An association between understanding cardinality and analog magnitude representations in preschoolers

The preschool years are a time of great advances in children’s numerical thinking, most notably as they master verbal counting. The present research assessed the relation between analog magnitude representations and cardinal number knowledge in preschool-aged children to ask two questions: (1) Is there a relationship between acuity in the analog magnitude system and cardinality proficiency? (2) Can evidence of the analog magnitude system be found within mappings of number words children have not successfully mastered? To address the first question, Study 1 asked three- to five-year-old children to discriminate side-by-side dot arrays with varying differences in numerical ratio, as well as to complete an assessment of cardinality. Consistent with the analog magnitude system, children became less accurate at discriminating dot arrays as the ratio between the two numbers approached one. Further, contrary to prior work with preschoolers, a significant correlation was found between cardinal number knowledge and non-symbolic numerical discrimination. Study 2 aimed to look for evidence of the analog magnitude system in mappings to the words in preschoolers’ verbal counting list. Based on a modified give-a-number task ( [Wynn, 1990] and [Wynn, 1992] ), three- to five-year-old children were asked to give quantities between 1 and 10 as many times as possible in order to assess analog magnitude variability within their developing cardinality understanding. In this task, even children who have not yet induced the cardinality principle showed signs of analog representations in their understanding of the verbal count list. Implications for the contribution of analog magnitude representations towards mastery of the verbal count list are discussed in light of the present work.     

Magnitude comparison in preschoolers: what counts? Influence of perceptual variables

This study examined numerosity comparison in 3-year-old children. Predictions derived from the analog numerical model and the object-file model were contrasted by testing the effects of size and ratio between numerosities to be compared. Different perceptual controls were also introduced to evaluate the hypothesis that comparison by preschoolers is based on correlated perceptual variables rather than on number per se. Finally, the relation between comparison performance and verbal counting knowledge was investigated. Results showed no evidence that preschoolers use an analog number magnitude or an object-file mechanism to compare numerosities. Rather, their inability to compare sets controlled for surface area suggests that they rely on perceptual cues. Furthermore, the development of numerosity-based representations seems to be related to some understanding of the cardinality concept.     

Children's mappings of large number words to numerosities

Previous studies have suggested that children's learning of the relation between number words and approximate numerosities depends on their verbal counting ability, and that children exhibit no knowledge of mappings between number words and approximate numerical magnitudes for number words outside their productive verbal counting range. In the present study we used a numerical estimation task to explore children's knowledge of these mappings. We classified children as Level 1 counters (those unable to produce a verbal count list up to 35), Level 2 counters (those who were able to count to 35 but not 60) and Level 3 counters (those who counted to 60 or above) and asked children to estimate the number of items on a card. Although the accuracy of children's estimates depended on counting ability, children at all counting skill levels produced estimates that increased linearly in proportion to the target number, for numerosities both within and beyond their counting range. This result was obtained at the group level (Experiment 1) and at the level of individual children (Experiment 2). These findings provide evidence that even the least skilled counters do exhibit some knowledge of the form of the mapping between large number words and approximate numerosities.     

Effects of impoverished environmental conditions on working memory performance

This cross-cultural study investigates the impact of background experience on four verbal and visuo-spatial working memory (WM) tasks. A total of 84 children from low-income families were recruited from the following groups: (1) Portuguese immigrant children from Luxembourg impoverished in terms of language experience; (2) Brazilian children deprived in terms of scholastic background; (3) Portuguese children from Portugal with no disadvantage in either scholastic or language background. Children were matched on age, gender, fluid intelligence, and socioeconomic status and completed four simple and complex span tasks of WM and a vocabulary measure. Results indicate that, despite large differences in their backgrounds and language abilities, the groups exhibited comparable performance on the visuo-spatial tasks dot matrix and odd-one-out and on the verbal simple span task digit recall. Group differences emerged on the verbal complex span task counting recall with children from Luxembourg and Portugal outperforming children from disadvantaged schools in Brazil. The study suggests that whereas contributions of prior knowledge to digit span, dot matrix, and odd-one-out are likely to be minimal, background experience can affect performance on counting recall. Implications for testing WM capacity in children growing up in poverty are discussed.     

