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.     

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.     

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.     

Predictors of early numeracy: Is there a place for mistakes when learning about number?

It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy‐two 5‐year‐olds were given a battery of nine tasks on each of three sessions (at 3‐monthly intervals). Tasks measured procedural proficiency, conceptual understanding (using an error‐detection paradigm) and the ability to compare sets using number knowledge. Procedural skills remained fairly stable over the 6‐month period, and preceded children's ability to detect another's violations to those procedures. Regression analysis revealed that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinal numbers for set comparisons. We suggest that although children's conceptual understanding of well‐rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.     

To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

Recent accounts of number word learning posit that when children learn to accurately count sets (i.e., become “cardinal principle” or “CP” knowers), they have a conceptual insight about how the count list implements the successor function – i.e., that every natural number n has a successor defined as n + 1 (Carey, 2004, 2009; Sarnecka & Carey, 2008). However, recent studies suggest that knowledge of the successor function emerges sometime after children learn to accurately count, though it remains unknown when this occurs, and what causes this developmental transition. We tested knowledge of the successor function in 100 children aged 4 through 7 and asked how age and counting ability are related to: (1) children’s ability to infer the successors of all numbers in their count list and (2) knowledge that all numbers have a successor. We found that children do not acquire these two facets of the successor function until they are about 5½ or 6 years of age – roughly 2 years after they learn to accurately count sets and become CP-knowers. These findings show that acquisition of the successor function is highly protracted, providing the strongest evidence yet that it cannot drive the cardinal principle induction. We suggest that counting experience, as well as knowledge of recursive counting structures, may instead drive the learning of the successor function.     

Children’s understandings of counting: Detection of errors and pseudoerrors by kindergarten and primary school children

In this study, the development of comprehension of essential and nonessential aspects of counting is examined in children ranging from 5 to 8 years of age. Essential aspects, such as logical rules, and nonessential aspects, including conventional rules, were studied. To address this, we created a computer program in which children watched counting errors (abstraction and order irrelevance errors) and pseudoerrors (with and without cardinal value errors) occurring during a detection task. The children judged whether the characters had counted the items correctly and were asked to justify their responses. In general, our data show that performance improved substantially with age in terms of both error and pseudoerror detection; furthermore, performance was better with regard to errors than to pseudoerrors as well as on pseudoerror tasks with cardinal values versus those without cardinal values. In addition, the children’s justifications, for both the errors and pseudoerrors, made possible the identification of conventional rules underlying the incorrect responses. A particularly relevant trend was that children seem to progressively ignore these rules as they grow older. Nevertheless, this process does not end at 8 years of age given that the conventional rules of temporal and spatial adjacency were present in their judgments and were primarily responsible for the incorrect responses.     

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.     

Why preschoolers are reluctant to count spontaneously

Two experiments are reported which examine children's counting and its role in reasoning about the relative numerosity of two arrays. In the first experiment, children's number judgements were compared under different conditions designed to evaluate the importance of three different cues to number—length and density of rows, small number perception and counting. Children were found to count very rarely unless specifically asked to do so. Experiment 2 investigated some possible reasons why children who count readily in some situations are reluctant to count spontaneously in this number judgement task. Spontaneous counting in 4-year-olds increased in one condition only: when they were given feedback as to the correctness of their previous judgements. This feedback showed that basing judgements on number as counted was always correct whereas length and density judgements were only sometimes correct. Preschoolers' preference for length as a cue to number may therefore be due to their belief that length is a more reliable cue than counting, rather than to their ignorance about the link between counting and numerical reasoning.     

