Nonsymbolic Numerical Processing Research Articles

How learning influences non-symbolic numerical processing: effects of feedback in the dot comparison task.

It has long been debated how humans estimate the numerosity of sets of elements and what role continuous visual properties play in this process. The dot comparison task, in which the more numerous of two dot arrays must be selected, is a dominant method to investigate this phenomenon. It has been shown that the visual properties of the two dot patterns strongly influence the comparison. This influence can be systematically investigated by manipulating visual properties congruently and incongruently with numerosity. However, it remains unclear how learning and prior experience affect the influence of the visual properties. To address this question, we introduced feedback into the classical dot comparison task: during the learning phase, participants in the experimental group received feedback after each trial indicating whether their answer was correct whereas participants in the control group did not. After the learning phase, neither group received feedback. The convex hull of the dot patterns and the average dot diameter were manipulated congruently and incongruently with numerosity. Our results show that feedback had no effect on overall performance. However, when manipulated separately, dot diameter no longer affected performance in the experimental group after the learning phase, but it did in the control group. Moreover, this effect remained visible even when diameter and convex hull were manipulated simultaneously. This pattern of results is consistent with the notion of sensory integration which proposes that weights are assigned to different visual cues and that numerical judgments depend on an additive combination of these weights. We also found a correlation between performance on an arithmetic task and performance on trials in which dot size was manipulated incongruently with numerosity. However, there were no correlations between an inhibition task and performance in the dot comparison task. Taken together, the current results suggest that learning with feedback may affect some visual properties but not others. Future studies should further investigate a wider range of visual properties to examine which of them can be influenced by learning and under what conditions learning occurs.

Frequency-based Dissociation of Symbolic and Nonsymbolic Numerical Processing during Numerical Comparison.

Recent evidence suggests that during numerical calculation, symbolic and nonsymbolic processing are functionally distinct operations. Nevertheless, both roughly recruit the same brain areas (spatially overlapping networks in the parietal cortex) and happen at the same time (roughly 250 msec poststimulus onset). We tested the hypothesis that symbolic and nonsymbolic processing are segregated by means of functionally relevant networks in different frequency ranges: high gamma (above 50 Hz) for symbolic processing and lower beta (12-17 Hz) for nonsymbolic processing. EEG signals were quantified as participants compared either symbolic numbers or nonsymbolic quantities. Larger EEG gamma-band power was observed for more difficult symbolic comparisons (ratio of 0.8 between the two numbers) than for easier comparisons (ratio of 0.2) over frontocentral regions. Similarly, beta-band power was larger for more difficult nonsymbolic comparisons than for easier ones over parietal areas. These results confirm the existence of a functional dissociation in EEG oscillatory dynamics during numerical processing that is compatible with the notion of distinct linguistic processing of symbolic numbers and approximation of nonsymbolic numerical information.

Numerical Processing Impairment in 22q11.2 (LCR22-4 to LCR22-5) Microdeletion: A Cognitive-Neuropsychological Case Study.

