Counting to Infinity: Does Learning the Syntax of the Count List Predict Knowledge That Numbers Are Infinite?

By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: Every natural number, n, has a successor, n + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language-specific counting routines (e.g., the rules in English that represent base-10 structure). We tested 4- and 5-year-old children’s knowledge of counting with three tasks, which we then related to (a) children’s belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end.     

Language access and theory of mind reasoning: evidence from deaf children in bilingual and oralist environments

This investigation examined whether access to sign language as a medium for instruction influences theory of mind (ToM) reasoning in deaf children with similar home language environments. Experiment 1 involved 97 deaf Italian children ages 4-12 years: 56 were from deaf families and had LIS (Italian Sign Language) as their native language, and 41 had acquired LIS as late signers following contact with signers outside their hearing families. Children receiving bimodal/bilingual instruction in LIS together with Sign-Supported and spoken Italian significantly outperformed children in oralist schools in which communication was in Italian and often relied on lipreading. Experiment 2 involved 61 deaf children in Estonia and Sweden ages 6-16 years. On a wide variety of ToM tasks, bilingually instructed native signers in Estonian Sign Language and spoken Estonian succeeded at a level similar to age-matched hearing children. They outperformed bilingually instructed late signers and native signers attending oralist schools. Particularly for native signers, access to sign language in a bilingual environment may facilitate conversational exchanges that promote the expression of ToM by enabling children to monitor others' mental states effectively.     

Signing environment in an institution: Teaching sign vocabulary to residents with mental retardation for communication with deaf peers

Two experiments were conducted so that deaf adults with mental retardation could develop their communication skills in an institution. Deaf and hearing residents were taught sign vocabulary using the following four procedures: sign language lessons in a large group, posters, review lessons, and distribution of a sign language book. Experiment 1 was a pilot study to assess the effect of a sign language lesson, and Experiment 2 was a systematic study to evaluate the 4 procedures using a multiple baseline design. Sign language use in natural settings was monitored for 10 months, starting with the first sign language lesson. Results showed that a sign language lesson was an effective way for most hearing residents to acquire expressive signs. Three out of 4 deaf residents learned expressive signs gradually during experiments. In natural settings, hearing residents were seen to use signs spontaneously with deaf residents in many different social contexts. Additional comprehensive assessment was conducted to confirm whether or not residents could understand one another's expressive signs. Seven out of 10 residents showed correct responses more than 80% of the time.     

What makes counting count? Verbal and visuo-spatial contributions to typical and atypical number development

Williams Syndrome (WS) is marked by a relative strength in verbal cognition coupled with a serious impairment in non-verbal cognition. A strong deficit in numerical cognition has been anecdotally reported in this disorder; however, its nature has not been systematically investigated. Here, we tested 14 children with WS (mean age=7 years 2 months), 14 typically developing controls individually matched on visuo-spatial ability (mean age=3 years 5 months) as well as a larger group of typically developing controls (mean age=3 years 4 months) on two tasks to assess their understanding that counting determines the exact quantity of sets (cardinality principle). The understanding of the cardinality principle in children with WS is extremely delayed and only at the level predicted by their visuo-spatial MA. In this clinical group, only language accounted for a significant amount of the variance in cardinality understanding, whereas in the normal comparison group only visuo-spatial competence predicted the variance. The present findings suggest that visuo-spatial ability plays a greater role than language ability in the actual development of cardinality understanding in typically developing children, whereas the opposite obtains for the clinical group.     

Temporal processing in deaf signers

The auditory and visual modalities differ in their capacities for temporal analysis, and speech relies on more rapid temporal contrasts than does sign language. We examined whether congenitally deaf signers show enhanced or diminished capacities for processing rapidly varying visual signals in light of the differences in sensory and language experience of deaf and hearing individuals. Four experiments compared rapid temporal analysis in deaf signers and hearing subjects at three different levels: sensation, perception, and memory. Experiment 1 measured critical flicker frequency thresholds and Experiment 2, two-point thresholds to a flashing light. Experiments 3-4 investigated perception and memory for the temporal order of rapidly varying nonlinguistic visual forms. In contrast to certain previous studies, specifically those investigating the effects of short-term sensory deprivation, no significant differences between deaf and hearing subjects were found at any level. Deaf signers do not show diminished capacities for rapid temporal analysis, in comparison to hearing individuals. The data also suggest that the deficits in rapid temporal analysis reported previously for children with developmental language delay cannot be attributed to lack of experience with speech processing and production.     

