Spontaneous analog number representations in 3‐year‐old children

When enumerating small sets of elements nonverbally, human infants often show a set‐size limitation whereby they are unable to represent sets larger than three elements. This finding has been interpreted as evidence that infants spontaneously represent small numbers with an object‐file system instead of an analog magnitude system ( Feigenson, Dehaene & Spelke, 2004 ). In contrast, non‐human animals and adult humans have been shown to rely on analog magnitudes for representing both small and large numbers ( Brannon & Terrace, 1998 ; Cantlon & Brannon, 2007 ; Cordes, Gelman, Gallistel & Whalen, 2001). Here we demonstrate that, like adults and non‐human animals, children as young as 3 years of age spontaneously employ analog magnitude representations to enumerate both small and large sets. Moreover, we show that children spontaneously attend to numerical value in lieu of cumulative surface area. These findings provide evidence of young children’s greater sensitivity to number relative to other quantities and demonstrate continuity in the process they spontaneously recruit to judge small and large values.     

4-5岁幼儿数数行为的规则性与策略化应用特点的研究

本研究的主要目的是考察4-5岁儿童数数行为的规则性和策略化应用特点。研究选取了104名幼儿园中班儿童(年龄范围4-5岁),男女各半,采用现场实验法进行了逐一测查,测查任务为标准计数任务和数字复制任务。结果发现:4-5岁儿童的数数行为已经遵循一些数数规则,如固定顺序原则、一一对应原则和基数原则;但该年龄段的儿童还不会自发地把数数作为问题解决的策略应用于比较两组物体数目的问题上。     

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.     

A model of exact small-number representation

To account for the size effect in numerical comparison, three assumptions about the internal structure of the mental number line (e.g., Dehaene, 1992) have been proposed. These are magnitude coding (e.g., Zorzi & Butterworth, 1999), compressed scaling (e.g., Dehaene, 1992), and increasing variability (e.g., Gallistel & Gelman, 1992). However, there are other tasks besides numerical comparison for which there is clear evidence that the mental number line is accessed, and no size effect has been observed in these tasks. This is contrary to the predictions of these three assumptions. Moreover, all three assumptions have difficulties explaining certain symmetries in priming studies of number naming and parity judgment. We propose a neural network model that avoids these three assumptions but, instead, uses place coding, linear scaling, and constant variability on the mental number line. We train the model on naming, parity judgment, and comparison and show that the size effect appears in comparison, but not in naming or parity judgment. Moreover, no asymmetries appear in primed naming or primed parity judgment with this model, in line with empirical data. Implications of our findings are discussed. This work was supported by Grant P5/04 from the Interuniversity Attraction Poles Program—Belgian Science Policy and by a GOA grant from the Ghent University Research Council to W.F.     

One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles

Since the publication of [Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.] seminal work on the development of verbal counting as a representation of number, the nature of the ontogenetic sources of the verbal counting principles has been intensely debated. The present experiments explore proposals according to which the verbal counting principles are acquired by mapping numerals in the count list onto systems of numerical representation for which there is evidence in infancy, namely, analog magnitudes, parallel individuation, and set-based quantification. By asking 3- and 4-year-olds to estimate the number of elements in sets without counting, we investigate whether the numerals that are assigned cardinal meaning as part of the acquisition process display the signatures of what we call "enriched parallel individuation" (which combines properties of parallel individuation and of set-based quantification) or analog magnitudes. Two experiments demonstrate that while "one" to "four" are mapped onto core representations of small sets prior to the acquisition of the counting principles, numerals beyond "four" are only mapped onto analog magnitudes about six months after the acquisition of the counting principles. Moreover, we show that children's numerical estimates of sets from 1 to 4 elements fail to show the signature of numeral use based on analog magnitudes - namely, scalar variability. We conclude that, while representations of small sets provided by parallel individuation, enriched by the resources of set-based quantification are recruited in the acquisition process to provide the first numerical meanings for "one" to "four", analog magnitudes play no role in this process.     