The development of ordinal numerical competence in young children

Two experiments assessed ordinal numerical knowledge in 2- and 3-year-old children and investigated the relationship between ordinal and verbal numerical knowledge. Children were trained on a 1 vs 2 comparison and then tested with novel numerosities. Stimuli consisted of two trays, each containing a different number of boxes. In Experiment 1, box size was held constant. In Experiment 2, box size was varied such that cumulative surface area was unrelated to number. Results show children as young as 2 years of age make purely numerical discriminations and represent ordinal relations between numerosities as large as 6. Children who lacked any verbal numerical knowledge could not make ordinal judgments. However, once children possessed minimal verbal numerical competence, further knowledge was entirely unrelated to ordinal competence. Number may become a salient dimension as children begin to learn to count. An analog magnitude representation of number may underlie success on the ordinal task.     

The impact of finger counting habits on arithmetic in adults and children

Here, we explored the impact of finger counting habits on arithmetic in both adults and children. Two groups of participants were examined, those that begin counting with their left hand (left-starters) and those that begin counting with their right hand (right-starters). For the adults, performance on an addition task in which participants added 2 two-digit numbers was compared. The results revealed that left-starters were slower than right-starters when adding and they had lower forward and backward digit-span scores. The children (aged 5–12) showed similar results on a single-digit timed addition task—right-starters outperformed left-starters. However, the children did not reveal differences in working memory or verbal and non-verbal intelligence as a function of finger counting habit. We argue that the motor act of finger counting influences how number is represented and suggest that left-starters may have a more bilateral representation that accounts for the slower processing.     

Verbal Counting Moderates Perceptual Biases Found in Children's Cardinality Judgments

A crucial component of numerical understanding is one's ability to abstract numerical properties regardless of varying perceptual attributes. Evidence from numerical match-to-sample tasks suggests that children find it difficult to match sets based on number in the face of varying perceptual attributes, yet it is unclear whether these findings are indicative of incomplete numerical abstraction abilities early in development or instead are driven by specific demands of the matching task. In this study, we explored whether perceptual biases would be found in data from a numerical task invoking verbal representations of number and whether these biases are moderated by verbal counting behavior. Three- to 6-year-old children classified as proficient counters (cardinal principle knowers) participated in a number cardinality task in which they were asked to identify which of 2 arrays—either perceptually homogeneous or heterogeneous in appearance—contained a specific number of animals (e.g., “12 animals”). Results revealed an overall performance bias for homogeneous trials in this cardinality task, such that children were better able to exactly identify the target cardinality when items within the sets were perceptually identical. Further analyses revealed that these biases were found only for those children who did not explicitly verbally count during the task. In contrast, performance was unaffected by the perceptual attributes of the array when the child spontaneously counted. Together, results reveal that cardinality judgments are negatively impacted by perceptual variation, but this relationship is muted in those children who engage in verbal counting.     

视知觉线索对幼儿小数离散量表征的影响

本研究采用数量辨别任务和数量比较的干扰任务考察了元素累积面积和元素特征丰富度两种视知觉线索对3～6岁幼儿表征4以内离散量的影响。两个任务的被试分别为80名和74名。结果表明,幼儿在辨别小数离散量时并未表现出对元素累积面积的偏好;且幼儿能同时关注累积面积和饼干个数两个维度,从而辨别抽象的数量关系;这种辨别能力有随年龄提高的趋势,不能辨别组以3、4岁幼儿居多,他们辨别方形饼干的数量关系比辨别圆形饼干更容易。当元素累积面积干扰小数量比较时,幼儿数量比较的正确率随数量比例增大而降低,此时幼儿小数量表征不再精确;而当单个元素的知觉特征更鲜明时,即使仍存在元素累积面积干扰,其正确率也不再受数量比例的影响。这说明单个元素的知觉线索与整体知觉线索共同影响小数离散量表征,也提示着幼儿的小数量表征精确性可能并非稳定存在,小数量表征在特定条件下也需要借助模拟数量系统。     

A common representational system governed by Weber's law: nonverbal numerical similarity judgments in 6-year-olds and rhesus macaques

