Arithmetic Fact Retrieval Research Articles

Representation and retrieval of arithmetical facts: developmental difficulties.

One form of developmental difficulty with arithmetic affects the storage or retrieval of arithmetical facts, such as tables, which are required to implement arithmetical computations (Temple, 1991, 1994). Such difficulties may arise because of impairment in a specialized system for the storage of arithmetical facts or as a result of causally linked impairment in another cognitive domain. This study explored issues concerning the representation and retrieval of arithmetical facts in children with number fact disorders (NF) and in normal children, in particular the status of hypothesized linked impairments: short-term memory (STM) spans, counting skills, speed of speech, and speed of number fact and lexical retrieval. There was no evidence that NF children had weak STM spans on any span measure or that STM spans related to arithmetical fact skills. There was also no evidence that NF children had weak counting abilities or free counting speeds. The NF children were slower in speeded counting, which also correlated with number fact skill. The significance or not of this is discussed. The NF children were also slower than controls in speed of speech and on some measures of speed of access. However, the absence of correlation with number fact skill, the absence of generality across tasks, and the possibility that delayed speeds in fact retrieval reflect the use of alternative strategies, together suggest that the increased speeds are not causally linked to number fact skill. The results are consistent with modular accounts, in which there is a specialized system for the storage and retrieval of arithmetical facts.

Arithmetic fact retrieval and working memory in schizophrenia

Despite its importance in every-day life and vocational rehabilitation, arithmetic ability has rarely been investigated in schizophrenic patients. Those few studies reporting arithmetic deficits in schizophrenia, however, administered complex calculation tasks which drew not only on arithmetic abilities, but also on working memory resources known to be impaired in schizophrenia. In the present study, arithmetic abilities and working memory functions were investigated in schizophrenic patients ( n=24) and healthy control subjects ( n=24). Arithmetic fact retrieval was assessed in single-digit multiplication and corresponding division problems using a result verification task which minimized working memory demands. Problem size and the disparity of the proposed result were manipulated. The storage component of working memory was tested with a digit span forward task and the executive control component with a digit span backward as well as with verbal fluency tasks. Schizophrenic patients performed worse than controls only in the executive tasks. Digit span forward was preserved. In the arithmetic tasks, groups did not differ from each other, and a similar pattern of task manipulations was obtained. Hence, despite the executive control deficit retrieval of arithmetic facts is preserved in schizophrenia. Moreover, the same underlying cognitive processes as in control subjects are involved.

Rehabilitation of number processing and calculation skills

The main purpose of this article is to present the research that has been done on the rehabilitation of number and calculation disorders. It is argued that theoretically based rehabilitation of arithmetical processing requires the formulation of a detailed functional diagnosis based on a theoretically driven evaluation of numerical processing and calculation skills. Up to now, the strategies that have been adopted in this rehabilitation field have mainly consisted in attempts to re-teach lost knowledge via extensive practice. These therapeutic programmes are described in two domains: the transcoding of numerals and the retrieval of arithmetical facts. Finally, the authors underline the necessity to develop in the near future programmes of rehabilitation adopting a more ecological perspective.

Dissociating neural correlates of cognitive components in mental calculation.

Mental calculation is a complex cognitive operation that is composed of a set of distinct functional processes. Using functional magnetic resonance imaging (fMRI), we mapped brain activity in healthy subjects performing arithmetical tasks and control tasks evoking a comparable load on visuo-constructive, linguistic, attentional and mnemonic functions. During calculation, as well as non-mathematical tasks, similar cortical networks consisting of bilateral prefrontal, premotor and parietal regions were activated, suggesting that most of these cortical areas do not exclusively represent modules for calculation but support more general cognitive operations that are instrumental but not specific to mental arithmetic. Significant differences between calculation and the non-mathematical tasks were found in parietal sub-regions, where non-arithmetic number or letter substitution tasks preferentially activated the superior parietal lobules whereas calculation predominantly elicited activation of the left dorsal angular gyrus and the medial parietal cortices. We interpret the latter activations to reflect sub-processes of mental calculation that are related to the processing of numerical representations during exact calculation and to arithmetical fact retrieval. Finally, we found that more complex calculation tasks involving the application of calculation rules increased activity in left inferior frontal areas that are known to subserve linguistic and working memory functions. Taken together, these findings help to embed the specific cognitive operation of calculation into a neural framework that provides the required set of instrumental components. This result may further inform the cognitive modeling of calculation and adds to the understanding of neuropsychological deficit patterns in patients.

