The Idea of an Exact Number: Children's Understanding of Cardinality and Equinumerosity

Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one‐to‐one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood.     

Elimination problems in logic: a brief history

A common aim of elimination problems for languages of logic is to express the entire content of a set of formulas of the language, or a certain part of it, in a way that is more elementary or more informative. We want to bring out that as the languages for logic grew in expressive power and, at the same time, our knowledge of their expressive limitations also grew, elimination problems in logic underwent some change. For languages other than that for monadic second-order logic, there remain important open problems.     

From grammatical number to exact numbers: early meanings of 'one', 'two', and 'three' in English, Russian, and Japanese

This study examined whether singular/plural marking in a language helps children learn the meanings of the words 'one,' 'two,' and 'three.' First, CHILDES data in English, Russian (which marks singular/plural), and Japanese (which does not) were compared for frequency, variability, and contexts of number-word use. Then young children in the USA, Russia, and Japan were tested on Counting and Give-N tasks. More English and Russian learners knew the meaning of each number word than Japanese learners, regardless of whether singular/plural cues appeared in the task itself (e.g., "Give two apples" vs. "Give two"). These results suggest that the learning of "one," "two" and "three" is supported by the conceptual framework of grammatical number, rather than that of integers.     

How counting represents number: what children must learn and when they learn it

This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowers) to those who do not (subset-knowers), in order to better characterize the knowledge itself. New results are that (1) Many children answer the question "how many" with the last word used in counting, despite not understanding how counting works; (2) Only children who have mastered the cardinal principle, or are just short of doing so, understand that adding objects to a set means moving forward in the numeral list whereas subtracting objects mean going backward; and finally (3) Only cardinal-principle-knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change.     

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.     

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.     

Cardinality without Enumeration

We show that the notion of cardinality of a set is independent from that of wellordering, and that reasonable total notions of cardinality exist in every model of ZF where the axiom of choice fails. Such notions are either definable in a simple and natural way, or non-definable, produced by forcing. Analogous cardinality notions exist in nonstandard models of arithmetic admitting nontrivial automorphisms. Certain motivating phenomena from quantum mechanics are also discussed in the Appendix.     

What makes counting count? Verbal and visuo-spatial contributions to typical and atypical number development

Williams Syndrome (WS) is marked by a relative strength in verbal cognition coupled with a serious impairment in non-verbal cognition. A strong deficit in numerical cognition has been anecdotally reported in this disorder; however, its nature has not been systematically investigated. Here, we tested 14 children with WS (mean age=7 years 2 months), 14 typically developing controls individually matched on visuo-spatial ability (mean age=3 years 5 months) as well as a larger group of typically developing controls (mean age=3 years 4 months) on two tasks to assess their understanding that counting determines the exact quantity of sets (cardinality principle). The understanding of the cardinality principle in children with WS is extremely delayed and only at the level predicted by their visuo-spatial MA. In this clinical group, only language accounted for a significant amount of the variance in cardinality understanding, whereas in the normal comparison group only visuo-spatial competence predicted the variance. The present findings suggest that visuo-spatial ability plays a greater role than language ability in the actual development of cardinality understanding in typically developing children, whereas the opposite obtains for the clinical group.     

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.