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

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Abstract

Learning 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.

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... There is wide agreement that the first phase unfolds in a stepwise manner in the order of magnitude-commonly called the n-knower levels (Condry & Spelke, 2008;Le Corre & Carey, 2007Le Corre et al., 2006;Sarnecka & Carey, 2008;Wynn, 1990Wynn, , 1992. A 1-knower can reliably recognize and label single items as "one" or can give one item upon request but cannot do so for larger numbers; a "2-knower" can reliably recognize and give sets of one and two but not larger numbers; and so forth up to the 4-knower level (but see Sella et al, 2021). ...
... In terms of theoretical implications, although the study clearly has limited external validity and needs to be replicated with a wider range of materials, more trials (including those just beyond five), and a larger and more chronologically, demographically, and linguistically diverse sample, its preliminary results support the HLP outline in Table 1. For example, future research should consider the caution that children who can successfully create a set of 5 may not be able to do so with larger sets-as was the case for two participants in the present study (Posid & Cordes, 2018;Sella et al., 2021). Such children may not know a cardinal-count concept but "successfully" create a set of five any way using a subitizingbased putting-out strategy instead. ...
... In terms of methodological implications, the present results are consistent with the growing concern that the give-n task may underestimate (pre-counting and) counting-based cardinality knowledge (Barner & Bachrach, 2010;Krajcsi, 2021;Mou et al., 2021;Sella et al., 2021;Wagner et al., 2019). If future results confirm that the give-n task requires the more advanced cardinal-count concept, assessing the CP-knower level might better involve a combination of tasks such as the how-many task and a how-many application task (e.g., the cardinal-identity task) and using a composite cardinality score as in the present study. ...
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Are there differential benefits of training sequential number knowledge versus spatial skills for children's numerical and spatial performance? Three- to five-year-old children (N = 84) participated in 1 session of either sequential training (e.g., what comes before and after the number 5?) or non-numerical spatial training (i.e., decomposition of shapes). Children who received sequential training showed near transfer to a number ordering task and far transfer to a number line task. Furthermore, these children showed more improvement on the version of the number line task where a midpoint reference was presented (i.e., at 5) than on the version without a midpoint. Before the training, the midpoint reference did not enhance performance. In contrast, although children who received non-numerical spatial training showed near transfer to a 2-D mental transformation task, they did not show transfer to number ordering or number line tasks, even though spatial skills were correlated with performance on these tasks. These results support the view that knowledge of sequential relations among successive numbers is an important aspect of children's early numeracy knowledge in tasks that require ordinal understanding of numbers from 1 to 10 and support the educational importance of developing numerical activities that enhance children's understanding of these relations. (PsycINFO Database Record
Article
Approximate number word knowledge-understanding the relation between the count words and the approximate magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of approximate number word knowledge-before, or only after, they have learned the cardinal principle. In two studies, children who had not yet learned the cardinal principle (subset-knowers) produced sets in response to number words (verbal comprehension task) and produced number words in response to set sizes (verbal production task). As evidence of approximate number word knowledge, we examined whether children's numerical responses increased with increasing numerosity of the stimulus. In Study 1, subset-knowers (ages 3.0-4.2years) showed approximate number word knowledge above their knower-level on both tasks, but this effect did not extend to numbers above 4. In Study 2, we collected data from a broader age range of subset-knowers (ages 3.1-5.6years). In this sample, children showed approximate number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted approximate number word knowledge (above 4) on the verbal production task when controlling for their knower-level, study (1 or 2), and parents' education, none of which predicted approximation ability. Thus, children can develop approximate knowledge of number words up to 10 before learning the cardinal principle. Furthermore, approximate number word knowledge increases with age and might not be closely related to the development of exact number word knowledge. Copyright © 2014 Elsevier Inc. All rights reserved.
Article
How do we map number words to the magnitudes they represent? While much is known about the developmental trajectory of number word learning, the acquisition of the counting routine, and the academic correlates of estimation ability, previous studies have yet to describe the mechanisms that link number words to nonverbal representations of number. We investigated two mechanisms: associative mapping and structure mapping. Four dot array estimation tasks found that adults' ability to match a number word to one of two discriminably different sets declined as a function of set size and that participants' estimates of relatively large, but not small, set sizes were influenced by misleading feedback during an estimation task. We propose that subjects employ structure mappings for linking relatively large number words to set sizes, but rely chiefly on item-by-item associative mappings for smaller sets. These results indicate that both inference and association play important roles in mapping number words to approximate magnitudes.
Article
While infants' ability to discriminate quantities has been extensively studied, showing that this competence is present even in neonates, the ability to compute ordinal relations between magnitudes has received much less attention. Here we show that the ability to represent ordinal information embedded in size-based sequences is apparent at 4months of age, provided that magnitude changes involve increasing relations. Infants in Experiments 1A and 1B discriminated changes in ordinal relations after habituation to ascending sequences, but did not show evidence of discrimination after habituation to descending sequences. In Experiment 2 we replicated this asymmetry in magnitude discrimination even when additional cues known to boost ordinal competence were provided. The presence of an asymmetry between ascending vs. descending order during infancy suggests a developmental continuity in the underlying code used to represent magnitude, whereby the reported addition advantage in children and adults' arithmetic performance emerges.
Article
