Core information processing deficits in developmental dyscalculia and low numeracy

There are two different conceptions of the innate basis for numerical abilities. On the one hand, it is claimed that infants possess a 'number module' that enables them to construct concepts of the exact numerosities of sets upon which arithmetic develops (e.g. Butterworth, 1999; Gelman & Gallistel, 1978). On the other hand, it has been proposed that infants are equipped only with a sense of approximate numerosities (e.g. Feigenson, Dehaene & Spelke, 2004), upon which the concepts of exact numerosities are constructed with the aid of language (Carey, 2004) and which forms the basis of arithmetic (Lemer, Dehaene, Spelke & Cohen, 2003). These competing proposals were tested by assessing whether performance on approximate numerosity tasks is related to performance on exact numerosity tasks. Moreover, performance on an analogue magnitude task was tested, since it has been claimed that approximate numerosities are represented as analogue magnitudes. In 8-9-year-olds, no relationship was found between exact tasks and either approximate or analogue tasks in normally achieving children, in children with low numeracy or in children with developmental dyscalculia. Low numeracy was related not to a poor grasp of exact numerosities, but to a poor understanding of symbolic numerals.     

Representations of numerical and non‐numerical magnitude both contribute to mathematical competence in children

A growing body of evidence suggests that non‐symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non‐numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non‐symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5‐ and 6‐year‐old children using comparison tasks of non‐symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non‐numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non‐symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non‐numerical magnitude.     

Developmental bias for number words in the intraparietal sulcus

Children and adults show behavioral evidence of psychological overlap between their early, non‐symbolic numerical concepts and their later‐developing symbolic numerical concepts. An open question is to what extent the common cognitive signatures observed between different numerical notations are coupled with physical overlap in neural processes. We show that from 8 years of age, regions of the intraparietal sulcus (IPS) that exhibit a numerical ratio effect during non‐symbolic numerical judgments also show a semantic distance effect for symbolic number words. In both children and adults, the IPS showed a semantic distance effect during magnitude judgments of number words (i.e. larger/smaller number) but not for magnitude judgments of object words (i.e. larger/smaller object size). The results provide novel evidence of conceptual overlap between neural representations of symbolic and non‐symbolic numerical values that cannot be explained by a general process, and present the first demonstration of an early‐developing dissociation between number words and object words in the human brain.     

手指的感知、运动以及数量表征对数字认知的促进作用

手指是儿童在习得数字符号之前最常使用的表征数量的工具,大量研究都表明手指在数字认知中具有促进作用。但是,目前仍不清楚手指在数字认知中的作用机制。综述从手指感知、手指运动以及手指数量表征三个方面总结了手指在数字认知中所起的作用。手指感知可能通过影响儿童的数量表征能力间接地影响其它数学能力;与表征量有关的手指运动可能促进了数量大小的加工。关于手指数量表征在数字认知中的作用存在两种有争议的观点：一种认为手指数量表征促进了儿童由非符号数量表征向符号数量表征的转化;另一种认为手指数量表征可能是一种数量语义表征方式。未来应该在发展、作用机制、性别差异等方向继续开展研究,进一步探讨手指在数字认知中所起的作用。     

The deterioration of semantic memory in Alzheimer's disease.

Semantic memory impairment is a common feature of dementia of the Alzheimer type (DAT). Recent research has shown that patients with DAT are more impaired (relative to non-demented controls) in generating exemplars from a particular semantic category (e.g., animals) than words beginning with a particular letter, exhibit an altered temporal dynamic during the production of category exemplars, are impaired on confrontation naming tasks and make predominantly superordinate or semantically related errors, consistently misidentify the same objects across a variety of semantic tasks, and have alterations in multi-dimensional scaling models of their semantic network that are indicative of a loss of concepts and associations. These results are consistent with the view that Alzheimer's disease results in a breakdown in the organization and structure of semantic knowledge as neurodegeneration spreads to the association cortices that presumably store semantic representations.     

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like , is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self‐reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task‐specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.     

Numerical estimation in blind subjects: evidence of the impact of blindness and its following experience

Vision was for a long time considered to be essential in the elaboration of the semantic numerical representation. However, early visual deprivation does not seem to preclude the development of a spatial continuum oriented from left to right to represent numbers (J. Castronovo & X. Seron, 2007; D. Szücs & V. Csépe, 2005). The authors investigated the impact of blindness and its following experience on a 3rd property of the mental number line: its obedience to Weber's law. A group of blind subjects and a group of sighted subjects were submitted to 2 numerical estimation tasks: (a) a keypress estimation task and (b) an auditory events estimation task. Blind and sighted subjects' performance obeyed Weber's law. However, blind subjects demonstrated better numerical estimation abilities than did sighted subjects, especially in contexts involving proprioception, indicating the existence of better mapping abilities between the symbolic representations of numbers and their corresponding magnitude representations, obeying Weber's law (e.g., J. S. Lipton & E. Spelke, 2005). These findings suggest that blindness and its following experience with numbers might result in better accuracy in numerical processing.     

