Nonsymbolic Number Processing Research Articles

Number Processing and Heterogeneity of Developmental Dyscalculia

This study investigated if developmental dyscalculia (DD) in children with different profiles of mathematical deficits has the same or different cognitive origins. The defective approximate number system hypothesis and the access deficit hypothesis were tested using two different groups of children with DD (11–13 years old): a group with arithmetic fact dyscalculia (AFD) and a group with general dyscalculia (GD). Several different aspects of number magnitude processing were assessed in these two groups and compared with age-matched typically achieving children. The GD group displayed weaknesses with both symbolic and nonsymbolic number processing, whereas the AFD group displayed problems only with symbolic number processing. These findings provide evidence that the origins of DD in children with different profiles of mathematical problems diverge. Children with GD have impairment in the innate approximate number system, whereas children with AFD suffer from an access deficit. These findings have implications for researchers’ selection procedures when studying dyscalculia, and also for practitioners in the educational setting.

Numerical predictors of arithmetic success in grades 1–6

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1-6. In grades 1-2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.

The Approximate Number System is not Predictive for Symbolic Number Processing in Kindergarteners

The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.

Neural connectivity patterns underlying symbolic number processing indicate mathematical achievement in children

In early childhood, humans learn culturally specific symbols for number that allow them entry into the world of complex numerical thinking. Yet little is known about how the brain supports the development of the uniquely human symbolic number system. Here, we use functional magnetic resonance imaging along with an effective connectivity analysis to investigate the neural substrates for symbolic number processing in young children. We hypothesized that, as children solidify the mapping between symbols and underlying magnitudes, important developmental changes occur in the neural communication between the right parietal region, important for the representation of non-symbolic numerical magnitudes, and other brain regions known to be critical for processing numerical symbols. To test this hypothesis, we scanned children between 4 and 6years of age while they performed a magnitude comparison task with Arabic numerals (numerical, symbolic), dot arrays (numerical, non-symbolic), and lines (non-numerical). We then identified the right parietal seed region that showed greater blood-oxygen-level-dependent signal in the numerical versus the non-numerical conditions. A psychophysiological interaction method was used to find patterns of effective connectivity arising from this parietal seed region specific to symbolic compared to non-symbolic number processing. Two brain regions, the left supramarginal gyrus and the right precentral gyrus, showed significant effective connectivity from the right parietal cortex. Moreover, the degree of this effective connectivity to the left supramarginal gyrus was correlated with age, and the degree of the connectivity to the right precentral gyrus predicted performance on a standardized symbolic math test. These findings suggest that effective connectivity underlying symbolic number processing may be critical as children master the associations between numerical symbols and magnitudes, and that these connectivity patterns may serve as an important indicator of mathematical achievement.

Ordinality and the Nature of Symbolic Numbers

The view that representations of symbolic and nonsymbolic numbers are closely tied to one another is widespread. However, the link between symbolic and nonsymbolic numbers is almost always inferred from cardinal processing tasks. In the current work, we show that considering ordinality instead points to striking differences between symbolic and nonsymbolic numbers. Human behavioral and neural data show that ordinal processing of symbolic numbers (Are three Indo-Arabic numerals in numerical order?) is distinct from symbolic cardinal processing (Which of two numerals represents the greater quantity?) and nonsymbolic number processing (ordinal and cardinal judgments of dot-arrays). Behaviorally, distance-effects were reversed when assessing ordinality in symbolic numbers, but canonical distance-effects were observed for cardinal judgments of symbolic numbers and all nonsymbolic judgments. At the neural level, symbolic number-ordering was the only numerical task that did not show number-specific activity (greater than control) in the intraparietal sulcus. Only activity in left premotor cortex was specifically associated with symbolic number-ordering. For nonsymbolic numbers, activation in cognitive-control areas during ordinal processing and a high degree of overlap between ordinal and cardinal processing networks indicate that nonsymbolic ordinality is assessed via iterative cardinality judgments. This contrasts with a striking lack of neural overlap between ordinal and cardinal judgments anywhere in the brain for symbolic numbers, suggesting that symbolic number processing varies substantially with computational context. Ordinal processing sheds light on key differences between symbolic and nonsymbolic number processing both behaviorally and in the brain. Ordinality may prove important for understanding the power of representing numbers symbolically.

