Mental Representation Of Numbers Research Articles

Mental Number Representations in 2D Space.

There is evidence both for mental number representations along a horizontal mental number line with larger numbers to the right of smaller numbers (for Western cultures) and a physically grounded, vertical representation where “more is up.” Few studies have compared effects in the horizontal and vertical dimension and none so far have combined both dimensions within a single paradigm where numerical magnitude was task-irrelevant and none of the dimensions was primed by a response dimension. We now investigated number representations over both dimensions, building on findings that mental representations of numbers and space co-activate each other. In a Go/No-go experiment, participants were auditorily primed with a relatively small or large number and then visually presented with quasi-randomly distributed distractor symbols and one Arabic target number (in Go trials only). Participants pressed a central button whenever they detected the target number and elsewise refrained from responding. Responses were not more efficient when small numbers were presented to the left and large numbers to the right. However, results indicated that large numbers were associated with upper space more strongly than small numbers. This suggests that in two-dimensional space when no response dimension is given, numbers are conceptually associated with vertical, but not horizontal space.

Vertical spatial representation of numbers across two cultures

There is strong evidence that numbers are spatially represented. However, much of the research has been concerned with investigating number representation within the horizontal rather than the vertical dimension. It has been suggested that the organisation of the writing system, by rows or by columns, plays a pivotal role in shaping the mental number representation. We investigated the vertical number representation in British and Japanese participants, who were asked to randomly generate numbers after turning their heads up or down. Previous works showed that this method facilitated the access to the corresponding portions of the horizontal mental number representation. We found that neither group seemed to be affected by the up/ down head turns. These results are discussed in terms of cultural crossover, differences in the horizontal vs. vertical spatial and numerical representation, and sensitivity of the random number generation task.

Incidental Counting: Speeded Number Naming Through Finger Movements.

The first steps in numerical cognition are usually done in conjunction with fingers. Following the assumption that abstract concepts stay associated with the sensory-motor information that was present during their acquisition and consolidation, mental number representations should always be associated with the respective finger counting components. We tested whether finger movements that imply finger counting actually prime the corresponding number concepts in adults. All participants counted number 1 with their thumb and incremented sequentially to number 5 with their pinky. In the experiment, participants sequentially and repeatedly pressed five buttons from thumb to pinky. Each button press triggered the visual presentation of a random number between 1 and 5 that had to be named aloud, resulting in 20% counting-congruent and 80% counting-incongruent finger-number mappings. Average naming latencies were significantly shorter for congruent than incongruent finger-number combinations. Furthermore, there was a distance effect where primes partly co-activated numerically close target numbers and with decreasing priming for more distant prime-target pairs. Overall, these results provide further evidence that number representations are strongly associated with finger counting experience, making fingers an effective tool for number comprehension.

Learning to focus on number

With age and education, children become increasingly accurate in processing numerosity. This developmental trend is often interpreted as a progressive refinement of the mental representation of number. Here we provide empirical and theoretical support for an alternative possibility, the filtering hypothesis, which proposes that development primarily affects the ability to focus on the relevant dimension of number and to avoid interference from irrelevant but often co-varying quantitative dimensions. Data from the same numerical comparison task in adults and children of various levels of numeracy, including Mundurucú Indians and western dyscalculics, show that, as predicted by the filtering hypothesis, age and education primarily increase the ability to focus on number and filter out potentially interfering information on the non-numerical dimensions. These findings can be captured by a minimal computational model where learning consists in the training of a multivariate classifier whose discrimination boundaries get progressively aligned to the task-relevant dimension of number. This view of development has important consequences for education.

Children's Understanding of the Natural Numbers' Structure.

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between "5" and "10" is larger than the distance between "75" and "80." This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, ; Siegler & Opfer, ). However, several investigators have questioned this argument (e.g., Barth & Paladino, ; Cantlon, Cordes, Libertus, & Brannon, ; Cohen & Blanc-Goldhammer, ). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as "5" or "80"). In Experiments 1 and 2, when 5- and 6-year-olds choose between number lines in a forced-choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1-100 are arranged. The bead placement of 4- and 5-year-olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number-line task may depend on strategies specific to the task.

Spatial-Numerical Associations Enhance the Short-Term Memorization of Digit Locations.

Little is known about how spatial-numerical associations (SNAs) affect the way individuals process their environment, especially in terms of learning and memory. In this study, we investigated the potential effects of SNAs in a digit memory task in order to determine whether spatially organized mental representations of numbers can influence the short-term encoding of digits positioned on an external display. To this aim, we designed a memory game in which participants had to match pairs of identical digits in a 9 × 2 matrix of cards. The nine cards of the first row had to be turned face up and then face down, one by one, to reveal a digit from 1 to 9. When a card was turned face up in the second row, the position of the matching digit in the first row had to be recalled. Our results showed that performance was better when small numbers were placed on the left side of the row and large numbers on the right side (i.e., congruent) as compared to the inverse (i.e., incongruent) or a random configuration. Our findings suggests that SNAs can enhance the memorization of digit positions and therefore that spatial mental representations of numbers can play an important role on the way humans process and encode the information around them. To our knowledge, this study is the first that reaches this conclusion in a context where digits did not have to be processed as numerical values.

