Numerical morphology supports early number word learning: Evidence from a comparison of young Mandarin and English learners.

Previous studies showed that children learning a language with an obligatory singular/plural distinction (Russian and English) learn the meaning of the number word for one earlier than children learning Japanese, a language without obligatory number morphology (Barner, Libenson, Cheung, & Takasaki, 2009; Sarnecka, Kamenskaya, Yamana, Ogura, & Yudovina, 2007). This can be explained by differences in number morphology, but it can also be explained by many other differences between the languages and the environments of the children who were compared. The present study tests the hypothesis that the morphological singular/plural distinction supports the early acquisition of the meaning of the number word for one by comparing young English learners to age and SES matched young Mandarin Chinese learners. Mandarin does not have obligatory number morphology but is more similar to English than Japanese in many crucial respects. Corpus analyses show that, compared to English learners, Mandarin learners hear number words more frequently, are more likely to hear number words followed by a noun, and are more likely to hear number words in contexts where they denote a cardinal value. Two tasks show that, despite these advantages, Mandarin learners learn the meaning of the number word for one three to six months later than do English learners. These results provide the strongest evidence to date that prior knowledge of the numerical meaning of the distinction between singular and plural supports the acquisition of the meaning of the number word for one.

Children's mappings between number words and the approximate number system.

Humans can represent number either exactly--using their knowledge of exact numbers as supported by language, or approximately--using their approximate number system (ANS). Adults can map between these two systems--they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate (i.e., ANS-to-Word), and can generate an approximation, for example by rapidly tapping, when provided with an exact verbal number (i.e., Word-to-ANS). Here we ask how these mappings are initially formed and whether one mapping direction may become functional before the other during development. In two experiments, we gave 2-5 year old children both an ANS-to-Word task, where they had to give a verbal number response to an approximate presentation (i.e., after seeing rapidly flashed dots, or watching rapid hand taps), and a Word-to-ANS task, where they had to generate an approximate response to a verbal number request (i.e., rapidly tapping after hearing a number word). Replicating previous results, children did not successfully generate numerically appropriate verbal responses in the ANS-to-Word task until after 4 years of age--well after they had acquired the Cardinality Principle of verbal counting. In contrast, children successfully generated numerically appropriate tapping sequences in the Word-to-ANS task before 4 years of age--well before many understood the Cardinality Principle. We further found that the accuracy of the mapping between the ANS and number words, as captured by error rates, continues to develop after this initial formation of the interface. These results suggest that the mapping between the ANS and verbal number representations is not functionally bidirectional in early development, and that the mapping direction from number representations to the ANS is established before the reverse.

Children acquire the later-greater principle after the cardinal principle.

Many have proposed that the acquisition of the cardinal principle (CP) is a result of the discovery of the numerical significance of the order of the number words in the count list. However, this need not be the case. Indeed, the CP does not state anything about the numerical significance of the order of the number words. It only states that the last word of a correct count denotes the numerosity of the counted set. Here, we test whether the acquisition of the CP involves the discovery of the later-greater principle - that is, that the order of the number words corresponds to the relative size of the numerosities they denote. Specifically, we tested knowledge of verbal numerical comparisons (e.g., Is 'ten' more than 'six'?) in children who had recently learned the CP. We find that these children can compare number words between 'six' and 'ten' only if they have mapped them onto non-verbal representations of numerosity. We suggest that this means that the acquisition of the CP does not involve the discovery of the correspondence between the order of the number words and the relative size of the numerosities they denote.

One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles.

Since the publication of [Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.] seminal work on the development of verbal counting as a representation of number, the nature of the ontogenetic sources of the verbal counting principles has been intensely debated. The present experiments explore proposals according to which the verbal counting principles are acquired by mapping numerals in the count list onto systems of numerical representation for which there is evidence in infancy, namely, analog magnitudes, parallel individuation, and set-based quantification. By asking 3- and 4-year-olds to estimate the number of elements in sets without counting, we investigate whether the numerals that are assigned cardinal meaning as part of the acquisition process display the signatures of what we call "enriched parallel individuation" (which combines properties of parallel individuation and of set-based quantification) or analog magnitudes. Two experiments demonstrate that while "one" to "four" are mapped onto core representations of small sets prior to the acquisition of the counting principles, numerals beyond "four" are only mapped onto analog magnitudes about six months after the acquisition of the counting principles. Moreover, we show that children's numerical estimates of sets from 1 to 4 elements fail to show the signature of numeral use based on analog magnitudes - namely, scalar variability. We conclude that, while representations of small sets provided by parallel individuation, enriched by the resources of set-based quantification are recruited in the acquisition process to provide the first numerical meanings for "one" to "four", analog magnitudes play no role in this process.

Re-visiting the competence/performance debate in the acquisition of the counting principles.

Advocates of the "continuity hypothesis" have argued that innate non-verbal counting principles guide the acquisition of the verbal count list (Gelman & Galistel, 1978). Some studies have supported this hypothesis, but others have suggested that the counting principles must be constructed anew by each child. Defenders of the continuity hypothesis have argued that the studies that failed to support it obscured children's understanding of counting by making excessive demands on their fragile counting skills. We evaluated this claim by testing two-, three-, and four-year-olds both on "easy" tasks that have supported continuity and "hard" tasks that have argued against it. A few noteworthy exceptions notwithstanding, children who failed to show that they understood counting on the hard tasks also failed on the easy tasks. Therefore, our results are consistent with a growing body of evidence that shows that the count list as a representation of the positive integers transcends pre-verbal representations of number.