What basic number processing measures in kindergarten explain unique variability in first-grade arithmetic proficiency?

Relations between children’s mathematics achievement and their basic number processing skills have been reported in both cross-sectional and longitudinal studies. Yet, some key questions are currently unresolved, including which kindergarten skills uniquely predict children’s arithmetic fluency during the first year of formal schooling and the degree to which predictors are contingent on children’s level of arithmetic proficiency. The current study assessed kindergarteners’ non-symbolic and symbolic number processing efficiency. In addition, the contribution of children’s underlying magnitude representations to differences in arithmetic achievement was assessed. Subsequently, in January of Grade 1, their arithmetic proficiency was assessed. Hierarchical regression analysis revealed that children’s efficiency to compare digits, count, and estimate numerosities uniquely predicted arithmetic differences above and beyond the non-numerical factors included. Moreover, quantile regression analysis indicated that symbolic number processing efficiency was consistently a significant predictor of arithmetic achievement scores regardless of children’s level of arithmetic proficiency, whereas their non-symbolic number processing efficiency was not. Finally, none of the task-specific effects indexing children’s representational precision was significantly associated with arithmetic fluency. The implications of the results are 2-fold. First, the findings indicate that children’s efficiency to process symbols is important for the development of their arithmetic fluency in Grade 1 above and beyond the influence of non-numerical factors. Second, the impact of children’s non-symbolic number processing skills does not depend on their arithmetic achievement level given that they are selected from a nonclinical population.

Stages of Nonsymbolic Number Processing in Occipitoparietal Cortex Disentangled by fMRI Adaptation

The neurobiological mechanisms of nonsymbolic number processing in humans are still unclear. Computational modeling proposed three successive stages: first, the spatial location of objects is stored in an object location map; second, this information is transformed into a numerical summation code; third, this summation code is transformed to a number-selective code. Here, we used fMRI-adaptation to identify these three stages and their relative anatomical location. By presenting the same number of dots on the same locations in the visual field, we adapted neurons of human volunteers. Occasionally, deviants with the same number of dots at different locations or different numbers of dots at the same location were shown. By orthogonal number and location factors in the deviants, we were able to calculate three independent contrasts, each sensitive to one of the stages. We found an occipitoparietal gradient for nonsymbolic number processing: the activation of the object location map was found in the inferior occipital gyrus. The summation coding map exhibited a nonlinear pattern of activation, with first increasing and then decreasing activation, and most activity in the middle occipital gyrus. Finally, the number-selective code became more pronounced in the superior parietal lobe. In summary, we disentangled the three stages of nonsymbolic number processing predicted by computational modeling and demonstrated that they constitute a pathway along the occipitoparietal processing stream.

Generating nonsymbolic number stimuli

Studies investigating nonsymbolic numbers (e.g., dot arrays) are confronted with the problem that changes in numerosity are always accompanied by changes in the visual properties of the stimulus. It is therefore debated whether the visual properties of the stimulus rather than number can explain the results obtained in studies investigating nonsymbolic number processing. In this report, we present a program (available at http://titiagebuis.eu/Materials.html ; note that the program is designed to work with the Psychophysics Toolbox in MATLAB) that exports information about the visual properties of stimuli that co-vary with number (area extended, item size, total surface, density, and circumference). Consequently, insight into the relation between the visual properties of the stimulus and numerical distance can be achieved, and posthoc analyses can be conducted to directly reveal whether numerical distance or (some combinations of) the visual properties of a stimulus could be the most likely candidate underlying the results. Here, we report data that demonstrate the program's usefulness for research on nonsymbolic number stimuli.

