Understanding Of Cardinality Research Articles

The influence of language on the formation of number concepts: Evidence from preschool children who are bilingual in English and Arabic

We asked whether grammatical number marking has specific influence on the formation of early number concepts. In particular, does comprehension of dual case marking support young children’s understanding of cardinality? We assessed number knowledge in 77 3-year-old Arabic–English bilingual children using the Give-a-Number task in both languages. Given recent concerns around the administration and scoring of the Give-a-Number task, we used two complementary approaches: one based on conceptual levels and the other based on overall test scores. We also tested comprehension of dual case marking in Arabic and number sequence knowledge in both languages. Regression analyses showed that dual case comprehension exerts a strong influence on cardinality tested in Arabic independent of age, general language skills, and number sequence knowledge. No such influence was found for cardinality tested in English, indicating a language-specific effect. Further analyses tested for transfer of cardinality knowledge between languages. These revealed, in addition to the findings outlined above, a powerful cross-linguistic transfer effect. Our findings are consistent with a model in which the direct effect of dual case marking is language specific, but concepts, once acquired, may be represented abstractly and transferred between languages.

Mapping skills between symbols and quantities in preschoolers: The role of finger patterns.

Mapping skills between different codes to represent numerical information, such as number symbols (i.e., verbal number words and written digits) and non-symbolic quantities, are important in the development of the concept of number. The aim of the current study is to investigate children's mapping skills by incorporating another numerical code that emerges at early stages in development, finger patterns. Specifically, the study investigates (i) the order in which mapping skills develop and the association with young children's understanding of cardinality; and (ii) whether finger patterns are processed similarly to symbolic codes or rather as non-symbolic quantities. Preschool children (3-year-olds, N=113, Mage=40.8months, SDage=3.6months; 4-year-olds, N=103, Mage=52.9months, SDage=3.4months) both cardinality knowers and subset-knowers, were presented with twelve tasks that assessed the mappings between number words, Arabic digits, finger patterns, and quantities. The results showed that children's ability to map symbolic numbers precedes the understanding that such symbols reflect quantities, and that children recognize finger patterns above their cardinality knowledge, suggesting that finger patterns are symbolic in essence. RESEARCH HIGHLIGHTS: Children are more accurate in mapping between finger patterns and symbols (number words and Arabic digits) than in mapping finger patterns and quantities, indicating that fingers are processed holistically as symbolic codes. Children can map finger patterns to symbols above their corresponding cardinality level even in subset-knowers. Finger patterns may play a role in the process by which children learn to map symbols to quantities. Fingers patterns' use in the classroom context may be an adequate instructional and diagnostic tool.

The how many and give-N tasks: Conceptually distinct measures of the cardinality principle

In the current study we investigated performance on the how many (how many?) and give-N tasks. We first investigated the relative performance of three-to-five-year-old children (N = 393; M = 4.75 years, SD = .75 years) on these tasks. Replicating prior work, we found that children performed worse on give-N compared to how many? and this performance gap increased as the set size increased. This performance gap remained even after controlling for children's counting skill. We next conducted a series of confirmatory factor analyses to identify the best fitting factor structure of three tasks that measure children's understanding of set sizes across both small and large sets: a subitizing measure, how many?, and give-N. We found that the best fitting factor structure contained separate factors for each task. This suggests that each measure provides task-specific variance not captured by the other cardinality measures. These results highlight the different aspects of children's understanding of cardinality.

Use them or lose them: Are manipulatives needed to assess numeracy and geometry performance in preschool?

AbstractIn two studies, we investigated whether using three‐dimensional (3D) manipulatives during assessment aided performance on a variety of preschool mathematics tasks compared to pictorial representations. On measures of children's understanding of counting and cardinality (n = 103), there was no difference in performance between manipulatives and pictures, with Bayes factors suggesting moderate evidence in favor of the null hypothesis. On a measure of children's shape identification (n = 93), there was no difference in performance between objects and pictures, with Bayes factors suggesting moderate evidence in favor of the null hypothesis. These results suggest flexibility in the materials that can be used during assessment. Pictures, or 2D renderings of 3D objects, which can be easily printed and reproduced, may be sufficient for assessing counting and shape knowledge without the need for more cumbersome concrete manipulatives.

