All for one but not one for all: How multiple number representations are recruited in one numerical task

Add, subtract and multiply: Meta-analyses of brain correlates of arithmetic operations in children and adults

Mathematical operations are cognitive actions we take to calculate relations among numbers. Arithmetic operations, addition, subtraction, multiplication, and division are elemental in education. Addition is the first one taught in school and is most popular in functional magnetic resonance imaging (fMRI) studies. Division, typically taught last is least studied with fMRI. fMRI meta-analyses show that arithmetic operations activate brain areas in parietal, cingulate and insular cortices for children and adults. Critically, no meta-analysis examines concordance across brain correlates of separate arithmetic operations in children and adults. We review and examine using quantitative meta-analyses data from fMRI articles that report brain coordinates separately for addition, subtraction, multiplication, and division in children and adults. Results show that arithmetic operations elicit common areas of concordance in fronto-parietal and cingulo-opercular networks in adults and children. Between operations differences are observed primarily for adults. Interestingly, higher within-group concordance, expressed in activation likelihood estimates, is found in brain areas associated with the cingulo-opercular network rather than the fronto-parietal network in children, areas also common between adults and children. Findings are discussed in relation to constructivist cognitive theory and practical directions for future research.

Examining the relevance of basic numerical skills for mathematical achievement in secondary school using a within-task assessment approach

Previous research repeatedly found basic numerical abilities (e.g., magnitude understanding, arithmetic fact knowledge, etc.) to predict young students' current and later arithmetic achievement as assessed by achievement tests - even when controlling for the influence of domain-general abilities (e.g., intelligence, working memory). However, to the best of our knowledge, previous studies hardly addressed this issue in secondary school students. Additionally, they primarily assessed basic numerical abilities in a between-task approach (i.e., using different tasks for different abilities). Finally, their relevance for real-life academic outcomes such as mathematics grades has only rarely been investigated. The present study therefore pursued an approach using one and the same task (i.e., a within-task approach) to reduce confounding effects driven by between-task differences. In particular, we evaluated the relevance of i) number magnitude understanding, ii) arithmetic fact knowledge, and iii) conceptual and procedural knowledge for the mathematics grades of 81 students aged between ten and thirteen (i.e., in Grades 5 and 6) employing the number bisection task. Results indicated that number magnitude understanding, arithmetic fact knowledge, and conceptual and procedural knowledge contributed to explaining mathematics grades even when controlling for domain-general cognitive abilities. Methodological and practical implications of the results are discussed.

Hemispheric Lateralization of Arithmetic Facts and Magnitude Processing for Two-Digit Numbers

In the human brain, a (relative) functional asymmetry (i.e., laterality; functional and performance differences between the two cerebral hemispheres) exists for a variety of cognitive domains (e.g., language, visual-spatial processing, etc.). For numerical cognition, both bi-lateral and unilateral processing has been proposed with the retrieval of arithmetic facts postulated as being lateralized to the left hemisphere. In this study, we aimed at evaluating this claim by investigating whether processing of multiplicatively related triplets in a number bisection task (e.g., 12_16_20) in healthy participants (n = 23) shows a significant advantage when transmitted to the right hemisphere only as compared to transmission to the left hemisphere. As expected, a control task revealed that stimulus presentation to the left or both visual hemifields did not increase processing disadvantages of unit-decade incompatible number pairs in magnitude comparison. For the number bisection task, we replicated the multiplicativity effect. However, in contrast to the hypothesis deriving from the triple code model, we did not observe significant hemispheric processing asymmetries for multiplicative items. We suggest that participants resorted to keep number triplets in verbal working memory after perceiving them only very briefly for 150 ms. Rehearsal of the three numbers was probably slow and time-consuming so allowing for interhemispheric communication in the meantime. We suggest that an effect of lateralized presentation may only be expected for early effects when the task is sufficiently easy.

Meta-analytic evidence for a core problem solving network across multiple representational domains

Problem solving is a complex skill engaging multi-stepped reasoning processes to find unknown solutions. The breadth of real-world contexts requiring problem solving is mirrored by a similarly broad, yet unfocused neuroimaging literature, and the domain-general or context-specific brain networks associated with problem solving are not well understood. To more fully characterize those brain networks, we performed activation likelihood estimation meta-analysis on 280 neuroimaging problem solving experiments reporting 3166 foci from 1919 individuals across 131 papers. The general map of problem solving revealed broad fronto-cingulo-parietal convergence, regions similarly identified when considering separate mathematical, verbal, and visuospatial problem solving domain-specific analyses. Conjunction analysis revealed a common network supporting problem solving across diverse contexts, and difference maps distinguished functionally-selective sub-networks specific to task type. Our results suggest cooperation between representationally specialized sub-network and whole-brain systems provide a neural basis for problem solving, with the core network contributing general purpose resources to perform cognitive operations and manage problem demand. Further characterization of cross-network dynamics could inform neuroeducational studies on problem solving skill development.

