Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia

Developmental dyscalculia is a learning disability that affects the acquisition of knowledge about numbers and arithmetic. It is widely assumed that numeracy is rooted on the “number sense”, a core ability to grasp numerical quantities that humans share with other animals and deploy spontaneously at birth. To probe the links between number sense and dyscalculia, we used a psychophysical test to measure the Weber fraction for the numerosity of sets of dots, hereafter called number acuity. We show that number acuity improves with age in typically developing children. In dyscalculics, numerical acuity is severely impaired, with 10-year-old dyscalculics scoring at the level of 5-year-old normally developing children. Moreover, the severity of the number acuity impairment predicts the defective performance on tasks involving the manipulation of symbolic numbers. These results establish for the first time a clear association between dyscalculia and impaired “number sense”, and they may open up new horizons for the early diagnosis and rehabilitation of mathematical learning deficits.     

Working memory and number line representations in single-digit addition: Approximate versus exact,nonsymbolic versus symbolic

How do kindergarteners solve different single-digit addition problem formats? We administered problems that differed solely on the basis of two dimensions: response type (approximate or exact), and stimulus type (nonsymbolic, i.e., dots, or symbolic, i.e., Arabic numbers). We examined how performance differs across these dimensions, and which cognitive mechanism (mental model, transcoding, or phonological storage) underlies performance in each problem format with respect to working memory (WM) resources and mental number line representations. As expected, nonsymbolic problem formats were easier than symbolic ones. The visuospatial sketchpad was the primary predictor of nonsymbolic addition. Symbolic problem formats were harder because they either required the storage and manipulation of quantitative symbols phonologically or taxed more WM resources than their nonsymbolic counterparts. In symbolic addition, WM and mental number line results showed that when an approximate response was needed, children transcoded the information to the nonsymbolic code. When an exact response was needed, however, they phonologically stored numerical information in the symbolic code. Lastly, we found that more accurate symbolic mental number line representations were related to better performance in exact addition problem formats, not the approximate ones. This study extends our understanding of the cognitive processes underlying children's simple addition skills.     

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.     

发展性计算障碍儿童的数量转换缺陷

本研究拟考察发展性计算障碍儿童的认知缺陷成因。实验1要求被试在三种形式(点/点,数/数,点/数)下进行数量比较,实验2仅将点集替换为汉字数字词。结果表明障碍组和正常组在数/数、点/数和汉字/汉字比较任务上的成绩存在显著差异,而在点/点和汉字/汉字比较上没有差异。据此推论,计算障碍儿童符号加工能力受到损伤,符号与非符号数量转换能力存在缺陷,但非符号加工能力和不同符号间数量转换没有缺陷,支持语义提取缺陷假设。     

Brief non-symbolic,approximate number practice enhances subsequent exact symbolic arithmetic in children

Recent research reveals a link between individual differences in mathematics achievement and performance on tasks that activate the approximate number system (ANS): a primitive cognitive system shared by diverse animal species and by humans of all ages. Here we used a brief experimental paradigm to test one causal hypothesis suggested by this relationship: activation of the ANS may enhance children’s performance of symbolic arithmetic. Over 2 experiments, children who briefly practiced tasks that engaged primitive approximate numerical quantities performed better on subsequent exact, symbolic arithmetic problems than did children given other tasks involving comparison and manipulation of non-numerical magnitudes (brightness and length). The practice effect appeared specific to mathematics, as no differences between groups were observed on a comparable sentence completion task. These results move beyond correlational research and provide evidence that the exercise of non-symbolic numerical processes can enhance children’s performance of symbolic mathematics.     

Drawing boundaries: From individual to common region – the development of spatial region attribution in children

The study investigated at what age children draw boundaries around pairs of objects that share either similarity or proximity. In two studies (N=132 and N=252) using a Wertheimer array, a clear age trend between 4 and 8 years showed that while young children were more likely to code objects into individual regions, older children were more likely to attribute common regions to pairs of dots. Unsystematic coding occurred as a transitional pattern until age 6, but except for the youngest group, at all ages a proportion of children drew boundaries of common region around arbitrary numbers of stimuli. Future research might show whether constructing exact, stimulus‐matched boundaries of common region is a performance predictor in other cognitive domains, such as visual memory.     

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.     

