Knowledge of counting principles: How relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality—that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children’s conceptual knowledge of counting.     

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。     

Children's understanding of counting

This study examines the abstractness of children's mental representation of counting, and their understanding that the last number word used in a count tells how many items there are (the cardinal word principle). In the first experiment, twenty-four 2- and 3-year-olds counted objects, actions, and sounds. Children counted objects best, but most showed some ability to generalize their counting to actions and sounds, suggesting that at a very young age, children begin to develop an abstract, generalizable mental representation of the counting routine. However, when asked "how many" following counting, only older children (mean age 3.6) gave the last number word used in the count a majority of the time, suggesting that the younger children did not understand the cardinal word principle. In the second experiment (the "give-a-number" task), the same children were asked to give a puppet one, two, three, five, and six items from a pile. The older children counted the items, showing a clear understanding of the cardinal word principle. The younger children succeeded only at giving one and sometimes two items, and never used counting to solve the task. A comparison of individual children's performance across the "how-many" and "give-a-number" tasks shows strong within-child consistency, indicating that children learn the cardinal word principle at roughly 3 1/2 years of age. In the third experiment, 18 2- and 3-year-olds were asked several times for one, two, three, five, and six items, to determine the largest numerosity at which each child could succeed consistently. Results indicate that children learn the meanings of smaller number words before larger ones within their counting range, up to the number three or four. They then learn the cardinal word principle at roughly 3 1/2 years of age, and perform a general induction over this knowledge to acquire the meanings of all the number words within their counting range.     

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.     

Number gestures predict learning of number words

When asked to explain their solutions to a problem, children often gesture and, at times, these gestures convey information that is different from the information conveyed in speech. Children who produce these gesture‐speech “mismatches” on a particular task have been found to profit from instruction on that task. We have recently found that some children produce gesture‐speech mismatches when identifying numbers at the cusp of their knowledge, for example, a child incorrectly labels a set of two objects with the word “three” and simultaneously holds up two fingers. These mismatches differ from previously studied mismatches (where the information conveyed in gesture has the potential to be integrated with the information conveyed in speech) in that the gestured response contradicts the spoken response. Here, we ask whether these contradictory number mismatches predict which learners will profit from number‐word instruction. We used the Give‐a‐Number task to measure number knowledge in 47 children (M_age = 4.1 years, SD = 0.58), and used the What's on this Card task to assess whether children produced gesture‐speech mismatches above their knower level. Children who were early in their number learning trajectories (“one‐knowers” and “two‐knowers”) were then randomly assigned, within knower level, to one of two training conditions: a Counting condition in which children practiced counting objects; or an Enriched Number Talk condition containing counting, labeling set sizes, spatial alignment of neighboring sets, and comparison of these sets. Controlling for counting ability, we found that children were more likely to learn the meaning of new number words in the Enriched Number Talk condition than in the Counting condition, but only if they had produced gesture‐speech mismatches at pretest. The findings suggest that numerical gesture‐speech mismatches are a reliable signal that a child is ready to profit from rich number instruction and provide evidence, for the first time, that cardinal number gestures have a role to play in number‐learning.     

Beginning kindergarteners' perception of number

97 children in 5 kindergarten classes when tested for perceptions of number could begin counting and 9 counted to 30, 84 counted a set of 6 correctly but when the blocks were covered 33 could say there were 6. 57 showed 1:1 correspondence but some had to count to decide. 14 showed conservation of number, but 94 recognized two sets were unequal and which one was larger. 30 pointed to the third car, but 24 could state an ordinal name for a target. Clearly the children varied greatly.     

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.     

Who Did What to Whom? Children Track Story Referents First in Gesture

Children achieve increasingly complex language milestones initially in gesture or in gesture+speech combinations before they do so in speech, from first words to first sentences. In this study, we ask whether gesture continues to be part of the language-learning process as children begin to develop more complex language skills, namely narratives. A key aspect of narrative development is tracking story referents, specifying who did what to whom. Adults track referents primarily in speech by introducing a story character with a noun phrase and then following the same referent with a pronoun—a strategy that presents challenges for young children. We ask whether young children can track story referents initially in communications that combine gesture and speech by using character viewpoint in gesture to introduce new story characters, before they are able to do so exclusively in speech using nouns followed by pronouns. Our analysis of 4- to 6-year-old children showed that children introduced new characters in gesture+speech combinations with character viewpoint gestures at an earlier age than conveying the same referents exclusively in speech with the use of nominal phrases followed by pronouns. Results show that children rely on viewpoint in gesture to convey who did what to whom as they take their first steps into narratives.     

4～5岁儿童一一对应任务中的比例效应

一一对应和数量比较是幼儿数概念发展的两个重要方面。在本研究中．40名4岁和39名5岁的儿童分别完成了不同比例数量的一一对应任务和数量比较任务。结果表明：（1）两种实验任务下均出现比例效应．说明儿童在解决一一对应问题时,仍然用数量比较的方法来进行判断;（2）任务类型的主效应显著,一一对应任务下的正确率要显著低于数量比较任务下的正确率;（3）知觉线索更多地影响数量比较任务。     

Number words in young children's conceptual and procedural knowledge of addition, subtraction and inversion

Three studies addressed children's arithmetic. First, 50 3- to 5-year-olds judged physical demonstrations of addition, subtraction and inversion, with and without number words. Second, 20 3- to 4-year-olds made equivalence judgments of additions and subtractions. Third, 60 4- to 6-year-olds solved addition, subtraction and inversion problems that varied according to the inclusion of concrete referents and number words. The results indicate that number words play a different role in conceptual and procedural development. Children have strong addition and subtraction concepts before they can translate the physical effects of these operations into number words. However, using number words does not detract from their calculation procedures. Moreover, consistent with iterative relations between conceptual and procedural development, the results suggest that inversion acquisition depends on children's calculation procedures and that inversion understanding influences these procedures.     

