Neural correlates of symbolic and non-symbolic arithmetic

Recent evidence suggests that areas in and around the intraparietal sulcus (IPS) represent magnitude in a stimulus-independent format. However, it has not been established whether the same is true for mental arithmetic or whether activation for higher level numerical processing diverges as a function of stimulus format. We addressed this question in a functional imaging study by presenting participants with simple addition problems using both symbolic (Arabic numerals) and non-symbolic (arrays of dots) stimuli. Conjunction analysis revealed common neural substrates for symbolic and non-symbolic addition in the anterior IPS bilaterally, left posterior IPS, medial frontal gyrus and left precentral gyrus. Right parietal and frontal cortex showed greater activation for non-symbolic addition. Our results demonstrate that mental arithmetic, studied using addition problems, is processed within the IPS independent of stimulus form. Additionally we examined whether exact and approximate addition conditions activated different neural substrates as a function of stimulus format. We did not find any differences between exact and approximate addition using symbolic and non-symbolic stimuli. This could be due to the inability of the participants to suppress exact calculation for single-digit addition problems. In contrast to recent findings, we found no significant activation for exact addition condition in left, language-related areas.     

Mathematical abilities in dyslexic children: a diffusion tensor imaging study

Dyslexia is characterized by a deficit in language processing which mainly affects word decoding and spelling skills. In addition, children with dyslexia also show problems in mathematics. However, for the latter, the underlying structural correlates have not been investigated. Sixteen children with dyslexia (mean age 9.8 years [0.39]) and 24 typically developing children (mean age 9.9 years [0.29]) group matched for age, gender, IQ, and handedness underwent 3 T MR diffusion tensor imaging as well as cognitive testing. Tract-Based Spatial Statistics were performed to correlate behavioral data with diffusion data. Children with dyslexia performed worse than controls in standardized verbal number tasks, such as arithmetic efficiency tests (addition, subtraction, multiplication, division). In contrast, the two groups did not differ in the nonverbal number line task. Arithmetic efficiency, representing the total score of the four arithmetic tasks, multiplication, and division, correlated with diffusion measures in widespread areas of the white matter, including bilateral superior and inferior longitudinal fasciculi in children with dyslexia compared to controls. Children with dyslexia demonstrated lower performance in verbal number tasks but performed similarly to controls in a nonverbal number task. Further, an association between verbal arithmetic efficiency and diffusion measures was demonstrated in widespread areas of the white matter suggesting compensatory mechanisms in children with dyslexia compared to controls. Taken together, poor fact retrieval in children with dyslexia is likely a consequence of deficits in the language system, which not only affects literacy skills but also impacts on arithmetic skills.     

Dyscalculia and the Calculating Brain

Dyscalculia, like dyslexia, affects some 5% of school-age children but has received much less investigative attention. In two thirds of affected children, dyscalculia is associated with another developmental disorder like dyslexia, attention-deficit disorder, anxiety disorder, visual and spatial disorder, or cultural deprivation. Infants, primates, some birds, and other animals are born with the innate ability, called subitizing, to tell at a glance whether small sets of scattered dots or other items differ by one or more item. This nonverbal approximate number system extends mostly to single digit sets as visual discrimination drops logarithmically to “many” with increasing numerosity (size effect) and crowding (distance effect). Preschoolers need several years and specific teaching to learn verbal names and visual symbols for numbers and school agers to understand their cardinality and ordinality and the invariance of their sequence (arithmetic number line) that enables calculation. This arithmetic linear line differs drastically from the nonlinear approximate number system mental number line that parallels the individual number-tuned neurons in the intraparietal sulcus in monkeys and overlying scalp distribution of discrete functional magnetic resonance imaging activations by number tasks in man. Calculation is a complex skill that activates both visual and spatial and visual and verbal networks. It is less strongly left lateralized than language, with approximate number system activation somewhat more right sided and exact number and arithmetic activation more left sided. Maturation and increasing number skill decrease associated widespread non-numerical brain activations that persist in some individuals with dyscalculia, which has no single, universal neurological cause or underlying mechanism in all affected individuals.     