Although progress has been made, the cognitive, biological and, particularly, the genetic underpinnings of math learning difficulties (MD) remain largely unknown. This difficulty stems from the heterogeneity of MD and from the large contribution of environmental factors to its etiology. Understanding endophenotypes, e.g., the role of the Approximate Number System (ANS), may help understanding the nature of MD. MD associated with ANS impairments has been described in some genetic conditions, e.g., 22q11.2 deletion syndrome (22q11.2DS or Velocardiofacial syndrome, VCFS). Recently, a girl with MD was identified in a school population screening. She has a new syndrome resulting from a microdeletion in 22q11.2 (LCR22-4 to LCR22-5), a region adjacent to but not overlapping with region 22q11.2 (LCR22-2 to LCR22-4), typically deleted in VCFS. Here, we describe her cognitive-neuropsychological and numerical-cognitive profiles. The girl was assessed twice, at 8 and 11 years. Her numerical-cognitive performance at both times was compared to demographically similar girls with normal intelligence in a single-case, quasi-experimental study. Neuropsychological assessment was normal, except for relatively minor impairments in executive functions. She presented severe and persistent difficulties in the simplest single-digit calculations. Difficulties in commutative operations improved from the first to the second assessment. Difficulties in subtraction persisted and were severe. No difficulties were observed in Arabic number writing. Difficulties in single-digit calculation co-occurred with basic numerical processing impairments in symbolic and non-symbolic (single-digit comparison, dot sets size comparison and estimation) tasks. Her difficulties suggest ANS impairment. No difficulties were detected in visuospatial/visuoconstructional and in phonological processing tasks. The main contributions of the present study are: (a) this is the first characterization of the neuropsychological phenotype in 22q11.2DS (LCR22-4 to LCR22.5) with normal intelligence; (b) mild forms of specific genetic conditions contribute to persistent MD in otherwise typical persons; (c) heterogeneity of neurogenetic underpinnings of MD is suggested by poor performance in non-symbolic numerical processing, dissociated from visuospatial/visuoconstructional and phonological impairments; (d) similar to what happens in 22q11.2DS (LCR22-2 to LCR22-4), ANS impairments may also characterize 22q11.2DS (LCR22-4 to LCR22-5).

Automatic non-symbolic numerosity processing in preschoolers.

There has recently been an increasing focus on the development of automatic processing of numerical magnitude. However, little effort has been made to explore automatic access to non-symbolic numerical magnitude in preschool children. In experiment 1, we used a non-symbolic physical size comparison task in 3- to 6-year-olds to examine developmental changes and the effect of ratio and counting principle knowledge. Results showed that the existence of automatic non-symbolic numerical processing began at age 3–4 years and size congruity effects tended to reduce with increasing age from 4 years old. The study also found that non-counting-principle knowers had a larger congruity effect, and in low ratio conditions the size congruity effect was more easily found. In addition, symbolic number comparison ability was negatively related to size congruity effect. In experiment 2, we explored the relationship between inhibition skill and size congruity effects, as well as interference and facilitatory components in children aged 4 years old. Results showed no correlation between inhibition skills and the size congruity effect and only interference effects were found. We also found a larger interference effect in low ratio conditions than in high ratio conditions.

Both non-symbolic and symbolic quantity processing are important for arithmetical computation but not for mathematical reasoning

ABSTRACTThis study investigated whether numerical processing was important for two types of mathematical competence: arithmetical computation and mathematical reasoning. Thousand eight hundred and fifty-seven Chinese primary school children in third through sixth grades took eight computerised tasks: numerical processing (numerosity comparison, digit comparison), arithmetical computation, number series completion, non-verbal matrix reasoning, mental rotation, choice reaction time, and word rhyming. Hierarchical regressions showed that both non-symbolic numerical processing (numerosity comparison) and symbolic numerical processing (digit comparison) were independent predictors of arithmetical computation but neither was a predictor of mathematical reasoning (assessed by number series completion). These findings suggest that the cognitive basis of mathematical performance varies depending on the type of mathematical competence measured.

Spontaneous focusing on numerosity and the arithmetic advantage

Children show individual differences in their tendency to focus on the numerical aspects of their environment. These individual differences in ‘Spontaneous Focusing on Numerosity’ (SFON) have been shown to predict both current numerical skills and later mathematics success. Here we investigated possible factors which may explain the positive relationship between SFON and symbolic number development. Children aged 4–5 years (N = 130) completed a battery of tasks designed to assess SFON and a range of mathematical skills. Results showed that SFON was positively associated with children's symbolic numerical processing skills and their performance on a standardised test of arithmetic. Hierarchical regression analyses demonstrated that the relationship between SFON and symbolic mathematics achievement can be explained, in part, by individual differences in children's nonsymbolic numerical processing skills and their ability to map between nonsymbolic and symbolic representations of number.