Sex differences in number processing: Differential systems for subtraction and multiplication were confirmed in men, but not in women

Neuroimaging studies suggest segregated neuronal systems underlying number magnitude processing (e.g. subtraction) and arithmetic fact retrieval (e.g. multiplication). While number magnitude processing is thought to rely on the intraparietal sulcus (IPS) bilaterally, arithmetic fact retrieval is thought to rely on the left angular gyrus (AG). However, evidence from brain damaged patients and brain stimulation challenges this view and suggests considerable overlap between the systems underlying number magnitude processing and arithmetic fact retrieval. This study investigates, whether sex differences in number processing can account for these conflicting findings. A subtraction and a multiplication task were administered to 40 men and 34 women in their luteal phase during functional MRI. Replicating previous studies in men, we found the IPS to be more strongly activated during subtraction than multiplication, and the AG to be more strongly activated during multiplication than subtraction. However, no differences between the two tasks were observed in women.

Transcranial Direct Current Stimulation over the Parietal Cortex Improves Approximate Numerical Averaging

The parietal cortex has been implicated in a variety of numerosity and numerical cognition tasks and was proposed to encompass dedicated neural populations that are tuned for analogue magnitudes as well as for symbolic numerals. Nonetheless, it remains unknown whether the parietal cortex plays a role in approximate numerical averaging (rapid, yet coarse computation of numbers' mean)—a process that is fundamental to preference formation and decision-making. To causally investigate the role of the parietal cortex in numerical averaging, we have conducted a transcranial direct current stimulation (tDCS) study, in which participants were presented with rapid sequences of numbers and asked to convey their intuitive estimation of each sequence's average. During the task, the participants underwent anodal (excitatory) tDCS (or sham), applied either on a parietal or a frontal region. We found that, although participants exhibit above-chance accuracy in estimating the average of numerical sequences, they did so with higher precision under parietal stimulation. In a second experiment, we have replicated this finding and confirmed that the effect is number-specific rather than domain-general or attentional. We present a neurocomputational model postulating population-coding underlying rapid numerical averaging to account for our findings. According to this model, stimulation of the parietal cortex elevates neural activity in number-tuned dedicated detectors, leading to increase in the system's signal-to-noise level and thus resulting in more precise estimations.

Behavioural evidence for sex differences in the overlap between subtraction and multiplication

The present study aims to identify factors that may influence the dissociability of number magnitude processing and arithmetic fact retrieval at the behavioural level. To that end, we assessed both subtraction and multiplication performance in a within-subject approach and evaluated the interdependence of unit-decade integration measures on the one hand as well as sex differences in the interdependence of performance measures on the other hand. We found that subtraction items requiring borrowing (e.g. 53-29 = 24, 3 < 9) are more error prone than subtraction items not requiring borrowing (e.g. 59-23 = 34, 9 > 3), thereby demonstrating a borrowing effect, which has been suggested as a measure of unit-decade integration in subtraction. Furthermore, we observed that multiplication items with decade-consistent distractors (e.g. 6 × 4 = 28 instead of 24) are more error prone that multiplication items with decade-inconsistent distractors (e.g. 6 × 4 = 30 instead of 24), thereby demonstrating a decade-consistency effect, which has been suggested as a measure of unit-decade integration in simple multiplication. However, the borrowing effect in subtraction was not correlated with the effect of decade consistency in simple multiplication in either men or women. This indicates that unit-decade integration arises from different systems in subtraction and multiplication. Nevertheless, men outperformed women not only in subtraction, but also in multiplication. Furthermore, subtraction and multiplication performance on correct solution probes were correlated in women, but unrelated in men. Thus, the view of differential systems for number magnitude processing and arithmetic fact retrieval may not be universal across sexes.