Gesture and early bilingual development

The relationship between speech and gestural proficiency was investigated longitudinally (from 2 years to 3 years 6 months, at 6-month intervals) in 5 French-English bilingual boys with varying proficiency in their 2 languages. Because of their different levels of proficiency in the 2 languages at the same age, these children's data were used to examine the relative contribution of language and cognitive development to gestural development. In terms of rate of gesture production, rate of gesture production with speech, and meaning of gesture and speech, the children used gestures much like adults from 2 years on. In contrast, the use of iconic and beat gestures showed differential development in the children's 2 languages as a function of mean length of utterance. These data suggest that the development of these kinds of gestures may be more closely linked to language development than other kinds (such as points). Reasons why this might be so are discussed.     

Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity—with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting.     

Preschool children master the logic of number word meanings

Although children take over a year to learn the meanings of the first three number words, they eventually master the logic of counting and the meanings of all the words in their count list. Here, we ask whether children's knowledge applies to number words beyond those they have mastered: Does a child who can only count to 20 infer that number words above 'twenty' refer to exact cardinal values? Three experiments provide evidence for this understanding in preschool children. Before beginning formal education or gaining counting skill, children possess a productive symbolic system for representing number.     

Early gesture selectively predicts later language learning

The gestures children produce predict the early stages of spoken language development. Here we ask whether gesture is a global predictor of language learning, or whether particular gestures predict particular language outcomes. We observed 52 children interacting with their caregivers at home, and found that gesture use at 18 months selectively predicted lexical versus syntactic skills at 42 months, even with early child speech controlled. Specifically, number of different meanings conveyed in gesture at 18 months predicted vocabulary at 42 months, but number of gesture+speech combinations did not. In contrast, number of gesture+speech combinations, particularly those conveying sentence‐like ideas, produced at 18 months predicted sentence complexity at 42 months, but meanings conveyed in gesture did not. We can thus predict particular milestones in vocabulary and sentence complexity at age by watching how children move their hands two years earlier.     

Gesture Reflects Language Development: Evidence From Bilingual Children

There is a growing awareness that language and gesture are deeply intertwined in the spontaneous expression of adults. Although some research suggests that children use gesture independently of speech, there is scant research on how language and gesture develop in children older than 2 years. We report here on a longitudinal investigation of the relation between gesture and language development in French-English bilingual children from 2 to 3 1/2 years old. The specific gesture types of iconics and beats correlated with the development of the children's two languages, whereas pointing types of gestures generally did not. The onset of iconic and beat gestures coincided with the onset of sentencelike utterances separately in each of the children's two languages. The findings show that gesture is related to language development rather than being independent from it. Contrasting theories about how gesture is related to language development are discussed.     

The metric matters: determining the extent of children's knowledge of morphological spelling regularities

All developmental research needs to carefully consider how children's knowledge is measured. The study of children's knowledge of spelling conventions, or the ways in which the English orthography encodes the roots and affixes and the sounds in words, is no exception. This experiment examined the extent of 7- to 9-year-old children's knowledge of the role of root morphemes in spelling words across different contexts and with different units of assessment. Different writing contexts did not appear to affect children's performance; children were better able to spell the first components of two- than of one-morpheme words (e.g. only free in freely and freeze), both when writing whole words and their first sections (e.g. completing__ or __ly for freely). A second analysis revealed that the unit of coding can influence conclusions. Children demonstrated similar abilities across ages 7 to 9 when only the first segments of words were coded; in contrast, there was evidence of age-related differences when whole word spelling accuracy was assessed. In combination, these results suggest that children's knowledge of the principle of root consistency is remarkably robust to changes in writing context, but that coding is key when drawing conclusions. These findings remind us that the metric matters in studies of spelling, as in other domains, and they offer a manner to reconcile previously conflicting data on spelling development.     

Young children use their hands to tell their mothers what to say

Children produce their first gestures before their first words, and their first gesture+word sentences before their first word+word sentences. These gestural accomplishments have been found not only to predate linguistic milestones, but also to predict them. Findings of this sort suggest that gesture itself might be playing a role in the language‐learning process. But what role does it play? Children's gestures could elicit from their mothers the kinds of words and sentences that the children need to hear in order to take their next linguistic step. We examined maternal responses to the gestures and speech that 10 children produced during the one‐word period. We found that all 10 mothers ‘translated’ their children's gestures into words, providing timely models for how one‐ and two‐word ideas can be expressed in English. Gesture thus offers a mechanism by which children can point out their thoughts to mothers, who then calibrate their speech to those thoughts, and potentially facilitate language‐learning.     

Hands in the air: using ungrounded iconic gestures to teach children conservation of quantity

Including gesture in instruction facilitates learning. Why? One possibility is that gesture points out objects in the immediate context and thus helps ground the words learners hear in the world they see. Previous work on gesture's role in instruction has used gestures that either point to or trace paths on objects, thus providing support for this hypothesis. The experiments described here investigated the possibility that gesture helps children learn even when it is not produced in relation to an object but is instead produced "in the air." Children were given instruction in Piagetian conservation problems with or without gesture and with or without concrete objects. The results indicate that children given instruction with speech and gesture learned more about conservation than children given instruction with speech alone, whether or not objects were present during instruction. Gesture in instruction can thus help learners learn even when those gestures do not direct attention to visible objects, suggesting that gesture can do more for learners than simply ground arbitrary, symbolic language in the physical, observable world.