Counting in sign language

Do the visuomanual modality and the structure of the sequence of numbers in sign language have an impact on the development of counting and its use by deaf children? The sequence of number signs in Belgian French Sign Language follows a base-5 rule while the number sequence in oral French follows a base-10 rule. The accuracy and use of sequence number string were investigated in hearing children varying in age from 3 years 4 months to 5 years 8 months and in deaf children varying in age from 4 years to 6 years 2 months. Three tasks were used: abstract counting, object counting, and creation of sets of a given cardinality. Deaf children exhibited age-related lags in their knowledge of the number sequence; they made different errors from those of hearing children, reflecting the rule-bound nature of sign language. Remarkably, their performance in object counting and creating sets of given cardinality was similar to that of hearing children who had a longer sequence number string, indicating a better use of counting than predicted by their knowledge of the linguistic sequence of numbers.     

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.     

One first? Acquisition of the cardinal and ordinal uses of numbers in preschoolers

We studied the acquisition of the ordinal meaning of number words and examined its development relative to the acquisition of the cardinal meaning. Three groups of 3-, 4-, and 5-year-old children were tested in two tasks requiring the use of number words in both cardinal and ordinal contexts. Understanding of the counting principles was also measured by asking the children to assess the correctness of a cartoon character's counting in both contexts. In general, the children performed cardinal tasks significantly better than ordinal ones. Tasks requiring the production of the number for a given quantity or position were solved more accurately than those testing the ability to select a set of n objects or the object in the nth position. Different profiles were obtained for the principles; those principles shared by the two contexts were mastered earlier in the cardinal context. Regarding order (ir)relevance, older children adhered to rigid ways of counting, producing better results in the ordinal context and incorrect rejections in the cardinal trials. Altogether, our data indicate that the acquisitions of cardinal and ordinal meanings of numbers are related, and cardinality precedes the development of ordinality.     

Infants recognize counting as numerically relevant

Children do not understand the meanings of count words like “two” and “three” until the preschool years. But even before knowing the meanings of these individual words, might they recognize that counting is “about” the dimension of number? Here in five experiments, we asked whether infants already associate counting with quantities. We measured 14‐ and 18‐month olds’ ability to remember different numbers of hidden objects that either were or were not counted by an experimenter before hiding. As in previous research, we found that infants failed to differentiate four hidden objects from two when the objects were not counted—suggesting an upper limit on the number of individual objects they could represent in working memory. However, infants succeeded when the objects were simply counted aloud before hiding. We found that counting also helped infants differentiate four hidden objects from six (a 2:3 ratio), but not three hidden objects from four (a 3:4 ratio), suggesting that counting helped infants represent the arrays’ approximate cardinalities. Hence counting directs infants’ attention to numerical aspects of the world, showing that they recognize counting as numerically relevant years before acquiring the meanings of number words.     

Piecing together numerical language: children's use of default units in early counting and quantification

When asked to ‘find three forks’, adult speakers of English use the noun ‘fork’ to identify units for counting. However, when number words (e.g. three) and quantifiers (e.g. more, every) are used with unfamiliar words (‘Give me three blickets’) noun‐specific conceptual criteria are unavailable for picking out units. This poses a problem for young children learning language, who begin to use quantifiers and number words by age 2, despite knowing a relatively small number of nouns. Without knowing how individual nouns pick out units of quantification – e.g. what counts as a blicket– how could children decide whether there are three blickets or four? Three experiments suggest that children might solve this problem by assigning ‘default units’ of quantification to number words, quantifiers, and number morphology. When shown objects that are broken into arbitrary pieces, 4‐year‐olds in Experiment 1 treated pieces as units when counting, interpreting quantifiers, and when using singular–plural morphology. Experiment 2 found that although children treat object‐hood as sufficient for quantification, it is not necessary. Also sufficient for individuation are the criteria provided by known nouns. When two nameable things were glued together (e.g. two cups), children counted the glued things as two. However, when two arbitrary pieces of an object were put together (e.g. two parts of a ball), children counted them as one, even if they had previously counted the pieces as two. Experiment 3 found that when the pieces of broken things were nameable (e.g. wheels of a bicycle), 4‐year‐olds did not include them in counts of whole objects (e.g. bicycles). We discuss the role of default units in early language acquisition, their origin in acquisition, and how children eventually acquire an adult semantics identifying units of quantification.     