Characterizing exact arithmetic abilities before formal schooling

Children appear to have some arithmetic abilities before formal instruction in school, but the extent of these abilities as well as the mechanisms underlying them are poorly understood. Over two studies, an initial exploratory study of preschool children in the U.S. (N = 207; Age = 2.89–4.30 years) and a pre-registered replication of preschool children in Italy (N = 130; Age = 3–6.33 years), we documented some basic behavioral signatures of exact arithmetic using a non-symbolic subtraction task. Furthermore, we investigated the underlying mechanisms by analyzing the relationship between individual differences in exact subtraction and assessments of other numerical and non-numerical abilities. Across both studies, children performed above chance on the exact non-symbolic arithmetic task, generally showing better performance on problems involving smaller quantities compared to those involving larger quantities. Furthermore, individual differences in non-verbal approximate numerical abilities and exact cardinal number knowledge were related to different aspects of subtraction performance. Specifically, non-verbal approximate numerical abilities were related to subtraction performance in older but not younger children. Across both studies we found evidence that cardinal number knowledge was related to performance on subtraction problems where the answer was zero (i.e., subtractive negation problems). Moreover, subtractive negation problems were only solved above chance by children who had a basic understanding of cardinality. Together these finding suggest that core non-verbal numerical abilities, as well as emerging knowledge of symbolic numbers provide a basis for some, albeit limited, exact arithmetic abilities before formal schooling.

The real preschoolers of Orange County: Early number learning in a diverse group of children

The authors assessed a battery of number skills in a sample of over 500 preschoolers, including both monolingual and bilingual/multilingual learners from households at a range of socio-economic levels. Receptive vocabulary was measured in English for all children, and also in Spanish for those who spoke it. The first goal of the study was to describe entailment relations among numeracy skills by analyzing patterns of co-occurrence. Findings indicated that transitive and intransitive counting skills are jointly present when children show understanding of cardinality and that cardinality and knowledge of written number symbols are jointly present when children successfully use number lines. The study’s second goal was to describe relations between symbolic numeracy and language context (i.e., monolingual vs. bilingual contexts), separating these from well-documented socio-economic influences such as household income and parental education: Language context had only a modest effect on numeracy, with no differences detectable on most tasks. However, a difference did appear on the scaffolded number-line task, where bilingual learners performed slightly better than monolinguals. The third goal of the study was to find out whether symbolic number knowledge for one subset of children (Spanish/English bilingual learners from low-income households) differed when tested in their home language (Spanish) vs. their language of preschool instruction (English): Findings indicated that children performed as well or better in English than in Spanish for all measures, even when their receptive vocabulary scores in Spanish were higher than in English.

Features in children's counting books that lead dyads to both count and label sets during shared book reading.

This study examined how book features influence talk during shared book reading. We used data from a study in which parent-child dyads (n= 157; child's Mage = 43.99 months; 88 girls, 69 boys; 91.72% of parents self-reported as white) were randomly assigned to read two number books. The focus was comparison talk (i.e., talk in which dyads count a set and also label its total), as this type of talk has been shown to promote children's understanding of cardinality. Replicating previous findings, dyads produced relatively low levels of comparison talk. However, book features influenced the talk. Books containing a greater number of numerical representations (e.g., number word, numeral, and non-symbolic set) and a greater word count elicited more comparison talk.