Two mental calculation systems: a case study of severe acalculia with preserved approximation.

We report the case of an aphasic and acalculic patient with selective preservation of approximation abilities. The patient's deficit was so severe that he judged 2 + 2 = 5 to be correct, illustrating a radical impairment in exact calculation. However, he easily rejected grossly false additions such as 2 + 2 = 9, therefore demonstrating a preserved knowledge of the approximate result. The dissociation between impaired exact processing and preserved approximation was identified in several numerical tasks: solving and verifying arithmetical operations, number reading, short-term memory, number comparison, parity judgement, and number knowledge. We suggest the existence of two distinct number-processing routes in the normal subject. One route permits exact number representation, memory and calculation using symbolic notation. The other route allows for approximate computations using an analog representation of quantities.     

Numerical magnitude processing in children with mild intellectual disabilities

The present study investigated numerical magnitude processing in children with mild intellectual disabilities (MID) and examined whether these children have difficulties in the ability to represent numerical magnitudes and/or difficulties in the ability to access numerical magnitudes from formal symbols. We compared the performance of 26 children with MID on a symbolic (digits) and a non-symbolic (dot-arrays) comparison task with the performance of two control groups of typically developing children: one group matched on chronological age and one group matched on mathematical ability level. Findings revealed that children with MID performed more poorly than their typically developing chronological age-matched peers on both the symbolic and non-symbolic comparison tasks, while their performance did not substantially differ from the ability-matched control group. These findings suggest that the development of numerical magnitude representation in children with MID is marked by a delay. This performance pattern was observed for both symbolic and non-symbolic comparison tasks, although difficulties on the former task were more prominent. Interventions in children with MID should therefore foster both the development of magnitude representations and the connections between symbols and the magnitudes they represent.     

Developmental dyscalculia and low numeracy in Chinese children

Children struggle with mathematics for different reasons. Developmental dyscalculia and low numeracy – two kinds of mathematical difficulties – may have their roots, respectively, in poor understanding of exact non-symbolic numerosities and of symbolic numerals. This study was the first to explore whether Chinese children, despite cultural and linguistic factors supporting their mathematical learning, also showed such mathematical difficulties and whether such difficulties have measurable impact on children's early school mathematical performance. First-graders, classified as dyscalculia, low numeracy, or normal achievement, were compared for their performance in various school mathematical tasks requiring a grasp of non-symbolic numerosities (i.e., non-symbolic tasks) or an understanding of symbolic numerals (i.e., symbolic tasks). Children with dyscalculia showed poorer performance than their peers in non-symbolic tasks but not symbolic ones, whereas those with low numeracy showed poorer performance in symbolic tasks but not non-symbolic ones. As hypothesized, these findings suggested that dyscalculia and low numeracy were distinct deficits and caused by deficits in non-symbolic and symbolic processing, respectively. These findings went beyond prior research that only documented generally low mathematical achievements for these two groups of children. Moreover, these deficits appeared to be persistent and could not be remedied simply through day-to-day school mathematical learning. The present findings highlighted the importance of tailoring early learning support for children with these distinct deficits, and pointed to future directions for the screening of such mathematical difficulties among Chinese children.     

The development of numerical magnitude processing and its association with working memory in children with mild intellectual disabilities

The present research examined numerical magnitude processing and its association with working memory in children with mild intellectual disabilities (MID). We investigated the performance of 8-year-old children with MID on a symbolic (Arabic digits) and non-symbolic (dot patterns) magnitude comparison task by means of a chronological-age/ability-level-match design. We also examined whether the predicted problems with numerical magnitude comparison could be explained by working memory by using three working memory tasks. Findings revealed that children with MID performed more poorly than their chronological age-matched peers on both the symbolic and non-symbolic magnitude comparison tasks, suggesting impairments in these children's ability to represent numerical magnitudes. They also performed more poorly on working memory compared to their typically developing age- and ability-matched peers, but when these differences in working memory performance were additionally controlled for, the group differences on the numerical magnitude comparison tasks remained. Both symbolic numerical magnitude processing and central executive functioning predicted addition performance in children with MID.     

