Toward exact number: young children use one-to-one correspondence to measure set identity but not numerical equality

How to evaluate the efficiency of rural preschool education resources and its regional differences in China

Rural preschool education is an integral part of rural society, and improving the efficiency system of evaluation of rural preschool education resource allocation is an important strategy for the implementation of the rural revitalization. This paper uses an input-oriented three-stage DEA model to analyze the efficiency of rural preschool education resource allocation in 30 provinces in China from 2012 to 2020. The results show that external factors such as the level of urbanization, birth rate, and the scale of kindergarten impacts the efficiency of rural preschool education resource allocation significantly. Without regard to the influence of environmental and random factors, the overall trend of the average efficiency of rural preschool education resource allocation in China has improved, showing a regional pattern of "central > eastern > western." Therefore, based on the relevant policies, this paper puts forward rational suggestions for the improvement of rural preschool resource allocation efficiency in China from the perspectives of human, financial and material resources.

The algorithmic origins of counting

The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics.

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 cultural origins of symbolic number

It is popular in psychology to hypothesize that representations of exact number are innately determined-in particular, that biology has endowed humans with a system for manipulating quantities which forms the primary representational substrate for our numerical and mathematical concepts. While this perspective has been important for advancing empirical work in animal and child cognition, here we examine six natural predictions of strong numerical nativism from a multidisciplinary perspective, and find each to be at odds with evidence from anthropology and developmental science. In particular, the history of number reveals characteristics that are inconsistent with biological determinism of numerical concepts, including a lack of number systems across some human groups and remarkable variability in the form of numerical systems that do emerge. Instead, this literature highlights the importance of economic and social factors in constructing fundamentally new cognitive systems to achieve culturally specific goals. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Easy or difficult? Children's understanding of how supply and demand affect goal completion

Three experiments examined children's understanding of how supply and demand affect the difficulty of completing goals. Participants were 368 predominantly White Canadians (52% female, 48% male) tested in 2017-2022. In Experiment 1, 3-year-olds recognized that obtaining resources is easier where supply exceeds demand than where demand exceeds supply. However, in Experiment 2, 3-year-olds were insensitive to supply and demand when comparing situations where demand exceeded supply to a greater or lesser degree. Finally, Experiment 3 revealed a developmental lag in 3- to 7-year-olds' understanding of how supply and demand affects goal completion: Children succeeded when contrasting a surplus and a shortage of supply relative to demand at 4;2. But they only succeeded when contrasting degrees of greater supply than demand at 5;10.

Children's attention to numerical quantities relates to verbal number knowledge: An introduction to the Build-A-Train task

Which dimension of a set of objects is more salient to young children: number or size? The 'Build-A-Train' task was developed and used to examine whether children spontaneously use a number or physical size approach on an un-cued matching task. In the Build-A-Train task, an experimenter assembles a train using one to five blocks of a particular length and asks the child to build the same train. The child's blocks differ in length from the experimenter's blocks, causing the child to build a train that matches based on either the number of blocks or length of the train, as it is not possible to match on both. One hundred and nineteen children between 2 years 2 months and 6 years 0 months of age (M = 4.05, SD = 0.84) completed the Build-A-Train task, and the Give-a-Number task, a classic task used to assess children's conceptual knowledge of verbal number words. Across train lengths and verbal number knowledge levels, children used a number approach more than a size approach on the Build-A-Train task. However, children were especially likely to use a number approach over a size approach when they knew the verbal number word that corresponded to the quantity of blocks in the train, particularly for quantities smaller than four. Therefore, children's attention to number relates to their knowledge of verbal number words. The Build-A-Train task and findings from the current study set a foundation for future longitudinal research to investigate the causal relationship between children's acquisition of symbolic mathematical concepts and attention to number.

Counting and the ontogenetic origins of exact equality

Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent large exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a "set-matching" task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children's ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number.

