Neuropsychological evidence for a dissociation in counting and subitizing

Developmental Dyscalculia in Relation to Individual Differences in Mathematical Abilities

There is still much debate about the exact nature and frequency of developmental dyscalculia, and about how it should be defined. This article examines several key questions in turn: Is developmental dyscalculia a distinct disorder, or should it be seen as the lower end of a continuum-or possibly more than one continuum-of numerical ability? Do individuals with developmental dyscalculia show atypical brain structure or function? Does the study of acquired dyscalculia have anything to teach us about developmental dyscalculia? In studying dyscalculia, should we look less at arithmetical ability as a single entity, and more at separable components of arithmetical ability? How heterogeneous is developmental dyscalculia, and how important is it to study individual profiles? To what extent is developmental dyscalculia influenced by domain-specific versus domain-general abilities? The conclusion is that, though a significant amount has been discovered through existing research, and though this has some important implications for screening and diagnosis of dyscalculia, there is much more research that still needs to be conducted if we are to answer all of these questions fully. In particular, the study of developmental dyscalculia must be more integrated with the study of individual differences in mathematics in the population as a whole.

The neuroscience basis and educational interventions of mathematical cognitive impairment and anxiety: a systematic literature review

Introduction

Mathematics is a fundamental subject with significant implications in education and neuroscience. Understanding the cognitive processes underlying mathematical cognition is crucial for enhancing educational practices. However, mathematical cognitive impairment and anxiety significantly hinder learning and application in this field. This systematic literature review aims to investigate the neuroscience basis and effective educational interventions for these challenges.

Methods

The review involved a comprehensive screening of 62 research articles that meet the ESSA evidence levels from multiple databases. The selection criteria focused on studies employing various methodologies, including behavioral experiments and neuroimaging techniques, to explore the neuroscience underpinnings and educational interventions related to mathematical cognitive impairment and anxiety.

Results

The review identified key themes and insights into the neuroscience basis of mathematical cognitive impairment and anxiety. It also examined their impact on educational practices, highlighting the interplay between cognitive processes and educational outcomes. The analysis of these studies revealed significant findings on how these impairments and anxieties manifest and can be addressed in educational settings.

Discussion

The review critically analyzes the shortcomings of existing research, noting gaps and limitations in current understanding and methodologies. It emphasizes the need for more comprehensive and diverse studies to better understand these phenomena. The discussion also suggests new directions and potential improvement strategies for future research, aiming to contribute to more effective educational interventions and enhanced learning experiences in mathematics.

Conclusion

This systematic review provides valuable insights into the neuroscience basis of mathematical cognitive impairment and anxiety, offering a foundation for developing more effective educational strategies. It underscores the importance of continued research in this area to improve educational outcomes and support learners facing these challenges.

The componential nature of arithmetical cognition: some important questions

Research on typically developing children and adults and people with developmental and acquired dyscalculia converges in indicating that arithmetical ability is not unitary but is made up of many different components. Categories of components include non-symbolic quantity representation and processing; symbolic quantity representation and processing; counting procedures and principles; arithmetic operations; arithmetical knowledge and understanding; multiple forms and applications of conceptual knowledge of arithmetic; and domain-general abilities such as attention, executive functions and working memory. There is much evidence that different components can and often do show considerable functional independence, not only in developmental and acquired dyscalculia, but in typically achieving children and adults. At the same time, it is possible to find complex interactions and bidirectional relationships between the different components, including between domain-specific and apparently domain-general abilities. There is a great deal that still needs to be discovered. In particular, we need to learn more about the origins in infancy of subitizing and approximate magnitude comparison, the extent to which these interact, the extent to which they may be further divisible, and the extent and ways in which they themselves may develop with age and the extent to which they may influence later-developing components. There also needs to be a lot more research on exactly how domain-general and domain-specific abilities contribute to mathematical development, and how they interact with one another.

Numerical Cognition after Brain Injury: Is There a Relationship between Subitizing and Arithmetical Abilities?

Subitizing is the ability to enumerate small quantities efficiently and automatically. Counting is a strategy adopted for larger numerosities resulting in a near linear increase in response time with each increase in quantity. Some developmental studies suggest that being able to subitize efficiently may be a predictor of later arithmetical ability. Being able to enumerate small quantities efficiently may be necessary for at least some aspects of arithmetical skill and understanding to develop. According to this view, arithmetic ability ultimately depends upon subitizing. If this were the case, when acquired brain injury results in impaired performance on subitizing tasks, mathematical performance may also be impaired. The following study tested eleven healthy control participants and nine chronic patients with acquired brain injury on tasks focused on visual enumeration, addition and multiplication to explore a potential relationship between subitizing ability and calculation performance. No overall correlations were found between subitizing and addition or multiplication speed. However, a very clear subitizing impairment was found in two patients who then demonstrated very different levels of preserved addition skills. The dissociations found and the large inter-individual variability supports a more componential view of arithmetical ability.

