Exact Equality and Successor Function: Two Key Concepts on the Path towards understanding Exact Numbers

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.

Aprendizaje del conteo y los números naturales en preescolar: una revisión sistemática de la literatura

Aprender a contar cantidades discretas de forma exacta constituye uno de los primeros hitos del desarrollo del conocimiento matemático infantil. En los últimos años, ha habido un extenso debate en torno a cómo ocurre este proceso de aprendizaje en preescolar. La actual investigación tuvo como objetivo conocer las temáticas y preguntas de investigación generales desarrolladas en los últimos cinco años en cuanto al aprendizaje del conteo y los números naturales en preescolar. Para ello, se realizó una revisión sistemática en la que se hizo una indagación en las bases de datos ScienceDirect, EBSCO, Web of Science, SpringerLink, JSTOR y Sage. Se obtuvieron 98 artículos de investigación que fueron examinados mediante análisis de conglomerados y mapas jerárquicos a través de NVIVO 11.0.  Se encontraron cuatro núcleos temáticos (Ideas sobre los procesos cognitivos implicados en la comprensión del número, Representación de magnitudes numéricas, Intervenciones para favorecer el desarrollo de habilidades matemáticas y Aspectos estructurales del número), que muestran el panorama actual de investigación sobre aprendizaje del conteo. Los resultados de este estudio son importantes para delimitar posibles programas futuros de investigación, y pueden ser usados por docentes como insumo para enriquecer los ambientes de aprendizaje de sus aulas de clase.

A language of thought for the mental representation of geometric shapes

In various cultures and at all spatial scales, humans produce a rich complexity of geometric shapes such as lines, circles or spirals. Here, we propose that humans possess a language of thought for geometric shapes that can produce line drawings as recursive combinations of a minimal set of geometric primitives. We present a programming language, similar to Logo, that combines discrete numbers and continuous integration to form higher-level structures based on repetition, concatenation and embedding, and we show that the simplest programs in this language generate the fundamental geometric shapes observed in human cultures. On the perceptual side, we propose that shape perception in humans involves searching for the shortest program that correctly draws the image (program induction). A consequence of this framework is that the mental difficulty of remembering a shape should depend on its minimum description length (MDL) in the proposed language. In two experiments, we show that encoding and processing of geometric shapes is well predicted by MDL. Furthermore, our hypotheses predict additive laws for the psychological complexity of repeated, concatenated or embedded shapes, which we confirm experimentally.

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).

Verbal counting and the timing of number acquisition in an indigenous Amazonian group

Children in industrialized cultures typically succeed on Give-N, a test of counting ability, by age 4. On the other hand, counting appears to be learned much later in the Tsimane’, an indigenous group in the Bolivian Amazon. This study tests three hypotheses for what may cause this difference in timing: (a) Tsimane’ children may be shy in providing behavioral responses to number tasks, (b) Tsimane’ children may not memorize the verbal list of number words early in acquisition, and/or (c) home environments may not support mathematical learning in the same way as in US samples, leading Tsimane’ children to primarily acquire mathematics through formalized schooling. Our results suggest that most of our subjects are not inhibited by shyness in responding to experimental tasks. We also find that Tsimane’ children (N = 100, ages 4-11) learn the verbal list later than US children, but even upon acquiring this list, still take time to pass Give-N tasks. We find that performance in counting varies across tasks and is related to formal schooling. These results highlight the importance of formal education, including instruction in the count list, in learning the meanings of the number words.

Environmental influences on mathematics performance in early childhood

Math skills relate to lifelong career, health, and financial outcomes. Individuals' own cognitive abilities predict math performance and there is growing recognition that environmental influences including differences in culture and variability in math engagement also impact math skills. In this Review, we summarize evidence indicating that differences between languages, exposure to math-focused language, socioeconomic status, attitudes and beliefs about math, and engagement with math activities influence young children's math performance. These influences play out at the community and individual level. However, research on the role of these environmental influences for foundational number skills, including understanding of number words, is limited. Future research is needed to understand individual differences in the development of early emerging math skills such as number word skills, examining to what extent different types of environmental input are necessary and how children's cognitive abilities shape the impact of environmental input.

