Subtraction bugs in an acalculic patient

Exploring the linguistic complexity of third-grade numerical literacy

Reading numbers aloud, a central aspect of numerical literacy, is a challenging skill to acquire, but the origins of this difficulty remain poorly understood. To investigate this matter, we examined the performance of 127 third- and fourth-grade children who read aloud, in Hebrew, numbers with 2-5 digits. We found several key observations. First, we observed a substantial variation among the 3rd graders-7% and 59% errors in the top and bottom deciles, respectively. Second, the task difficulty stemmed from syntactic processing: Most errors were distortions of the number's syntax, as opposed to digit substitutions or transpositions, and the main factor affecting a specific number's difficulty was not its magnitude, as is commonly assumed, but rather its syntactic structure. Third, number reading performance was not predicted by a school-like task that assessed syntactic-conceptual knowledge of the decimal system structure, but rather by knowledge of specific syntactic-verbal rules, suggesting that the syntactic-verbal knowledge is separate from the syntactic-conceptual knowledge. Last, there was a double dissociation between 4-digit numbers and 5-digit numbers, which in Hebrew have completely different syntactic structures: Half of the children showed a significant advantage in one number length compared to the other, with equal numbers of children preferring either length. This indicates that the different syntactic-verbal rules are learned relatively independently of each other, with little or no generalization from one rule to another. In light of these findings, we propose that schools should specifically teach number reading, with focus on specific syntactic-verbal rules.

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.

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.

Flexible transfer of knowledge in mental arithmetic--an fMRI study

Recent imaging studies could show that fact acquisition in arithmetic is associated with decreasing activation in several frontal and parietal areas, and relatively increasing activation within the angular gyrus, indicating a switch from direct calculation to retrieval of a learned fact from memory. So far, however, little is known about the transfer of learned facts between arithmetic operations. The aim of the present fMRI study was to investigate whether and how newly acquired arithmetic knowledge might transfer from trained multiplication problems to related division problems. On the day before scanning, ten complex multiplication problems were trained. Within the scanner, trained multiplication problems were compared with untrained multiplication problems, and division problems related to multiplication (transfer condition) were compared with unrelated division problems (no-transfer condition). Replicating earlier results, untrained multiplication problems activated several frontal and parietal brain areas more strongly than trained multiplication problems, while trained multiplication problems showed relatively stronger activation in the left angular gyrus than untrained multiplication problems. Concerning division, an ROI analysis indicated that activation in the left angular gyrus was relatively stronger for the transfer condition than for the no-transfer condition. We also observed distinct inter-individual differences with regard to transfer that modulated activation within the left angular gyrus. Activation within the left angular gyrus was generally higher for participants who showed a transfer effect for division problems. In conclusion, the present study yielded some evidence that successful transfer of knowledge between arithmetic operations is accompanied by modifications of brain activation patterns. The left angular gyrus seems not only to be involved in the retrieval of stored arithmetic facts, but also in the transfer between arithmetic operations.

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.

EFNS guidelines on cognitive rehabilitation: report of an EFNS task force

Disorders of language, spatial perception, attention, memory, calculation and praxis are a frequent consequence of acquired brain damage [in particular, stroke and traumatic brain injury (TBI)] and a major determinant of disability. The rehabilitation of aphasia and, more recently, of other cognitive disorders is an important area of neurological rehabilitation. We report here a review of the available evidence about effectiveness of cognitive rehabilitation. Given the limited number and generally low quality of randomized clinical trials (RCTs) in this area of therapeutic intervention, the Task Force considered, besides the available Cochrane reviews, evidence of lower classes which was critically analysed until a consensus was reached. In particular, we considered evidence from small group or single cases studies including an appropriate statistical evaluation of effect sizes. The general conclusion is that there is evidence to award a grade A, B or C recommendation to some forms of cognitive rehabilitation in patients with neuropsychological deficits in the post-acute stage after a focal brain lesion (stroke, TBI). These include aphasia therapy, rehabilitation of unilateral spatial neglect (ULN), attentional training in the post-acute stage after TBI, the use of electronic memory aids in memory disorders, and the treatment of apraxia with compensatory strategies. There is clearly a need for adequately designed studies in this area, which should take into account specific problems such as patient heterogeneity and treatment standardization.

Does the Left Inferior Parietal Lobule Contribute to Multiplication Facts?

