Arithmetic Facts Research Articles

Event-related potentials findings differ between children and adults during arithmetic-fact retrieval

Some cognitive abilities of arithmetical calculation depend on retrieval of arithmetic facts from long-term memory. Arithmetic-fact retrieval has been studied in adults through Event-Related Potentials (ERP) experiments. Such information in children, however, has been scarce. It has been reported that from the age of 9 years, children employ a memory retrieval strategy for solving simple multiplication problems. The present study compared arithmetical-fact retrieval in children and adults while they were being subjected to ERP recording. The subjects were asked to make judgments about solutions to simple multiplication problems. Both groups of participants displayed the so-called arithmetic N400 effect for incorrect solutions relative to correct solutions. Adults showed a posterior N400 effect, while children showed a widely distributed N400 effect. Children displayed a larger amplitude and longer latency arithmetic N400 component than adults; this observation could be due to children exerting greater effort involving more widespread cortical activation than adults to solve the experimental problems. The Late Positive Component (LPC), which follows the arithmetic N400 and has been described previously in adult subjects, was observed in the present adult subjects, but was present in children only for correct solutions. These results may indicate that, relative to adults, children showed slower memory retrieval and a different pattern of a verification mechanism for correct and incorrect solutions.

The mathematical strengths and weaknesses of children with dyslexia

This paper reports the findings of two related studies that examined the mathematical strengths and weaknesses of children with dyslexia. In study one, dyslexic children were compared to children without special educational needs on tests that assessed arithmetic fact recall, place value understanding and counting speed. Study two used the same methodology, but matched the dyslexic children with the non‐dyslexic children on a number of factors, including intellectual ability. This excluded the possibility that confounding variables accounted for the differences between the groups in study one. Both studies indicated that the dyslexic children had slower and less accurate number fact recall than the non‐dyslexic children, but had unimpaired place value understanding. The results for counting speed were more equivocal, but there was a trend for the dyslexic children to be slower counters. The results suggest that dyslexia is not normally associated with a general mathematical impairment, but rather an uneven profile of skills. This profile can be explained in terms of the phonological processing weaknesses associated with dyslexia. The finding that dyslexic children have a specific difficulty recalling arithmetic facts suggests that a teaching programme that emphasises mental methods may disadvantage them.

Event-related potentials for simple arithmetic in Arabic digits and Chinese number words: a study of the mental representation of arithmetic facts through notation and operation effects

To test whether the retrieval of arithmetic facts is independent of numerical notations, this study investigated the event-related potentials elicited by single-digit addition and multiplication problems in Arabic digits and Chinese number words. The results showed that, in comparison with addition, multiplication elicited a greater N300-like component at the left anterior electrodes and greater late positive potentials at the right posterior electrodes, regardless of numerical notations. The operation effects lasted from 250 to 900 ms for Arabic digits, but from 250 to 1400 ms for Chinese number words when participants were asked to respond only to false arithmetic equations (experiment one), and lasted from 350 to 1400 ms for Arabic digits and Chinese number words when participants were asked to respond to both true and false arithmetic equations (experiment two). The consistency in the operation effects in ERPs (i.e., a dissociation of brain organization for different arithmetic operations) for different number notations suggests that mental representation and retrieval of arithmetic facts may be relatively independent of numerical notations.

Conceptual knowledge‐based strategy training in single‐digit calculation: a single case intervention study in a child with specific language impairment

The aim of this single‐case intervention study was to examine whether a conceptual knowledge‐based strategy training would improve the accuracy of single‐digit arithmetical calculation in a child with specific language impairment. The intervention concerned an11‐year‐old Finnish‐speaking child with specific language impairment. He was trained individually once a week for three months. On the basis of his baseline performance single‐digit arithmetical problems were divided into ‘well‐known arithmetical problems’ and ‘error‐prone arithmetical problems’. Error‐prone problems were taught in meaningful relation to well‐known problems, which he repeatedly solved correctly by using arithmetical fact knowledge. An intervention study design was used consisting of six baseline and three follow‐up assessments. The results show that the participant calculated additions, subtractions and multiplications more accurately after the intervention was applied. In subtraction he also used more retrieval‐based strategies after than before the intervention. The results suggest that an intervention programme targeted at improving calculation skill based on conceptual knowledge is beneficial, especially when children have weaknesses in use of counting procedures and in direct memory retrieval.

