Representation Of Number Magnitude Research Articles

Age-related effects in magnitude and place-value processing

While general cognitive skills decline during aging, numerical skills seem to be mainly preserved. Such skills are essential for an independent life up to old age, e.g., when dealing with money or time. Operating with numbers usually requires number magnitude and place-value processing. The question is whether these processes are negatively affected by aging due to the general cognitive decline or positively affected due to lifelong experience with numbers. Therefore, we investigated age-related changes in the distance and compatibility effects in single-digit, two-digit, and four-digit number comparison. On the one hand, older adults took longer for number processing and showed a smaller distance effect, indicating altered number magnitude representations. On the other hand, older adults were better in place-value processing as indicated by a smaller compatibility effect than in younger adults. We conclude that aging differentially affects basic numerical skills.

Arabic digit processing in adults with mathematical learning disability.

The processing of Arabic digits is a core difficulty of children suffering from mathematical learning disability (MLD). Dominant accounts assume a semantic impairment affecting either the magnitude representation per se or its access from numerical symbols. But recent data have raised the hypothesis that the impaired processing of Arabic digits may be explained by a selective deficit of digit visual recognition (i.e., recognising a symbol as one of the digits, no matter its identity or numerical meaning). This study aims at testing whether the difficulty to process Arabic digits remains prevalent in adults with MLD and whether it is effectively associated with a digit visual recognition deficit. To do so, we compared 19 adults with MLD to 19 matched controls in an Arabic digit comparison task that required to identify the largest of two digits, and in an Arabic digit lexical decision task that required to decide whether a visual stimulus is a digit or not. The results showed that MLD participants took more time than control participants to perform the comparison task. In contrast, their performance in the digit lexical decision task was within the range of the control participants. Overall, this finding indicates that adults with MLD continue to experience difficulties to process the magnitude of Arabic digits efficiently, and this cannot be explained by a visual recognition deficit for Arabic digits. We conclude that their difficulties are best explained by an impaired representation of number magnitude or by an impaired access to this representation.

Visuospatial Cognitive Dysfunction in Patients with Vestibular Loss.

To characterize visuospatial and nonvisuospatial cognitive domains affected by vestibular loss and determine whether patient-reported outcomes measures (PROMs) correlate with performance on neuropsychological tests. Cross-sectional study. University-based tertiary medical center. Sixty-nine age-matched subjects: 25 patients with bilateral vestibular loss (BVL), 14 patients with unilateral vestibular loss (UVL), and 30 normal controls (NC). Neuropsychological tests used to assess visuospatial and auditory short-term and working memory, number magnitude representation, executive function, and attention. Validated PROMs used to evaluate quality of life and subjective cognitive impairment. Performance on neuropsychological tests and scores on PROM surveys. BVL and UVL patients performed significantly worse than NC subjects on tasks requiring visuospatial representation compared with NC subjects ( p < 0.01). BVL patients demonstrated decreased performance on spatial representation tasks compared with UVL and NC subjects ( p < 0.05 and p < 0.05, respectively). All subject groups performed similarly on tasks assessing nonvisuospatial cognitive domains, such as auditory short-term and working memory, executive function, and attention. PROMs did not seem to correlate with performance on neuropsychological tasks. Patients with vestibular loss exhibit impairments in tasks requiring visuospatial representation but perform similarly to NC subjects in tasks of auditory working memory, executive function, or attention. Currently available questionnaires may be insufficient to screen patients for cognitive deficits.