听障人群的工作记忆机制

听障人群听觉通道受损,使用手语交流,提供了独特的切入点来探讨工作记忆的结构和功能。研究表明,听障人群具备在功能上与正常人的语音环路平行的手语复述机制。通过发声训练,听障人群也可采用语音编码,即语音环路可被通达。听障人群具有与正常人相当的语言工作记忆资源,但是这种资源在具体使用时受视觉通道处理特性的限制。越来越多的研究支持互补理论,认为手语的使用增强了听障人群非语言的视空间处理能力。     

Memory for Faces and Objects by Deaf and Hearing Signers and Hearing Nonsigners

The memory of 11 deaf and 11 hearing British Sign Language users and 11 hearing nonsigners for pictures of faces of and verbalizable objects was measured using the game Concentration. The three groups performed at the same level for the objects. In contrast the deaf signers were better for faces than the hearing signers, who in turn were superior to the hearing nonsigners, who were the worst. Three hypotheses were made: That there would be no significant difference in terms of the number of attempts between the three groups on the verbalizable object task, that the hearing and deaf signers would demonstrate superior performance to that of the hearing nonsigners on the matching faces task, and that the hearing and deaf signers would exhibit similar performance levels on the matching faces task. The first two hypotheses were supported, but the third was not. Deaf signers were found to be superior for memory for faces to hearing signers and hearing nonsigners. Possible explanations for the findings are discussed, including the possibility that deafness and the long use of sign language have additive effects.     

Understanding of friendship and social rules in deaf and hearing adolescents

Children who are profoundly and prelingually deaf and live in residential schools lack ready access to the general language environment and some opportunities of societial interactions. How does this affect their understanding of the social world To probe this question 60 deaf and 60 hearing children, ages 9, 13, and 17, were interviewed regarding friendship and social rules (a game rule, a school rule, driving law). Summary scores for each of the 4 areas yielded age and hearing differences. The major findings were: (1) the developmental pattern of responses was similar for deaf and hearing children; (2) deaf children lagged behind hearing children in their social understanding, most in game rules, least in friendship. The results were related to a constructivist developmental perspective and to previous research on socialization of deaf children.     

Deaf and hearing children's plural noun spelling

The present study examines deaf and hearing children's spelling of plural nouns. Severe literacy impairments are well documented in the deaf, which are believed to be a consequence of phonological awareness limitations. Fifty deaf (mean chronological age 13;10 years, mean reading age 7;5 years) and 50 reading-age-matched hearing children produced spellings of regular, semiregular, and irregular plural nouns in Experiment 1 and nonword plurals in Experiment 2. Deaf children performed reading-age appropriately on rule-based (regular and semiregular) plurals but were significantly less accurate at spelling irregular plurals. Spelling of plural nonwords and spelling error analyses revealed clear evidence for use of morphology. Deaf children used morphological generalization to a greater degree than their reading-age-matched hearing counterparts. Also, hearing children combined use of phonology and morphology to guide spelling, whereas deaf children appeared to use morphology without phonological mediation. Therefore, use of morphology in spelling can be independent of phonology and is available to the deaf despite limited experience with spoken language. Indeed, deaf children appear to be learning about morphology from the orthography. Education on more complex morphological generalization and exceptions may be highly beneficial not only for the deaf but also for other populations with phonological awareness limitations.     

Is the Chinese number-naming system transparent? Evidence from Chinese-English bilingual children.

Chinese-speaking children have been shown to have an advantage over English-speaking children in a variety of mathematical areas, including counting. One possible explanation for the advantage in counting is that the Chinese number-naming system is relatively transparent, compared to English, in that number names typically are directly indicative of base-10 structure (e.g., 12 is named "ten-two" rather than "twelve"). To determine whether the transparency of the Chinese number-naming system influences counting in bilingual children, we tested 25 Chinese-English bilingual children between the ages of 3 and 5 years, both in English and in Chinese. Children were asked to count as high as they could (abstract counting) and also to count objects in small, medium, and large arrays (object counting). No evidence was found for transparency or for transfer from one language to the other. Instead, relative proficiency in the two languages influenced counting skill. These results are discussed in terms of linguistic and cultural variables that might account for cross-linguistic differences in counting.     

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.     

中国幼儿数数过程信息加工活动研究

探讨中国幼儿数数发展特点。实验1探讨3－5岁幼儿数数的终结点分布状况,结果表明,最大的密集终结点是“19”。实验2进一步考察幼儿在其数数范围内与范围外的起点进行数数的成绩,结果表明,数数范围为11－19的幼儿就开始有部分人可以在其数数范围外的起点进行数数,此表明他们开始应用了数列规则。据此可以认为,中国幼儿的数数活动同样包括联结学习与数列规则学习两种信息加工活动,但中国幼儿数列规则的学习活动在“11－19”数数过程开始。     

Linguistic flexibility in signed and written language productions of deaf children

Linguistic flexibility of deaf and hearing children was compared by examining the relative frequencies of their nonliteral constructions in stories written and signed (by the deaf) or written and spoken (by the hearing). Seven types of nonliteral constructions were considered: novel figurative language, frozen figurative language, gestures, pantomime, linguistic modifications, linguistic inventions, and lexical substitutions. Among the hearing 8- to 15-year-olds, oral and written stories contained comparable numbers of nonliteral constructions. Among their age-matched deaf peers, however, nonliteral constructions were significantly stories contained comparable numbers of nonliteral constructions. Among their age-matched deaf peers, however, nonliteral constructions were significantly more common in signed than written stories. Overall, hearing students used more nonliteral constructions in their written stories than did their deaf peers (who used very few), whereas deaf students used more nonliteral constructions in their signed stories than their hearing peers did in their spoken stories. The results suggest that deaf children are linguistically and cognitively more competent than is generally assumed on the basis of evaluations in English. Although inferior to hearing age-mates in written expression, they are comparable to, and in some ways better than those peers when evaluated using their primary mode of communication.     