Three-systems for visual numerosity: A single case study

Humans possess the remarkable capacity to assess the numerosity of a set of items over a wide range of conditions, from a handful of items to hundreds of them. Recent evidence is starting to show that judgments over such a large range is possible because of the presence of three mechanisms, each tailored to specific stimulation conditions. Previous evidence in favour of this theory comes from the fact that discrimination thresholds and estimation reaction times are not constants across numerosity levels. Likewise, attention is capable of dissociating the three mechanisms: when healthy adult observers are asked to perform concurrently a taxing task, the judgments of low numerosities (<4 dots) or of high numerosities is affected greatly, not so however for intermediate numerosities. Here we bring evidence from a neuropsychological perspective. To this end we measured perceptual performance in PA, a 41 year-old patient who suffers simultanagnosia after a hypoxic brain injury. PA showed a profound deficit in attentively tracking objects over space and time (multiple object tracking), even in very simple conditions where controls made no errors. PA also showed a massive deficit on sensory thresholds when comparing dot-arrays containing extremely low (3 dots) or extremely high (64, 128 dots) numerosities as well as in comparing dot-distances. Surprisingly, PA discrimination thresholds were relatively spared for intermediate numerosity (12 and 16 dots). Overall his deficit on the numerosity task results in a U-shape function across numerosity which, combined with the attentional deficit and the inability to judge dot-distances, confirms previously suggested three-systems for numerosity judgments.

Numerical processing profiles in children with varying degrees of arithmetical achievement

Recent studies show basic cognitive abilities such as the rapid and precise apprehension of small numerosities in object sets ("subitizing"), verbal counting and numerical magnitude comparison significantly influence the acquisition of arithmetic and continues to modulate more advanced stages of mathematical cognition. Additionally, children with low arithmetic achievement (LAA) and Developmental Dyscalculia (DD) exhibit significant deficits in these cognitive processes. Nevertheless, the different cognitive profiles of children with varying degrees of numerical and arithmetic processing deficits have not been sufficiently characterized, despite its potential relevance to the stimulation of numerical cognition and the design of appropriate intervention strategies. Here, the cognitive profiles of groups of typically developing children, children with low arithmetical achievement and DD, exhibiting typical and atypical subitizing ability were contrasted. The results suggest that relatively independent neurocognitive mechanisms may produce distinct profiles at the behavioral level and suggest children with low arithmetic performance exhibiting atypical subitizing abilities are not only significantly slower, but rely on compensatory mechanisms and strategies compared to typical subitizers. The role of subitizing as a correlate of arithmetic fluency is revised in the light of the present findings.

Neuroemergentism: A Framework for Studying Cognition and the Brain

There has been virtual explosion of studies published in cognitive neuroscience primarily due to increased accessibility to neuroimaging methods, which has led to different approaches in interpretation. This review seeks to synthesize both developmental approaches and more recent views that consider neuroimaging. The ways in which Neuronal Recycling, Neural Reuse, and Language as Shaped by the Brain perspectives seek to clarify the brain bases of cognition will be addressed. Neuroconstructivism as an additional explanatory framework which seeks to bind brain and cognition to development will also be presented. Despite sharing similar goals, the four approaches to understanding how the brain is related to cognition have generally been considered separately. However, we propose that all four perspectives argue for a form of Emergentism in which combinations of smaller elements can lead to a greater whole. This discussion seeks to provide a synthesis of these approaches that leads to the emergence of a theory itself. We term this new synthesis Neurocomputational Emergentism (or Neuromergentism for short).

Interventions for Primary School Children With Difficulties in Mathematics

Difficulty with arithmetic is a common problem for children and adults, though there has been some work on the topic for a surprisingly long time. This chapter will review some of the research that has been done over the years on interventions with primary school children. Interventions can be of various levels of intensiveness, ranging from whole-class approaches that take account of individual differences through small-group and limited-time individual interventions to extended-time individual interventions. Interventions discussed here include those involving peer tuition and group collaboration; those involving board and computer games; and those that involve assessing children's strengths and weaknesses in different components of mathematics; and targeting remedial activities to the assessed weaknesses. Most of the interventions discussed in this chapter specifically involve mathematics (usually mainly arithmetic), but there is also some discussion of attempts to improve mathematics by training children in domain-general skills, including Piagetian operations, metacognition, and executive functions.