Number space is made by response space: Evidence from left spatial neglect

Whether the semantic representation of numbers is endowed with an intrinsic spatial component, so that smaller numbers are inherently represented to the left of larger ones on a Mental Number Line (MNL), is a central matter of debate in numerical cognition. To gain an insight into this issue, we investigated the performance of right brain damaged patients with left spatial neglect (N+) in a bimanual Magnitude Comparison SNARC task and in a uni-manual Magnitude Comparison Go/No-Go task (i.e. “is the number smaller or larger than 5?“). While the first task requires the use of contrasting left/right spatial codes for response selection, the second task does not require the use of these codes. In line with previous evidence, in the SNARC task N+ patients displayed a significant asymmetry in Reaction Times (RTs), with slower RTs to number “4”, that was immediately precedent to the numerical reference “5”, with respect to the number “6”, that immediately followed the same reference. This RTs asymmetry was correlated with lesion of white matter tracts, i.e. Fronto-Occipital-Fasciculus, that allows prefrontal Ba 8 and 46 to regulate the distribution of attention on sensory and memory traces in posterior occipital, temporal and parietal areas. In contrast, no similar RTs asymmetry was found in the Go/No-Go task. These findings suggest that while in the SNARC task numbers get mentally organised from left-to-right as a function of their increasing magnitude, so that N+ patients display a delay in the processing of number-magnitudes that are immediately smaller than a given numerical reference, in the Go/No-Go task no left-to-right organization is activated. These results support the idea that it is the use of contrasting left/right spatial codes, whether motor or conceptual, that triggers the generation of a spatially left-to-right organised MNL and that the representation of number magnitude is not endowed with an inherent spatial component.

The Implicit Contribution of Fine Motor Skills to Mathematical Insight in Early Childhood.

Understanding number magnitude is an important prerequisite for children’s mathematical development. One early experience that contributes to this understanding is the common practice of finger counting. Recent research suggested that through repeated finger counting, children internalize their fingers as representations of number magnitude. Furthermore, finger counting habits have been proposed to predict concurrent and future mathematical performance. However, little is known about how finger-based number representations are formed and by which processes they could influence mathematical development. Regarding the emergence of finger-based number representations, it is likely that they result from repeated practice of finger counting. Accordingly, children need sufficient fine motor skills (FMS) to successfully count on their fingers. However, the role that different types of FMS (such as dexterity and graphomotor skills) might play in the development of finger-based number representations is still unknown. In the current study, we investigated (a) whether children’s FMS (dexterity and graphomotor skills) are associated with their emerging finger-based number representations (ordinal and cardinal), (b) whether FMS explain variance in children’s finger-based number representations beyond the influence of general cognitive skills, and (c) whether the association between FMS and numerical skills is mediated by finger-based representations. We tested associations between preschool children’s (N = 80) FMS (dexterity and graphomotor skills), finger-based number representations, and numerical skills. Furthermore, visuo-spatial working memory and nonverbal intelligence were controlled for. Dexterity was related to children’s finger-based number representations as well as numerical skills after controlling for chronological age, but not after also controlling for cognitive skills. Moreover, the relationship between dexterity and numerical skills was mediated by finger-based number representations. No such associations were observed for graphomotor skills. These results suggest that dexterity plays a role in children’s development of finger-based number representations, which in turn contribute to their numerical skills. Possible explanations are discussed.

Time Reading in Middle and Secondary School Students: The Influence of Basic-Numerical Abilities

Time reading skills are central for the management of personal and professional life. However, little is known about the differential influence of basic numerical abilities on analog and digital time reading in general and in middle and secondary school students in particular. The present study investigated the influence of basic numerical skills separately for analog and digital time reading in N = 709 students from 5th to 8th grade. The present findings suggest that the development of time reading skills is not completed by the end of primary school. Results indicated that aspects of magnitude manipulation and arithmetic fact knowledge predicted analog time reading significantly over and above the influence of age. Furthermore, results showed that spatial representations of number magnitude, magnitude manipulation, arithmetic fact knowledge, and conceptual knowledge were significant predictors of digital time reading beyond general cognitive ability and sex. To the best of our knowledge, the present study is the first to show differential effects of basic numerical abilities on analog and digital time reading skills in middle and secondary school students. As time readings skills are crucial for everyday life, these results are highly relevant to better understand basic numerical processes underlying time reading.