PRESCHOOL ORIGINS OF CROSS-NATIONAL DIFFERENCES IN MATHEMATICAL COMPETENCE:

Abstract— Differences in mathematical competence between U S and Chinese children first emerge during the preschool years, favor Chinese children, and are limited to specific aspects of mathematical competence The base-10 structure of number names is less obvious in English than in Chinese, differences between these languages are reflected in children's difficulties learning count Language differences do not affect other aspects of early mathematics including counting small sets and solving simple numerical problems Because later mathematics increasingly involves manipulation of symbols, this early deficit in apprehending the base-10 structure of number names may provide a basis for previously reported differences in mathematical competence favoring Chinese schoolchildren     

Using cardinality to compare quantities: the role of social-cognitive conflict in early numeracy

A key question in early number development is how 4- and 5-year-olds learn the roles that counting and cardinal numbers play when comparing quantities. Children who wrongly used length to identify numerosity were assigned to five experimental groups and trained to judge whether a puppet--who sometimes miscounted--created equivalent sets. Over three training sessions, children who were asked to compare sets after they were counted learned to base their judgments on cardinal numbers when the puppet counted accurately by being given feedback. However, only the groups who were also asked to explain either their own or the experimenter's reasoning made progress in identifying the puppet's miscounts. This ability to recognize the importance of counting accuracy for quantitative comparisons predicted whether children would spontaneously count to compare sets on a post-test. The importance of asking children to identify miscounts is discussed alongside the social factors that influence children's recognition of the relationship between procedural counting, cardinality and relative number.     

Signs of Resistance: Peer Learning of Sign Languages Within ‘Oral’ Schools for the Deaf

This article explores the role of the Deaf child as peer educator. In schools where sign languages were banned, Deaf children became the educators of their Deaf peers in a number of contexts worldwide. This paper analyses how this peer education of sign language worked in context by drawing on two examples from boarding schools for the deaf in Nicaragua and Thailand. The argument is advanced that these practices constituted a child-led oppositional pedagogy. A connection is drawn to Freire’s (1972) theory of critical pedagogy. Deaf children’s actions as peer educators are framed as an act of resistance towards the oppression of their language and culture. A contrast is drawn between oralist pedagogy that is historically associated with punitive practices and didactic methods and the experiential and dialogic interaction that characterised peer learning of sign languages. The argument is made that the peer teaching and learning processes enabled the self-actualisation of the Deaf children whereas the oralist methods were based on a deficit model that focused on modifying deaf children according to the norms of hearing society. The implications of this for current policy and practice are inferred to be about access to sign languages and the importance of Deaf communities in deaf children’s education. The argument is made that space needs to be created for deaf children to engage in peer learning.     

聋人手语视觉表象生成能力的实验研究

通过视觉表象判断实验,对聋手语使用者和听力正常人两类被试视觉表象生成的能力进行了比较。实验发现：与听力正常的人相比,聋手语使用者学习和记忆大写字母的时间短于听力正常的被试,并且两组被试记忆复杂字母的时间都较长;聋被试和听力正常被试采用了相同的字母表征方式。但是,习得手语的年龄对聋手语者生成表象的能力没有明显的影响。     

Phonological coding in reading of deaf children: Pseudohomophone effects in lexical decision

The purpose of this study was to find new evidence for phonological coding in written word recognition among deaf Dutch children. A lexical decision task was presented to 48 severely and profoundly deaf children aged from 6 years 8 months to 13 years 5 months, and a control group of Grade 1 hearing children matched on written word recognition. Sixteen pseudohomophones were introduced, closely matched on orthographic similarity with 16 control pseudo-words. Both hearing children and deaf children made significantly more mistakes on pseudohomophones than on control pseudo-words. Although pseudohomophony effects were smaller for deaf than for hearing participants, the findings were taken as evidence that deaf children also used phonological coding during written word recognition.     

聋童执行功能发展：聋童与正常儿童的比较

利用标准Dimensional Change Card Sort任务(DCCS),对76名智力正常的3~8岁聋童和78名3~5.5岁的正常儿童进行了对比测试,旨在考察聋童执行功能发展的年龄特征与发展水平。结果发现,3岁组的聋童和正常儿童在DCCS任务上的表现没有显著性差异,但正常儿童在4~4.5岁时进入一个迅速发展期,而聋童要在6岁时才有快速的发展,到7岁后才相当于正常儿童5岁的发展水平,大约滞后2年。研究认为,造成聋童执行功能发展滞后的原因主要有：（1）语言符号系统和聋童特有的符号系统之间可能存在的差异;（2）聋童可能存在计划和灵活性的缺陷;（3）聋童可能存在命名和标识策略上的困难和注意机制的缺陷。结合关于聋童心理理论发展滞后于正常儿童7年以上的报道,心理理论发展和执行功能发展在聋童身上表现出较大的不一致性。