Decision over Time as a By-Product of a Measure of Utility: A Reappraisal of Paul Samuelson’s “A Note on Measurement of Utility” (1937)

This contribution aims to highlight a neglected aspect of Samuelson’s famous 1937 paper “A Note on Measurement of Utility”. Although the 1937 paper is usually regarded as the foundation of discounted utility theory, and rightly so, it is primarily concerned with utility measurement and deals only indirectly with decision over time – intertemporal issues appearing as a by-product of the realisation of a unique utility measure. But the treatment of discounted utility in turn influenced Samuelson’s understanding of cardinality. Cardinality appears here as the result of a cognitive ability that manifests when agents face a decision experiment over time in which they are compelled to cardinalize their utility functions. The result is the weak plausibility of cardinality in a more general context, such that, contrary to the usual views, we may say that Samuelson’s ordinalist approach was already in the making in 1937.

The emergence of children’s natural number concepts: Current theoretical challenges

AbstractLearning the meaning of number words is a lengthy and error‐prone process. In this review, we highlight outstanding issues related to current accounts of children’s acquisition of symbolic number knowledge. We maintain that, despite the ability to identify and label small numerical quantities, children do not understand initially that number words refer only to sets of discrete countable items, not to other nonnumerical dimensions. We question the presence of a sudden change in children’s understanding of cardinality, and we report the limits of the give‐a‐number task. We also highlight that children are still learning the directional property of the counting list, even after acquiring the cardinality principle. Finally, we discuss the role that the Approximate Number System may have in supporting the acquisition of symbolic numbers. We call for improvements in methodological tools and refinement in theoretical understanding of how children learn natural numbers.

Get to the point: Preschoolers’ spontaneous gesture use during a cardinality task

Gesture is a powerful learning tool for language, mathematics, and general problem solving, although the specific function of gesture is subject to debate. To address this question, we characterized predictors of preschoolers’ spontaneous gestures while completing a cardinality task. We analyzed video data from 343 preschoolers (158 male, Mage = 4.07y) who participated in the Give-N task (Wynn, 1992), which is used to assess children's number knowledge and understanding of cardinality. Analyses revealed that children’s number knowledge positively predicted gesture use, even when controlling for age, suggesting that as children gain better understanding of the skills necessary to complete the task, they are more likely to use gesture to solve numerical tasks. Moreover, those who had not yet acquired an understanding of cardinality (subset knowers, who are said to not use pointing gestures) were shown to use an effective gestural strategy (pointing and counting). Spontaneous gesture use across all knower-levels was also found to increase as a function of trial difficulty (trials involving larger set sizes), such that children were most likely to gesture on trials that challenged their abilities. Overall, findings indicate that children rely upon gestures as a scaffold during cognitively difficult tasks, and it is their relevant knowledge, not their age, that predicts their likelihood of engaging in gesture.

Improved set-size labeling mediates the effect of a counting intervention on children's understanding of cardinality.

How does improving children's ability to label set sizes without counting affect the development of understanding of the cardinality principle? It may accelerate development by facilitating subsequent alignment and comparison of the cardinal label for a given set and the last word counted when counting that set (Mix et al., 2012). Alternatively, it may delay development by decreasing the need for a comprehensive abstract principle to understand and label exact numerosities (Piantadosi et al., 2012). In this study, preschoolers (N=106, Mage =4;8) were randomly assigned to one of three conditions: (a) count-and-label, wherein children spent 6weeks both counting and labeling sets arranged in canonical patterns like pips on a die; (b) label-first,wherein children spent the first 3weeks learning to label the set sizes without counting before spending 3weeks identical to the count-and-label condition; (c) print referencing control. Both counting conditions improved understanding of cardinality through increases in children's ability to label set sizes without counting. In addition to this indirect effect, there was a direct effect of the count-and-label condition on progress toward understanding of cardinality. Results highlight the roles of set labeling and equifinality in the development of children's understanding of number concepts.