The equality of quantity

Do disparate dimensions of magnitude share an underlying mental representation? Two recent papers offer suggestive evidence that participants' discrimination thresholds are identical across domains. Brannon, Lutz and Cordes showed that six-month-old infants' area discriminations match their number discriminations. VanMarle and Wynn demonstrated the same pattern for six-month-olds' discrimination of temporal duration. These parallels across infants' responses to number, area and time raise questions about the fundamental nature of quantity processing.     

Common Representations of Abstract Quantities

ABSTRACT— Representations of abstract quantities such as time and number are essential for survival. A number of studies have revealed that both humans and nonhuman animals are able to nonverbally estimate time and number; striking similarities in the behavioral data suggest a common magnitude-representation system shared across species. It is unclear, however, whether these representations provide animals with a true concept of time and number, as posited by Gallistel and Gelman (2000) . In this article, we review the prominent cognitive and neurobiological models of timing and counting and explore the current evidence suggesting that nonhuman animals represent these quantities in a modality-independent (i.e., abstract) and ordered manner. Avenues for future research in the area of temporal and mathematical cognition are also discussed.     

Memory for verbal and nonverbal stimuli in learning disability subgroups: analysis by selective reminding

Memory for verbal and nonverbal stimuli was evaluated using selective reminding procedures in normal achieving children and four groups of disabled learners: (1) reading-spelling disabled (R-S); (2) reading-spelling-arithmetic disabled (R-S-A); (3) spelling-arithmetic disabled (S-A); and (4) arithmetic disabled (A). Each child received two analogous free-list memory tasks, one for verbal material (animal names) and the other for nonverbal material (random dot patterns). These tasks were administered using selective reminding procedures that permit separation of storage and retrieval aspects of memory by reminding children only of those words not recalled on previous trials. Results revealed that relative to controls, the A and S-A children had significantly lower storage and retrieval scores on the nonverbal task, but did not differ on the verbal task; the R-S children differed only on retrieval scores from the verbal task; and the R-S-A children on retrieval scores on the verbal task and storage and retrieval scores on the nonverbal task. Thus, results indicate that the memory performance of disabled learners varies according to (1) the type of learning problem (arithmetic vs reading), (2) the nature of the stimuli (verbal vs nonverbal), and (3) the aspect of memory being assessed (storage vs retrieval). This study provides external validation for the classification of disabled learners according to patterns of academic achievement, demonstrating a useful procedure for dealing with the intrasubject variability characteristic of disabled learners.     

Number-processing skills in adults with dyslexia

The present study investigated basic numerical skills and arithmetic in adults with developmental dyslexia. Participants performed exact and approximate calculation, basic numerical tasks (e.g., counting; symbolic number comparison; spatial–numerical association of response codes, SNARC), and visuospatial tasks (mental rotation and visual search tasks). The group with dyslexia showed a marginal impairment in counting compared to age- and IQ-matched controls, and they were impaired in exact addition, in particular with respect to speed. They were also significantly slower in multiplication. In basic number processing, however, there was no significant difference in performance between those with dyslexia and controls. Both groups performed similarly on subtraction and approximate addition tasks. These findings indicate that basic number processing in adults with dyslexia is intact. Their difficulties are restricted to the verbal code and are not associated with deficits in nonverbal magnitude representation, visual Arabic number form, or spatial cognition.     

More precisely defining and measuring the order-irrelevance principle

R. Gelman and C. Gallistel (1978, Young Children's Understanding of Number, Cambridge, MA: Harvard Univ. Press) use two definitions of the order-irrelevance principle interchangeably: (1) count tags do not have to be assigned in a fixed order and (2) the order in which elements of a set are enumerated does not affect the cardinal designation of the set. A study involving 107 kindergarten and first grade children indicates that the two are actually distinct concepts. Apparently, a willingness to arbitrarily assign tags is a developmentally less sophisticated ability than--and hence does not necessarily imply--an ability to predict that differently ordered counts produce the same cardinal designation. Thus it appears that evidence of the second ability is necessary to infer a full understanding of the order-irrelevance principle. The first ability alone implies what might better be termed an "order-indifferent tagging scheme." Suggestions for measuring and further researching the order-irrelevance principle are discussed.     