What Is the Role of Manual Preference in Hand-Digit Mapping During Finger Counting? A Study in a Large Sample of Right- and Left-Handers

The goal of the present study was to test whether there is a relationship between manual preference and hand-digit mapping in 369 French adults with similar numbers of right- and left-handers. Manual laterality was evaluated with the finger tapping test to evaluate hand motor asymmetry, and the Edinburgh handedness inventory was used to assess manual preference strength (MPS) and direction. Participants were asked to spontaneously "count on their fingers from 1 to 10" without indications concerning the hand(s) to be used. The results indicated that both MPS and hand motor asymmetry affect the hand-starting preference for counting. Left-handers with a strong left-hand preference (sLH) or left-hand motor asymmetry largely started to count with their left hand (left-starter), while right-handers with a strong right-hand preference (sRH) or right-hand motor asymmetry largely started to count with their right hand (right-starter). Notably, individuals with weak MPS did not show a hand-starting preference. These findings demonstrated that manual laterality contributes to finger counting directionality. Lastly, the results showed a higher proportion of sLH left-starter individuals compared with sRH right-starters, indicating an asymmetric bias of MPS on hand-starting preference. We hypothesize that the higher proportion of sLH left-starters could be explained by the congruence between left-to-right hand-digit mapping and left-to-right mental number line representation that has been largely reported in the literature. Taken together, these results indicate that finger-counting habits integrate biological and cultural information.

Adaptive Spontaneous Transitions between Two Mechanisms of Numerical Averaging

We investigated the mechanism with which humans estimate numerical averages. Participants were presented with 4, 8 or 16 (two-digit) numbers, serially and rapidly (2 numerals/second) and were instructed to convey the sequence average. As predicted by a dual, but not a single-component account, we found a non-monotonic influence of set-size on accuracy. Moreover, we observed a marked decrease in RT as set-size increases and RT-accuracy tradeoff in the 4-, but not in the 16-number condition. These results indicate that in accordance with the normative directive, participants spontaneously employ analytic/sequential thinking in the 4-number condition and intuitive/holistic thinking in the 16-number condition. When the presentation rate is extreme (10 items/sec) we find that, while performance still remains high, the estimations are now based on intuitive processing. The results are accounted for by a computational model postulating population-coding underlying intuitive-averaging and working-memory-mediated symbolic procedures underlying analytical-averaging, with flexible allocation between the two.

Number Processing in Arabic and Hebrew Bilinguals

In the current study, a direct assessment of the effect of presentation language and format on the compatibility effect of two-digit numbers was made by contrasting performance of Arabic/Hebrew bilinguals in a digital (Hindi digits/Arabic digits) and verbal numerical comparison task (Arabic an inverted language: units-decades and Hebrew a non-inverted language: decades-units). Our data revealed in digital presentation format a regular compatibility effect in Hindi digits and Arabic digits characterized by lower reaction-time (RT) means for compatible number pairs than incompatible ones with no difference in the RT means of participants in the two languages, Arabic language–Hindi digits as a mother tongue and Hebrew language–Arabic digits as a second language. However, in verbal presentation format, different patterns of compatibility effect were found in Arabic and Hebrew verbal numbers. In Arabic number words, a regular compatibility effect was found, while in Hebrew number words, no compatibility effect was found. This reflects the influence and modulation of the lexical-syntactic structure of the language in two-digit numbers comparison. Evidently, these differences in the compatibility effect advocate and strengthen the claim that two-digit numbers comparison is influenced by the numbers presentation format. Different modes of presentation of two-digit numbers (digital vs. verbal) can lead to different number comparison styles. The parallel model accounts for the numerical comparison in digital presentation, while for the verbal numbers presentation, a revised sequential-syntactic model is preferable.

Mathematics anxiety reduces default mode network deactivation in response to numerical tasks

Mathematics anxiety is negatively related to mathematics performance, thereby threatening the professional success. Preoccupation with the emotional content of the stimuli may consume working memory resources, which may be reflected in decreased deactivation of areas associated with the default mode network (DMN) activated during self-referential and emotional processing. The common problem is that math anxiety is usually associated with poor math performance, so that any group differences are difficult to interpret. Here we compared the BOLD-response of 18 participants with high (HMAs) and 18 participants with low mathematics anxiety (LMAs) matched for their mathematical performance to two numerical tasks (number comparison, number bisection). During both tasks, we found stronger deactivation within the DMN in LMAs compared to HMAs, while BOLD-response in task-related activation areas did not differ between HMAs and LMAs. The difference in DMN deactivation between the HMA and LMA group was more pronounced in stimuli with additional requirement on inhibitory functions, but did not differ between number magnitude processing and arithmetic fact retrieval.