Neural Correlates of Simple and Complex Mental Calculation

Some authors proposed that exact mental calculation is based on linguistic representations and relies on the perisylvian language cortices, while the understanding of proximity relations between numerical quantities implicates the parietal cortex. However, other authors opposed developmental arguments to suggest that number sense emerges from nonspecific visuospatial processing areas in the parietal cortex. Within this debate, the present study aimed at revealing the functional anatomy of the two basic resolution strategies involved in mental calculation, namely arithmetical fact retrieval and actual computation, questioning in particular the respective role of language and/or visuospatial cerebral areas. Regional cerebral blood flow was measured with positron emission tomography while subjects were at rest (Rest), read digits (Read), retrieved simple arithmetic facts from memory (i.e., 2 × 4, Retrieve), and performed mental complex calculation (i.e., 32 × 24, Compute). Compared to Read, Retrieve engaged a left parieto-premotor circuit representing a developmental trace of a finger-counting representation that mediates, by extension, the numerical knowledge in adult. Beside this basic network, Retrieve involved a naming network, including the left anterior insula and the right cerebellar cortex, while it did not engage the perisylvian language areas, which were deactivated as compared to Rest. In addition to this retrieval network, Compute specifically involved two functional networks: a left parieto-frontal network in charge of the holding of the multidigit numbers in visuospatial working memory and a bilateral inferior temporal gyri related to the visual mental imagery resolution strategy. Overall, these results provide strong evidence of the involvement of visuospatialrepresentations in different levels of mental calculation.

Mental calculation in a prodigy is sustained by right prefrontal and medial temporal areas.

Calculating prodigies are individuals who are exceptional at quickly and accurately solving complex mental calculations. With positron emission tomography (PET), we investigated the neural bases of the cognitive abilities of an expert calculator and a group of non-experts, contrasting complex mental calculation to memory retrieval of arithmetic facts. We demonstrated that calculation expertise was not due to increased activity of processes that exist in non-experts; rather, the expert and the non-experts used different brain areas for calculation. We found that the expert could switch between short-term effort-requiring storage strategies and highly efficient episodic memory encoding and retrieval, a process that was sustained by right prefrontal and medial temporal areas.

Cortical activation during retrieval of arithmetical facts and actual calculation: a functional magnetic resonance imaging study.

By using functional magnetic resonance imaging (fMRI), the neural substrates involved in mental recitation of the single-digit multiplication table and serial subtraction were studied. The former depends mostly on well-learned arithmetical facts, while the latter requires arithmetic processing. Activation during each task was compared with that in a number counting control. During the recitation of single-digit multiplication, the activated regions included the area lying along the left intraparietal sulcus, the premotor and supplementary motor areas, and the posterior portion of the left inferior frontal gyrus. The areas activated during serial subtraction included these areas as well as the bilateral prefrontal and right parietal areas. From the results obtained during retrieval of the multiplication table in this study and previous studies, it was concluded that semantic memory of the multiplication table is stored in the area lying along the intraparietal sulcus and that the frontal areas play an executive role in utilizing the semantic memory of arithmetical facts. It was assumed that the arithmetical facts requiring actual calculation are also stored in the same region. The additional activation during serial subtraction compared with the activation during retrieval of the multiplication table is probably due to the processes of actual calculation. These processes include proper alignment of digits, which may have caused the right parietal activation, and maintaining digits needed for the mental serial subtractions, which may have caused the bilateral prefrontal activation.