Testing the role of symbols in preschool numeracy: An experimental computer-based intervention study

Numeracy is of critical importance for scholastic success and modern-day living, but the precise mechanisms that drive its development are poorly understood. Here we used novel experimental training methods to begin to investigate the role of symbols in the development of numeracy in preschool-aged children. We assigned pre-school children in the U.S. and Italy (N = 215; Mean age = 49.15 months) to play one of five versions of a computer-based numerical comparison game for two weeks. The different versions of the game were equated on basic features of gameplay and demands but systematically varied in numerical content. Critically, some versions included non-symbolic numerical comparisons only, while others combined non-symbolic numerical comparison with symbolic aids of various types. Before and after training we assessed four components of early numeracy: counting proficiency, non-symbolic numerical comparison, one-to-one correspondence, and arithmetic set transformation. We found that overall children showed improvement in most of these components after completing these short trainings. However, children trained on numerical comparisons with symbolic aids made larger gains on assessments of one-to-one correspondence and arithmetic transformation compared to children whose training involved non-symbolic numerical comparison only. Further exploratory analyses suggested that, although there were no major differences between children trained with verbal symbols (e.g., verbal counting) and non-verbal visuo-spatial symbols (i.e., abacus counting), the gains in one-to-one correspondence may have been driven by abacus training, while the gains in non-verbal arithmetic transformations may have been driven by verbal training. These results provide initial evidence that the introduction of symbols may contribute to the emergence of numeracy by enhancing the capacity for thinking about exact equality and the numerical effects of set transformations. More broadly, this study provides an empirical basis to motivate further focused study of the processes by which children’s mastery of symbols influences children’s developing mastery of numeracy.

What we count dictates how we count: A tale of two encodings

We argue that what we count has a crucial impact on how we count, to the extent that even adults may have difficulty using elementary mathematical notions in concrete situations. Specifically, we investigate how the use of certain types of quantities (durations, heights, number of floors) may emphasize the ordinality of the numbers featured in a problem, whereas other quantities (collections, weights, prices) may emphasize the cardinality of the depicted numerical situations. We suggest that this distinction leads to the construction of one of two possible encodings, either a cardinal or an ordinal representation. This difference should, in turn, constrain the way we approach problems, influencing our mathematical reasoning in multiple activities. This hypothesis is tested in six experiments (N = 916), using different versions of multiple-strategy arithmetic word problems. We show that the distinction between cardinal and ordinal quantities predicts problem sorting (Experiment 1), perception of similarity between problems (Experiment 2), direct problem comparison (Experiment 3), choice of a solving algorithm (Experiment 4), problem solvability estimation (Experiment 5) and solution validity assessment (Experiment 6). The results provide converging clues shedding light into the fundamental importance of the cardinal versus ordinal distinction on adults' reasoning about numerical situations. Overall, we report multiple evidence that general, non-mathematical knowledge associated with the use of different quantities shapes adults' encoding, recoding and solving of mathematical word problems. The implications regarding mathematical cognition and theories of arithmetic problem solving are discussed.

The Challenge of Modeling the Acquisition of Mathematical Concepts

As a full-blown research topic, numerical cognition is investigated by a variety of disciplines including cognitive science, developmental and educational psychology, linguistics, anthropology and, more recently, biology and neuroscience. However, despite the great progress achieved by such a broad and diversified scientific inquiry, we are still lacking a comprehensive theory that could explain how numerical concepts are learned by the human brain. In this perspective, I argue that computer simulation should have a primary role in filling this gap because it allows identifying the finer-grained computational mechanisms underlying complex behavior and cognition. Modeling efforts will be most effective if carried out at cross-disciplinary intersections, as attested by the recent success in simulating human cognition using techniques developed in the fields of artificial intelligence and machine learning. In this respect, deep learning models have provided valuable insights into our most basic quantification abilities, showing how numerosity perception could emerge in multi-layered neural networks that learn the statistical structure of their visual environment. Nevertheless, this modeling approach has not yet scaled to more sophisticated cognitive skills that are foundational to higher-level mathematical thinking, such as those involving the use of symbolic numbers and arithmetic principles. I will discuss promising directions to push deep learning into this uncharted territory. If successful, such endeavor would allow simulating the acquisition of numerical concepts in its full complexity, guiding empirical investigation on the richest soil and possibly offering far-reaching implications for educational practice.