The influence of number line estimation precision and numeracy on risky financial decision making.

This study examined whether different aspects of mathematical proficiency influence one's ability to make adaptive financial decisions. "Numeracy" refers to the ability to process numerical and probabilistic information and is commonly reported as an important factor which contributes to financial decision-making ability. The precision of mental number representation (MNR), measured with the number line estimation (NLE) task has been reported to be another critical factor. This study aimed to examine the contribution of these mathematical proficiencies while controlling for the influence of fluid intelligence, math anxiety and personality factors. In our decision-making task, participants chose between two options offering probabilistic monetary gain or loss. Sensitivity to expected value was measured as an index for the ability to discriminate between optimal versus suboptimal options. Partial correlation and hierarchical regression analyses revealed that NLE precision better explained EV sensitivity compared to numeracy, after controlling for all covariates. These results suggest that individuals with more precise MNR are capable of making more rational financial decisions. We also propose that the measurement of "numeracy," which is commonly used interchangeably with general mathematical proficiency, should include more diverse aspects of mathematical cognition including basic understanding of number magnitude.

How a brain says: Fingermath for Empowering Children’s Creativity

Children typically learn basic numerical and arithmetic principles using finger-based representations. However, whether or not reliance on finger-based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education. From the neurocognitive perspective, finger counting provides multisensory input, which conveys both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis, or “finger sense,” enhances mathematical skills. Therefore neurocognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting. More precisely, mathematics education recommends first using finger counting, then concrete structured representations and, finally, mental representations of numbers to perform numerical operations. Taken together, these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be discussed.

Des doigts et des nombres

In this literature review, we describe the state of the art of the scientific knowledge about the relationship between the use of fingers in numerical tasks and individuals’ performance in those tasks. One of the questions we want to answer is related with the universality of finger use in order to represent quantities on fingers or to achieve arithmetic calculations. In a first part of the paper, we report the results of studies highlighting the conscious and unconscious use of fingers in adults when they perform simple numerical tasks. In the second part of the manuscript, we adopt a more developmental point of view and our main question is whether finger use in children shapes their mental representations of numbers. In order to answer this question, we review the literature concerning the use of fingers for counting collection of objects through pointing, for representing quantities through finger rising and for calculating through finger manipulations. One crucial issue addressed in this part of the paper is the role of finger use for a possible transition between non-symbolic and symbolic number mental representations. In the last part of the paper, we discuss whether fingers are important, necessary or useless for the construction of numerical knowledge and conclude that, despite their possible influence on adults’ and children's behaviors and number understanding, they are not indispensable for developing accurate number mental representations. Finally, our review allows us to conclude that an innate view of the relationship between fingers and numbers is not fully supported in the literature and that a more functionalist view wherein fingers can support numerical acquisitions is more plausible.

Spatial-numerical associations in first-graders: evidence from a manual-pointing task.

The current study investigated whether children's mental representations of numbers are organized spatially at the onset of formal schooling using a manual-pointing task. First-graders (N=77) saw four numbers (1, 3, 7, 9) presented randomly in four spatial positions (extreme left, left, right, extreme right) on a touch screen. In a Go/No-Go task, children were asked to press the appearing numbers as fast and accurately as possible, but only when the numbers were "smaller" (or "larger" in a different block) than 5. Results indicated that response times were significantly affected by the spatial position in which the different numbers were presented. Response times for small numbers (1 and 3) increased and response times for large numbers (7 and 9) decreased, the more they were presented towards the right side of the screen. These findings suggested that first-graders spontaneously employed a spatial number representation that was oriented from left to right. Furthermore, this left-to-right organization could not be easily changed by priming a different direction. Our findings indicate that even young children map numbers continuously onto space.

Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks

How children’s understanding of numerical magnitudes changes over the course of development remains a key question in the study of numerical cognition. In an ongoing debate about the source of developmental change, some argue that children maintain and access different mental representations of number, with evidence coming largely from common number-line estimation tasks. In contrast, others argue that a theoretical framework based on psychophysical models of proportion estimation accounts for typical performance on these tasks. The current study explored children’s (n=71) and adults’ (n=27) performance on two number-line tasks: the “number to position” (or NP) task and the inverse “position to number” (or PN) task. Estimates on both tasks are consistent with the predictions of the proportion estimation account and do not support the hypothesis that a fundamental shift in mental representations underlies developmental change in numerical estimation and, in turn, mathematical ability. Converging evidence across the tasks also calls into question the utility of bounded number-line tasks as an evaluation of mental representations of number.