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.     

Cognitive phenotype of mathematical learning disabilities: What can we learn from siblings?

The sensitivity of number sense as cognitive phenotype for mathematical learning disabilities (MLD) was assessed in siblings of children with MLD (n = 9) and age matched children without family members with MLD (n = 63). A number line estimation paradigm was used as a measure of children's early number sense. In line with the triple code model of Dehaene (1992), three different presentation formats were presented. The results of the study confirmed that number line estimation was related to early arithmetic achievement in kindergarten. In addition siblings were less proficient in number line placements compared to non-siblings, with a larger effect size for symbolic and especially number word estimation compared to the non-symbolic results. Siblings also differed from non-siblings on procedural and conceptual counting skills and logical thinking in kindergarten. Moreover MLD had a familial aggregation, since about three out of five sibling girls had clinical scores on an Early Numeracy Test in kindergarten, pointing to a risk to develop MLD themselves. Implications of the study to our understanding of MLD are discussed.     

Aberrant functional activation in school age children at-risk for mathematical disability: A functional imaging study of simple arithmetic skill

We used functional magnetic resonance imaging (fMRI) to explore the patterns of brain activation associated with different levels of performance in exact and approximate calculation tasks in well-defined cohorts of children with mathematical calculation difficulties (MD) and typically developing controls. Both groups of children activated the same network of brain regions; however, children in the MD group had significantly increased activation in parietal, frontal, and cingulate cortices during both calculation tasks. A majority of the differences occurred in anatomical brain regions associated with cognitive resources such as executive functioning and working memory that are known to support higher level arithmetic skill but are not specific to mathematical processing. We propose that these findings are evidence that children with MD use the same types of problem solving strategies as TD children, but their weak mathematical processing system causes them to employ a more developmentally immature and less efficient form of the strategies.     

Sulcation of the intraparietal sulcus is related to symbolic but not non-symbolic number skills

Understanding the constraints, including biological ones, that may influence mathematical development is of great importance because math ability is a key predictor of career success, income and even psychological well-being. While research in developmental cognitive neuroscience of mathematics has extensively studied the key functional regions for processing numbers, particularly the horizontal segment of intraparietal sulcus (HIPS), few studies have investigated the effects of early cerebral constraints on later mathematical abilities. In this pre-registered study, we investigated whether variability of the sulcal pattern of the HIPS, a qualitative feature of the brain determined in-utero and not affected by brain maturation and learning, accounts for individual difference in symbolic and non-symbolic number abilities. Seventy-seven typically developing school-aged children and 21 young adults participated in our study. We found that the HIPS sulcal pattern, (a) explains part of the variance in participant’s symbolic number comparison and math fluency abilities, and (b) that this association between HIPS sulcal pattern and symbolic number abilities was found to be stable from childhood to young adulthood. However, (c) we did not find an association between participant’s non-symbolic number abilities and HIPS sulcal morphology. Our findings suggest that early cerebral constraints may influence individual difference in math abilities, in addition to the well-established neuroplastic factors.     

Principles underlying the design of "The Number Race", an adaptive computer game for remediation of dyscalculia

Background  

Adaptive game software has been successful in remediation of dyslexia. Here we describe the cognitive and algorithmic principles underlying the development of similar software for dyscalculia. Our software is based on current understanding of the cerebral representation of number and the hypotheses that dyscalculia is due to a "core deficit" in number sense or in the link between number sense and symbolic number representations.     