近似数量系统敏锐度与数学能力的关系

近年来,来自认知发展、比较认知、跨文化认知和神经生物学的研究证据都表明近似数量系统的存在,并且相较于一般认知能力,它更可能是决定个体数学能力差异最为重要的因素。本文综述了有关近似数量系统敏锐度与数学能力相互关系的横断研究、纵向研究、训练研究及认知神经科学的研究成果,分析了影响二者关系的因素,包括个体年龄、数学能力高低、抑制控制等,并总结了多种理论对二者间显著正相关关系的解释。未来研究需要在确定更具信效度的测量范式的基础上探讨近似数量系统与数学能力各维度的关系,以及这种相互关系背后的原因,并将研究结论运用于数学教学及计算障碍个体的干预。     

The role of part—whole information in reasoning about relative size

Models of comparative judgment have assumed that relative magnitude is computed from knowledge about absolute magnitude rather than retrieved directly. In Experiment 1, participants verified the relative size of part-whole pairs (e.g., tree-leaf) and unrelated controls (e.g., tree-penny). The symbolic distance effect was much smaller for part-whole pairs than for unrelated controls. In two subsequent experiments, participants determined either which of two objects was closer in size to a third object or which of two pairs had a greater difference in the size of its constituents. In contrast to the paired comparison task in Experiment 1, judgments of part-whole items were more sensitive to the influence of symbolic distance than were unrelated controls. The fact that the part-whole relation attenuates the effects of symbolic distance in a paired comparison task but not in tasks that require an explicit comparison of size differences suggests that the part-whole relation provides a source of information about relative magnitude that does not depend on knowledge about absolute magnitude.     

Linking somatic and symbolic representation in semantic memory: the dynamic multilevel reactivation framework

Biological plausibility is an essential constraint for any viable model of semantic memory. Yet, we have only the most rudimentary understanding of how the human brain conducts abstract symbolic transformations that underlie word and object meaning. Neuroscience has evolved a sophisticated arsenal of techniques for elucidating the architecture of conceptual representation. Nevertheless, theoretical convergence remains elusive. Here we describe several contrastive approaches to the organization of semantic knowledge, and in turn we offer our own perspective on two recurring questions in semantic memory research: (1) to what extent are conceptual representations mediated by sensorimotor knowledge (i.e., to what degree is semantic memory embodied)? (2) How might an embodied semantic system represent abstract concepts such as modularity, symbol, or proposition? To address these questions, we review the merits of sensorimotor (i.e., embodied) and amodal (i.e., disembodied) semantic theories and address the neurobiological constraints underlying each. We conclude that the shortcomings of both perspectives in their extreme forms necessitate a hybrid middle ground. We accordingly propose the Dynamic Multilevel Reactivation Framework—an integrative model predicated upon flexible interplay between sensorimotor and amodal symbolic representations mediated by multiple cortical hubs. We discuss applications of the dynamic multilevel reactivation framework to abstract and concrete concept representation and describe how a multidimensional conceptual topography based on emotion, sensation, and magnitude can successfully frame a semantic space containing meanings for both abstract and concrete words. The consideration of ‘abstract conceptual features’ does not diminish the role of logical and/or executive processing in activating, manipulating and using information stored in conceptual representations. Rather, it proposes that the materials upon which these processes operate necessarily combine pure sensorimotor information and higher-order cognitive dimensions involved in symbolic representation.     

The neural consequences of semantic richness: when more comes to mind, less activation is observed

Some concepts have richer semantic representations than others. That is, when considering the meaning of concepts, subjects generate more information (more features, more associates) for some concepts than for others. This variability in semantic richness influences responses in speeded tasks that involve semantic processing, such as lexical decision and semantic categorization tasks. It has been suggested that concepts with richer semantic representations build stronger attractors in semantic space, allowing faster settling of activation patterns and thus faster responding. Using event-related functional magnetic resonance imaging, we examined the neural activation associated with semantic richness by contrasting activation for words with high and low numbers of associates in a semantic categorization task. Results were consistent with faster semantic settling for words with richer representations: Words with a low number of semantic associates produced more activation than words with a high number of semantic associates in a number of regions, including left inferior frontal and inferior temporal gyri.     