The interplay of arithmetic fact training and mathematical competence on parietal brain activation during mental calculation

The interplay of arithmetic fact training and mathematical competence on parietal brain activation during mental calculation

Mapping arithmetic problem solving strategies in the brain: The role of the left angular gyrus in arithmetic fact retrieval

Mapping arithmetic problem solving strategies in the brain: The role of the left angular gyrus in arithmetic fact retrieval

The importance of LoLA: Brain and behavior indices of the organization of arithmetic facts in bilinguals

The importance of LoLA: Brain and behavior indices of the organization of arithmetic facts in bilinguals

Dysfunctional neural network of spatial working memory contributes to developmental dyscalculia

The underlying neural mechanisms of developmental dyscalculia (DD) are still far from being clearly understood. Even the behavioral processes that generate or influence this heterogeneous disorder are a matter of controversy. To date, the few studies examining functional brain activation in children with DD mainly focus on number and counting related tasks, whereas studies on more general cognitive domains that are involved in arithmetical development, such as working memory are virtually absent. There are several studies showing a close relationship between DD and spatial working memory [Camos, V. (2008). Low working memory capacity impedes both efficiency and learning of number transcoding in children. Journal of Experimental Child Psychology, 99(1), 37-57; McLean, J. F., & Hitch, G. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74(3), 240-260; Rosselli, M., Matute, E., Pinto, N., & Ardila, A. (2006). Memory abilities in children with subtypes of dyscalculia. Developmental Neuropsychology, 30(3), 801-818; Siegel, L. S., & Ryan, E. B. (1989). The development of working memory in normally achieving and subtypes of learning disabled children. Child Development, 60(4), 973-980]. The relationship between these two mechanisms is still matter of debate, but this study follows the assumption that poor spatial working memory capacity may hinder the acquisition of spatial number representations in children with DD [Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114(2), 345-362; von Aster, M., & Shalev, R. S. (2007). Number development and developmental dyscalculia. Developmental Medicine and Child Neurology, 49(11), 868-873]. Using functional MRI the current study compares brain activity associated with spatial working memory processes in 8-10-year-old children with DD and normally achieving controls. Both groups showed significant spatial working memory related activity in a network including occipital and parietal regions. Children with DD showed weaker neural activation compared to the control group during a spatial working memory task in the right intraparietal sulcus (IPS), the right insula and the right inferior frontal lobe. Performance tests outside the scanner showed impaired working memory proficiency in children with DD. Bringing behavioral performance and neural activity together we found significant correlations of right IPS activity with performance on the verbal digit span forward and the spatial Corsi Block Tapping test. Our findings demonstrate for the first time an involvement of spatial working memory processes in the neural underpinnings of DD. These poor spatial working memory processes may inhibit the formation of spatial number representations (mental numberline) as well as the storage and retrieval of arithmetical facts.

Oscillatory EEG correlates of arithmetic strategy use in addition and subtraction

Adults use different strategies in mental arithmetic. For example, they directly retrieve the answer from memory or calculate by means of procedural strategies. Despite growing insight into the hemodynamic and electrophysiological correlates of these strategies, the functional changes in the oscillatory brain dynamics during the use of these strategies remain unknown. In the present high-resolution electroencephalography (EEG) study, we analysed event-related synchronisation (ERS) and desynchronisation (ERD) in the theta and alpha bands while participants solved addition and subtraction problems, which displayed a high probability of retrieval or procedural strategy use. Findings revealed that arithmetic fact retrieval is reflected in left-hemispheric ERS in the theta band, whereas the application of procedural strategies is accompanied by bilateral parietooccipital ERD in the alpha band. The topographical and frequency specificity of the strategy effects provides a start for the development of electrophysiological indices of strategy use in arithmetic.