Mental additions and verbal-domain interference in children with developmental dyscalculia

This study examined the involvement of verbal and visuo-spatial domains in solving addition problems with carrying in a sample of children diagnosed with developmental dyscalculia (DD) divided into two groups: (i) those with DD alone and (ii) those with DD and dyslexia. Age and stage matched typically developing (TD) children were also studied. The addition problems were presented horizontally or vertically and associated with verbal or visuo-spatial information.Study results showed that DD children's performance on mental calculation tasks was more impaired when they tackled horizontally presented addition problems compared to vertically presented ones that are associated to verbal domain involvement. The performance pattern in the two DD groups was found to be similar. The theoretical, clinical and educational implications of these findings are discussed.     

Effect of language switching on arithmetic: a bilingual FMRI study

The role of language in performing numerical computations has been a topic of special interest in cognition. The "Triple Code Model" proposes the existence of a language-dependent verbal code involved in retrieving arithmetic facts related to addition and multiplication, and a language-independent analog magnitude code subserving tasks such as number comparison and estimation. Neuroimaging studies have shown dissociation between dependence of arithmetic computations involving exact and approximate processing on language-related circuits. However, a direct manipulation of language using different arithmetic tasks is necessary to assess the role of language in forming arithmetic representations and in solving problems in different languages. In the present study, 20 English-Chinese bilinguals were trained in two unfamiliar arithmetic tasks in one language and scanned using fMRI on the same problems in both languages (English and Chinese). For the exact "base-7 addition" task, language switching effects were found in the left inferior frontal gyrus (LIFG) and left inferior parietal lobule extending to the angular gyrus. In the approximate "percentage estimation" task, language switching effects were found predominantly in the bilateral posterior intraparietal sulcus and LIFG, slightly dorsal to the LIFG activation seen for the base-7 addition task. These results considerably strengthen the notion that exact processing relies on verbal and language-related networks, whereas approximate processing engages parietal circuits typically involved in magnitude-related processing.     

Patterns of Developmental Dyscalculia With or Without Dyslexia

This study has been conducted in order to investigate the extent to which some characteristics of dyscalculia may be common to dyslexia. Seven multiple single-cases were studied: two children with dyslexia only, two with dyscalculia only, and three more children with comorbidity of dyslexia and dyscalculia. Each participant was assessed with a standardized comprehensive battery of arithmetical, reading, and cognitive tests. We observed that a clinical impairment in mental and written calculations, arithmetical facts retrieval, number comparison, number alignment, and identification of arithmetical signs may appear with a normal reading capacity and independently of a short-term verbal memory deficit. These findings add convergent support to the evidence mainly obtained from group comparisons that the more distinctive characteristics of dyscalculia are functionally independent of dyslexia.     

Abrupt and ramped flicker-defined form shows evidence for a large magnocellular impairment in dyslexia

Controversy still exists over whether there is a magnocellular deficit associated with developmental dyslexia. Here we utilised a magnocellular system-biased phantom contour form discrimination task defined by high temporal frequency contrast reversals to compare contrast sensitivity in a group of children with dyslexia and an age- and nonverbal intelligence-matched control group (9-14 years). Stimuli were either abruptly presented for 4 refresh frames (34 ms), or in two reduced transience conditions had contrast progressively ramped on and off over either 4 frames or 10 frames (86 ms). Children in the dyslexia group showed increased contrast thresholds compared with the control group in all three conditions, and thus strong evidence for a magnocellular deficit. Although the absolute size of the differences in threshold scores between control and dyslexic groups increased dramatically between the abrupt and the 4 and 10 frame ramped onset stimuli, the similar effect size across all tasks, and also the similar range of contrast change at the first frame of stimulus presentation across all tasks between groups suggests that a similar neural mechanism could provide the locus of the apparent magnocellular deficit in children with dyslexia for all tasks tested. These results suggest that threshold discrimination of stimuli with low contrast and high temporal frequencies designed to target the magnocellular system, and has great potential for early screening for children at risk of visually derived reading difficulties.     