This study compared nonverbal numerical processing in 6-year-olds with that in nonhuman animals using a numerical bisection task. In the study, 16 children were trained on a delayed match-to-sample paradigm to match exemplars of two anchor numerosities. Children were then required to indicate whether a sample intermediate to the anchor values was closer to the small anchor value or the large anchor value. For two sets of anchor values with the same ratio, the probability of choosing the larger anchor value increased systematically with sample number, and the psychometric functions superimposed when plotted on a logarithmic scale. The psychometric functions produced by the children also superimposed with the psychometric functions produced by rhesus monkeys in an analogous previous experiment. These examples of superimposition demonstrate that nonverbal number representations, even in children who have acquired the verbal counting system, are modulated by Weber's law.     

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.     

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.     

Count out your intrusions: Effects of verbal encoding on intrusive memories

Peri-traumatic information processing is thought to affect the development of intrusive trauma memories. This study aimed to replicate and improve the study by Holmes, Brewin, and Hennessy (2004, Exp. 3) on the role of peri-traumatic verbal processing in analogue traumatic intrusion development. Participants viewed an aversive film under one of three conditions: counting backwards in 3s (“verbal interference”), verbalising emotions and thoughts (“verbal enhancement”), or without an extra task. A dual-process account of PTSD would predict that verbal interference would increase intrusion frequency compared to no task, whereas verbal enhancement would lead to a decrease. In contrast, mainstream memory theory predicts a decrease in intrusion frequency from any concurrent task that diverts attention away from the trauma film. The main finding was that the verbal interference task led to a decrease in intrusive memories of the film compared to the other two conditions. This finding does not support a dual-process account of PTSD, but is in line with general theories of memory and attention.     

数字记忆广度对策略应用模式的影响──中美儿童计算能力差异的比较研究

本研究旨在了解中美儿童在策略运用模式上的差异是否与数字记忆广度有关。为此,分别测验了中美幼儿园儿童的加法技能,数字记忆广度,并通过一组计算机呈现的加法作业评估儿童使用的解题策略。结果表明,中国儿童的基本算术技能和数学记忆广度均超过同龄的美国儿童。在解题时,中国儿童多使用效率较高的言语计算策略,美国儿童则多采用计数手指策略,在言语计数策略和检索策略的加工速度上中国儿童也超过美国儿童。相关分析表明．策略应用模式的文化差异与数字记忆广度有关。比较大的工作记忆容量增加了数字线索可利用的资源,为儿童早期言语计数能力的发展提供了有利条件。     

The development of the SNARC effect: evidence for early verbal coding

In a recent study, Gevers and colleagues (2010, Journal of Experimental Psychology: General, Vol. 139, pp. 180-190) showed that the SNARC (spatial numerical association of response codes) effect in adults results not only from spatial coding of magnitude (e.g., mental number line hypothesis) but also from verbal coding. Because children are surrounded by rulers, number lines, and the like in the classroom, it is intuitively appealing to assume that they first use their mental number line to represent numbers and that only later in development a verbal recoding of magnitude information takes place. However, this hypothesis has never been tested. The goal of the current study was to define the developmental pattern of both accounts (spatial and verbal) in explaining the SNARC effect. To this end, 9- and 11-year-olds were tested in a magnitude comparison task. Surprisingly, clear and robust evidence for verbal coding of magnitude information was observed in both age groups. Our results imply that the ability to use verbal coding of magnitude information is robustly present early in formal schooling.     

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction–decimal mixed pairs and decimal–fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.     

Two Wrongs May Make a Right ... If They Argue Together!