The Relationship Between Medical Impairments and Arithmetic Development in Children With Cerebral Palsy

Arithmetic ability was tested in children with cerebral palsy without severe intellectual impairment (verbal IQ >or= 70) attending special (n = 41) or mainstream education (n = 16) as well as control children in mainstream education (n = 16) throughout first and second grade. Children with cerebral palsy in special education did not appear to have fully automatized arithmetic facts by the end of second grade. Their lower accuracy and consistently slower (verbal) response times raise important concerns for their future arithmetic development. Differences in arithmetic performance between children with cerebral palsy in special or mainstream education were not related to localization of cerebral palsy or to gross motor impairment. Rather, lower accuracy and slower verbal responses were related to differences in nonverbal intelligence and the presence of epilepsy. Left-hand impairment was related to slower verbal responses but not to lower accuracy.

Numerical deficits in a single case of basal ganglia dysfunction

The present investigation assesses specific numerical difficulties in a patient (SJ) with basal ganglia (BG) dysfunction. While previous studies on number processing in BG disorders typically tested arithmetic facts by production tasks, the present study uses production, recognition (verification, multiple-choice) and indirect (number-matching) arithmetic tasks. Patient SJ was severely impaired in production and to a lesser extent in verification and multiple-choice tasks. In number-matching, an abnormal latency pattern was found. This study extends previous research by indicating that BG dysfunction may not only affect production processes and sequencing, as was found in previous investigations, but may lead to a breakdown of semantic relationships of arithmetic facts.

Learning basic addition facts from choosing between alternative answers

The acquisition of addition facts was investigated in a practice study. Participants were 103 Grade 1 children who practiced simple addition problems with three different methods: (a) writing down the answer, (b) choosing between two alternative answers, and (c) filling in the second missing addend. On a test with simple addition problems, children who practiced with the Choice method showed positive transfer: choosing between two answers was about as effective for learning addition facts as the conventional method of writing down the answer. There was no transfer effect for children who practiced with the Missing-Addend method. The results are in accordance with network theories on arithmetic fact learning and specifically the Identical Elements (IE) model of arithmetic fact representation.

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.

Dissociating the solution processes of small, large, and zero multiplications by means of fMRI

The present study investigated the neuroanatomical basis of different solution processes in single-digit multiplication by means of fMRI. Sixteen participants silently produced the solution of three distinct types of multiplication, i.e., problems involving zero (e.g., 3 ⁎ 0), small (e.g., 2 ⁎ 4), or large operands (e.g., 8 ⁎ 7). Zero and small problems are assumed to be solved by rule application and fact retrieval, respectively, and problems with large operands sometimes involve backup strategies when direct retrieval is not sufficient. Small problems, when compared with a high-level baseline not requiring any kind of calculation, activated a network of parietal, subcortical, and frontal areas. This activation pattern supports the hypothesis that arithmetic fact retrieval is mediated by a verbal processing loop including the angular gyrus and the basal ganglia. Problems with larger operands showed increased activation in the anterior cingulate cortex, SMA, and the left inferior frontal gyrus, which could reflect increased conflict during the fact-retrieval process, but also higher demands for controlling and coordinating multiple processing steps when a problem cannot be solved by direct retrieval. Zero problems, in comparison to multiplications with small operands, activated the caudate nucleus and the right inferior frontal cortex, showing that rule application is separable from fact retrieval on a neuroanatomical level, too.

Dissociation of Exact and Approximate Calculation in Severe Global Aphasia

We report a 68-year-old patient with severe global aphasia secondary to a large left hemisphere infarction including the parietal lobe. In addition to language and neuroradiological evaluation, the patient was given specifically designed arithmetic and clock tasks requiring either exact calculation or approximate calculation. Despite severe language impairment, the patient showed relatively well-preserved abilities for numerical comprehension and arithmetic operations. Further analyses using specifically designed arithmetic and clock tasks demonstrated a clear dissociation of the patient’s abilities between impaired exact calculation and well-preserved approximate calculation. The results support the notion that numerical and arithmetic abilities are heterogeneous in that rote verbal arithmetic facts and quantitative numerical knowledge can be separable. Implications of the present findings for neural correlates of numerical and arithmetic processing suggest that the right hemisphere plays a crucial role in approximate calculation.