Number Transcoding and Number Word Spelling in a Left-Brain-Damaged Non-aphasic Acalculic Patient

Abstract We present a patient with selective dyscalculia following left-hemisphere damage. GC was a cognitively unimpaired woman except for mild verbal and spatial learning disorders. Within the domain of cognitive numerical processing mechanisms, GC presented with multiple and severe disorders, the two most important being impaired number transcoding and impaired arithmetical fact retrieval. Compared to previously reported cases, GC's dissociation between preserved language and impaired mathematical skills is more marked. GC also committed many misspellings in writing number names, but only a few errors in writing words. This dissociation supports the hypothesis of categorical organization to a very peripheral level of language processing.

Number transcoding and number word spelling in a left-brain-damaged non-aphasic acalculic patient

We present a patient with selective dyscalculia following left-hemisphere damage. GC was a cognitively unimpaired woman except for mild verbal and spatial learning disorders. Within the domain of cognitive numerical processing mechanisms, GC presented with multiple and severe disorders, the two most important being impaired number transcoding and impaired arithmetical fact retrieval. Compared to previously reported cases, GC's dissociation between preserved language and impaired mathematical skills is more marked. GC also committed many misspellings in writing number names, but only a few errors in writing words. This dissociation supports the hypothesis of categorical organization to a very peripheral level of language processing.

Noncommutability of The N + 0 Arithmetical Rule: A Case Study of Dissociated Impairment

It has been shown that some arithmetical problems are stored in the form of individual facts representations (e.g., 3 × 4 = 12) whereas others are solved by general stored rules (e.g., 0 × N = 0). We describe the performance of a brain-damaged subject who presented a mild impairment in arithmetical fact retrieval. Although her performance was almost perfect for rule-based problems in all arithmetical operations, she was severely impaired for 0 + n problems in contrast with her relatively good performance for the corresponding n + 0 problems. This dissociation extends to addition the noncommutability of arithmetical rules described in multiplication.

N400 Effects Reflect Activation Spread During Retrieval of Arithmetic Facts

Arithmetic facts are stored in densely interconnected memory networks, and retrieval errors may occur because activation spreads to associated results. We studied the extension of activation spread by means of the so-called N400 effect of the event-related brain potential (ERP). With semantic stimuli, N400 amplitude has proved to be inversely proportional to the amount of activation that originates from a priming context. ERPs were recorded from 61 scalp positions while 16 subjects verified 600 multiplication problems ( a × b = c). The solution to each problem could be correct or incorrect. Incorrect solutions were either table related to one of the operands (e.g., 5 × 8 = 32, 24, or 16) or unrelated (e.g., 5 × 8 = 34, 26, or 18), and were either a small, medium, or large numerical distance from the correct product. Our findings suggest that activation spread in an arithmetic memory network is restricted to numbers that are table related to one of the operands and that are numerically plausible.

Do specific EEG frequencies indicate different processes during mental calculation?

EEG recordings during mental calculation and a control task (with presentation of stimuli with similar physical characteristics to the arithmetic symbols) were obtained in 10 subjects. Narrow band analyses of the EEG and distributed sources for each EEG frequency were calculated using variable resolution electromagnetic tomography. Significant differences between the sources for arithmetic and control tasks were observed at 3.9 Hz within Broca's and left parietotemporal cortices, suggesting that this frequency may be related with the production of internal speech, storage and rehearsal of verbal working memory. Differences at 5.46 Hz within the right dorsolateral prefrontal cortex were considered to be associated to sustained attention. The decrease at 12.46 Hz within the left parietal cortex was interpreted as a sign of retrieval of arithmetic facts from long term memory.

Arithmetic Reasoning and Implicit Memory: A Neuropsychological Study on Amnesia

The present study investigates implicit memory and arithmetic reasoning in a group of severely amnesic patients. Priming effects have been observed in the retrieval of arithmetic facts in amnesic patients (Delazer, Ewen and Benke, 1997), but no studies have so far investigated the effect of previous experience in a pure reasoning task. Findings indicate that answering to a first numerical series (e.g, 1 3 5 7, answer “9”) facilitates answering to a second numerical series (2 4 6 8, answer “10”) including different numbers but sharing the same underlying algorithm (+ 2). Within the model of Langdon and Warrington (1997) priming effects are attributed to the stage of abstract formula generation.

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.