One-to-one correspondence without language

A logical rule important in counting and representing exact number is one-to-one correspondence, the understanding that two sets are equal if each item in one set corresponds to exactly one item in the second set. The role of this rule in children's development of counting remains unclear, possibly due to individual differences in the development of language. We report that non-human primates, which do not have language, have at least a partial understanding of this principle. Baboons were given a quantity discrimination task where two caches were baited with different quantities of food. When the quantities were baited in a manner that highlighted the one-to-one relation between those quantities, baboons performed significantly better than when one-to-one correspondence cues were not provided. The implication is that one-to-one correspondence, which requires intuitions about equality and is a possible building block of counting, has a pre-linguistic origin.

Cross-modal attention modulates tactile subitizing but not tactile numerosity estimation

Debate remains about whether the same attentional mechanism subserves subitizing (with number of items less than or equal to 4) and numerosity estimation (with number of items equal to or larger than 5), and evidence is scarce from the tactile modality. Here, we examined tactile numerosity perception. Using tactile Braille displays, participants completed the following three main tasks: (1) Unisensory task with focused attention: Participants reported the number (1~12) of the tactile pins. (2) Unisensory task with divided attention: Participants compared the numbers of pins across the upper and lower area of their left index fingers, in addition to reporting the number of tactile pins on their right index fingers. (3) Cross-modal task with divided attention: Participants reported the number of tactile pins and compared the numbers of visual dots across the upper and lower part of a (illusory) rectangle that overlaid the tactile stimuli. We found that performance of subitizing rather than estimation was interfered with in dual tasks, regardless of whether distractor events were from the same modality (tactile modality) or from a different modality (visual modality). Moreover, a further test of visual/tactile working memory capacity revealed that the precision of tactile subitizing, in the presence of a visual distractor, was correlated with the capacity of visual working memory, not of tactile working memory. Overall, our study revealed that tactile numerosity perception is accounted for by amodal attentional modulation yet by differential attentional mechanisms in terms of subitizing and estimation.

Contrast and entailment: Abstract logical relations constrain how 2- and 3-year-old children interpret unknown numbers

Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations - contrast and entailment - to reason about the meanings of 'unknown' number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of contrasting alternatives, children assign them a meaning distinct from some. In the "Don't Give-a-Number task", children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast - plus knowledge of a number's membership in a count list - enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children's interpretation of unknown numbers is further constrained by understanding numerical entailment relations - that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number in-between - who either has or doesn't have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold the exact meanings of unknown number words.

Explaining early moral hypocrisy: Numerical cognition promotes equal sharing behavior in preschool-aged children

Recent work has documented that despite preschool-aged children's understanding of social norms surrounding sharing, they fail to share their resources equally in many contexts. Here we explored two hypotheses for this failure: an insufficient motivation hypothesis and an insufficient cognitive resources hypothesis. With respect to the latter, we specifically explored whether children's numerical cognition-their understanding of the cardinal principle-might underpin their abilities to share equally. In Experiment 1, preschoolers' numerical cognition fully mediated age-related changes in children's fair sharing. We found little support for the insufficient motivation hypothesis-children stated that they had shared fairly, and failures in sharing fairly were a reflection of their number knowledge. Numerical cognition did not relate to children's knowledge of the norms of equality (Experiment 2). Results suggest that the knowledge-behavior gap in fairness may be partly explained by the differences in cognitive skills required for conceptual and behavioral equality.

Cognitive Development Is a Reconstruction Process that May Follow Different Pathways: The Case of Number

Some cognitive functions shared by humans and certain animals were acquired early in the course of phylogeny and, in humans, are operational in their primitive form shortly after birth. This is the case for the quantification of discrete objects. The further phylogenetic evolution of the human brain allows such functions to be reconstructed in a much more sophisticated way during child development. Certain functional characteristics of the brain (plasticity, multiple cognitive processes involved in the same response, interactions, and substitution relationships between those processes) provide degrees of freedom that open up the possibility of different pathways of reconstruction. The within- and between-individual variability of these developmental pathways offers an original window on the dynamics of development. Here, I will illustrate this theoretical approach to cognitive development-which can be called "reconstructivist" and "pluralistic"-using children's construction of number as an example.

