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Numerous studies have shown that number word learning is a protracted process. One challenge facing children learning the meaning of number word such as “one”, “two”, or “three” is that number words refer to a property of a set and not to individual objects. In this study, we focused on a sample of children who have not learned the meaning of small number words such as “two” and “three” and tested whether children could learn number words from examples of sets that help them focus on set size. Specifically, the experimental training condition included examples that highlight a common relational structure between sets through varying object properties in the sets (e.g., three yellow stars and three red hearts are both “three”), whereas the control condition did not vary object properties(e.g., two sets of three yellow stars with different spatial arrangement). We trained two- and three-knowers (N = 65) on the next number (i.e., three or four) and assessed their learning with a Two-Alternative-Forced-Choice task and Give-a-Number task. Overall, we found weak effects of training. We discuss our findings in the broader literature on number word learning and explore the possibility of analogical reasoning as a mechanism of number word learning.

Very large numbers words such as "hundred," "thousand," "million," "billion," and "trillion" pose a learning problem for children because they are sparse in everyday speech and children's experience with extremely large quantities is scarce. In this study, we examine when children acquire the relative ordering of very large number words as a first step toward understanding their acquisition. In Study 1, a hundred and twenty-five 5-8-year-olds participated in a verbal number comparison task involving very large number words. We found that children can judge which of two very large numbers is more as early as age 6, prior to entering first grade. In Study 2, we provided a descriptive analysis on the usage of very large number words using the CHILDES database. We found that the relative frequency of large number words does not change across the years, with "hundred" uttered more frequently than others by an order of magnitude. We also found that adults were more likely to use large number words to reference units of quantification for money, weight, and time, than for discrete, physical entities. Together, these results show that children construct a numerical scale for large number words prior to learning their precise cardinal meanings, and highlight how frequency and context may support their acquisition. Our results have pedagogical implications and highlight a need to investigate how children acquire meanings for number words that reference quantities beyond our everyday experience.

Studies on children's understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn's (1990) Give-N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal-principle-knowers) and those who cannot as lacking knowledge of it (subset-knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP-knowers on the Give-N Task. Specifically, we test the claim that subset-knowers lack cardinal principle knowledge by separating children's understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel's (1978) other how-to-count principles emerge in development. We analyzed how often children violated the three how-to-count principles in a secondary analysis of Give-N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP-knowers, and that understanding of the stable-order and word-object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children's acquisition of the counting principles and that learning to coordinate all three principles represents an additional step beyond learning them individually.

"Place value," which underlies the meanings of multidigits, encompasses the principle of position and base-10 rules. To understand 65, one needs to know that the digits 6 and 5 occupy different positions and thus represent ordered values of different magnitudes (i.e., the "principle of position") and that the value of each position is determined by base-10 rules (i.e., the rightmost position is 10[superscript 0], followed by 10[superscript 1], 10[superscript 2], etc.). Without the principle of position, children cannot construct meanings for multidigits. Previous studies have shown that children do not know the exact value of digit positions until the early elementary school years, but less is known about the acquisition of positional knowledge for multidigits. To study when and how children construct a relationship between position and value, we explore when children begin to know that the leftmost digit represents the largest value and whether such knowledge relates to learning number names. Children ages 4 to 7 years, from primarily Caucasian, middle-class families were asked to compare different pairs of multidigits. Some comparisons (e.g., 12 vs. 21) required knowledge of positional property, and some did not (e.g., 35 vs. 36). We found that as a group, 6-year-old children could recruit positional knowledge to compare multidigits. We also found that children who knew the number names of both multidigits in a comparison pair were above chance on multidigit comparison. Our results shed light on the developmental steps toward acquiring place-value notation and highlight a role of learning number names for learning positional property of the place-value notation.

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.

We suggest that prior to school entry, our earliest "teachers" and "learning settings" --that is, our parents, caregivers, and homes--provide signals about our environmental conditions. In turn, our brains may interpret this information as cues indicating the types of environments we will likely face and adapt accordingly. We discuss ways in which two such early-life cues--bilingual exposure and sensitive caregiving quality, influence "domain general" neurocircuitry and associated functioning (e.g., temperament and emotional reactivity, emotion regulation, relational memory, exploratory play, and executive functioning), as well as pre-academic outcomes. We conclude by discussing the need for early upstream intervention programmes, as well as the need for additional research including our upcoming "BE POSITIVE" study, designed to help bridge the gap between the community, home, and school environments.

The current study investigated the development of numericalestimation in 3- to 5-year-old children sampled monthly for sixmonths. At each session, children completed a task thatassesses verbal number knowledge (Give-N task) and anumerical estimation task that assesses approximate numberknowledge (Fast Cards). Results showed that children whoacquired the cardinal principle (CP) during the course of thestudy showed marked improvement on the estimation task.Following CP acquisition, estimation became more accurateoverall but also fluctuated widely. We discuss the implicationsof our findings for number word learning, particularly themapping between verbal number and the approximate numbersystem (ANS).