Children's Emotional Reactivity to Interadult Nonverbal Conflict Expressions

The authors investigated children's responses to nonverbal expressions of conflict. Reactions of 3 groups of children (ranging in age from 6 to 16 years) to multiple forms of nonverbal conflict behaviors expressed in videotaped simulations of interadult disputes were examined. Results indicated that children make few discriminations between different forms of nonverbal conflict behaviors and that their reactions to nonverbal conflict are similar to their reactions to verbal conflict. Adults' expressions of fear elicited the most negative emotional responses from children, suggesting that children react to the meaning of conflict expressions and that expressions of fear may represent the greatest emotional security risks to children. Implications of these results for a theoretical model of the effects of forms of marital conflict on children are discussed (P. T. Davies & E. M. Cummings, 1994).     

"Constructive" enumeration by chimpanzees (Pan troglodytes) on a computerized task

Two chimpanzees used a joystick to collect dots, one at a time, on a computer monitor (see video-clip in the electronic supplementary material), and then ended a trial when the number of dots collected was equal to the Arabic numeral presented for the trial. Both chimpanzees performed substantially and reliably above chance in collecting a quantity of dots equal to the target numeral, one chimpanzee for the numerals 1–7, and the second chimpanzee for the numerals 1–6. Errors that were made were seldom discrepant from the target by more than one dot quantity, and the perceptual process subitization was ruled out as an explanation for the performance. Additionally, analyses of trial duration data indicated that the chimpanzees were responding based on the numerosity of the constructed set rather than on the basis of temporal cues. The chimpanzees' decreasing performance with successively larger target numerals, however, appeared to be based on a continuous representation of magnitude rather than a discrete representation of number. Therefore, chimpanzee counting in this type of experimental task may be a process that represents magnitudes with scalar variability in that the memory for magnitudes associated with each numeral is imperfect and the variability of responses increases as a function of the numeral's value. Accepted after revision: 11 June 2001 Electronic Publication     

Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

Humans appear to share with animals a nonverbal counting process. In a nonverbal counting condition, subjects pressed a key a numeral-specified number of times, while saying “the” at every press. The mean number of presses increased as a power function of the target number, with a constant coefficient of variation (c.v.), both within and beyond the proposed subitizing range (1–4 or 5), suggesting small numbers are represented on the same continuum as larger numbers and subject to the same noise process (scalar variability). By contrast, when subjects counted their presses out loud as fast as they could, the c.v. decreased as the inverse square root of the target value (binomial variability instead of scalar variability). The unexpected power-law relation between target value and mean number of presses in nonverbal counting suggests a new hypothesis about the development of the function relating number symbols to mental magnitudes.     

Children's emotional reactivity to interadult nonverbal conflict expressions

The authors investigated children's responses to nonverbal expressions of conflict. Reactions of 3 groups of children (ranging in age from 6 to 16 years) to multiple forms of nonverbal conflict behaviors expressed in videotaped simulations of interadult disputes were examined. Results indicated that children make few discriminations between different forms of nonverbal conflict behaviors and that their reactions to nonverbal conflict are similar to their reactions to verbal conflict. Adults' expressions of fear elicited the most negative emotional responses from children, suggesting that children react to the meaning of conflict expressions and that expressions of fear may represent the greatest emotional security risks to children. Implications of these results for a theoretical model of the effects of forms of marital conflict on children are discussed (P. T. Davies & E. M. Cummings, 1994).     