Children's understanding of the abstract logic of counting

When children learn to count, do they understand its logic independent of the number list that they learned to count with? Here we tested CP-knowers' (ages three to five) understanding of how counting reveals a set's cardinality, even when non-numerical lists are used to count. Participants watched an agent count unobservable objects in two boxes and were asked to identify the larger set. Participants successfully identified the box with more objects when the agent counted using their familiar number list (Experiment 1) and when the agent counted using a non-numeric ordered list, as long as the items in the list were not linguistically used as number words (Experiments 2-3). Additionally, children's performance was strongly influenced by visual cues that helped them link the list's order to representations of magnitude (Experiment 4). Our findings suggest that three- to six-year-olds who can count also understand how counting reveals a set's cardinality, but they require additional time to understand how symbols on any arbitrary ordered list can be used as numerals.

What Counts? Sources of Knowledge in Children's Acquisition of the Successor Function

Although many U.S. children can count sets by 4 years, it is not until 5½-6 years that they understand how counting relates to number-that is, that adding 1 to a set necessitates counting up one number. This study examined two knowledge sources that 3½- to 6-year-olds (N = 136) may leverage to acquire this "successor function": (a) mastery of productive rules governing count list generation; and (b) training with "+1" math facts. Both productive counting and "+1" math facts were related to understanding that adding 1 to sets entails counting up one number in the count list; however, even children with robust successor knowledge struggled with its arithmetic expression, suggesting they do not generalize the successor function from "+1" math facts.

The impact of set size on cumulative area judgments

The ability to track number has long been considered more difficult than tracking continuous quantities. Evidence for this claim comes from work revealing that continuous properties (specifically cumulative area) influence numerical judgments, such that adults perform worse on numerical tasks when cumulative area is incongruent with number. If true, then continuous extent tracking abilities should be unimpeded by number. The aim of the present study was to determine the precision with which adults track cumulative area and to uncover the process by which they do so. Across two experiments, we presented adults with arrays of dots and asked them to judge the relative cumulative area of the displays. Participants performed worse and were slower on incongruent trials, in which the more numerous array had the smaller cumulative area. These findings suggest that number interferes with continuous quantity judgments, and that number is at least as salient as continuous variables, undermining claims in the literature that continuous properties are easier to represent, and more salient to adults. Our primary research question, however, pertained to how cumulative area representations were impacted by set size. Results revealed that the area of a single item was tracked much faster and with greater precision than the area of multiple items. However, for sets with more than one item, results revealed less accurate, yet faster responses, as set size increased, suggesting a speed-accuracy trade-off in judgments of cumulative area. Results are discussed in the context of two distinct theories regarding the process of tracking cumulative area.

A Study on Congruency Effects and Numerical Distance in Fraction Comparison by Expert Undergraduate Students

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school, but many students fail to master them, and misconceptions frequently persist into adulthood. In this study, we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean engineering undergraduate students answered a computerized fraction comparison task, while their answers and response times were recorded. Task items were selected according to a number of mathematically and/or cognitively relevant characteristics: (a) whether the fractions to be compared shared a common component, (b) the numerical distance between fractions, and (c) the applicability of two strategies to answer successfully: a congruency strategy (a fraction is larger if it has larger natural number components than another) and gap thinking (a fraction is larger if it is missing fewer pieces than another to complete the whole). In line with previous research, our data indicated that the congruency strategy is inadequate to describe participants’ performance, as congruent items turned out to be more difficult than incongruent ones when fractions had no common component. Although we hypothesized that this lower performance for congruent items would be explained by the use of gap thinking, this turned out not to be the case: evidence was insufficient to show that the applicability of the gap thinking strategy modulated either participants’ accuracy rates or response times (although individual-level data suggest that there is an effect for response times). When fractions shared a common component, instead, our data display a more complex pattern that expected: an advantage for congruent items is present in the first experimental block but fades as the experiment progresses. Numerical distance had an effect in fraction comparison that was statistically significant for items without common components only. Altogether, our results from experts’ reasoning reveal nuances in the fraction comparison task with respect to previous studies and contribute to future models of reasoning in this task.

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.

When one-two-three beats two-one-three: Tracking the acquisition of the verbal number sequence

Learning how to count is a crucial step in cognitive development, which progressively allows for more elaborate numerical processing. The existing body of research consistently reports how children associate the verbal code with exact quantity. However, the early acquisition of this code, when the verbal numbers are encoded in long-term memory as a sequence of words, has rarely been examined. Using an incidental assessment method based on serial recall of number words presented in ordered versus non-ordered sequences (e.g., one-two-three vs. two-one-three), we tracked the progressive acquisition of the verbal number sequence in children aged 3-6 years. Results revealed evidence for verbal number sequence knowledge in the youngest children even before counting is fully mastered. Verbal numerical knowledge thus starts to be organized as a sequence in long-term memory already at the age of 3 years, and this numerical sequence knowledge is assessed in a sensitive manner by incidental rather than explicit measures of number knowledge.