Continuous visual properties of number influence the formation of novel symbolic representations.

Numerical symbols are thought to be mapped onto preexisting nonsymbolic representations of number. A growing body of evidence suggests that nonsymbolic numerical processing is significantly influenced by the associated visual properties of continuous quantity (e.g., surface area, density), but their role in the acquisition of novel symbols is unknown. Forty undergraduate students were trained to associate novel abstract symbols with numerical magnitudes. Half of the symbols were associated with nonsymbolic arrays in which total surface area and numerosity were correlated ("congruent"), and the other symbols were associated with arrays in which total surface area was equated across numerosities ("incongruent"). As numbers are represented in multiple formats (words, digits, nonsymbolic arrays), we also tested whether providing auditory nonword labels facilitated symbol learning. Following training, participants engaged in speeded comparisons of the newly learnt symbols. Comparisons were affected by the ratio between the numerosities associated with each symbol, a characteristic marker of numerical processing. Furthermore, comparisons were hardest for large-ratio comparisons of symbols associated with incongruent area and numerosity pairing during learning. In turn, these findings call for the further investigation of visual parameters on the development of numerical cognition.

Aging and the number sense: preserved basic non-symbolic numerical processing and enhanced basic symbolic processing.

Aging often leads to general cognitive decline in domains such as memory and attention. The effect of aging on numerical cognition, particularly on foundational numerical skills known as the number sense, is not well-known. Early research focused on the effect of aging on arithmetic. Recent studies have begun to investigate the impact of healthy aging on basic numerical skills, but focused on non-symbolic quantity discrimination alone. Moreover, contradictory findings have emerged. The current study aimed to further investigate the impact of aging on basic non-symbolic and symbolic numerical skills. A group of 25 younger (18–25) and 25 older adults (60–77) participated in non-symbolic and symbolic numerical comparison tasks. Mathematical and spelling abilities were also measured. Results showed that aging had no effect on foundational non-symbolic numerical skills, as both groups performed similarly [RTs, accuracy and Weber fractions (w)]. All participants showed decreased non-symbolic acuity (accuracy and w) in trials requiring inhibition. However, aging appears to be associated with a greater decline in discrimination speed in such trials. Furthermore, aging seems to have a positive impact on mathematical ability and basic symbolic numerical processing, as older participants attained significantly higher mathematical achievement scores, and performed significantly better on the symbolic comparison task than younger participants. The findings suggest that aging and its lifetime exposure to numbers may lead to better mathematical achievement and stronger basic symbolic numerical skills. Our results further support the observation that basic non-symbolic numerical skills are resilient to aging, but that aging may exacerbate poorer performance on trials requiring inhibitory processes. These findings lend further support to the notion that preserved basic numerical skills in aging may reflect the preservation of an innate, primitive, and embedded number sense.

An electrophysiological investigation of non-symbolic magnitude processing: Numerical distance effects in children with and without mathematical learning disabilities

IntroductionThe aim of the present study was to probe electrophysiological effects of non-symbolic numerical processing in 20 children with mathematical learning disabilities (mean age = 99.2 months) compared to a group of 20 typically developing matched controls (mean age = 98.4 months). MethodsEEG data were obtained while children were tested with a standard non-symbolic numerical comparison paradigm that allowed us to investigate the effects of numerical distance manipulations for different set sizes, i.e., the classical subitizing, counting and estimation ranges. Effects of numerical distance manipulations on event-related potential (ERP) amplitudes as well as activation patterns of underlying current sources were analyzed. ResultsIn typically developing children, the amplitudes of a late parietal positive-going ERP component showed systematic numerical distance effects that did not depend on set size. For the group of children with mathematical learning disabilities, ERP distance effects were found only for stimuli within the subitizing range. Current source density analysis of distance-related group effects suggested that areas in right inferior parietal regions are involved in the generation of the parietal ERP amplitude differences. ConclusionOur results suggest that right inferior parietal regions are recruited differentially by controls compared to children with mathematical learning disabilities in response to non-symbolic numerical magnitude processing tasks, but only for stimuli with set sizes that exceed the subitizing range.