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.     

Levels of number knowledge during early childhood

Researchers have long disagreed about whether number concepts are essentially continuous (unchanging) or discontinuous over development. Among those who take the discontinuity position, there is disagreement about how development proceeds. The current study addressed these questions with new quantitative analyses of children’s incorrect responses on the Give-N task. Using data from 280 children, ages 2 to 4 years, this study showed that most wrong answers were simply guesses, not counting or estimation errors. Their mean was unrelated to the target number, and they were lower-bounded by the numbers children actually knew. In addition, children learned the number-word meanings one at a time and in order; they treated the number words as mutually exclusive; and once they figured out the cardinal principle of counting, they generalized this principle to the rest of their count list. Findings support the ‘discontinuity’ account of number development in general and the ‘knower-levels’ account in particular.     

Individual differences in numerical abilities in preschoolers

This study investigated individual differences in different aspects of early number concepts in preschoolers. Eighty 4-year-olds from Oxford nursery classes took part. They were tested on accuracy of counting sets of objects; the cardinal word principle; the order irrelevance principle; and predicting the results of repeated addition and subtraction by 1 from a set of objects. There were marked individual differences for most tasks. Most children were reasonably proficient at counting and 70% understood the cardinal word principle. Based on the results of a repeated addition and subtraction by 1 task, the children were divided into three approximately equal groups: those who were already able to use an internalized counting sequence for the simplest forms of addition and subtraction; those who relied on a repeated 'counting-all' procedure for such tasks; and those who were as yet unable to cope with such tasks. In each group, significant relationships between some, but not all, of the numerical tasks were found. However, for almost any two tasks, it was possible to find individuals who could carry out either one of the tasks but not the other. Thus, even before formal instruction, arithmetical cognition is not unitary but is made up of many components.     

The role of gesture in children's learning to count.

The role of spontaneous gesture was examined in children's counting and in their assessment of counting accuracy. Eighty-five 2-, 3-, and 4-year-olds counted 6 sets of 2-, 4-, and 6-object arrays. In addition, children assessed the counting accuracy of a puppet whose gestures varied as he counted (i.e., gesture matched the number words, gesture mismatched the number words, no gesture at all). Results showed that the correspondence of children's speech and gesture varied systematically across the age groups and that children adhered to the one-to-one correspondence principle in gesture prior to speech. Moreover, children's correspondence of speech and gesture, adherence to the one-to-one principle in gesture, and assessment of the puppet's counting accuracy were related to children's counting accuracy. Findings are discussed in terms of the role that gesture may play in children's understanding 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.     

幼儿在集合比较中运用数数的实验研究

本研究中运用了两个实验探讨数数干预和测查条件对儿童在集合比较中运用数数的影响。干预对3岁儿童(M=3：9)没有影响。在平均年龄为4岁4个月时．干预组儿童比控制组儿童更倾向于用数数比较集合．自然组儿童也比传统组更倾向于用数数。许多4岁儿童在无干预时不用数数可能是因为,1)不知数数比视觉性比较更有效,或2)他们在集合比较中的数数极易受测查情景因素的影响。儿童在集合比较中的数数运用与他们的数数水平密切关联。     

Units of counting: Developmental changes

Insofar as counting is directed toward a definite quantification goal, only items that qualify as valid instances of what is being quantified should be included in the count. Thus, the choice of what to treat as a unit to be counted depends upon one's quantification goals. The present research examined developmental changes in the way children define units for counting. In the first experiment, children were shown arrays of toy animals and asked to count either the number of families or the number of individuals within a family. In the second and third experiments, the stimuli were objects that came apart into two pieces. Children were shown arrays composed of some intact objects and some objects that were separated into their parts, and they were asked either to count the number of wholes or the number of pieces in the entire array. Virtually all the counts children generated were based on some type of common unit, even if it was only defined by physical discreteness. However, marked age differences emerged in children's adaptation of their counting units to what they were asked to count.