A million is more than a thousand: Children's acquisition of very large number words.

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.

Non-symbolic Ratio Reasoning in Kindergarteners: Underlying Unidimensional Heuristics and Relations With Math Abilities.

Although it is thought that young children focus on the magnitude of the target dimension across ratio sets during binary comparison of ratios, it is unknown whether this is the default approach to ratio reasoning, or if such approach varies across representation formats (discrete entities and continuous amounts) that naturally afford different opportunities to process the dimensions in each ratio set. In the current study, 132 kindergarteners (Mage = 68 months, SD = 3.5, range = 62-75 months) performed binary comparisons of ratios with discrete and continuous representations. Results from a linear mixed model revealed that children followed an additive strategy to ratio reasoning-i.e., they focused on the magnitude of the target dimension across ratio sets as well as on the absolute magnitude of the ratio set. This approach did not vary substantially across representation formats. Results also showed an association between ratio reasoning and children's math problem-solving abilities; children with better math abilities performed better on ratio reasoning tasks and processed additional dimensions across ratio sets. Findings are discussed in terms of the processes that underlie ratio reasoning and add to the extant debate on whether true ratio reasoning is observed in young children.

Acquisition of the counting principles during the subset-knower stages: Insights from children's errors.

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.

Cracking the code of place value: The relationship between place and value takes years to master.

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 100, followed by 101, 102, 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. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

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.

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 6years of age - roughly 2years 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.

Why is number word learning hard? Evidence from bilingual learners.

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.

Cross-linguistic relations between quantifiers and numerals in language acquisition: evidence from Japanese.

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.

Syntactic Cues to Individuation in Mandarin Chinese.

When presented with an entity (e.g., a wooden honey-dipper) labeled with a novel noun, how does a listener know that the noun refers to an instance of an object kind (honey-dipper) rather than to a substance kind (wood)? While English speakers draw upon count-mass syntax for clues to the noun's meaning, linguists have proposed that classifier languages, which lack count-mass syntax, provide other syntactic cues. Three experiments tested Mandarin-speakers' sensitivity to the diminutive suffix -zi and the general classifier ge when interpreting novel nouns. Experiment 1 found that -zi occurs more frequently with nouns that denote object kinds. Experiment 2 demonstrated Mandarin-speaking adults' sensitivity to ge and -zi when inferring novel word meanings. Experiment 3 tested Mandarin three- to six-year-olds' sensitivity to ge. We discuss differences in the developmental course of these cues relative to cues in English, and the impact of this difference to children's understanding of individuation.