Word problems: a review of linguistic and numerical factors contributing to their difficulty

Word problems (WPs) belong to the most difficult and complex problem types that pupils encounter during their elementary-level mathematical development. In the classroom setting, they are often viewed as merely arithmetic tasks; however, recent research shows that a number of linguistic verbal components not directly related to arithmetic contribute greatly to their difficulty. In this review, we will distinguish three components of WP difficulty: (i) the linguistic complexity of the problem text itself, (ii) the numerical complexity of the arithmetic problem, and (iii) the relation between the linguistic and numerical complexity of a problem. We will discuss the impact of each of these factors on WP difficulty and motivate the need for a high degree of control in stimuli design for experiments that manipulate WP difficulty for a given age group.

On the Relation between the Mental Number Line and Arithmetic Competencies

In this study, we aimed at investigating whether it is indeed the spatial magnitude representation that links number line estimation performance to other basic numerical and arithmetic competencies. Therefore, estimations of 45 fourth-graders in both a bounded and a new unbounded number line estimation task (with only a start-point and a unit given) were correlated with their performance in a variety of tasks including addition, subtraction, and number magnitude comparison. Assuming that both number line tasks assess the same underlying mental number line representation, unbounded number line estimation should also be associated with other basic numerical and arithmetic competencies. However, results indicated that children's estimation performance in the bounded but not the unbounded number line estimation task was correlated significantly with numerical and arithmetic competencies. We conclude that unbounded and bounded number line estimation tasks do not assess the same underlying spatial–numerical representation. Rather, the observed association between bounded number line estimation and numerical/arithmetic competencies may be driven by additional numerical processes (e.g., proportion judgement, addition/subtraction) recruited to solve the task.

Bilateral bi-cephalic tDCS with two active electrodes of the same polarity modulates bilateral cognitive processes differentially [corrected]

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.

Differentiating core and co-opted mechanisms in calculation: the neuroimaging of calculation in aphasia

The role of language in exact calculation is the subject of debate. Some behavioral and functional neuroimaging investigations of healthy participants suggest that calculation requires language resources. However, there are also reports of individuals with severe aphasic language impairment who retain calculation ability. One possibility in resolving these discordant findings is that the neural basis of calculation has undergone significant reorganization in aphasic calculators. Using fMRI, we examined brain activations associated with exact addition and subtraction in two patients with severe agrammatic aphasia and retained calculation ability. Behavior and brain activations during two-digit addition and subtraction were compared to those of a group of 11 healthy, age-matched controls. Behavioral results confirmed that both patients retained calculation ability. Imaging findings revealed individual differences in processing, but also a similar activation pattern across patients and controls in bilateral parietal cortices. Patients differed from controls in small areas of increased activation in peri-lesional regions, a shift from left fronto-temporal activation to the contralateral region, and increased activations in bilateral superior parietal regions. Our results suggest that bilateral parietal cortex represents the core of the calculation network and, while healthy controls may recruit language resources to support calculation, these mechanisms are not mandatory in adult cognition.

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.

Object-based neglect in number processing

Recent evidence suggests that neglect patients seem to have particular problems representing relatively smaller numbers corresponding to the left part of the mental number line. However, while this indicates space-based neglect for representational number space little is known about whether and - if so - how object-based neglect influences number processing.

To evaluate influences of object-based neglect in numerical cognition, a group of neglect patients and two control groups had to compare two-digit numbers to an internally represented standard. Conceptualizing two-digit numbers as objects of which the left part (i.e., the tens digit should be specifically neglected) we were able to evaluate object-based neglect for number magnitude processing.

Object-based neglect was indicated by a larger unit-decade compatibility effect actually reflecting impaired processing of the leftward tens digits. Additionally, faster processing of within- as compared to between-decade items provided further evidence suggesting particular difficulties in integrating tens and units into the place-value structure of the Arabic number system.

In summary, the present study indicates that, in addition to the spatial representation of number magnitude, also the processing of place-value information of multi-digit numbers seems specifically impaired in neglect patients.

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.