Children's representation of symbolic and nonsymbolic magnitude examined with the priming paradigm

How people process and represent magnitude has often been studied using number comparison tasks. From the results of these tasks, a comparison distance effect (CDE) is generated, showing that it is easier to discriminate two numbers that are numerically further apart (e.g., 2 and 8) compared with numerically closer numbers (e.g., 6 and 8). However, it has been suggested that the CDE reflects decisional processes rather than magnitude representation. In this study, therefore, we investigated the development of symbolic and nonsymbolic number processes in kindergartners and first, second, and sixth graders using the priming paradigm. This task has been shown to measure magnitude and not decisional processes. Our findings revealed that a priming distance effect (PDE) is already present in kindergartners and that it remains stable across development. This suggests that formal schooling does not affect magnitude representation. No differences were found between the symbolic and nonsymbolic PDE, indicating that both notations are processed with comparable precision. Finally, a poorer performance on a standardized mathematics test seemed to be associated with a smaller PDE for both notations, possibly suggesting that children with lower mathematics scores have a less precise coding of magnitude. This supports the defective number module hypothesis, which assumes an impairment of number sense.     

Mental number line disruption in a right-neglect patient after a left-hemisphere stroke

A right-neglect patient with focal left-hemisphere damage to the posterior superior parietal lobe was assessed for numerical knowledge and tested on the bisection of numerical intervals and visual lines. The semantic and verbal knowledge of numbers was preserved, whereas the performance in numerical tasks that strongly emphasize the visuo-spatial layout of numbers (e.g. number bisection) was impaired. The behavioral pattern of error in the two bisection tasks mirrored the one previously described in left-neglect patients. In other words, our patient misplaced the subjective midpoint (numerical or visual) to the left as function of the interval size. These data, paired with the patient's lesion site are strictly consistent with the tripartite organization of number-related processes in the parietal lobes as proposed by Dehaene and colleagues. According to these authors, the posterior superior parietal lobe on both hemispheres underpins the attentional orientation on the putative mental number line, the horizontal segment of the intraparietal sulcus is bilaterally related to the semantic of the numerical domain, whereas the left angular gyrus subserves the verbal knowledge of numbers. In summary, our results suggest that the processes involved in the navigation along the mental number line, which are related to the parietal mechanisms for spatial attention, and the processes involved in the semantic and verbal knowledge of numbers, are dissociable.     

The Role of Co-Occurrence Statistics in Developing Semantic Knowledge

The organization of our knowledge about the world into an interconnected network of concepts linked by relations profoundly impacts many facets of cognition, including attention, memory retrieval, reasoning, and learning. It is therefore crucial to understand how organized semantic representations are acquired. The present experiment investigated the contributions of readily observable environmental statistical regularities to semantic organization in childhood. Specifically, we investigated whether co-occurrence regularities with which entities or their labels more reliably occur together than with others (a) contribute to relations between concepts independently and (b) contribute to relations between concepts belonging to the same taxonomic category. Using child-directed speech corpora to estimate reliable co-occurrences between labels for familiar items, we constructed triads consisting of a target, a related distractor, and an unrelated distractor in which targets and related distractors consistently co-occurred (e.g., sock-foot), belonged to the same taxonomic category (e.g., sock-coat), or both (e.g., sock-shoe). We used an implicit, eye-gaze measure of relations between concepts based on the degree to which children (N = 72, age 4–7 years) looked at related versus unrelated distractors when asked to look for a target. The results indicated that co-occurrence both independently contributes to relations between concepts and contributes to relations between concepts belonging to the same taxonomic category. These findings suggest that sensitivity to the regularity with which different entities co-occur in children's environments shapes the organization of semantic knowledge during development. Implications for theoretical accounts and empirical investigations of semantic organization are discussed.     

Knowledge on the line: Manipulating beliefs about the magnitudes of symbolic numbers affects the linearity of line estimation tasks

It has been suggested that differences in performance on number-line estimation tasks are indicative of fundamental differences in people’s underlying representations of numerical magnitude. However, we were able to induce logarithmic-looking performance in adults for magnitude ranges over which they can typically perform linearly by manipulating their familiarity with the symbolic number formats that we used for the stimuli. This serves as an existence proof that individuals’ performances on number-line estimation tasks do not necessarily reflect the functional form of their underlying numerical magnitude representations. Rather, performance differences may result from symbolic difficulties (i.e., number-to-symbol mappings), independently of the underlying functional form. We demonstrated that number-line estimates that are well fit by logarithmic functions need not be produced by logarithmic functions. These findings led us to question the validity of considering logarithmic-looking performance on number-line estimation tasks as being indicative that magnitudes are being represented logarithmically, particularly when symbolic understanding is in question.     