A strategy to improve arithmetical performance in four day-old domestic chicks (Gallus gallus)

A large body of literature shows that non-human animals master numerical discriminations, but a limit has been reported in a variety of species in the comparison 3vs.4. Little is known regarding the possibility of using “cognitive strategies” to enable this discrimination. The aims of this study were to investigate: whether domestic chicks discriminated 3vs.4, and if changes in stimuli presentation could improve chicks’ numerical performance. Newly hatched chicks were reared with seven identical objects. On day 4, they underwent 20 consecutive testing trials to assess their capability to discriminate 3vs.4. The objects were presented, one-by-one, to the chicks and hidden behind one of two identical panels. As expected, the chicks did not discriminate (Experiment 1). When objects were presented and hidden in groups comprising one or two objects (2 + 1)vs.(2 + 2), the chicks succeeded (Experiment 2). The grouping strategy did not help in the case of a harder discrimination of (3 + 1)vs.(3 + 2) (Experiment 3), unless chicks were allowed to rest for two hours between testing sessions (Experiment 4). Our results suggest that in some cases, the limits reported for numerical performance in animals do not depend on cognitive limitations but on attentional or motivational factors, which can be overcome employing simple procedural adjustments.

Configured-groups hypothesis: fast comparison of exact large quantities without counting

Our innate number sense cannot distinguish between two large exact numbers of objects (e.g., 45 dots vs 46). Configured groups (e.g., 10 blocks, 20 frames) are traditionally used in schools to represent large numbers. Previous studies suggest that these external representations make it easier to use symbolic strategies such as counting ten by ten, enabling humans to differentiate exactly two large numbers. The main hypothesis of this work is that configured groups also allow for a differentiation of large exact numbers, even when symbolic strategies become ineffective. In experiment 1, the children from grade 3 were asked to compare two large collections of objects for 5 s. When the objects were organized in configured groups, the success rate was over .90. Without this configured grouping, the children were unable to make a successful comparison. Experiments 2 and 3 controlled for a strategy based on non-numerical parameters (areas delimited by dots or the sum areas of dots, etc.) or use symbolic strategies. These results suggest that configured grouping enables humans to distinguish between two large exact numbers of objects, even when innate number sense and symbolic strategies are ineffective. These results are consistent with what we call "the configured group hypothesis": configured groups play a fundamental role in the acquisition of exact numerical abilities.

Language, procedures, and the non-perceptual origin of number word meanings

Perceptual representations of objects and approximate magnitudes are often invoked as building blocks that children combine to acquire the positive integers. Systems of numerical perception are either assumed to contain the logical foundations of arithmetic innately, or to supply the basis for their induction. I propose an alternative to this framework, and argue that the integers are not learned from perceptual systems, but arise to explain perception. Using cross-linguistic and developmental data, I show that small (~1-4) and large (~5+) numbers arise both historically and in individual children via distinct mechanisms, constituting independent learning problems, neither of which begins with perceptual building blocks. Children first learn small numbers using the same logic that supports other linguistic number marking (e.g. singular/plural). Years later, they infer the logic of counting from the relations between large number words and their roles in blind counting procedures, only incidentally associating number words with approximate magnitudes.

Primitive Concepts of Number and the Developing Human Brain

Counting is an evolutionarily recent cultural invention of the human species. In order for humans to have conceived of counting in the first place, certain representational and logical abilities must have already been in place. The focus of this review is the origins and nature of those fundamental mechanisms that promoted the emergence of the human number concept. Five claims are presented that support an evolutionary view of numerical development: 1) number is an abstract concept with an innate basis in humans, 2) maturational processes constrain the development of humans' numerical representations between infancy and adulthood, 3) there is evolutionary continuity in the neural processes of numerical cognition in primates, 4) primitive logical abilities support verbal counting development in humans, and 5) primitive neural processes provide the foundation for symbolic numerical development in the human brain. We support these claims by examining current evidence from animal cognition, child development, and human brain function. The data show that at the basis of human numerical concepts are primitive perceptual and logical mechanisms that have evolutionary homologs in other primates and form the basis of numerical development in the human brain. In the final section of the review, we discuss some hypotheses for what makes human numerical reasoning unique by drawing on evidence from human and non-human primate neuroimaging research.