The Importance of Central-Visual Perception Disorders for Dyslexia and Dyscalculia

Visual perception disorders can lead to different learning disorders in children. These are often difficulties in reading and writing, as well as in calculation and number processing. Dyslexia, the specific reading and spelling disorder, is also associated with various deficits of visual perception which, for example, make it difficult to distinguish between individual letters or impair the text-dependent eye movements. From a neural point of view, the visual word form area (VWFA) in the fusiform gyrus, which is responsible for the visual processing of orthographic information must be highlighted, since the VWFA is often disturbed in dyslexics. Dyscalculia, the specific calculation disorder, is often associated with difficulties in visual-spatial processing. This is not surprising since our mental representation of numbers and magnitudes is spatially organized. This means we have a kind of mental number line on which we store numbers according to their magnitude. In the case of dyscalculia, the construction and the automated access to this mental number line are often disturbed. This mental number line is located in the intraparietal sulcus (IPS) of the brain, the region which shows increased abnormalities in dyscalculia, such as volume differences of gray matter, reduced activation or reduced connection to other cortical areas. Although central-visual perception disorders can clearly lead to learning disabilities of reading and writing, as well as calculation, dyslexia or dyscalculia is defined only when organic causes can be ruled out, which is why a differentiation between classical dyslexia or dyscalculia and problems in reading/writing and calculation due to visual deficits is necessary.

The SNARC Effect in Number Memorization and Retrieval. What is the Impact of Congruency, Magnitude and the Exact Position of Numbers in Short-Term Memory Processing?

Mental representations of numbers are spatially organized along a Mental Number Line (MNL). One widely proven manifestation of this relationship is the Spatial Numerical Association of Response Codes (SNARC) effect. It refers to the phenomenon of faster responses to numbers when there is congruency between the reaction side and the number position on the MNL . Although long-term memory is considered to house the MNL, short-term memory (STM) load may also modulate responses to numbers and the SN ARCRC effect. Our question, however, was not how STM content modulates the SNARC effect observed in responses to digits, but rather how the MNLNL representation affects the number retrieval from ST M. Each trial began with four digits presented horizontally in a spatial sequence (prime stimuli), which were then replaced by one of the priming digits as a single target. The task required participants to recall the exact location of the target. The SN ARCRC effect occurred only in the retrieval of left-sided digits, most likely because of the generally better processing of right-sided ones, as well as in reaction to digits presented more laterally. Moreover, memory processing was more efficient with low-magnitude numbers, which may suggest that they trigger attention shifting. We conclude that the MNL affects not only the responses to numbers obtained in typical SNARC-induction tasks, such as number detection, parity judgment or magnitude comparison, but also memorization and retrieval of them. Importantly, this effect seems to be dependent on the exact position of a digit in STM.

Acute peripheral vestibular deficit increases redundancy in random number generation

Unilateral peripheral vestibular deficit leads to broad cognitive difficulties and biases in spatial orientation. More specifically, vestibular patients typically show a spatial bias toward their affected ear in the subjective visual vertical, head and trunk orientation, fall tendency, and walking trajectory. By means of a random number generation task, we set out to investigate how an acute peripheral vestibular deficit affects the mental representation of numbers in space. Furthermore, the random number generation task allowed us to test if patients with peripheral vestibular deficit show evidence of impaired executive functions while keeping the head straight and while performing active head turns. Previous research using galvanic vestibular stimulation in healthy people has shown no effects on number space, but revealed increased redundancy of the generated numbers. Other studies reported a spatial bias in number representation during active and passive head turns. In this experiment, we tested 43 patients with acute vestibular neuritis (18 patients with left-sided and 25 with right-sided vestibular deficit) and 28 age-matched healthy controls. We found no bias in number space in patients with peripheral vestibular deficit but showed increased redundancy in patients during active head turns. Patients showed worse performance in generating sequences of random numbers, which indicates a deficit in the updating component of executive functions. We argue that RNG is a promising candidate for a time- and cost-effective assessment of executive functions in patients suffering from a peripheral vestibular deficit.

The temporary nature of number-space interactions.

It is commonly accepted that the mental representation and processing of numbers and of space are tightly linked. This is evident from studies that have shown relations between math ability and visuospatial skill. Also, math instruction and education rely strongly on visuospatial tools and strategies. The dominant explanation for these number-space interactions is that the mental representation of numbers takes the form of a mental number line with numbers positioned in ascending order according to our reading habits. A long-standing debate is whether the link between numbers and space can be considered as evidence for a spatial number representation in long-term semantic memory, or whether this spatial frame is a temporary representation that emerges in working memory (WM) during task execution. We summarise our recent work that suggests basic number processing tasks do not operate on a long-term spatial memory representation, but on a representation constructed in serial order WM, where the elements are spatially coded as a function of their ordinal position in the memorised sequence. Implications for a new theoretical framework linking serial order WM and basic number processing are discussed.