The Effectiveness of Teaching Geometry to Enhance Mathematical Understanding in Children with Down Syndrome

ABSTRACTIt is widely known that people with Down syndrome have difficulties transitioning from a basic understanding of counting and cardinality to more advanced arithmetic skills. This is commonly addressed by resorting to the mechanical use of algorithms, which hinders the acquisition of mathematical concepts. For this reason some authors have recently proposed a shift in the focus of learning from arithmetic to more fertile fields, in terms of understanding. In this paper we claim geometry fits this profile, especially suited for initiating children with Down syndrome into mathematics. To support this we resort to historical, epistemological, and cognitive reasons: the work of Séguin and his intuition on the central role of geometry in the development of abstract thinking in the so-called idiot children, the ideas of René Thom about the role of continuum intuition in the emergence of conscious thinking, and finally the two strengths people with Down syndrome display: visual learning abilities and interest in abstract symbols. To support these ideas we present the main findings of qualitative research on elementary mathematics teaching to a group of seven children (3–8) with Down syndrome in Spain. The didactic method used, naturally enhance their naïve geometrical conceptions.

Early Understanding of Cardinal Number Value: Semiotic, Social, and Pragmatic Dimensions in a Case Study with a Child from 2 to 3Years Old.

Studies on cardinality date back many years, and it remains a current research issue today. Indeed, despite the many findings on the topic, there is still controversy surrounding when and how children understand cardinality. The main studies on this understanding employs the "how many" question in situations involving the quantification of objects, analyzing the relationship between counting and cardinality. In the present study, we argue that it is essential to consider how cardinality is used in the context in which it arises, including the in- depth consideration of semiotic, social, and pragmatic dimensions, in order to fully comprehend the topic. We analyze in microgenesis the interaction between a two-year-old girl and her mother when they are playing a game requiring the understanding and use of cardinality through five sessions conducted over the course of a year. Our findings suggest that, in the proposed situation, cardinal understanding develops slowly and gradually requires an integrated body of resources (such as gestures and the use of objects). In highlighting the role of semiotics and social interactions in the development of cardinal understanding, this research underscores the fundamental role that early education should play in its development.

Differential Development of Children's Understanding of the Cardinality of Small Numbers and Zero.

Counting and the understanding of cardinality are important steps in children’s numerical development. Recent studies have indicated that language and visuospatial abilities play an important role in the development of children’s cardinal knowledge of small numbers. However, predictors for the knowledge about zero were usually not considered in these studies. Therefore, the present study investigated whether the acquisition of cardinality knowledge on small numbers and the concept of zero share cross-domain and domain-specific numerical predictors. Particular interest was paid to the question whether visuospatial abilities – in addition to language abilities – were associated with children’s understanding of small numbers and zero. Accordingly, we assessed kindergarteners aged 4 to 5 years in terms of their understanding of small numbers and zero as well as their visuospatial, general language, counting, Arabic number identification abilities, and their finger number knowledge. We observed significant zero-order correlations of vocabulary, number identification, finger knowledge, and counting abilities with children’s knowledge about zero as well as understanding of the cardinality of small numbers. Subsequent regression analyses substantiated the influences of counting abilities on knowledge about zero and the influences of both counting abilities and finger knowledge on children’s understanding of the cardinality of small numbers. No significant influences of cross-domain predictors were observed. In sum, these results indicate that domain-specific numerical precursor skills seem to be more important for children’s development of an understanding of the cardinality of small numbers as well as of the concept of zero than the more proximal cross-domain abilities such as language and visuospatial abilities.

Children’s early understanding of number predicts their later problem-solving sophistication in addition

Previous studies suggest that the sophistication of the strategies children use to solve arithmetic problems is related to a more basic understanding of number, but they have not examined the relation between number knowledge in preschool and strategy choices at school entry. Accordingly, the symbolic and nonsymbolic quantitative knowledge of 134 children (65 boys) was assessed at the beginning of preschool and in kindergarten, and the sophistication of the strategies they used to solve addition problems was assessed at the beginning of first grade. Using a combination of Bayes and standard regression models, we found that children’s understanding of the cardinal value of number words at the beginning of preschool predicted the sophistication of their strategy choices 3 years later, controlling for other factors. The relation between children’s early understanding of cardinality and their strategy choices was mediated by their symbolic and nonsymbolic quantitative knowledge in kindergarten. The results suggest that sophisticated strategy choices emerge from children’s developing understanding of the relations among numbers, in keeping with the overlapping waves model.