Relationships between number and space processing in adults with and without dyscalculia

A large body of evidence indicates clear relationships between number and space processing in healthy and brain-damaged adults, as well as in children. The present paper addressed this issue regarding atypical math development. Adults with a diagnosis of dyscalculia (DYS) during childhood were compared to adults with average or high abilities in mathematics across two bisection tasks. Participants were presented with Arabic number triplets and had to judge either the number magnitude or the spatial location of the middle number relative to the two outer numbers. For the numerical judgment, adults with DYS were slower than both groups of control peers. They were also more strongly affected by the factors related to number magnitude such as the range of the triplets or the distance between the middle number and the real arithmetical mean. By contrast, adults with DYS were as accurate and fast as adults who never experienced math disability when they had to make a spatial judgment. Moreover, number-space congruency affected performance similarly in the three experimental groups. These findings support the hypothesis of a deficit of number magnitude representation in DYS with a relative preservation of some spatial mechanisms in DYS. Results are discussed in terms of direct and indirect number-space interactions.

Menstrual cycle variations in the BOLD-response to a number bisection task: implications for research on sex differences

Numerical processing involves either number magnitude processing, which has been related to spatial abilities and relies on superior parietal regions, or arithmetic fact retrieval, which has been related to verbal abilities and involves the inferior parietal lobule. Since men score better in spatial and women in verbal tasks, we assume that women have advantages in fact retrieval, while men have benefits in number magnitude processing. According to findings on menstrual cycle variations in spatial and verbal abilities, fact retrieval should improve during the luteal phase and magnitude processing during the follicular phase. To dissociate sex- and menstrual cycle-dependent effects on fact retrieval and number magnitude processing, we applied a number bisection task in 15 men and 15 naturally cycling women. Multiplicative items (e.g. 12_15_18) are part of a multiplication series and can be solved by fact retrieval, while non-multiplicative items (e.g. 11_14_17) are not part of a multiplication series and require number magnitude processing. In men and women in their luteal phase, error rates were higher and deactivation of the medial prefrontal cortex and the bilateral inferior parietal lobules was stronger for non-multiplicative compared to multiplicative items (positive multiplicativity effect), while in the follicular phase women showed higher error rates and stronger deactivation in multiplicative compared to non-multiplicative items (negative multiplicativity effect). Thus, number magnitude processing improves, while arithmetic fact retrieval impairs during the follicular phase. While a female superiority in arithmetic fact retrieval could not be confirmed, we observed that sex differences are significantly modulated by menstrual cycle phase.

Representation of multiplication facts--evidence for partial verbal coding

Background

The current view in numerical cognition research is that multiplication facts are stored and retrieved in a phonological code. Consistent with this view, it was found that multiplication could be impaired by a phonological but not by a visuo-spatial loading task. However, because the authors used an active production task, it remained unclear whether concurrent articulation impaired either access to multiplication facts or their retrieval.

Methods

In the current study, we investigated the influence of concurrent articulation on multiplication fact knowledge without active production of multiplication results.

Results

In a number bisection task, number triplets, which are part of a multiplication table, were classified faster as being correctly bisected than other triplets. Interestingly, concurrent articulation led to a relative slowing of the multiplicative triplets which reduced the multiplicativity effect.

Conclusions

This result indicates that concurrent articulation modulates access to phonologically stored multiplication facts and corroborates the notion of multiplication facts being represented in an at least partially verbal code.

Extending the Mental Number Line

Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single-digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language-related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi-digit number processing research may or should be heading in following years.

Complex span tasks and hippocampal recruitment during working memory

The working memory (WM) system is vital to performing everyday functions that require attentive, non-automatic processing of information. However, its interaction with long term memory (LTM) is highly debated. Here, we used fMRI to examine whether a popular complex WM span task, thought to force the displacement of to-be-remembered items in the focus of attention to LTM, recruited medial temporal regions typically associated with LTM functioning to a greater extent and in a different manner than traditional neuroimaging WM tasks during WM encoding and maintenance. fMRI scans were acquired while participants performed the operation span (OSPAN) task and an arithmetic task. Results indicated that performance of both tasks resulted in significant activation in regions typically associated with WM function. More importantly, significant bilateral activation was observed in the hippocampus, suggesting it is recruited during WM encoding and maintenance. Right posterior hippocampus activation was greater during OSPAN than arithmetic. Persitimulus graphs indicate a possible specialization of function for bilateral posterior hippocampus and greater involvement of the left for WM performance. Recall time-course activity within this region hints at LTM involvement during complex span.

Numerosity impairment in corticobasal syndrome

OBJECTIVE

We assessed the representation of numerosity in corticobasal syndrome (CBS), a neurodegenerative condition affecting the parietal lobe.