Adult developmental trajectories of pseudoneglect in the tactile, visual and auditory modalities and the influence of starting position and stimulus length

Pseudoneglect is a tendency to pay more attention to the left side of space, typically demonstrated on tasks like visuo-spatial line bisection, tactile rod bisection and the mental representation of numbers. The developmental trajectory of this bias on these three tasks is not fully understood. In the current study younger participants aged between 18 and 40years of age and older participants aged between 55 and 90years conducted three spatial tasks: (1) visuospatial line bisection – participants were asked to bisect visually presented lines of different lengths at the perceived midpoint; (2) touch-driven tactile rod bisection in the absence of vision – participants were asked to feel the length of a wooden rod with their index finger and bisect the rod at the perceived centre; and (3) mental number line bisection in the absence of vision – participants were asked to listen to a pair of numbers and respond with the numerical midpoint between the pair. The results showed that both younger and older participants demonstrated pseudoneglect (leftward biases) in the visual, tactile and mental number line tasks and that the magnitude of pseudoneglect for each group was influenced by physical or mentally represented starting side (start left versus start right) and stimulus length. We provide an exploration of pseudoneglect in younger and older adults in different tasks that vary in the degree to which mental representations are accessed and argue that pseudoneglect is a result of a right hemisphere attentional orienting process that is retained throughout adulthood. Our results indicate that, contrary to some current models of cognitive ageing, asymmetrical patterns of hemispheric activity may occur in older age.

Numerical distance effect in patients with schizophrenia

There is growing evidence showing that mental representation of numbers is impaired in patients with schizophrenia. Yet, no study has examined the distance effect in the patients.We assessed the distance effect using two number size comparison tasks, with different number references (5 and 7) in 23 patients and 28 healthy individuals. Response times and error rates significantly increased when the distances between the centered references and the targets decreased in both groups. However, patients responded significantly slower and had more error rates compared to controls. Our finding indicates distance effect in patients is similar to the controls, indicating an automatic numerical processing is preserved in patients with schizophrenia.

How feedback improves children's numerical estimation.

Developmental change in children's number-line estimation has been thought to reveal a categorical logarithmic-to-linear shift in mental representations of number. Some have claimed that the broad and rapid change in estimation patterns that occurs with corrective feedback provides strong evidence for this shift. However, quantitative models of proportion judgment may provide a better account of children's estimation patterns while also predicting broad and rapid change following feedback. Here we test the hypothesis that local corrective feedback provides children with additional reference points, rather than catalyzing a shift to a different mental representation of number. We tested 117 children from several second-grade classrooms in a number-line feedback study. Data indicate that the proportion-judgment framework accounts for individual differences in estimation patterns, and that the effects of feedback are consistent with the unique quantitative predictions of the framework. They do not provide evidence supporting the representational shift hypothesis or, more broadly, for the proposal that cognitive change can occur rapidly at the level of entire mental representations.

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.

Interaction of numbers and spatial attention in Iranian people

Abstract Introduction Previous studies on mental representation of numbers have shown an interaction between numbers and space. This representation is called mental number line with small numbers in left and big numbers in right. It is affected by writing direction. The aim of this study was to investigate number-space interaction and its nature considering the characteristics of Persian writing system. Method In the first experiment 30 undergraduate students volunteered to complete two symbolic and non-symbolic Posner’s attentional tasks. In second experiment 37 new students completed the symbolic task and their ERPs were recorded meanwhile. Results Data were analyzed by repeated-ANOVAs. There was no interaction between magnitude and side (F=0.437, p=0.51). The main effect of magnitude was significant (F=5.21, p=0.03). Dot detection after small numbers were responded faster (468±13 ms) than bigger numbers (475±14). Response times for symbolic quantities (464±13) were faster than non-symbolic quantities (479±14) (F=5.12, p=0.03). Similarly, in the second experiment, main effect of magnitude was significant (F=9.612, p=0.005). No main effect or interaction for latency of P300 was significant. Conclusion Behavioral results showed the size effect. However no interaction of magnitude by side (SNARC effect) was seen. In other words magnitude did not shift the mental attention to left or right. Behavioral findings were not verified by ERP results. In summary, affected by Persian writing system, there was no interaction between numbers and space contrary to other languages except Hebrew. In addition, this study indicated that the source of this variation is not at perceptual level and it should be investigated at other levels. Keywords Number, mental representation, attention