Patterns of developmental dyscalculia with or without dyslexia

This study has been conducted in order to investigate the extent to which some characteristics of dyscalculia may be common to dyslexia. Seven multiple single-cases were studied: two children with dyslexia only, two with dyscalculia only, and three more children with comorbidity of dyslexia and dyscalculia. Each participant was assessed with a standardized comprehensive battery of arithmetical, reading, and cognitive tests. We observed that a clinical impairment in mental and written calculations, arithmetical facts retrieval, number comparison, number alignment, and identification of arithmetical signs may appear with a normal reading capacity and independently of a short-term verbal memory deficit. These findings add convergent support to the evidence mainly obtained from group comparisons that the more distinctive characteristics of dyscalculia are functionally independent of dyslexia.     

Similar alterations in brain function for phonological and semantic processing to visual characters in Chinese dyslexia

Dyslexia in alphabetic languages has been extensively investigated and suggests a central deficit in orthography to phonology mapping in the left hemisphere. Compared to dyslexia in alphabetic languages, the central deficit for Chinese dyslexia is still unclear. Because of the logographic nature of Chinese characters, some have suggested that Chinese dyslexia should have larger deficits in the semantic system. To investigate this, Chinese children with reading disability (RD) were compared to typically developing (TD) children using functional magnetic resonance imaging (fMRI) on a rhyming judgment task and on a semantic association judgment task. RD children showed less activation for both tasks in right visual (BA18, 19) and left occipito-temporal cortex (BA 37), suggesting a deficit in visuo-orthographic processing. RD children also showed less activation for both tasks in left inferior frontal gyrus (BA44), which additionally showed significant correlations with activation of bilateral visuo-orthographic regions in the RD group, suggesting that the abnormalities in frontal cortex and in posterior visuo-orthographic regions may reflect a deficit in the connection between brain regions. Analyses failed to reveal larger differences between groups for the semantic compared to the rhyming task, suggesting that Chinese dyslexia is similarly impaired in the access to phonology and to semantics from the visual orthography.     

Spatial and numerical abilities without a complete natural language

We studied the cognitive abilities of a 13-year-old deaf child, deprived of most linguistic input from late infancy, in a battery of tests designed to reveal the nature of numerical and geometrical abilities in the absence of a full linguistic system. Tests revealed widespread proficiency in basic symbolic and non-symbolic numerical computations involving the use of both exact and approximate numbers. Tests of spatial and geometrical abilities revealed an interesting patchwork of age-typical strengths and localized deficits. In particular, the child performed extremely well on navigation tasks involving geometrical or landmark information presented in isolation, but very poorly on otherwise similar tasks that required the combination of the two types of spatial information. Tests of number- and space-specific language revealed proficiency in the use of number words and deficits in the use of spatial terms. This case suggests that a full linguistic system is not necessary to reap the benefits of linguistic vocabulary on basic numerical tasks. Furthermore, it suggests that language plays an important role in the combination of mental representations of space.     

Number development and developmental dyscalculia

There is a growing consensus that the neuropsychological underpinnings of developmental dyscalculia (DD) are a genetically determined disorder of 'number sense', a term denoting the ability to represent and manipulate numerical magnitude nonverbally on an internal number line. However, this spatially-oriented number line develops during elementary school and requires additional cognitive components including working memory and number symbolization (language). Thus, there may be children with familial-genetic DD with deficits limited to number sense and others with DD and comorbidities such as language delay, dyslexia, or attention-deficit-hyperactivity disorder. This duality is supported by epidemiological data indicating that two-thirds of children with DD have comorbid conditions while one-third have pure DD. Clinically, they differ according to their profile of arithmetic difficulties. fMRI studies indicate that parietal areas (important for number functions), and frontal regions (dominant for executive working memory and attention functions), are under-activated in children with DD. A four-step developmental model that allows prediction of different pathways for DD is presented. The core-system representation of numerical magnitude (cardinality; step 1) provides the meaning of 'number', a precondition to acquiring linguistic (step 2), and Arabic (step 3) number symbols, while a growing working memory enables neuroplastic development of an expanding mental number line during school years (step 4). Therapeutic and educational interventions can be drawn from this model.