Effects of cortical polarization on mental arithmetic

A biofeedback procedure was used to influence participants' cortical polarity before the presentation of single digit multiplication problems. To ensure that participants could solve the problems by direct memory retrieval of arithmetic facts, only nine different multiplication problems were used, and participants received extensive pre-experimental practice on these problems. After biofeedback training, cortical positivity before problem presentation was associated with faster response initiation times, a measure of mental calculation time, for correctly solved problems and with more multiplication errors than cortical negativity. Response execution time, a measure of motor speed, was not affected by cortical polarity. The shift in positivity from the amplitude of the slow cortical potential (SCP) before problem presentation to the amplitude of the positive slow wave (pSW) after problem presentation was less in positivity-required compared to negativity-required trials, mainly because of the manipulation of SCP starting points by biofeedback. We assume that cortical positivity is associated with direct retrieval of arithmetic facts, more precisely with the inhibition of incorrect answers. Cortical positivity before task presentation results in a reduced positive shift, a sign of less inhibition before answer production. The effect of less inhibition during positivity-required trials is to produce faster responses and more errors. The opposite might to be true of cortical negativity.

Individual Differences in Cognitive Addition

This study addresses the problem of individual differences in solving simple addition problems by comparing the result of group to individual data. We found that if the group data are better explained by the currently dominant model (retrieval of arithmetic facts from memory), individual data are explained by different models at the level of the process used.

Bilingualism and Numeric Cognition

This article is a critical synthesis of the work which has been done in the field of bilingualism and calculation. The theoretical frameworks dealing with the question of the possible influence of the format or of the language on the retrieval of arithmetical facts are described. The results of the research are discussed in detail by examining the possible alternative interpretations. Encoding and production processes, as well as the possible influence of short-term memory (which may vary according to the subject’s language), are the main considerations.

Localizing Arithmetic Processes in the Brain: Evidence from a Transient Deficit During Cortical Stimulation

Although substantial progress has been made in characterizing the cognitive processes involved in simple arithmetic, the localization of these processes in the brain is not yet well understood. In this article we consider the localization of a specific arithmetic process, the retrieval of arithmetic table facts from memory. We report a single-patient study in which cortical stimulation was used to create transient disruption of brain activity in localized regions of the cortex. We show that stimulation at a left parietal site impaired performance on simple multiplication problems and further that the impairment reflected stimulation-induced disruption of arithmetic fact retrieval. Our findings support the hypothesis (e.g., Warrington, 1982) that the left parietal lobe is implicated in the arithmetic fact retrieval process.

Priming arithmetic facts in amnesic patients

In this study, amnesic patients showed significant repetition priming effects in arithmetic fact retrieval tasks. The results indicate that repetition priming effects in arithmetic depend not on explicit recognition, but on the activation of specific long-term representations of arithmetic facts. Processing dissociations between easy and difficult items suggest that the priming effects result from the stage of fact retrieval and not from peripheral activation. This claim is also supported by encoding and naming tasks, which showed only slight priming effects as compared to the priming found in calculation tasks. Significant priming was found for identical (5×6 and 5×6) and complement problems (5×6 and 6×5), the latter showing a smaller magnitude of priming.

Cerebral Pathways for Calculation: Double Dissociation between Rote Verbal and Quantitative Knowledge of Arithmetic

We describe two acalculic patients, one with a left subcortical lesion and the other with a right inferior parietal lesion and Gerstmann's syndrome. Both suffered from “pure anarithmetia”: they could read arabic numerals and write them to dictation, but experienced a pronounced calculation deficit. On closer analysis, however, distinct deficits were found. The subcortical case suffered from a selective deficit of rote verbal knowledge, including but not limited to arithmetic tables, while her semantic knowledge of numerical quantities was intact. Conversely the inferior parietal case suffered from a category-specific impairment of quantitative numerical knowledge, particularly salient in subtraction and number bissection tasks, with preserved knowledge of rote arithmetic facts. This double dissociation suggests that numerical knowledge is processed in different formats within distinct cerebral pathways. We suggest that a left subcortical network contributes to the storage and retrieval of rote verbal arithmetic facts, while a bilateral inferior parietal network is dedicated to the mental manipulation of numerical quantities.