The strategy matters: Bounded and unbounded number line estimation in secondary school children

Changes in number line estimation (NLE) performance are frequently used as an indicator for the development of the number magnitude representation. For this purpose, two different task versions have been applied: a traditional bounded and a relatively new unbounded NLE task. Previous studies mainly assessed primary school children or adults showing that these tasks differ in terms of i) estimation accuracy and solution strategies employed as well as ii) with respect to their relation to other basic numerical/arithmetic skills. So far, data from secondary school children are scarce for bounded NLE, and even no data is available for unbounded NLE. Thus, we assessed bounded and unbounded NLE in grade levels 5–7 to evaluate a) developmental as well as strategic influences, and b) the relation of bounded and unbounded NLE performance with basic arithmetic skills. Our results show that children employed the use of different solution strategies for bounded (i.e., proportion-judgement) and unbounded (i.e., magnitude-estimation based) NLE. Moreover, only for bounded NLE, estimation accuracy increased with age. Furthermore, estimation performance for bounded but not unbounded NLE was strongly associated with basic arithmetic (addition, subtraction, multiplication, and division). Our findings indicate that the differential results for bounded and unbounded NLE obtained in primary school children seem to generalize to older secondary school children. Presented results substantially contribute to the knowledge about the (consecutive) development of skills pertaining to bounded and unbounded number line estimation.

Unbounded number line estimation as a measure of numerical estimation.

Number magnitude estimation has been investigated over the last decades using different tasks including non-symbolic numerosity but also number line estimation tasks. Recently, a bi-directional mapping process was suggested for numerosity estimation accounting for underestimation in a perception version of the task (i.e., indicating the number of non-symbolic dots in a set) and overestimation in the corresponding production task (i.e., produce the number of dots indicated by a symbolic number). In the present study, we evaluated the generalizability of these estimation biases in perception and production tasks to bounded and unbounded number line estimation. Importantly, target numbers were underestimated/overestimated by participants in the perception/production version of numerosity estimation as well as unbounded number line estimation. However, this pattern was reversed for bounded number line estimation. Thereby, the present data indicate a conceptual similarity of unbounded number line estimation and the established non-symbolic numerosity estimation task as a measure of numerical estimation. Accordingly, this corroborates the notion that unbounded number line estimation may reflect a purer measure of number magnitude representation than the bounded task version. Furthermore, our findings strengthen the bi-directional mapping hypothesis for numerical estimation by providing evidence for its generalizability to unbounded number line estimation for the first time.

Multiple left-to-right spatial representations of number magnitudes? Evidence from left spatial neglect.

The SNARC effect reflects the observation that when healthy observers with left-to-right reading habits are asked to compare the magnitude or to judge the parity of numbers, they provide faster reaction times (RT) to small numbers with left-sided responses and faster RTs to large numbers with right-sided responses. In magnitude comparison (MC), right brain damaged patients with left-sided neglect typically show a pathologically enlarged SNARC for large numbers and selective slowing to numbers that are immediately lower than the numerical reference (e.g. 4 for reference 5). This asymmetry has been taken as evidence that small numbers are mentally positioned to the left of the reference and, therefore, are processed less efficiently by patients neglecting the left side of space. In parity judgement (PJ), on the other hand, the size of the SNARC effect is unaffected by neglect. This dissociation is typically attributed to the disturbed explicit processing of number magnitude in MC and preserved implicit processing of magnitude in PJ. Before accepting this interpretation, however, it remains to be investigated whether neglect patients show the same RT pattern that characterizes the performance of healthy participants (i.e. left-side RTs that increase linearly as a function of number magnitude and right-side RTs that decrease linearly as a function of magnitude). Clarifying this point is crucial, because an equally sized SNARC can originate from different RT patterns. Here we demonstrate that the RT pattern of neglect patients during PJ is entirely comparable to those of patients without neglect and healthy controls, while the same neglect patients show selective slowing to numbers that are immediately lower than the numerical reference in MC. These findings suggest the existence of multiple left-to-right spatial representations of number magnitude and provides an explanation of the functional dissociation between MC and PJ tasks.

The unique and shared contributions of arithmetic operation understanding and numerical magnitude representation to children’s mathematics achievement

The current study examined the unique and shared contributions of arithmetic operation understanding and numerical magnitude representation to children’s mathematics achievement. A sample of 124 fourth graders was tested on their arithmetic operation understanding (as reflected by their understanding of arithmetic principles and the knowledge about the application of arithmetic operations) and their precision of rational number magnitude representation. They were also tested on their mathematics achievement and arithmetic computation performance as well as the potential confounding factors. The findings suggested that both arithmetic operation understanding and numerical magnitude representation uniquely predicted children’s mathematics achievement. The findings highlight the significance of arithmetic operation understanding in mathematics learning.