Simultanagnosia and object individuation

Simultanagnosic patients have difficulty in perceiving multiple objects when presented simultaneously. In this review article, I discuss how neuropsychological research on simultanagnosia has been inspirational for two interconnected lines of research related to the core mechanisms by which the visual system processes cluttered scenes. First, I review previous studies on enumeration tasks indicating that, despite their inability to identify multiple objects, simultanagnosic patients can enumerate up to 2-3 elements as efficiently as healthy individuals (the so-called "subitizing" phenomenon). This intriguing observation is one of the first results to support the existence of an "object individuation" mechanism that can spatially tag a limited set of objects simultaneously, and resonates with recent research on the brain dynamics of enumeration in healthy individuals. Second, I further develop the implications of the dissociation between object identification and object enumeration in simultanagnosia specifically for the distinction between object identification and individuation. The latter distinction has been the subject of recent neuroimaging research that has provided fine-grained information on the spatial as well as temporal aspects of object individuation and recognition. The lessons learned from neuropsychological research on exact enumeration in simultanagnosia can be generalized to the normal functioning of the human mind, and have provided insightful clues for cognitive neuroscience.

Early occipital injury affects numerosity counting but not simple arithmetic

This study investigated the effects of early occipital injury on the development of counting and simple arithmetic abilities in an occipital epileptic patient. This patient had obvious softening lesions in the bilateral occipital regions due to viral encephalitis at the age of 1.5 years. Results showed that she could perform subitizing and simple arithmetic very well, but could not perform numerosity counting tasks. These results suggest that the occipital cortex plays an important role in the development of numerosity counting skills, but not in the development of subitizing and simple arithmetic.

Reliability and validity of the Leuven Perceptual Organization Screening Test (L-POST)

Neuropsychological tests of visual perception mostly assess high-level processes like object recognition. Object recognition, however, relies on distinct mid-level processes of perceptual organization that are only implicitly tested in classical tests. Furthermore, the psychometric properties of the existing instruments are limited. To fill this gap, the Leuven perceptual organization screening test (L-POST) was developed, in which a wide range of mid-level phenomena are measured in 15 subtests. In this study, we evaluated reliability and validity of the L-POST. Performance on the test is evaluated relative to a norm sample of more than 1,500 healthy control participants. Cronbach's alpha of the norm sample and test-retest correlations for 20 patients provide evidence for adequate reliability of L-POST performance. The convergent and discriminant validity of the test was assessed in 40 brain-damaged patients, whose performance on the L-POST was compared with standard clinical tests of visual perception and other measures of cognitive function. The L-POST showed high sensitivity to visual dysfunction and decreased performance was specific to visual problems. In conclusion, the L-POST is a reliable and valid screening test for perceptual organization. It offers a useful online tool for researchers and clinicians to get a broader overview of the mid-level processes that are preserved or disrupted in a given patient.

Common and dissociated mechanisms for estimating large and small dot arrays: value-specific fMRI adaptation

An fMRI pair-adaptation paradigm was used to identify the brain regions linked to the apprehension of small and large numbers of items. Participants classified stimuli on the basis of their numerosities (fewer or more than five dots). We manipulated the type of repetition within pairs of dot arrays. Overall processing of pairs with small as opposed to large quantities was associated with a decreased BOLD response in the midline structures and inferior parietal cortex. The opposite pattern was observed in middle cingulate cortex. Pairs in which the same numerosity category was repeated, were associated with a decreased signal in the left prefrontal and the left inferior parietal cortices, compared with when numerosities changed. Repetitions of exact numerosities irrespective of sample size were associated with decreased responses in bi-lateral prefrontal, sensory-motor regions, posterior occipital and left intraparietal sulcus (IPS). More importantly, we found value-specific adaptation specific to repeated small quantity in the left lateral occipito-temporal cortex, irrespective of whether the exact same stimulus pattern repeated. Our results indicate that a large network of regions (including the IPS) support visual quantity processing independent of the number of items present; however assimilation of small quantities is associated with additional support from regions within the left occipito-temporal cortex. We propose that processing of small quantities is aided by a subitizing-specific network. This network may account for the increased processing efficiency often reported for numerosities in the subitizing range.