Subitizing and counting depend on different attentional mechanisms: Evidence from visual enumeration in afterimages

Two experiments showed that, when selective eye movements were disabled by the presentation of stimuli in the form of afterimages, increased inspection time and facilitative stimulus configurations failed to increase the subitizing limit of 4 objects. Afterimages of two to eight dots induced by a photographic flashgun were shown to 3 adult subjects. For more than 4 objects, enumeration errors occurred at a rate of 20%–30%. Enumeration was effectively perfect for 2–4 linearly configured dots, with occasional errors surprisingly occurring in that range when dots appeared in groups of up to 3 items. No errors occurred in nonafterimage control conditions. Enumeration errors were attributed to failures of individuating dots to be counted due to the deactivation of selective eye movements in afterimages. A third experiment supported this interpretation by disabling eye movements with briefly presented stimuli and producing results much like those of the afterimage conditions.     

Enumeration of dots: An eye movement analysis

The present study reports the measurement of response latencies and the recording of eye movements in a task in which adults had to enumerate dots in figures that differed in number of dots (n^d = 19–23) and grouping of dots. The functional relationship between latencies per dot and mean group size was in agreement with earlier findings (van Oeffelen & Vos, 1982). Temporal information from eye movement data indicated that the relative contribution of fixation durations to overall latency was far larger than the contribution of saccades, which superseded the contribution from eyeblinks. Spatial information in the form of eye movement trajectories indicated that, in general, there occurred one or two fixations at the starting position. From this position onward, eye movements were directed toward areas of dots rather than to each dot in particular. Scanning behavior was sometimes reiterative, in the sense that groups of dots were visited more than once. The results are discussed with respect to the nature of strategies employed during a dot-enumeration task.     

How Do Opportunities to View Objects Together in Time Influence Children's Memory for Location?

We conducted three experiments to investigate how opportunities to view objects together in time influence memory for location. Children and adults learned the locations of 20 objects marked by dots on the floor of an open, square box. During learning, participants viewed the objects either simultaneously or in isolation. At test, participants replaced the objects without the aid of the dots. Experiment 1 showed that when the box was divided into quadrants and the objects in each quadrant were categorically related, 7-, 9-, and 11-year-olds and adults in the simultaneous viewing condition exhibited categorical bias, but only 11-year-olds and adults in the isolated viewing condition exhibited categorical bias. Experiment 2 showed that when the objects were categorically related but no boundaries were present, 11-year-olds and adults in the simultaneous viewing condition exhibited categorical bias, but only adults showed bias in the isolated viewing condition. Experiment 3 revealed that adults exhibited bias in both simultaneous and isolated viewing conditions when boundaries were present but the objects were not related. These findings suggest that opportunities to see objects together in time interact with cues available for grouping objects to help children form spatial groups.     

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.     

Configurational effects on the enumeration of dots: Counting by groups

The processing time for quantifying numerosity of two-dimensional dot patterns was investigated as a function of both number of dots and relative proximity between dots. A cluster algorithm (CODE) was first developed as a formal model of how human subjects organize neighboring dots into groups. CODE-based predictions of grouping effects on number processing latencies were then tested with patterns consisting of n dots (range n = 13–23). The results largely confirmed CODE-based predictions and thereby indicated that large collections of dots are preferably counted by groups. Small (n ≤ 5) groups are subitized and their partial results are summed to a running total. Based on criteria other than dot proximity, large (n > 5), proximity-based groups are subdivided into smaller groups of two or three dots, which are again subitized.     

Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude ( Butterworth, 1999 , 2005 ; A. J. Wilson & Dehaene, 2007 ) and/or to access that number magnitude representation from numerical symbols ( Rousselle & No?l, 2007 ). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.     

Finding one's meaning: a test of the relation between quantifiers and integers in language development

We explored children’s early interpretation of numerals and linguistic number marking, in order to test the hypothesis (e.g., Carey (2004). Bootstrapping and the origin of concepts. Daedalus, 59-68) that children’s initial distinction between one and other numerals (i.e., two, three, etc.) is bootstrapped from a prior distinction between singular and plural nouns. Previous studies have presented evidence that in languages without singular-plural morphology, like Japanese and Chinese, children acquire the meaning of the word one later than in singular-plural languages like English and Russian. In two experiments, we sought to corroborate this relation between grammatical number and integer acquisition within English. We found a significant correlation between children’s comprehension of numerals and a large set of natural language quantifiers and determiners, even when controlling for effects due to age. However, we also found that 2-year-old children, who are just acquiring singular-plural morphology and the word one, fail to assign an exact interpretation to singular noun phrases (e.g., a banana), despite interpreting one as exact. For example, in a Truth-Value Judgment task, most children judged that a banana was consistent with a set of two objects, despite rejecting sets of two for the numeral one. Also, children who gave exactly one object for singular nouns did not have a better comprehension of numerals relative to children who did not give exactly one. Thus, we conclude that the correlation between quantifier comprehension and numeral comprehension in children of this age is not attributable to the singular-plural distinction facilitating the acquisition of the word one. We argue that quantifiers play a more general role in highlighting the semantic function of numerals, and that children distinguish between numerals and other quantifiers from the beginning, assigning exact interpretations only to numerals.     