Fact learning in complex arithmetic and figural‐spatial tasks: The role of the angular gyrus and its relation to mathematical competence

Neuroimaging studies have revealed a strong link between mental calculation and the angular gyrus (AG) which has been interpreted to reflect arithmetic fact retrieval. Moreover, a stronger AG activation in individuals with higher mathematical competence has been reported. The present fMRI study investigates the specificity of the AG for arithmetic fact learning and the interplay between training and mathematical competence on brain activation. Adults of lower and higher mathematical competence underwent a five-day training on sets of complex multiplication and figural-spatial problems. In the following fMRI test session, trained and untrained problems were presented. Similar training effects were observed in both problem types, consisting of AG activation increases bilaterally and wide-spread activation decreases in frontal and parietal regions. This finding indicates that the AG is not specifically involved in the learning of arithmetic facts. Competence-related differences in the AG only emerged in untrained but not in trained multiplication problems. The relation between AG activation and mathematical competence in arithmetic problem solving therefore seems to be due to differences in arithmetic fact retrieval which can be attenuated through training.

Early development of quantity to number-word linkage as a precursor of mathematical school achievement and mathematical difficulties: Findings from a four-year longitudinal study

This article reports results of a four-year longitudinal study that investigated the impact of specific and non-specific precursors on mathematical school achievement. Preschool quantity-number competencies (QNC) predicted mathematical achievement in primary school. Furthermore, basic arithmetic fact retrieval in Grade 1 had an impact on early mathematics school achievement. The influence of socioeconomic status and number naming speed, assessed in kindergarten, became especially important at the end of Grade 4. Particularly, a subgroup of mathematically low-achieving children in Grade 4 had already performed more poorly than normal children in tasks assessing preschool QNC, number naming speed, and basic arithmetic fact retrieval, as well as nonverbal intelligence and socioeconomic status.

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.

To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving

While there is consistent evidence from neuropsychological and brain imaging studies for an association between the left angular gyrus and mental arithmetic, its specific role in calculation has remained poorly understood. It has been speculated that the angular gyrus mediates the retrieval of arithmetic facts during problem solving, but this hypothesis has not been directly tested. In the present functional Magnetic Resonance Imaging study comprising 28 adults, we used trial-by-trial strategy self-reports to identify brain regions underpinning different strategies in arithmetic problem solving. Analyses revealed stronger activation of the left angular gyrus while solving arithmetic problems for which participants reported fact retrieval whereas the application of procedural strategies was accompanied by widespread activation in a fronto-parietal network. These data directly link the left angular gyrus with arithmetic fact retrieval and show that strategy self-reports can be used to predict differential patterns of brain activation.

Rehabilitation of arithmetic fact retrieval via extensive practice: A combined fMRI and behavioural case-study

The present study investigates the effects of a training of arithmetic fact retrieval in a patient suffering from particular difficulties with multiplication facts. Over a period of four weeks simple multiplication facts were trained extensively. The outcome of the training was assessed behaviourally and changes in cerebral activation patterns were investigated using fMRI. The training led to a change in calculation strategies: Prior to training, the patient used predominantly time-consuming back-up strategies, after training he relied increasingly on the direct retrieval of arithmetic facts from long-term memory. Regarding the fMRI results, prefrontal activations were observed for untrained problems, which can be attributed to the application of back-up strategies strongly relying on fronto-executive functions. Interestingly, significant foci of activation for both trained and untrained items were found in the angular gyrus of the right hemisphere, which, however, differed in their exact localisation. For the trained condition, activations were observed in anterior parts of the angular gyrus which may be related to the training-based automatisation in fact retrieval. Activations in the untrained condition were found in a more posterior portion of the angular gyrus, that might be attributable to one of the patient's back-up strategies, namely to recite a whole multiplication row to get to the correct answer.