Enumeration skills in Down syndrome

Individuals with Down syndrome (DS) exhibit various math difficulties which can be ascribed both to global intelligence level and/or to their atypical cognitive profile. In this light, it is crucial to investigate whether DS display deficits in basic numerical skills. In the present study, individuals with DS and two groups of typically developing (TD) children matched for mental and chronological age completed two delayed match-to-sample tasks in order to evaluate the functioning of visual enumeration skills. Children with DS showed a specific deficit in the discrimination of small numerosities (within the subitizing range) with respect to both mental and chronological age matched TD children. In contrast, the discrimination of larger numerosities, though lower than that of chronological age matched controls, was comparable to that of mental age matched controls. Finally, counting was less fluent but the understanding of cardinality seemed to be preserved in DS. These results suggest a deficit of the object tracking system underlying the parallel individuation of small numerosities and a typical - but developmentally delayed - acuity of the approximate number system for discrimination of larger numerosities.

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.

Rethinking the role of counting in mathematics learning

This paper challenges the emphasis on counting in New Zealand’s Numeracy Development Project (NDP), arguing that subitizing provides an alternative pathway to quantification. Longitudinal data is presented showing that children’s subitizing skills at the age of five years were a strong predictor of their later success in mathematics at the age of nine years. Sophian’s work of comparison of continuous quantities is explored. Data from students whose teachers participated in NDP professional development programmes are compared with the expectations documented in the Mathematics Standards. The analysis shows that the percentages of students who reach the expected level for their year group is well short of the Standards. It is suggested that the large number of micro-stages at the lower end of the Number Framework together with the positioning of part-whole strategies as the fifth stage on the framework may give the impression that teachers should not focus on the relationship between a whole and its parts until students are able to count on (stage 4). The paper concludes by suggesting that a dual focus on subitizing and counting right from the beginning might help students to develop a deeper understanding of cardinality and of the relationship between a whole and its parts, resulting in them reaching expected levels earlier.

Supporting children’s development of cardinality understanding

In this study, we use children’s prior knowledge to support their development of cardinality understanding, based on Bermejo’s (1996) model of cardinality understanding and on the distinction between cardinality (the number of items in a set is represented by a number-word) and the cardinality principle (the last number-word used in counting determines the cardinal number of a set) . Two groups of 24 children each were selected by means of a pretest. The experimental group completed a learning program centered on cognitive conflict, whereas the control group did not receive any special intervention. Results show that most of children in the experimental group reached full understanding of cardinality. In addition, we provide evidence that cardinality is not a component of counting, but that counting and cardinality are different things, just as forms and functions or means and goal are different; and that it is number which gives meaning to subitizing and counting.

Counting and cardinal understanding in children with Down syndrome and typically developing children.

This study compares the procedural counting ability (independently and with parental support) and conceptual understanding of cardinality of a group of children with Down syndrome and a group of typically developing children, matched for non-verbal mental age. Participants were 23 children with Down syndrome (chronological age range: 3.5-7 years; mental age range: 2.5-4 years) and 20 typically developing children (chronological age range: 2-4 years; mental age range: 2.5-4 years), and their main caregiver. The children were asked to count sets of toys (assessing procedural counting skills) and to give sets of toys (assessing understanding of cardinality), with set sizes between 2 and 18 items. The counting task was performed in two conditions, with and without parental support. The children were also asked to say the count word sequence aloud, to assess sequence production independent from object counting. The typically developing children produced significantly more number words altogether, longer standard number sequences and could count larger sets than the children with Down syndrome. Support from an adult improved performance on the count task significantly for both groups of children, and there was no significant difference between the groups in the degree of improvement, i.e. the zone of proximal development. No significant differences were found between the frequency of children (approximately one third) in each group who used counting to solve the give task, indicating an understanding of cardinality.