We report a single case, who presents with a selective and severe impairment for multiplication and division facts. His ability to retrieve subtraction and addition facts was entirely normal. His brain lesion affected the left superior temporal and to lesser extent in the left middle temporal gyri and the left precentral gyrus extending inferiorly to the pars opercularis of the left frontal lobe. Interestingly, the left supramarginal and angular gyri (SMG/AG) were spared. This finding realised a double dissociation with a previously reported patient, who despite lesions in the SMG/AG did not have a multiplication impairment (van Harskamp et al., 2002). The previously suggested crucial role of the SMG/AG in the retrieval of simple multiplication facts is therefore poorly supported (Cohen et al., 2000; Lee, 2000).

Number processing and basal ganglia dysfunction: a single case study

Numerical processing has never been investigated in a case of Fahr's disease (FD) and only rarely in cases of basal ganglia dysfunction. The study describes the cognitive decline of a pre-morbidly high-functioning patient (medical doctor) affected by FD and his difficulties in number processing. A MRI scan revealed bilateral calcifications in the basal ganglia and a brain PET showed a massive reduction of glucose metabolism in the basal ganglia and both frontal lobes, but no other brain abnormalities. The patient's cognitive deficits included impairments in problem solving, in cognitive set shifting and in mental flexibility, as well as in verbal memory. These deficits are attributed to the disruption of the dorsolateral prefrontal circuit involving the basal ganglia. In number processing, the patient showed a severe deficit in the retrieval of multiplication facts, deficits in all tasks of numerical problem solving and in the execution of complex procedures. Importantly, he also showed a dense deficit in conceptual knowledge, which concerned all test conditions and all operations. The findings confirm the predictions of the triple code model in so far, as a disruption of cortico-subcortical loops involving the basal-ganglia may lead to specific deficits in fact retrieval. However, no verbal deficit, as assumed in the triple code model and reported in similar cases, could be observed. The present findings further add to current knowledge on numerical processing, showing how fronto-executive dysfunction may disrupt conceptual understanding of arithmetic. This study shows that not only parietal lesions may lead to severe deficits in conceptual understanding, but that basal ganglia lesions leading to frontal dysfunction may have a devastating effect.

Learning complex arithmetic—an fMRI study

Aim of the present functional magnet resonance imaging (fMRI) study was to detect modifications of cerebral activation patterns related to learning arithmetic. Thirteen right-handed subjects were extensively trained on a set of 18 complex multiplication problems. In the following fMRI session, trained and untrained problems (closely matched for difficulty) were presented in blocked order alternating with a number matching task and a fact retrieval task. Importantly, left hemispheric activations were dominant in the two contrasts between untrained and trained condition, suggesting that learning processes in arithmetic are predominantly supported by the left hemisphere. Contrasting untrained versus trained condition, the left intraparietal sulcus showed significant activations, as well as the inferior parietal lobule. A further significant activation was found in the left inferior frontal gyrus. This activation may be accounted for by higher working memory demands in the untrained as compared to the trained condition. Contrasting trained versus untrained condition a significant focus of activation was found in the left angular gyrus. Following the triple-code model [Science 284 (1999) 970], the shift of activation within the parietal lobe from the intraparietal sulcus to the left angular gyrus suggests a modification from quantity-based processing to more automatic retrieval. The present study shows that the left angular gyrus is not only involved in arithmetic tasks requiring simple fact retrieval, but may show significant activations as a result of relatively short training of complex calculation.

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

We report the case study of a severe fluent aphasic patient, who showed relatively preserved numerical abilities. A detailed investigation of number processing indicated good numerical comprehension and a relative sparing of addition and subtraction abilities; on the other hand, multiplication and division were severely impaired. A further study of multi-digit operations showed that the patient's performance was characterized by a selective impairment of the borrowing procedure, in which she applied the so-called Smaller-from-Larger bug, typically observed in children learning to calculate. The present case provides further evidence for the dissociation between operations based on verbal sequences and on quantity manipulation, respectively impaired and preserved in patients with severe aphasia. Moreover, it provides evidence indicating that procedures may be dissociated from conceptual knowledge within a single arithmetical operation.

Approximate quantities and exact number words: dissociable systems

Numerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolic representation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending on whether the deficit affects the verbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularly the presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion of the left parietal lobe and Gerstmann's syndrome and another with semantic dementia with predominantly left temporal hypometabolism. As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slowness in approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and with arrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intact approximation abilities, and showed preserved processing of non-symbolic numerosities.

