The residual calculation abilities of a patient with severe aphasia: evidence for a selective deficit of subtraction procedures

Deafness and early language deprivation influence arithmetic performances

It has been consistently reported that deaf individuals experience mathematical difficulties compared to their hearing peers. However, the idea that deafness and early language deprivation might differently affect verbal (i.e., multiplication) vs. visuospatial (i.e., subtraction) arithmetic performances is still under debate. In the present paper, three groups of 21 adults (i.e., deaf signers, hearing signers, and hearing controls) were therefore asked to perform, as fast and as accurately as possible, subtraction and multiplication operations. No significant group effect was found for accuracy performances. However, reaction time results demonstrated that the deaf group performed both arithmetic operations slower than the hearing groups. This group difference was even more pronounced for multiplication problems than for subtraction problems. Weaker language-based phonological representations for retrieving multiplication facts, and sensitivity to interference are two hypotheses discussed to explain the observed dissociation.

Numerical cognition: A meta-analysis of neuroimaging, transcranial magnetic stimulation and brain-damaged patients studies

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We review neuroimaging, TMS, and patients studies on numerical cognition.

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We focused on the predictions derived from the Triple Code Model (TCM).

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Our findings generally agree with TCM predictions.

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Our results open avenues for the study of the neural bases of numerical cognition.

This article offers the first comprehensive review examining the neurocognitive bases of numerical cognition from neuroimaging, Transcranial Magnetic Stimulation (TMS) and brain-damaged patients studies. We focused on the predictions derived from the Triple Code Model (TCM), particularly the assumption that the representation of numerical quantities rests on a single format-independent representation (i.e., the analogical code) involving both intraparietal sulci (IPS). To do so, we conducted a meta-analysis based on 28 neuroimaging, 12 TMS and 12 brain-damaged patients studies, including arithmetic and magnitude tasks in symbolic and non-symbolic formats. Our findings generally agree with the TCM predictions indicating that both IPS are engaged in all tasks. Nonetheless, the results of brain-damaged patients studies conflicted with neuroimaging and TMS studies, suggesting a right hemisphere lateralization for non-symbolic formats. Our findings also led us to discuss the involvement of brain regions other than IPS in the processing of the analogical code as well as the neural substrate of other codes underlying numerical cognition (i.e., the auditory-verbal code).

The neural bases of the multiplication problem-size effect across countries

Multiplication problems involving large numbers (e.g., 9 × 8) are more difficult to solve than problems involving small numbers (e.g., 2 × 3). Behavioral research indicates that this problem-size effect might be due to different factors across countries and educational systems. However, there is no neuroimaging evidence supporting this hypothesis. Here, we compared the neural correlates of the multiplication problem-size effect in adults educated in China and the United States. We found a greater neural problem-size effect in Chinese than American participants in bilateral superior temporal regions associated with phonological processing. However, we found a greater neural problem-size effect in American than Chinese participants in right intra-parietal sulcus (IPS) associated with calculation procedures. Therefore, while the multiplication problem-size effect might be a verbal retrieval effect in Chinese as compared to American participants, it may instead stem from the use of calculation procedures in American as compared to Chinese participants. Our results indicate that differences in educational practices might affect the neural bases of symbolic arithmetic.

When solving 22-7 is much more difficult than 99-12

We describe the case of a 69-year-old professor of mathematics (GV) who was examined 2 years after left-hemispheric capsular-thalamic haemorrhage. GV showed disproportionate impairment in subtractions requiring borrowing (22 - 7). For large subtraction problems without borrowing (99 - 12) performance was almost flawless. Subtractions with borrowing mostly relied on inadequate attempts to invert subtractions into the corresponding additions (22 - 7 = x as 7 + x = 22). The hypothesis is advanced that difficulty in the inhibitory components of attention tasks (Stroop test, go-no-go task) might be the responsible factor of his calculation impairment. A deficit in subtractions with borrowing might be related to left-hemispheric damage involving thalamo-cortical connections.

Common substrate for mental arithmetic and finger representation in the parietal cortex

The history of mathematics provides several examples of the use of fingers to count or calculate. These observations converge with developmental data showing that fingers play a critical role in the acquisition of arithmetic knowledge. Further studies evidenced specific interference of finger movements with arithmetic problem solving in adults, raising the question of whether or not finger and number manipulations rely on common brain areas. In the present study, functional magnetic resonance imaging (fMRI) was used to investigate the possible overlap between the brain areas involved in mental arithmetic and those involved in finger discrimination. Solving subtraction and multiplication problems was found to increase cerebral activation bilaterally in the horizontal part of the intraparietal sulcus (hIPS) and in the posterior part of the superior parietal lobule (PSPL). Finger discrimination was associated with increased activity in a bilateral occipito-parieto-precentral network extending from the extrastriate body area to the primary somatosensory and motor cortices. A conjunction analysis showed common areas for mental arithmetic and finger representation in the hIPS and PSPL bilaterally. Voxelwise correlations further showed that finger discrimination and mental arithmetic induced a similar pattern of activity within the parietal areas only. Pattern similarity was more important for the left than for the right hIPS and for subtraction than for multiplication. These findings provide the first evidence that the brain circuits involved in finger representation also underlie arithmetic operations in adults.