This paper examines how and when children come to understand the way in which counting determines numerosity and learn the meanings of the number words. A 7-month longitudinal study of 2 and 3 year olds shows that, very early on, children already know that the counting words each refer to a distinct, unique numerosity, though they do not yet know to which numerosity each word refers. It is possible that children learn this in part from the syntax of the number words. Despite this early knowledge, however, it takes children a long time (on the order of a year) to learn how the counting system represents numerosity. This suggests that our initial concept of number is represented quite differently from the way the counting system represents number, making it a difficult task for children to map the one Onto the Other.
Article
In acquiring number words, children exhibit a qualitative leap in which they transition from understanding a few number words, to possessing a rich system of interrelated numerical concepts. We present a computational framework for understanding this inductive leap as the consequence of statistical inference over a sufficiently powerful representational system. We provide an implemented model that is powerful enough to learn number word meanings and other related conceptual systems from naturalistic data. The model shows that bootstrapping can be made computationally and philosophically well-founded as a theory of number learning. Our approach demonstrates how learners may combine core cognitive operations to build sophisticated representations during the course of development, and how this process explains observed developmental patterns in number word learning.
Article
We tested the hypothesis that, when children learn to correctly count sets, they make a semantic induction about the meanings of their number words. We tested the logical understanding of number words in 84 children that were classified as "cardinal-principle knowers" by the criteria set forth by Wynn (1992). Results show that these children often do not know (1) which of two numbers in their count list denotes a greater quantity, and (2) that the difference between successive numbers in their count list is 1. Among counters, these abilities are predicted by the highest number to which they can count and their ability to estimate set sizes. Also, children's knowledge of the principles appears to be initially item-specific rather than general to all number words, and is most robust for very small numbers (e.g., 5) compared to larger numbers (e.g., 25), even among children who can count much higher (e.g., above 30). In light of these findings, we conclude that there is little evidence to support the hypothesis that becoming a cardinal-principle knower involves a semantic induction over all items in a child's count list.
Article
Attaching meaning to arbitrary symbols (i.e. words) is a complex and lengthy process. In the case of numbers, it was previously suggested that this process is grounded on two early pre-verbal systems for numerical quantification: the approximate number system (ANS or 'analogue magnitude'), and the object tracking system (OTS or 'parallel individuation'), which children are equipped with before symbolic learning. Each system is based on dedicated neural circuits, characterized by specific computational limits, and each undergoes a separate developmental trajectory. Here, I review the available cognitive and neuroscientific data and argue that the available evidence is more consistent with a crucial role for the ANS, rather than for the OTS, in the acquisition of abstract numerical concepts that are uniquely human.
Article
How do children as young as 2 years of age know that numerals, like one, have exact interpretations, while quantifiers and words like a do not? Previous studies have argued that only numerals have exact lexical meanings. Children could not use scalar implicature to strengthen numeral meanings, it is argued, since they fail to do so for quantifiers [Papafragou, A., & Musolino, J. (2003). Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282]. Against this view, we present evidence that children's early interpretation of numerals does rely on scalar implicature, and argue that differences between numerals and quantifiers are due to differences in the availability of the respective scales of which they are members. Evidence from previous studies establishes that (1) children can make scalar inferences when interpreting numerals, (2) children initially assign weak, non-exact interpretations to numerals when first acquiring their meanings, and (3) children can strengthen quantifier interpretations when scalar alternatives are made explicitly available.
Article
This study examines the abstractness of children's mental representation of counting, and their understanding that the last number word used in a count tells how many items there are (the cardinal word principle). In the first experiment, twenty-four 2- and 3-year-olds counted objects, actions, and sounds. Children counted objects best, but most showed some ability to generalize their counting to actions and sounds, suggesting that at a very young age, children begin to develop an abstract, generalizable mental representation of the counting routine. However, when asked "how many" following counting, only older children (mean age 3.6) gave the last number word used in the count a majority of the time, suggesting that the younger children did not understand the cardinal word principle. In the second experiment (the "give-a-number" task), the same children were asked to give a puppet one, two, three, five, and six items from a pile. The older children counted the items, showing a clear understanding of the cardinal word principle. The younger children succeeded only at giving one and sometimes two items, and never used counting to solve the task. A comparison of individual children's performance across the "how-many" and "give-a-number" tasks shows strong within-child consistency, indicating that children learn the cardinal word principle at roughly 3 1/2 years of age. In the third experiment, 18 2- and 3-year-olds were asked several times for one, two, three, five, and six items, to determine the largest numerosity at which each child could succeed consistently. Results indicate that children learn the meanings of smaller number words before larger ones within their counting range, up to the number three or four. They then learn the cardinal word principle at roughly 3 1/2 years of age, and perform a general induction over this knowledge to acquire the meanings of all the number words within their counting range.
Article
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.
Article
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.
Article
Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.
How reliable is the give-a-number task
  • E Marchand
  • D Barner
Marchand, E., & Barner, D. (2021). How reliable is the give-a-number task?. PsyArXiv. https://doi.org/10.31234/ osf.io/ur3z5
Learning to count: New insights on the acquisition of symbolic numerical knowledge
  • E Slusser
  • P Cravalho
Slusser, E., & Cravalho, P. (2020). Size vs. number: Assigning number words to discrete and continuous quantities. In F. Sella (Chair), Learning to count: New insights on the acquisition of symbolic numerical knowledge. Proceedings of the Mathematical Cognition and Learning Society, Dublin, Ireland. https://doi.org/10.17605/ osf.io/uyh7g
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