Native Amazonian children forego egalitarianism in merit-based tasks when they learn to count

Cooperation often results in a final material resource that must be shared, but deciding how to distribute that resource is not straightforward. A distribution could count as fair if all members receive an equal reward (egalitarian distributions), or if each member's reward is proportional to their merit (merit-based distributions). Here, we propose that the acquisition of numerical concepts influences how we reason about fairness. We explore this possibility in the Tsimane', a farming-foraging group who live in the Bolivian rainforest. The Tsimane' learn to count in the same way children from industrialized countries do, but at a delayed and more variable timeline, allowing us to de-confound number knowledge from age and years in school. We find that Tsimane' children who can count produce merit-based distributions, while children who cannot count produce both merit-based and egalitarian distributions. Our findings establish that the ability to count - a non-universal, language-dependent, cultural invention - can influence social cognition.

The relationship between non-verbal systems of number and counting development: a neural signatures approach

Two non-verbal cognitive systems, an approximate number system (ANS) for extracting the numerosity of a set and a parallel individuation (PI) system for distinguishing between individual items, are hypothesized to be foundational to symbolic number and mathematics abilities. However, the exact role of each remains unclear and highly debated. Here we used an individual differences approach to test for a relationship between the spontaneously evoked brain signatures (using event-related potentials) of PI and the ANS and initial development of symbolic number concepts in preschool children as displayed by counting. We observed that individual differences in the neural signatures of the PI system, but not the ANS, explained a unique portion of variance in counting proficiency after extensively controlling for general cognitive factors. These results suggest that differences in early attentional processing of objects between children are related to higher-level symbolic number concept development.

Mastery of the logic of natural numbers is not the result of mastery of counting: evidence from late counters

To master the natural number system, children must understand both the concepts that number words capture and the counting procedure by which they are applied. These two types of knowledge develop in childhood, but their connection is poorly understood. Here we explore the relationship between the mastery of counting and the mastery of exact numerical equality (one central aspect of natural number) in the Tsimane', a farming-foraging group whose children master counting at a delayed age and with higher variability than do children in industrialized societies. By taking advantage of this variation, we can better understand how counting and exact equality relate to each other, while controlling for age and education. We find that the Tsimane' come to understand exact equality at later and variable ages. This understanding correlates with their mastery of number words and counting, controlling for age and education. However, some children who have mastered counting lack an understanding of exact equality, and some children who have not mastered counting have achieved this understanding. These results suggest that understanding of counting and of natural number concepts are at least partially distinct achievements, and that both draw on inputs and resources whose distribution and availability differ across cultures.

On the relation between grammatical number and cardinal numbers in development

This mini-review focuses on the question of how the grammatical number system of a child’s language may help the child learn the meanings of cardinal number words (e.g., “one” and “two”). Evidence from young children learning English, Russian, Japanese, Mandarin, Slovenian, or Saudi Arabic suggests that trajectories of number-word learning differ for children learning different languages. Children learning English, which distinguishes between singular and plural, seem to learn the meaning of the cardinal number “one” earlier than children learning Japanese or Mandarin, which have very little singular/plural marking. Similarly, children whose languages have a singular/dual/plural system (Slovenian and Saudi Arabic) learn the meaning of “two” earlier than English-speaking children. This relation between grammatical and cardinal number may shed light on how humans acquire cardinal-number concepts. There is an ongoing debate about whether mental symbols for small cardinalities (concepts for “oneness,” “twoness,” etc.) are innate or learned. Although an effect of grammatical number on number-word learning does not rule out nativist accounts, it seems more consistent with constructivist accounts, which portray the number-learning process as one that requires significant conceptual change.