Several studies have investigated the cognitive development of interacting peers. This study focuses on a phenomenon that has not yet been studied: the cognitive gains of 2 children with low levels of competence who fail to solve a task individually but who improve when working in peer interaction. We show that this phenomenon (which we call the two-wrongs-make-a-right phenomenon) may occur when (a) the 2 wrongs disagree, (b) they have different strategies, and (c) active hypothesis testing is made possible. In a preliminary study, 30 Grade 10 low-achieving students were tested about the rules they use to compare 2 decimal fractions in a questionnaire. The students who were diagnosed as wrongs were invited to solve a task (the 6-cards task) with peers. Three kinds of pairs were formed: 7 W1-W2 pairs in which the 2 wrongs have different conceptual bugs; 4 W1-W1 pairs in which the 2 wrongs have the same conceptual bugs; 4 R-W pairs in which a wrong interacted with a right student. The 6-cards task was designed to create conflicts between students with different conceptual bugs and between wrong and right students. Two days after solving the 6-cards task, the students were asked to answer a similar questionnaire individually. The preliminary study revealed the two-wrongs-make-a-right phenomenon: Among the 7 W1-W2 pairs, at least 1 wrong became right. In contrast, in the 4 R-W pairs, only 1 wrong became right, and in the 3 W1-W1 pairs, no change was detected. In a case study that replicated the phases of the preliminary study, disagreement, argumentative operations (such as challenge and concession), hypothesis testing (with a calculator), and the internalization of social interactions mediated the change of peers from wrongs to rights. We then replicated the initial study with 72 low-achieving Grade 10 and 11 students, confirming the two-wrongs-make-a-right effect.     

学前儿童近似数量系统敏锐度与符号数学能力的关系

选取杭州市122名学前儿童(3~6岁)为被试,以点数比较任务及点数异同任务测量幼儿的近似数量系统敏锐度,以数数测验、基数测验、符号数字知识测验及简单计算来测量幼儿的符号数学能力,以此考察学前儿童近似数量系统敏锐度的发展及与符号数学能力的关系。结果发现:(1)随年龄增长,学前儿童的近似数量加工的敏锐度逐渐提高;(2)点数比较任务与点数异同任务均适合测量学前儿童近似数量系统敏锐度,但儿童完成点数比较任务的正确率要高于点数异同任务的正确率;(3)在抑制控制、短时记忆、工作记忆和言语测验成绩被控制后,根据点数比较任务计算的韦伯系数能显著预测学前儿童的基数和符号数字知识测验分数,总正确率能显著预测学前儿童的数数、基数、符号数字知识测验分数;(4)点数异同任务中只有点数不同试次下的正确率能显著预测学前儿童的符号数字知识测验分数。     

Development of strategies for recalling letter sequences

Kindergarten, second-grade, and fourth-grade children (6, 8, and 10 years of age, respectively) were asked to recall letter sequences on a task that required the use of: (a) a verbal strategy; (b) a positional strategy; or (c) either strategy. Additional trials, allowing either strategy, were interspersed among those requiring a verbal or positional strategy; strategy use was determined from performance on these interspersed trials. The results show that children at all grade levels attempted to use a task-appropriate strategy. Performance on the verbal task showed a more marked developmental increase than did performance on the positional task. A second experiment replicated the first with the interspersed trials removed. A third experiment was designed to determine whether letter names interfered with older children's efficient use of a positional strategy. Even with this source of interference removed, the developmental pattern of Experiments 1 and 2 was replicated. There was a Sex by Condition interaction in Experiments 2 and 3 which showed that males do better in the condition requiring a positional strategy, and females do better in the condition requiring a verbal strategy.     

The effect of working memory load on the SNARC effect: Maybe tasks have a word to say

We investigated the effect of working memory load on the SNARC (spatial–numerical association of response codes) effect under different number judgment tasks (parity judgment and magnitude comparison), using a novel dual task. Instead of exerting load over the whole block of number judgment trials, in this dual task, number judgment trials were inserted into each interstimulus interval of an n-back task, which served as the working memory load. We varied both load type (verbal and spatial) and amount (1-load, 2-load, and 3-load). The results indicated that the SNARC effect disappeared even under the 1-load condition for a parity judgment, regardless of the type of load. However, during the magnitude comparison task, the SNARC effect increased with increasing load amounts under spatial load conditions; under verbal load conditions, the SNARC effect decreased with increasing amounts of load, and disappeared during the 3-load task. The difference between the parity and magnitude tasks was not attributable to the interval stimuli or task switching. These findings confirm that different spatial–numerical associations for comparing numerical magnitudes and judgments of parity have different needs with respect to working memory resources.