METHOD

Patients judged whether a target numerosity (e.g., "3") falls between two bounding numerosities (e.g., "1" and "5"). We manipulated the format for representing numerosity (Arabic numerals or dot arrays), the size of the gap between the two bounding numerosities, the absolute magnitude of the numerosities, and the order for presenting the bounding numerosities. In a subset of patients with available imaging, we related performance to cortical atrophy using voxel-based morphometry.

RESULTS

CBS patients were significantly impaired overall (65.7% +/- 16.2 correct) compared to healthy seniors (96.6% +/- 2.4 correct), and required three times longer than controls to judge correct stimuli. This deficit was equally evident for Arabic numeral and dot array formats. Controls were significantly slower with smaller gaps than larger gaps, consistent with the greater challenge distinguishing between numerosities that are more similar to each other than very different numerosities. However, CBS patients were equally slow and inaccurate for all gap sizes. Controls also were significantly slower with larger numerosities than smaller numerosities, but CBS patients were equally slow and inaccurate with all numerosity magnitudes. Voxel-based morphometry revealed significant cortical atrophy in parietal and frontal regions in CBS compared to controls, including the intraparietal sulcus.

CONCLUSIONS

These observations are consistent with the claim that the representation of numerosity is degraded in CBS.

On the neuro-cognitive foundations of basic auditory number processing: an fMRI study

Background

It is widely agreed that numbers automatically activate a magnitude representation. Nevertheless, so far no systematic evaluation of the neuro-cognitive correlates has been provided for the case of auditorily presented numbers.

Methods

To address this question, we presented spoken number words in three different tasks (passive listening, magnitude comparison, parity judgement) as well as spoken pseudowords in an fMRI study.

Results

We found IPS activation typically associated with magnitude processing in all tasks with numerical stimuli only. Interestingly, directly contrasting the two semantic tasks magnitude comparison (magnitude-relevant) and parity judgement (magnitude-irrelevant) revealed a left lateralized predominance within the IPS for the processing of parity information as compared to a right lateralization for number magnitude for auditorily presented number words.

Conclusions

In summary, our results suggest a highly automatic activation of number magnitude for spoken number words similar to previous observations for visually presented numbers, but also indicate that the issue of hemispheric asymmetries deserves specific consideration.

Developmental trajectories of magnitude processing and interference control: an FMRI study

Neurodevelopmental changes regarding interference and magnitude processing were assessed in 3 age groups (children, n = 10; young adults, n = 11; elderly participants, n = 9) by using an functional magnetic resonance imaging version of the numerical Stroop task. Behaviorally, comparable distance and size congruity effects were found in all 3 age groups. Distance effects were most pronounced in the more difficult numerical task, whereas size congruity effects were comparable across tasks. In response to interference, an age-linear trend in the pattern of activation in left and right prefrontal and left middle temporal regions of the brain was observed. This implicates that with increasing age interference control requires increasing effort (possible explanations for children's relatively lower interference effects are provided). In contrast, the distance effect produced a negative linear trend in right prefrontal, supplementary motor area, and intraparietal cortex. This suggests that relative to old adults, children and young adults had to recruit a larger network upon processing magnitude. The latter findings are even more remarkable considering that the behavioral effects were similar across groups. In summary, the developmental trajectories of interference control and magnitude processing differ, although these cognitive functions activate partially overlapping brain regions.

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.

Using parametric regressors to disentangle properties of multi-feature processes

FMRI data observed under a given experimental condition may be decomposed into two parts: the average effect and the deviation of single replications from this average effect. The average effect is represented by the mean activation over a specific condition. The deviation from this average effect may be decomposed into two components as well: systematic variation due to known empirical factors and pure measurement error. In most fMRI designs deviations from mean activation may be treated as measurement error. Nevertheless, often deviation from the average also may contain systematic variation that can be distinguished from simple measurement error. In these cases, the average fMRI signal may provide only a coarse picture of real brain activation. The larger the variation within-condition, the coarser the average effect and the more relevant is the impact of deviations from it. Systematic deviation from the mean activation may be examined by defining a set of parametric regressors. Here, the applicability of parametric methods to refine the evaluation of fMRI studies is discussed with special emphasis on (i) examination of the impact of continuous predictors on the fMRI signal, (ii) control for variation within each experimental condition and (iii) isolation of specific contributions by different features of a single complex stimulus, especially in the case of a sampled stimulus. The usefulness and applicability of this method are discussed and an example with real data is presented.