Symbolic and non-symbolic number processing in children with developmental dyslexia

This study examined number processing in 10-year-olds with developmental dyslexia (DD). The phonological deficit and double deficit hypotheses imply that children with DD might have a connection deficit that affects their ability to establish links between number symbols and magnitude representations. The double deficit hypothesis also posits that symbolic number difficulties may emerge due to difficulties with processes underlying rapid automatic naming (RAN). The DD group displayed difficulties with symbolic number processing but not with non-symbolic number processing. However, the underlying processes of this access or connection deficit appeared not to be related to phonological awareness or RAN. The DD group displayed impaired arithmetic fluency and calculation that were accounted for by defective processes underlying RAN. In view of the triple-code model, children with DD have impaired verbal number codes or defective access to verbal number codes but an intact core magnitude representation.

Delaware Longitudinal Study of Fraction Learning: Implications for Helping Children With Mathematics Difficulties

The goal of the present article is to synthesize findings to date from the Delaware Longitudinal Study of Fraction Learning. The study followed a large cohort of children (N = 536) between Grades 3 and 6. The findings showed that many students, especially those with diagnosed learning disabilities, made minimal growth in fraction knowledge and that some showed only a basic grasp of the meaning of a fraction even after several years of instruction. Children with low growth in fraction knowledge during the intermediate grades were much more likely to fail to meet state standards on a broad mathematics measure at the end of Grade 6. Although a range of general and mathematics-specific competencies predicted fraction outcomes, the ability to estimate numerical magnitudes on a number line was a uniquely important marker of fraction success. Many children with mathematics difficulties have deep-seated problems related to whole number magnitude representations that are complicated by the introduction of fractions into the curriculum. Implications for helping students with mathematics difficulties are discussed.

Sex differences in number line estimation: The role of numerical estimation.

Sex differences in mathematical performance have frequently been examined over the last decades indicating an advantage for males especially when numerical problems cannot be solved by (classroom-)learnt strategies and/or estimation. Even in basic numerical tasks such as number line estimation, males were found to outperform females - with sex differences argued to emerge from different solution strategies applied by males and females. We evaluated the latter using two versions of the number line estimation task: a bounded and an unbounded task version. Assuming that women tend more strongly to apply known procedures, we expected them to be at a particular disadvantage in the unbounded number line estimation task, which is less prone to be solved by specific strategies such as proportion judgement but requires numerical estimation. Results confirmed more pronounced sex differences for unbounded number line estimation with males performing significantly more accurately in this task version. This further adds to recent evidence suggesting that estimation performance in the bounded task version may reflect solution strategies rather than numerical estimation. Additionally, it indicates that sex differences regarding the spatial representation of number magnitude may not be universal, but associated with spatial-numerical estimations in particular.

Space, time, and numbers in the right posterior parietal cortex: Differences between response code associations and congruency effects

The mental representations of space, time, and number magnitude are inherently linked. The right posterior parietal cortex (PPC) has been suggested to contain a general magnitude system that underlies the overlap between various perceptual dimensions. However, comparative studies including spatial, temporal, and numerical dimensions are missing. In a unified paradigm, we compared the impact of right PPC inhibition on associations with spatial response codes (i.e., Simon, SNARC, and STARC effects) and on congruency effects between space, time, and numbers. Prolonged cortical inhibition was induced by continuous theta-burst stimulation (cTBS), a protocol for transcranial magnetic stimulation (TMS), at the right intraparietal sulcus (IPS).Our results show that congruency effects, but not response code associations, are affected by right PPC inhibition, indicating different neuronal mechanisms underlying these effects. Furthermore, the results demonstrate that interactions between space and time perception are reflected in congruency effects, but not in an association between time and spatial response codes. Taken together, these results implicate that the congruency between purely perceptual dimensions is processed in PPC areas along the IPS, while the congruency between percepts and behavioral responses is independent of this region.