Moving along the number line: operational momentum in nonsymbolic arithmetic

Can human adults perform arithmetic operations with large approximate numbers, and what effect, if any, does an internal spatial-numerical representation of numerical magnitude have on their responses? We conducted a psychophysical study in which subjects viewed several hundred short videos of sets of objects being added or subtracted from one another and judged whether the final numerosity was correct or incorrect. Over a wide range of possible outcomes, the subjects' responses peaked at the approximate location of the true numerical outcome and gradually tapered off as a function of the ratio of the true and proposed outcomes (Weber's law). Furthermore, an operational momentum effect was observed, whereby addition problems were overestimated and subtraction problems were underestimated. The results show that approximate arithmetic operates according to precise quantitative rules, perhaps analogous to those characterizing movement on an internal continuum.     

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.     

Evidence for the Syndrome of Nonverbal Learning Disabilities in Children with Brain Tumors

This study examined the predicted utility of the Nonverbal Learning Disabilities syndrome (NLD) (Rourke, 1995) for characterizing neurocognitive and psychosocial outcomes in 123 children with brain tumors. Children with brain tumors were found to be at high risk of having a specific academic deficit, particularly in arithmetic. Children with arithmetic deficit evidenced a higher rate of impairment on nonverbal tasks than on verbal tasks, whereas children with reading deficit evidenced a higher rate of impairment on verbal tasks than on nonverbal tasks. However, significant differences between children with arithmetic and reading deficits were not found for all of the component features of the NLD syndrome, and arithmetic deficit was not related to treatment with irradiation.     

Sensori-motor spatial training of number magnitude representation

An adequately developed spatial representation of number magnitude is associated with children's general arithmetic achievement. Therefore, a new spatial-numerical training program for kindergarten children was developed in which presentation and response were associated with a congruent spatial numerical representation. In particular, children responded by a full-body spatial movement on a digital dance mat in a magnitude comparison task. This spatial-numerical training was more effective than a non-spatial control training in enhancing children's performance on a number line estimation task and a subtest of a standardized mathematical achievement battery (TEDI-MATH). A mediation analysis suggested that these improvements were driven by an improvement of children's mental number line representation and not only by unspecific factors such as attention or motivation. These results suggest a benefit of spatial numerical associations. Rather than being a merely associated covariate, they work as an independently manipulated variable which is functional for numerical development.     

Two core systems of numerical representation in infants

Two nonverbal representation systems, the analog magnitude system (AMS) and the object tracking system (OTS), have been proposed to explain how humans and nonhuman animals represent numerosities. There has long been debate about which of the two systems is responsible for representing small numerosities (<4). This review focuses on findings with human infants to inform that debate. We argue that the empirical data cannot all be explained by a single system, and in particular, infants’ failures to compare small and large numerosities – the boundary effect – undermines the claim that the AMS can account for infants’ numerical abilities in their entirety. We propose that although the two systems coexist throughout the lifespan, competition between the systems is primarily a developmental phenomenon. Potential factors that drive the engagement of each system in infancy, such as stimulus features and task demands, are discussed, and directions for future research are suggested.     

Brief report—Phonological awareness and visual-spatial sketchpad functioning predict early arithmetic attainment: Evidence from a longitudinal study

This study examines the relationships between phonological awareness, visual-spatial sketchpad (VSSP) functioning and arithmetic attainment in young children. A sample of 42 children had their VSSP functioning and phonological awareness assessed when they were 5 years old. Approximately 12 months later their nonverbal reasoning, vocabulary, arithmetic, and reading attainment were assessed. Together, VSSP functioning, phonological awareness, vocabulary, and nonverbal reasoning predicted 41% of the variation in the children's arithmetic attainment. Only phonological awareness and VSSP functioning were significant independent predictors. In contrast, only phonological awareness was a significant independent predictor of reading attainment. These findings are consistent with phonological awareness influencing both the development of reading and arithmetic, whilst VSSP functioning only impacts on arithmetic development.