For arithmetic facts (i.e., table facts, such as 2 x 3 = 6), there is strong evidence that cerebral damage can cause selective impairments within basic arithmetic operations. However, to understand the brain underpinnings of arithmetic abilities, it is necessary to distinguish the process of retrieval of arithmetic facts from calculation procedures. Actually, several neuropsychological reports described isolated deficits within calculation procedures and some studies documented a double dissociation between performance in arithmetic facts retrieval and calculation procedures, even within the same operation. However, brain functional systems subsiding calculation procedures are still poorly characterized in comparison to functional systems of basic arithmetic facts.
Dehaene and Cohen offered some explanations how calculation processes map onto brain anatomy (the “triple code” hypothesis). Regarding arithmetic procedures, the authors suggested that working memory, processed by dorsolateral prefrontal cortical-subcortical systems, is an essential component for problems that require the temporary storage of intermediate results, for instance during ”carrying over” or borrowing operations. A visuo-spatial store, supported by the same brain regions, is complementary functioning to keep on-line spatial layouts and digits of ongoing multi-digit calculation. Both phonological and visual working memory systems are part of a large network of areas in the left (for the verbal working memory) and in the right hemisphere (for the visuo-spatial working memory system) that includes temporal-parietal and frontal-subcortical regions.
This view is different from the one proposed by McCloskey and colleagues that conceives the existence of three serial modules for calculation: input number processing, calculation procedures and output number processing. Within the module of calculation procedures, neural processes, whcih are specific and selective for each operation and independent from the retrieval of arithmetic facts, are presumed to exist. Based on a cognitive neuropsychological framework, McCloskey et al. distinguish between disorders of number processing (reading, writing, producing, comprehending, or repeating numbers) and disorders of calculation (arithmetic facts, knowledge of procedures). Recent studies also suggest that number representation, arithmetic fact knowledge and procedural aspects of calculation should be studied with a specific within-task approach more than a between task approach and that individual differences in competences, training and in strategy use (procedural vs retrieval) can strongly influence arithmetic problem solving.
Me and other authors published (Neurocase 2013;19(1):54-66) a case of a 69-year-old professor of mathematics, examined two years after a subcortical hemorrhagic stroke, who presented with a persistent form of dyscalculia exclusively limited to the procedure of subtractions with borrowing (i.e. great difficulties for operations as 22-7 when operations as 99-12 were easy to perform). Patient’s errors in subtractions with borrowing mostly relied on his automatic and inadequate attempts to invert subtractions into the corresponding additions (e.g. he used to transform 22-7=x into 7+x=22). The hypothesis is that difficulties with the inhibitory components of executive and working memory tasks (he had problems with the Stroop test and go-nogo tasks) could be responsible of this specific impairment for subtractions with borrowing.
Clinical and experimental findings of this single-case study are compatible with both the Dehaene and Cohen’s model (as per the role of working memory and executive functions in calculation procedures) and the McCloskey’s model (assuming that brain lesions can involve selectively neural pathways for different calculation procedures). We further speculated, based on MRI findings of that case, that a deficit in subtractions with borrowing could be related to left-hemispheric subcortical damage involving thalamo-cortical connections.
Do you assess calculation routinely in cognitive impaired patients? How do you do that?
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