A general number-to-space mapping deficit in developmental dyscalculia

Previous research on developmental dyscalculia (DD) suggested that deficits in the number line estimation task are related to a failure to represent number magnitude linearly. This conclusion was derived from the observation of logarithmically shaped estimation patterns. However, recent research questioned this idea of an isomorphic relationship between estimation patterns and number magnitude representation. In the present study, we evaluated an alternative hypothesis: impairments in the number line estimation task are due to a general deficit in mapping numbers onto space.Adults with DD and a matched control group had to learn linear and non-linear layouts of the number line via feedback. Afterwards, we assessed their performance how well they learnt the new number-space mappings. We found irrespective of the layouts worse performance of adults with DD. Additionally, in case of the linear layout, we observed that their performance did not differ from controls near reference points, but that differences between groups increased as the distance to reference point increased.We conclude that worse performance of adults with DD in the number line task might be due a deficit in mapping numbers onto space which can be partly overcome relying on reference points.

Spatial displacement of numbers on a vertical number line in spatial neglect.

Previous studies that investigated the association of numbers and space in humans came to contradictory conclusions about the spatial character of the mental number magnitude representation and about how it may be influenced by unilateral spatial neglect. The present study aimed to disentangle the debated influence of perceptual vs. representational aspects via explicit mapping of numbers onto space by applying the number line estimation paradigm with vertical orientation of stimulus lines. Thirty-five acute right-brain damaged stroke patients (6 with neglect) were asked to place two-digit numbers on vertically oriented lines with 0 marked at the bottom and 100 at the top. In contrast to the expected, nearly linear mapping in the control patient group, patients with spatial neglect overestimated the position of numbers in the lower middle range. The results corroborate spatial characteristics of the number magnitude representation. In neglect patients, this representation seems to be biased towards the ipsilesional side, independent of the physical orientation of the task stimuli.

Assessing Young Children's Number Magnitude Representation: A Comparison Between Novel and Conventional Tasks

Previously, researchers have relied on asking young children to plot a given number on a 0-to-10 number line to assess their mental representation of numbers 1 to 9. However, such a (“conventional”) number-to-position (N-P) task may underestimate the accuracy of young children's magnitude estimates and misrepresent the nature of their number representation. The purpose of this study was to compare young children's performance on the conventional N-P task and a “modified” N-P task that is more consistent with a discrete-quantity view of number and with measures of theoretically related mathematical competencies. Participants (n = 45), ranging in age from 4;0 to 6;0, were administered both versions of the N-P task twice during 4 sessions in 1 of 2 randomly assigned and counterbalanced orders. Between and within conditions, children were significantly more accurate on the modified version than on the conventional task. The results indicate that the conventional task, in particular, may be confusing and that several simple modifications can make it more understandable for young children. However, when performance on theoretically related number tasks is taken into account, both the conventional and the modified N-P tasks appeared to underestimate competence.

Considering digits in a current model of numerical development.

Numerical cognition has long been considered the perfect example of abstract information processing. Nevertheless, there is accumulating evidence in recent years suggesting that the representation of number magnitude may not be entirely abstract but may present a specific case of embodied cognition rooted in the sensory and bodily experiences of early finger counting and calculating. However, so far none of the existing models of numerical development considers the influence of finger-based representations. Therefore, we make first suggestions on (i) how finger-based representations may be integrated into a current model of numerical development; and (ii) how they might corroborate the acquisition of basic numerical competencies at different development levels.

Numerical development-from cognitive functions to neural underpinnings.

Numerical development-from cognitive functions to neural underpinnings.

Dissociating Number Line Estimations from Underlying Numerical Representations

Estimation patterns in the number line task are usually interpreted to indicate the (logarithmic or linear) nature of the underlying mental number line. However, indicators of the to-be-achieved linear representation may also be confounded with task requirements or strategies to achieve optimal task performance. In this study, we dissociated correct task performance from indices of a linear representation. Therefore, we designed an experiment in which adults and first graders had to learn number-to-space mappings of nonlinear functions. For adults, we found better fits of the corresponding functions than a linear function after just a few minutes of training. For most first graders, estimation patterns were not fitted better by a logarithmic function when they had to learn a logarithmic layout. Thus, estimation patterns produced in the number line estimation task do not necessarily allow for valid inferences on the underlying representation of number magnitude.