Individual differences in working memory, nonverbal IQ, and mathematics achievement and brain mechanisms associated with symbolic and nonsymbolic number processing

Individual differences in mathematics performance may stem from domain-general factors like working memory and intelligence. Parietal and frontal brain areas have been implicated in number processing, but the influence of such cognitive factors on brain activity during mathematics processing is not known. The relationship between brain mechanisms during symbolic and nonsymbolic number comparison and individual differences in working memory, nonverbal IQ, and mathematics achievement were investigated to determine possible associations with behavior and brain function. A number of brain areas showed correlations with working memory and number processing. For symbolic digits, working memory showed a positive relationship with brain activity in a network of bilateral parietal, temporal, and right frontal regions. For nonsymbolic dot arrays, working memory showed a negative relationship with several parietal and frontal brain areas. This relationship indicates differences between behavioral and brain function measures and points to the importance of working memory and basic number processing.

Common and segregated neural pathways for the processing of symbolic and nonsymbolic numerical magnitude: An fMRI study

Numbers are everywhere in modern life. Looking out a window, one might see both symbolic numbers, like the numerals on a thermometer, and nonsymbolic quantities, such as the number of chickadees at a bird feeder. Although differences between symbolic and nonsymbolic numbers appear very salient, most research on numerical cognition has focused on similarities rather than differences between numerical stimulus formats. Thus, little is known about differences in the processing of symbolic and nonsymbolic numerical magnitudes. A recent computational model proposed that symbolic and nonsymbolic quantities undergo distinct encoding processes which then converge on a common neural representation of numerical magnitude (Verguts, T., Fias, W., 2004. Representation of number in animals and humans: a neural model. J. Cogn. Neurosci. 16 (9), 1493–1504.). Moreover, this model predicted that discrete brain regions underlie these encoding processes. Using functional magnetic resonance imaging, the present study tested the predictions of this model by examining the functional neuroanatomy of symbolic and nonsymbolic number processing. Nineteen adults compared the relative numerical magnitude of symbolic and nonsymbolic stimuli. An initial conjunction analysis revealed the right inferior parietal lobule to be significantly active in both symbolic and nonsymbolic numerical comparison. A contrast of the activation associated with symbolic and nonsymbolic stimuli revealed that both the left angular and superior temporal gyri were more activated for symbolic compared to nonsymbolic numerical magnitude judgments. The reverse comparison (nonsymbolic>symbolic) revealed several regions including the right posterior superior parietal lobe. These results reveal both format-general and format-specific processing of numerical stimuli in the brain. The potential roles of these regions in symbolic and nonsymbolic numerical processing are discussed.

Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children’s mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6- to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.

A developmental fMRI study of nonsymbolic numerical and spatial processing

This functional magnetic resonance imaging (fMRI) study systematically investigates whether there is a neurofunctional overlap of nonsymbolic numerical and spatial cognition in (intra)parietal regions in children and adults. The study also explores the association between finger use and (nonsymbolic) number processing across development. Twenty-four healthy individuals (12 children, 12 adults) were asked to make nonsymbolic numerical and spatial (experimental tasks) as well as color discriminations (control task). Using identical stimulus material across the three tasks disentangled nonsymbolic number representations from general attentional mechanisms, visual–spatial processing and response selection requirements. In both age groups, behavioral distance effects were obtained upon processing numerical (but not spatial and/or color) stimuli. Baseline imaging effects revealed age-dependent, partly overlapping activations of nonsymbolic numerical and spatial processing in the right posterior superior parietal lobe (PSPL) in adults only. Interestingly, regions more activated in children relative to adults were centred on bilateral supramarginal gyrus (SMG) and lateral portions of the anterior intraparietal sulcus (IPS), further extending to adjacent right post- and precentral gyrus, the latter of which has been reported to support grasping previously (Simon et al., 2002). Overall, our results are first evidence for an age-dependent neurofunctional link between areas supporting finger use and nonsymbolic number processing and furthermore, might be suggestive of a special role of fingers for the development of number magnitude representations and early arithmetic.

A Neural Seat for Math?