The role of pattern recognition in children's exact enumeration of small numbers

Enumeration can be accomplished by subitizing, counting, estimation, and combinations of these processes. We investigated whether the dissociation between subitizing and counting can be observed in 4- to 6-year-olds and studied whether the maximum number of elements that can be subitized changes with age. To detect a dissociation between subitizing and counting, it is tested whether task manipulations have different effects in the subitizing than in the counting range. Task manipulations concerned duration of presentation of elements (limited, unlimited) and configuration of elements (random, line, dice). In Study 1, forty-nine 4- and 5-year-olds were tested with a computerized enumeration task. Study 2 concerned data from 4-, 5-, and 6-year-olds, collected with Math Garden, a computer-adaptive application to practice math. Both task manipulations affected performance in the counting, but not the subitizing range, supporting the conclusion that children use two distinct enumeration processes in the two ranges. In all age groups, the maximum number of elements that could be subitized was three. The strong effect of configuration of elements suggests that subitizing might be based on a general ability of pattern recognition.

Impaired texture segregation but spared contour integration following damage to right posterior parietal cortex

We examined the relations between texture segregation and contour integration in patients with deficits in spatial attention leading to left or right hemisphere extinction. Patients and control participants were presented with texture and contour stimuli consisting of oriented elements. We induced regularity in the stimuli by manipulating the element orientations resulting in an implicit texture border or explicit contour. Participants had to discriminate curved from straight shapes without making eye movements, while the stimulus presentation time was varied using a QUEST procedure. The results showed that only patients with right hemisphere extinction had a spatial bias, needing a longer presentation time to determine the shape of the border or contour on the contralesional side, especially for borders defined by texture. These results indicate that texture segregation is modulated by attention-related brain areas in the right posterior parietal cortex.

Effects of number magnitude and notation at 7T: separating the neural response to small and large, symbolic and nonsymbolic number

We examined the effects of number magnitude (within vs. outside the subitizable range) and notation (symbolic vs. nonsymbolic number) on neural responses to visual displays in the human brain using fMRI at 7T. We found that the right temporoparietal junction (rTPJ) responded more strongly to small than to larger numbers (2, 4 > 6, 8), while there was greater activity bilaterally within and around the intraparietal sulcus (IPS) as number magnitude increased (6, 8 > 2, 4). The effects of number magnitude were greatest for nonsymbolic stimuli. In addition, there was striking overlap between rTPJ regions responding to small numbers and those most strongly activated by symbolic stimuli, and between IPS regions responding to large numbers and those most activated by nonsymbolic stimuli. The results are consistent with distinct neural processes recruited for the processing of small- and large-number magnitudes. Contributions due to differences in representing exact number (small nonsymbolic arrays and all symbolic numbers, in rTPJ) and overall magnitude (particularly with large nonsymbolic arrays, in IPS), and the associated theoretical implications of the findings, are discussed.

Challenging the use of adult neuropsychological models for explaining neurodevelopmental disorders: developed versus developing brains

In this paper, I contrast approaches from adult neuropsychology that seek selective, domain-specific deficits with approaches aimed at understanding the dynamics of developmental trajectories in children with genetic disorders. I stress the crucial difference between developed brains damaged in their mature state, and atypically developing brains. I also challenge the search for single genes to explain selective cognitive-level outcomes. Throughout, the paper argues that it is critical to trace cognitive-level deficits back to their basic-level processes in infancy, where genes are likely to exert their early influences, if we are to understand both the impairments and proficiencies displayed in children with neurodevelopmental disorders.

Genetic and environmental vulnerabilities in children with neurodevelopmental disorders

One might expect that children with varying genetic mutations or children raised in low socioeconomic status environments would display different deficits. Although this expectation may hold for phenotypic outcomes in older children and adults, cross-syndrome comparisons in infancy reveal many common neural and sociocognitive deficits. The challenge is to track dynamic trajectories over developmental time rather than focus on end states like in adult neuropsychological studies. We contrast the developmental and adult approaches with examples from the cognitive and social domains, and we conclude that static models of adult brain lesions cannot be used to account for the dynamics of change in genetic and environmentally induced disorders in children.