Role of distinct parietal areas in arithmetic: an fMRI-guided TMS study

Although several parietal areas are known to be involved in number processing, their possible role in arithmetic operations remains debated. It has been hypothesized that the horizontal segment of the intraparietal sulcus (hIPS) and the posterior superior parietal lobule (PSPL) contribute to operations solved by calculation procedures, such as subtraction, but whether these areas are also involved in operations solved by memory retrieval, such as multiplication, is controversial. In the present study, we first identified the parietal areas involved in subtraction and multiplication by means of functional magnetic resonance imaging (fMRI) and we found an increased activation, bilaterally, in the hIPS and PSPL during both arithmetic operations. In order to test whether these areas are causally involved in subtraction and multiplication, we used transcranial magnetic stimulation (TMS) to create, in each participant, a virtual lesion of either the hIPS or PSPL, over the sites corresponding to the peaks of activation gathered in fMRI. When compared to a control site, we found an increase in response latencies in both operations after a virtual lesion of either the left or right hIPS, but not of the PSPL. Moreover, TMS over the hIPS increased the error rate in the multiplication task. The present results indicate that even operations solved by memory retrieval, such as multiplication, rely on the hIPS. In contrast, the PSPL seems to underlie processes that are nonessential to solve basic subtraction and multiplication problems.

Acalculia in a patient with severe language disturbances: how do we test it?

The present paper describes a case of a patient with severe Wernicke aphasia, which when tested with a number processing and calculation battery adapted to his difficulties showed remarkable arithmetic skills. These findings suggest that the patients with severe cognitive impairments (e.g., aphasia, apraxia) should be tested with batteries adapted to their disturbances because using a standard test may bias the results.

Arithmetic procedural knowledge: a cortico-subcortical circuit

The disturbances of arithmetic procedural knowledge form a heterogeneous picture, in which we can distinguish "memory" impairments and "monitoring" problems. Patients with "memory" disturbances reported in the literature present left parietal lesions, while "monitoring" impairments have been assumed to be due to frontal damage. Procedural knowledge has been less investigated in basal ganglia lesions, in which there has been no analysis of procedural impairments. The present study investigates and compares the patterns of acalculia in two patients, one with a left parietal lesion and the other with a left basal ganglia lesion. The patients were tested on a broad range of neuropsychological abilities, with the main focus on number processing and calculation. The results show many similarities between their deficits, with some difficulties in simple arithmetic, arithmetical rules and mental and written complex calculations. The errors made in complex mental and written calculations were due to memory-based procedural impairments in both patients. These findings, corroborated with other studies reported in the literature, suggest the existence of a fronto-parieto-subcortical circuit responsible for arithmetic complex calculations and that procedural knowledge relies on a visuo-spatial sketchpad that contains a representation of each sub-step of the procedure.

A case of acalculia due to impaired procedural knowledge

The present paper describes the single-case of a patient presenting acalculia with preserved arithmetic facts but impaired procedural knowledge, being unable to resolve mental or written complex calculations. The implications of these findings are discussed in the context of the different theoretical models that have been proposed for the cognitive mechanisms underlying calculation skills.

Acalculia from a right hemisphere lesion dealing with "where" in multiplication procedures

The present study describes in detail, for the first time, a case of failure with multiplication procedures in a right hemisphere damaged patient (PN). A careful, step-by-step, error analysis made possible to show that an important portion of PN's errors could be better explained as spatial in nature and specifically related to the demands of a multi-digit multiplication. These errors can be distinguished from other types of errors, including those, expected after a right hemisphere lesion, determined by a generic inability to deal with spatial material, or from other deficits, like neglect, affecting cognitive capacities across the board. The best explanation for PN's problems is that he might have difficulties in relying on a visuo-spatial store containing a layout representation specific to multiplication. As a consequence, while knowing what, when and how to carry out the various steps, PN does not know where. What he may thus lack is a spatial schema of multiplication. Such schema is thought to help normal calculators in overcoming working memory demands of complex calculation by representing the information of where exactly each sub-step should be placed.

The role of working memory in carrying and borrowing

The present study analyzed the role of phonological and executive components of working memory in the borrow operation in complex subtractions (Experiments 1 and 2) and in the carry operation in complex multiplications (Experiments 3 and 4). The number of carry and borrow operations as well as the value of the carry were manipulated. Results indicated that both the number of carry/borrow operations and the value of the carry increased problem difficulty, resulting in higher reliance on phonological and executive working-memory components. Present results are compared with those obtained for the carry operation in complex addition and are further discussed in the broader framework of working-memory functions.

The differential involvement of inferior parietal lobule in number comparison: a rTMS study

Number processing is known to involve several sites within the posterior regions of parietal cortex. Here, we investigated whether neural activity in the inferior parietal lobule (IPL) is essential for number processing, by observing the effects of interfering with its activity during the execution of a standard number comparison task. Subjects performance on the task was significantly slowed down when we delivered trains of repetitive transcranial magnetic stimuli (rTMS) to the posterior parietal scalp site overlying the left IPL, while rTMS did not affect the number comparison task if delivered to homologous, contralateral (right) IPL. In conclusion, the present findings add support to a growing body of evidence from neuropsychology and neuroimaging studies that the left inferior parietal lobule is an important component of the networks subserving the representation of quantity.