EFNS guidelines on cognitive rehabilitation: report of an EFNS task force

In 1999, a Task Force was set up under the auspices of the European Federation of Neurological Societies with the aim to evaluate the existing evidence for the clinical effectiveness of cognitive rehabilitation. This review led to the development of a set of guidelines to be used in the management of adult patients with cognitive disorders due to acquired focal neurological damage.

The selective impairment of arithmetical procedures

The theoretical distinction between arithmetic facts and procedures was first made by Groen and Parkman (1972). This was confirmed with a neuropsychological single case described by Warrington (1982) who had impaired arithmetical facts but well preserved arithmetical procedures. Since this time there have been several patients described who showed a selective impairment of arithmetic facts. There have also been reports of cases with impaired arithmetical procedures. However, there has not yet been a case reported with the selective impairment of procedures in the context of intact arithmetic facts. This paper describes a patient, SR, with probable Alzheimer's dementia who had well preserved addition, multiplication and subtraction facts but who nevertheless had severe difficulties with a range of arithmetical procedures such as multidigit sums, decimals and fractions. The implications of this case for current theoretical models are discussed.

Selective impairments for addition, subtraction and multiplication. implications for the organisation of arithmetical facts

This study reports for the first time a selective impairment for simple addition in patient FS. Moreover, patient VP presented with a selective impairment for simple multiplication and patient DT with a selective impairment for simple subtraction. These findings are discussed in the context of two of the most influential models for the organisation of arithmetical facts in memory (Dehaene and Cohen, 1995, 1997, and Dagenbach and McCloskey, 1992). Dehaene and Cohen (1995, 1997) have proposed that dissociation between arithmetical facts result from a selective impairment to two different types of processing: rote verbal memory for multiplication and simple addition vs. quantity processing for subtraction and division. Dagenbach and McCloskey (1992) suggest dissociation between arithmetical facts result from a selective damage to segregated memory networks specific for each operation. We will argue that our findings are problematic for Dehaene's model and in good accord with McCloskey's view.

Neuropsychological differentiation of subnormal arithmetic abilities in children

For the neuropsychological diagnosis of a Specific disorder of arithmetic skills the standard is defined worldwide by the diagnostic guidelines of the two major classification systems of psychiatric disorders, the ICD- 10 and the DSM-IV, both of which use the discrepancy criterion as the key feature of the diagnosis. It becomes clear that following such guidelines can only mean an extensive diagnostic assessment of patients with subnormal arithmetic abilities, including the social and previous medical history followed by the exclusion of a present disorder or disability, often necessitating laboratory, neuroimaging and neurophysiological tests. In the neuropsychological diagnosis of a Specific disorder of arithmetical skills we found a stepwise approach including previously obtained test results to be most practical and economical. The assessment instruments such as intelligence tests, neuropsychological test batteries, academic achievement tests, specific neuropsychological tests, test batteries for arithmetic abilities or error analysis are discussed. An overview of case studies in the literature is given; however, the overwhelming majority of case reports on patients with arithmetic disabilities are casuistics of adult patients with an acquired loss of arithmetic abilities. The importance of a differentiated neuropsychological diagnosis is demonstrated by four of our own cases: of borderline deficient intelligence, a combined disorder of academic skills, congenital brain dysfunction, and a specific disorder of arithmetic skills. Especially the new quality in assessing arithmetic abilities with the NUCALC battery is demonstrated; diagnostic and therapeutic consequences are discussed.

The breakdown of calculation procedures in Alzheimer's disease

The study investigates calculation abilities in 12 patients with probable Alzheimer's disease (AD) and compares them to calculation abilities of healthy control subjects (NC) and patients with focal left hemisphere lesions (LHL). AD patients scored significantly lower than NC in all calculation tasks and lower than LHL patients in the execution of complex written calculation, but not in the retrieval of arithmetic facts. In the AD group a subject-by-subject error analysis on the complex written calculation showed a low consistency and a high variability of error types. It is suggested that AD patients' difficulties in complex calculation arise from a monitoring deficit and not from incomplete or distorted calculation algorithms. Overall, deficits in monitoring calculation procedures may be an early and common symptom of AD.

A deficit for arithmetical procedures: lack of knowledge or lack of monitoring?

A patient is described with a specific deficit for arithmetical procedures. Unlike in previously described cases, where the observed problems could be attributed to the systematic application of disturbed algorithms, this patient's difficulty seems to stem from an inability to monitor the sequence of operations that calculation procedures specify. Criteria are provided for distinguishing impairments in written calculation due to the application of defective knowledge of the procedures from those determined by lack of monitoring. The role of monitoring and control processes in different calculation components is also discussed.