Intentional subitizing: exploring the role of automaticity in enumeration

Subitizing is traditionally described as the rapid, preattentive and automatic enumeration of up to four items. Counting, by contrast, describes the enumeration of larger sets of items and requires slower serial shifts of attention. Although recent research has called into question the preattentive nature of subitizing, whether or not numerosities in the subitizing range can be automatically accessed is yet to be empirically tested. In the current study, participants searched for two pre-defined digits in a circular visual-search array. Distractor dots of various set sizes were placed at the centre of the array. Despite the relevance of the distractor numerosities to the target detection task, the distractors did not influence target detection, thereby suggesting that their numerosities were not automatically accessed in Experiment 1. In Experiment 2, participants were explicitly instructed to enumerate the distractor dots. Here, congruent and incongruent distractor numerosities influenced the target detection task, thereby revealing that the distractor dots were capable of generating interference. Experiment 3 ensured that dots were attended by asking participants to detect the luminance of dots. Data confirmed that subitizing was not automatic. The present study also supported the alleged discontinuity between the subitizing and counting ranges because an examination of reaction time gradients in Experiment 2 found the counting gradient to be significantly steeper than the subitizing gradient. In sum, the results suggest that subitizing is a distinct but non-automatic style of enumeration.

Enumeration of Small and Large Numerosities: The Effect of Element Visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1–8) or by tens (10–80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.

The Neuroanatomy of Visual Enumeration: Differentiating Necessary Neural Correlates for Subitizing versus Counting in a Neuropsychological Voxel-based Morphometry Study

This study is the first to assess lesion–symptom relations for subitizing and counting impairments in a large sample of neuropsychological patients (41 patients) using an observer-independent voxel-based approach. We tested for differential effects of enumerating small versus large numbers of items while controlling for hemianopia and visual attention deficits. Overall impairments in the enumeration of any numbers (small or large) were associated with an extended network, including bilateral occipital and fronto-parietal regions. Within this network, severe impairments in accuracy when enumerating small sets of items (in the subitizing range) were associated with damage to the left posterior occipital cortex, bilateral lateral occipital and right superior frontal cortices. Lesions to the right calcarine extending to the precuneus led to patients serially counting even small numbers of items (indicated by a steep response slope), again demonstrating an impaired subitizing ability. In contrast, impairments in counting large numerosities were associated with damage to the left intraparietal sulcus. The data support the argument for some distinctive processes and neural areas necessary to support subitization and counting with subitizing relying on processes of posterior occipital cortex and with counting associated with processing in the parietal cortex.

Evidence for two numerical systems that are similar in humans and guppies

BACKGROUND

Humans and non-human animals share an approximate non-verbal system for representing and comparing numerosities that has no upper limit and for which accuracy is dependent on the numerical ratio. Current evidence indicates that the mechanism for keeping track of individual objects can also be used for numerical purposes; if so, its accuracy will be independent of numerical ratio, but its capacity is limited to the number of items that can be tracked, about four. There is, however, growing controversy as to whether two separate number systems are present in other vertebrate species.

METHODOLOGY/PRINCIPAL FINDINGS

In this study, we compared the ability of undergraduate students and guppies to discriminate the same numerical ratios, both within and beyond the small number range. In both students and fish the performance was ratio-independent for the numbers 1-4, while it steadily increased with numerical distance when larger numbers were presented.

CONCLUSIONS/SIGNIFICANCE

Our results suggest that two distinct systems underlie quantity discrimination in both humans and fish, implying that the building blocks of uniquely human mathematical abilities may be evolutionarily ancient, dating back to before the divergence of bony fish and tetrapod lineages.

Ontogeny of numerical abilities in fish

BACKGROUND

It has been hypothesised that human adults, infants, and non-human primates share two non-verbal systems for enumerating objects, one for representing precisely small quantities (up to 3-4 items) and one for representing approximately larger quantities. Recent studies exploiting fish's spontaneous tendency to join the larger group showed that their ability in numerical discrimination closely resembles that of primates but little is known as to whether these capacities are innate or acquired.

METHODOLOGY/PRINCIPAL FINDINGS

We used the spontaneous tendency to join the larger shoal to study the limits of the quantity discrimination of newborn and juvenile guppies. One-day old fish chose the larger shoal when the choice was between numbers in the small quantity range, 2 vs. 3 fish, but not when they had to choose between large numbers, 4 vs. 8 or 4 vs. 12, although the numerical ratio was larger in the latter case. To investigate the relative role of maturation and experience in large number discrimination, fish were raised in pairs (with no numerical experience) or in large social groups and tested at three ages. Forty-day old guppies from both treatments were able to discriminate 4 vs. 8 fish while at 20 days this was only observed in fish grown in groups. Control experiments showed that these capacities were maintained after guppies were prevented from using non numerical perceptual variables that co-vary with numerosity.

CONCLUSIONS/SIGNIFICANCE

Overall, our results suggest the ability of guppies to discriminate small numbers is innate and is displayed immediately at birth while discrimination of large numbers emerges later as a result of both maturation and social experience. This developmental dissociation suggests that fish like primates might have separate systems for small and large number representation.