Differences between literates and illiterates on symbolic but not nonsymbolic numerical magnitude processing

The study of numerical magnitude processing provides a unique opportunity to examine interactions between phylogenetically ancient systems of semantic representations and those that are the product of enculturation. While nonsymbolic representations of numerical magnitude are processed similarly by humans and nonhuman animals, symbolic representations of numerical magnitude (e.g., Hindu–Arabic numerals) are culturally invented symbols that are uniquely human. Here, we report a comparison of symbolic and nonsymbolic numerical magnitude processing in two groups of participants who differ substantially in their level of literacy. In this study, level of literacy is used as an index of level of school-based numeracy skill. The data from these groups demonstrate that while the processing of nonsymbolic numerical magnitude (numerical distance effect) is unaffected by an individual’s level of literacy, the processing of Hindu–Arabic numerals differs between literate and illiterate individuals who live in a literature culture and have limited symbolic recognition skills. These findings reveal that nonsymbolic numerical magnitude processing is unaffected by enculturation, while the processing of numerical symbols is modulated by literacy.     

Eye fixation behaviour in the number bisection task: Evidence for temporal specificity

Together with magnitude representations, knowledge about multiplicativity and parity contributes to numerical problem solving. In the present study, we used eye tracking to document how and when multiplicativity and parity are recruited in the number bisection task. Fourteen healthy adults evaluated whether the central number of a triplet (e.g., 21_24_27) corresponds to the arithmetic integer mean of the interval defined by the two outer numbers. We observed multiplicativity to specifically affect gaze duration on numbers, indicating that the information of multiplicative relatedness is activated at early processing stages. In contrast, parity only affected total reading time, suggesting involvement in later processing stages. We conclude that different representational features of numbers are available and integrated at different processing stages within the same task and outline a processing model for these temporal dynamics of numerical cognition.     

Arithmetic split effects reflect strategy selection: an adult age comparative study in addition comparison and verification tasks.

We tested whether split effects in arithmetic (i.e., better performance on large-split problems, like 3 + 8 = 16, than on small-split problems, like 3 + 8 = 12) reflect decision processing or strategy selection. To achieve this end, we tested performance of younger and older adults, matched on arithmetic skills, on two arithmetic tasks: the addition/number comparison task (e.g., 4 + 8, 13; which item is the larger?) and in the inequality verification task (e.g., 4 + 8 < 13; Yes/No?). In both tasks, split between additions and proposed numbers were manipulated. We also manipulated the difficulty of the additions, which represents an index of arithmetic fact calculation (i.e., hard problems, like 6 + 8 < 15, are solved more slowly than easy problems, like 2 + 4 < 07, suggesting that calculation takes longer). Analyses of latencies revealed three main results: First, split effects were of smaller magnitude in older adults compared to younger adults, whatever the type of arithmetic task; second, split effects were of smaller magnitude on easy problems; and third, calculation processes were well maintained in older adults with high level of arithmetic skills. This set of results improves our understanding of cognitive aging and strategy selection in arithmetic.     

Improving arithmetic performance with number sense training: An investigation of underlying mechanism

A nonverbal primitive number sense allows approximate estimation and mental manipulations on numerical quantities without the use of numerical symbols. In a recent randomized controlled intervention study in adults, we demonstrated that repeated training on a non-symbolic approximate arithmetic task resulted in improved exact symbolic arithmetic performance, suggesting a causal relationship between the primitive number sense and arithmetic competence. Here, we investigate the potential mechanisms underlying this causal relationship. We constructed multiple training conditions designed to isolate distinct cognitive components of the approximate arithmetic task. We then assessed the effectiveness of these training conditions in improving exact symbolic arithmetic in adults. We found that training on approximate arithmetic, but not on numerical comparison, numerical matching, or visuo-spatial short-term memory, improves symbolic arithmetic performance. In addition, a second experiment revealed that our approximate arithmetic task does not require verbal encoding of number, ruling out an alternative explanation that participants use exact symbolic strategies during approximate arithmetic training. Based on these results, we propose that nonverbal numerical quantity manipulation is one key factor that drives the link between the primitive number sense and symbolic arithmetic competence. Future work should investigate whether training young children on approximate arithmetic tasks even before they solidify their symbolic number understanding is fruitful for improving readiness for math education.     

Understanding and using principles of arithmetic: operations involving negative numbers

Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers ( Dixon, Deets, & Bangert, 2001 ). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation. Participants ( N = 44) completed two tasks. In the Evaluation task, participants rated how well sets of equations were solved. Some sets violated principles of arithmetic and others did not. Participants rated non-violation sets higher than violation sets for two different principles for subtraction with a negative number. In the Word Problem task, participants read word problems and set up equations that could be used to solve them. Participants who displayed greater knowledge of principles of arithmetic with a negative number were more likely to set up equations that involved negative numbers. Thus, participants' knowledge of arithmetic principles was related to their problem representations.