-term memory. Regarding the fMRI results, prefrontal activations were observed for untrained problems, which can be attributed to the application of back-up strategies strongly relying on fronto-executive functions. Interestingly, significant foci of activation for both trained and untrained items were found in the angular gyrus of the right hemisphere, which, however, differed in their exact localisation. For the trained condition, activations were observed in anterior parts of the angular gyrus which may be related to the training-based automatisation in fact retrieval. Activations in the untrained condition were found in a more posterior portion of the angular gyrus, that might be attributable to one of the patients back-up strategies, namely to recite a whole multiplication row to get to the correct answer.

Categorical and continuous - disentangling the neural correlates of the carry effect in multi-digit addition

BackgroundRecently it was suggested that the carry effect observed in addition involves both categorical and continuous processing characteristics.MethodsIn the present study, we aimed at identifying the specific neural correlates associated with processing either categorical or continuous aspects of the carry effect in an fMRI study on multi-digit addition.ResultsIn line with our expectations, we observed two distinct parts of the fronto-parietal network subserving numerical cognition to be associated with either one of these two characteristics. On the one hand, the categorical aspect of the carry effect was associated with left-hemispheric language areas and the basal ganglia probably reflecting increased demands on procedural and problem solving processes. Complementarily, the continuous aspect of the carry effect was associated with increased intraparietal activation indicating increasing demands on magnitude processing as well as place-value integration with increasing unit sum.ConclusionsIn summary, the findings suggest representations and processes underlying the carry effect in multi-digit addition to be more complex and interactive than assumed previously.