Abstract mathematical reasoning is often treated as a uniquely human endeavor. But many species, from pigeons to primates, show some ability to grasp the concept of number, suggesting that these numerical abilities represent the evolutionary building blocks of higher math in humans. Sophisticated symbolic number processing in adults recruits a region of the brain called the intraparietal sulcus (IPS). While children can grasp basic math concepts relating to size and number before they know the words that describe them, very little is known about the neural basis of these abilities. Does the IPS support the relatively simple numerical tasks of childhood as well as the sophisticated numerical calculations that adults learn to perform? Behavioral studies show that adults respond similarly to nonsymbolic numerical stimuli (arrays of dots) and symbolic numerical stimuli (Arabic numerals), suggesting that a common pathway supports both tasks. But neuroimaging studies have not resolved whether the same brain pathway is involved in both symbolic and nonsymbolic number processing. In a new study, Jessica Cantlon, Elizabeth Brannon, Elizabeth Carter, and Kevin Pelphrey at Duke University used functional magnetic resonance imaging (fMRI) to investigate how the IPS responds to nonsymbolic numerical values in adults and preschool-aged children. They show that the brain circuitry governing nonsymbolic number processing is already in place very early in human development. In the study, adults and four-year-olds lay in a scanner while passively viewing a continuous stream of visual arrays on a computer screen. The arrays were designed to elicit differences in brain response to stimuli that were either novel in number or novel in shape. This study design operates under the assumption that neurons tuned to a particular stimuli (numbers, for example) will stop responding when exposed to a standard stimulus (16 circles) over and over, but will respond to stimuli that deviate from the norm (six or 32 circles). Every so often, a deviant number or shape (a triangle or square in place of a circle) was mixed in with the standard stimuli. Participants pressed a button when a crossbar in the center of the visual display turned red to maintain focus. Cantlon et al. analyzed the fMRI data to determine which brain regions responded to both types of deviant stimuli in the adults and children. Number deviants produced a much greater bilateral response in the IPS of adults compared with shape deviants, with activity extending into the inferior and superior parietal lobules (SPL). This response was confirmed by an alternate measure of brain activation based on blood oxygen level, which rose significantly three to 7.5 seconds after the number of elements changed. Brain regions that responded to shape deviants were concentrated in the ventral temporal-occipital cortex. fMRI results for the children showed that number deviants produced a significant response in and around the right IPS and the right SPL. Brain response to shape deviants was similar to that observed in adults. The location and pattern of brain activity in the preschoolers resembled that reported in studies of nonsymbolic numerical processing and basic math ability in adults. By four years old, children's brains already selectively respond to nonsymbolic numerical values, suggesting that the neural networks for number processing are established early in life. How to explain the finding that IPS activity was bilateral in adults and concentrated in the right hemisphere in the four-year-olds? It could be that the left hemisphere acquires more sophisticated math-related functions over time while the right remains relatively stable. But since some children showed more activity in the left IPS, the researchers warn that future study will have to determine whether this pattern is unique to kids. Overall, these results indicate that the brain dedicates a region to cultivating numerical abilities early in development. The IPS provides the neurobiological platform for nonsymbolic numerical processing in young children, then supports the expanding capacity for higher-math operations in adulthood. Six-month-olds also have an abstract numerical sense, suggesting that the IPS may even underlie numerical processing in infancy. Much remains to be learned about how children learn to count and match words with symbolic representations of numbers, but these results suggest that focusing on the IPS might help relate biology to behavior to answer some of these questions, and perhaps shed light on the evolution of numerical cognition.

Numerical Magnitude in the Human Parietal Lobe

Behavioral evidence suggests that human adults have a single system for representing the numerical magnitude of both symbolic numbers (e.g., Arabic digits) and nonsymbolic number stimuli (e.g., dot arrays). Brain imaging studies have implicated a specific parietal region in symbolic number processing, leading to the influential hypothesis that this region is the locus of a dedicated, domain-specific number system. Here we evaluated a prediction of this hypothesis, that this region should be activated not only by symbolic but also nonsymbolic number processing. Using nonsymbolic stimuli, we tested for higher parietal activations for number than for nonnumber comparison tasks (experiment 1), fMRI adaptation for numerosity repetition (experiment 2), and greater fMRI increases with increasing task difficulty for number than nonnumber tasks (experiment 3). None of these predictions were supported by the data, posing a serious challenge to the hypothesis that a single, domain-specific parietal region underlies both symbolic and nonsymbolic number representation.

Symbolic and non-symbolic number processing: an fMRI study

Symbolic and non-symbolic number processing: an fMRI study