The development of the first symbolic uses in Mexican children from the pragmatics of object / Desarrollo de los primeros usos simbólicos en niños mexicanos desde la pragmática del objeto

This paper longitudinally studies the emergence and evolution of the first symbolic uses of objects in Mexican children. We observed eight children in triadic interaction with one of their parents and 10 objects in a semi-structured situation at nine, 12, 15 and 18 months old. The children began to use objects symbolically at 12 months, and the duration and frequency increased with age. The highest percentage of the total frequency of symbolic uses, of the four levels identified, was level 1. The frequency of level 4, where two or more symbolic actions occur one after the other, giving rise to ‘symbolic narratives’, increased according to age. These data confirm that knowledge of the rules of canonical uses of objects are the meanings which first symbols base themselves on and that children use the symbolic uses that they are aware of to create ‘symbolic narratives’.     

Non‐symbolic halving in an Amazonian indigene group

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non‐human animals to generate coarse, non‐symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large‐number arrays. The Mundurucu children effected this transformation even when non‐numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.     

The comparative psychology of same-different judgments by humans (Homo sapiens) and monkeys (Macaca mulatta)

The authors compared the performance of humans and monkeys in a Same-Different task. They evaluated the hypothesis that for humans the Same-Different concept is qualitative, categorical, and rule-based, so that humans distinguish 0-disparity pairs (i.e., same) from pairs with any discernible disparity (i.e., different); whereas for monkeys the Same-Different concept is quantitative, continuous, and similarity-based, so that monkeys distinguish small-disparity pairs (i.e., similar) from pairs with a large disparity (i.e., dissimilar). The results supported the hypothesis. Monkeys, more than humans, showed a gradual transition from same to different categories and an inclusive criterion for responding Same. The results have implications for comparing Same-Different performances across species--different species may not always construe or perform even identical tasks in the same way. In particular, humans may especially apply qualitative, rule-based frameworks to cognitive tasks like Same-Different.     

Activation of Real-World Knowledge in the Solution of Word Problems

This article describes two studies that examine factors influencing children's access to real-world knowledge during the solution of word problems. In the first study, based on work in Brazil by Carraher, Carraher, and Schliemann (1987), children were asked to solve arithmetic problems presented in three contexts: (a) as word problems, (b) in simulated store situations, and (c) as symbolic computations. Brazilian children were both more successful and more likely to use mental, informal strategies when solving word problems than when solving symbolic computations. We did not find the same results with our U.S. sample; no effects of context were found in either strategy use or success. Comparison of U.S. and Brazilian children's responses suggested that children may tend to access real-world content when the numbers in a word problem match the problem content, and a second study was conducted to test this interpretation. Children were presented with word problems in which the problem content either matched or did not match the numbers in the problem. It was found that when the numbers matched the problem content, children were more successful in solving the problems and more likely to access their domain knowledge during problem solution, as evidenced by the strategies they used to solve problems in the matched condition. These findings suggest ways in which activation of real-world knowledge might be facilitated during the solution of word problems in school.     

Common-fate grouping as feature selection

The visual system groups elements within the visual field that are physically separated yet similar to each other. Although grouping processes have been intensely studied for a century, the mechanisms of grouping remain elusive. We propose that a primary mechanism for grouping by common fate is attentional selection of a direction of motion. A unique prediction follows from this account: that the visual system must be limited to forming only a single common-fate group at a time, and that attempts to find a particular common-fate group among other groups, or among nongroups, should therefore be highly inefficient. We show that this is true in searches for vertically oriented groups of moving dots among horizontally oriented groups (Experiment 1) and in searches for motion-linked groups among nonlinked objects (Experiment 2). Feature selection may limit the visual system to the construction of only one common-fate group at a time, and thus the experience of simultaneous grouping may be an illusion.     

Learning overhypotheses with hierarchical Bayesian models

Inductive learning is impossible without overhypotheses, or constraints on the hypotheses considered by the learner. Some of these overhypotheses must be innate, but we suggest that hierarchical Bayesian models can help to explain how the rest are acquired. To illustrate this claim, we develop models that acquire two kinds of overhypotheses--overhypotheses about feature variability (e.g. the shape bias in word learning) and overhypotheses about the grouping of categories into ontological kinds like objects and substances.