Complex numerals (e.g., four hundred) have a multiplicativestructure (four hundred = 4 x 100). This paper investigateswhether children are sensitive to the meaning of themultiplicative structure. We designed a novel word learningparadigm and taught 4- to 6-year-old children the meaning ofa novel numeral phrase (e.g., ‘one gobi houses’ to mean agroup of three houses). We then asked whether they couldgeneralize it to a novel context (e.g., ‘two gobi butterflies’ tomean two groups of three). Experiment 1 showed that onlyEnglish-speaking children who received multiplier syntaxtraining were able to generalize. Experiment 2 extendedfindings from Experiment 1 to Cantonese-speaking childrenand found that they could also generalize a novel multiplier tonovel contexts. These results suggest that children as youngas 4 can create a mapping between the structure of complexnumerals and a multiplicative meaning.

Young children typically take between 18 months and 2 years to learn the meanings of number words. In the present study, we investigated this developmental trajectory in bilingual preschoolers to examine the relative contributions of two factors in number word learning: (1) the construction of numerical concepts, and (2) the mapping of language specific words onto these concepts. We found that children learn the meanings of small number words (i.e., one, two, and three) independently in each language, indicating that observed delays in learning these words are attributable to difficulties in mapping words to concepts. In contrast, children generally learned to accurately count larger sets (i.e.,five or greater) simultaneously in their two languages, suggesting that the difficulty in learning to count is not tied to a specific language. We also replicated previous studies that found that children learn the counting procedure before they learn its logic – i.e., that for any natural number, n, the successor of n in the count list denotes the cardinality n + 1. Consistent with past studies, we found that children’s knowledge of successors is first acquired incrementally. In bilinguals, we found that this knowledge exhibits item-specific transfer between languages, suggesting that the logic of the positive integers may not be stored in a language-specific format. We conclude that delays in learning the meanings of small number words are mainly due to language-specific processes of mapping words to concepts, whereas the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.

The current study asks when children begin to understand thatwhen an object is added to a set, the numerosity of the set hasincreased regardless of set size. This knowledge can beexpressed algebraically as ‘x + 1 > x’. In Experiment 1, 3- to5-year-old children were asked to reason abouttransformations (i.e., addition, subtraction, rearrangement)performed on a visible set of objects. We found that 5-yearoldswere able to reason about how each transformationaffected numerosity, and 4-year-olds showed limitedunderstanding. In Experiment 2, children were asked toreason about transformations performed on a hidden set ofobjects. Similar results were found. Together, we showed thatthe ability to reason about number algebraically developsgradually between the ages of 3 and 5. Implications fornumber word acquisition were discussed

A study of 104 Japanese-speaking 2- to 5-year-olds tested the relation between numeral and quantifier acquisition. A first study assessed Japanese children's comprehension of quantifiers, numerals, and classifiers. Relative to English-speaking counterparts, Japanese children were delayed in numeral comprehension at 2 years of age but showed no difference at 3 and 4 years of age. Also, Japanese 2-year-olds had better comprehension of quantifiers, indicating that their delay was specific to numerals. A second study examined the speech of Japanese and English caregivers to explore the syntactic cues that might affect integer acquisition. Quantifiers and numerals occurred in similar syntactic positions and overlapped to a greater degree in English than in Japanese. Also, Japanese nouns were often dropped, and both quantifiers and numerals exhibited variable positions relative to the nouns they modified. We conclude that syntactic cues in English facilitate bootstrapping numeral meanings from quantifier meanings and that such cues are weaker in classifier languages such as Japanese. (Contains 2 tables and 8 figures.)

Very large numbers words such as "hundred," "thousand," "million," "billion," and "trillion" pose a learning problem for children because they are sparse in everyday speech and children's experience with extremely large quantities is scarce. In this study, we examine when children acquire the relative ordering of very large number words as a first step toward understanding their acquisition. In Study 1, a hundred and twenty‐five 5–8‐year‐olds participated in a verbal number comparison task involving very large number words. We found that children can judge which of two very large numbers is more as early as age 6, prior to entering first grade. In Study 2, we provided a descriptive analysis on the usage of very large number words using the CHILDES database. We found that the relative frequency of large number words does not change across the years, with "hundred" uttered more frequently than others by an order of magnitude. We also found that adults were more likely to use large number words to reference units of quantification for money, weight, and time, than for discrete, physical entities. Together, these results show that children construct a numerical scale for large number words prior to learning their precise cardinal meanings, and highlight how frequency and context may support their acquisition. Our results have pedagogical implications and highlight a need to investigate how children acquire meanings for number words that reference quantities beyond our everyday experience. [ABSTRACT FROM AUTHOR]

Studies on children's understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn's (1990) Give‐N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal‐principle‐knowers) and those who cannot as lacking knowledge of it (subset‐knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP‐knowers on the Give‐N Task. Specifically, we test the claim that subset‐knowers lack cardinal principle knowledge by separating children's understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel's (1978) other how‐to‐count principles emerge in development. We analyzed how often children violated the three how‐to‐count principles in a secondary analysis of Give‐N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP‐knowers, and that understanding of the stable‐order and word‐object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children's acquisition of the counting principles and that learning to coordinate all three principles represents an additional step beyond learning them individually. [ABSTRACT FROM AUTHOR]