Subitizing and visual short-term memory in human and non-human species: a common shared system?

Subitizing and visual short-term memory in human and non-human species: a common shared system?

Time and Number Sense Develop in Tandem?

Time and Number Sense Develop in Tandem?

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.

Learning by Numbers in the Primate Cortex

When a two-year-old child learns that a short vertical line can represent the number one, and that the number one represents a single item—whatever that item is—he or she has taken a great stride in the development of symbolic thinking. Although many animals can judge numerosity (for example, which of two groups of animals is larger) and can use such judgements for important decisions, such as whether to fight a smaller group or retreat from a larger one, only humans have developed the ability to use numbers as symbols to represent numerosity in a precise manner. In addition, of course, we use symbols as language more generally, both written and verbal. So how does the human brain learn to associate inherently meaningless symbols with important, meaningful concepts such as quantities? Diester and Nieder investigated this question by teaching two macaque monkeys to associate Arabic numerals (1, 2, and so on) with the numerosity of dot displays—for example, the numeral “1” was paired with a single dot, whereas the numeral “4” was paired with four dots. The authors then recorded the activity of individual neurons in the monkeys' brains to investigate how the brain represented this association. In humans, two parts of the brain seem to be particularly important for processing numerical information, whether symbolic or not—the prefrontal cortex (PFC) and the intraparietal sulcus (IPS). The same areas of the monkey brain have been implicated in nonsymbolic numerical processing. It has therefore been proposed that symbolic numerical abilities in humans build on the same neural systems that are used by other animals for judgements of numerosity. Diester and Nieder recorded from these two brain areas in their macaques to test whether the PFC and IPS were used by monkeys to associate visual symbols (numbers) with specific quantities. Over several months, the monkeys learned to use a lever to indicate whether two stimuli, shown before and after a short delay, represented the same quantity—even if one of the stimuli was an Arabic numeral and the other was a group of dots. When the authors recorded from neurons in the monkeys' brains while the monkeys were performing the task, the results showed that many neurons in both the IPS and the PFC had a preference for a specific number of dots. For example, a particular PFC neuron might respond most strongly to a group of three dots, regardless of their arrangement. The neurons also showed a characteristic “tuning function,” in which their responses became weaker as the number of dots became further from the “preferred” number. So, the neuron that responded strongly to three dots would respond less strongly to two or four dots, and still less strongly to only one dot. Prefrontal neurons link numerical categories to visual signs. Many neurons in both the PFC and the IPS also showed a preference for specific Arabic numerals. However, only in the PFC did the authors find a significant number of neurons that were “tuned” to both a preferred quantity and a preferred numeral. In these neurons, the preferred numeral and quantity were correlated—that is, a PFC neuron that responded most strongly to a group of three dots was also likely to respond most strongly to the numeral “3.” Strikingly, these neurons also showed a similar tuning function, with a graded reduction in response to numerals that represented quantities progressively further from the preferred numeral. These neurons represent the association between symbols (numerals) and numerical categories, so the authors called them “association neurons.” Their abundance in the PFC indicates that the PFC is particularly important for establishing links between symbols and categories. Interestingly, young children who are just learning about numbers show increased activity in the PFC when they are carrying out numerical activities that require associations with symbols. In older humans, who are more proficient at using numbers as symbols, there is greater activity in parietal areas such as the IPS. The authors therefore suggest that the PFC is capable of establishing symbolic associations, but that established associations are relocated to the IPS, freeing up the PFC for more demanding cognitive processes. These results provide exciting insights into how symbolic numerical abilities might have evolved in the human brain, building on existing precursor systems that are used by other animals for numerical judgements.

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.