Symbol grounding of number words in the subitization range

In three experiments, we explored whether number words are grounded in a nonsymbolic representation of numerosity. We used a sentence-picture verification task, where participants are required to check whether the concept given in a sentence corresponds to the subsequently presented object. We concurrently manipulated numerical congruency by orthogonally varying the number word attached to the concept and the quantity of objects. The number words and numerosities varied from one to four in Experiment 1 and from six to nine in Experiment 2. In Experiment 3, we employed number words six and eight with the constraint that, in the incongruent condition, a constant number-to-numerosity ratio of 2:1 was used. In Experiment 1, we found that participants were faster and more efficient when concept-object matches were accompanied by numerical congruency relative to incongruency. On the other hand, no such difference was observed in Experiments 2 and 3 for numbers falling outside of the subitization range. The results are consistent with the hypothesis that number words from one to four are grounded in a nonsymbolic representation of the size of small sets.

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.

The Enculturated Move From Proto-Arithmetic to Arithmetic

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation.

The statistical shape of geometric reasoning

Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to perfect, idealized entities, whereas its use in the physical world relies on dynamic perception of objects. How do abstract Euclidean concepts, dynamics, and statistics come together to support our intuitive geometric reasoning? Here, we address this question using a simple geometric task – planar triangle completion. An analysis of the distribution of participants’ errors in localizing a fragmented triangle’s missing corner reveals scale-dependent deviations from a deterministic Euclidean representation of planar triangles. By considering the statistical physics of the process characterized via a correlated random walk with a natural length scale, we explain these results and further predict participants’ estimates of the missing angle, measured in a second task. Our model also predicts the results of a categorical reasoning task about changes in the triangle size and shape even when such completion strategies need not be invoked. Taken together, our findings suggest a critical role for noisy physical processes in our reasoning about elementary Euclidean geometry.

The Use of Concrete Experiences in Early Childhood Mathematics Instruction

Addressed are four key issues regarding concrete instruction: What is concrete? What is a worthwhile concrete experience? How can concrete experiences be used effectively in early childhood mathematics instruction? Is there evidence such experiences work? I argue that concrete experiences are those that build on what is familiar to a child and can involve objects, verbal analogies, or virtual images. The use of manipulatives or computer games, for instance, does not in itself guarantee an educational experience. Such experiences are worthwhile if they target and further learning (e.g., help children extend their informal knowledge or use their informal knowledge to understand and learn formal knowledge). A crucial guideline for the effective use of concrete experience is Dewey's principle of interaction-external factors (e.g., instructional activities) need to mesh with internal factors (readiness, interest). Cognitive views of concrete materials, such as the cognitive alignment perspective and dual-representation hypothesis, provide useful guidance about external factors but do not adequately take into account internal factors and their interaction with external factors. Research on the effectiveness of concrete experience is inconclusive because it frequently overlooks internal factors.

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.

Representation of different exact numbers of prey by a spider-eating predator

Our objective was to use expectancy-violation methods for determining whether Portia africana, a salticid spider that specializes in eating other spiders, is proficient at representing exact numbers of prey. In our experiments, we relied on this predator's known capacity to gain access to prey by following pre-planned detours. After Portia first viewed a scene consisting of a particular number of prey items, it could then take a detour during which the scene went out of view. Upon reaching a tower at the end of the detour, Portia could again view a scene, but now the number of prey items might be different. We found that, compared with control trials in which the number was the same as before, Portia's behaviour was significantly different in most instances when we made the following changes in number: 1 versus 2, 1 versus 3, 1 versus 4, 2 versus 3, 2 versus 4 or 2 versus 6. These effects were independent of whether the larger number was seen first or second. No significant effects were evident when the number of prey changed between 3 versus 4 or 3 versus 6. When we changed prey size and arrangement while keeping prey number constant, no significant effects were detected. Our findings suggest that Portia represents 1 and 2 as discrete number categories, but categorizes 3 or more as a single category that we call 'many'.

The Faculty of Language Integrates the Two Core Systems of Number

Only humans possess the faculty of language that allows an infinite array of hierarchically structured expressions (Hauser et al., 2002; Berwick and Chomsky, 2015). Similarly, humans have a capacity for infinite natural numbers, while all other species seem to lack such a capacity (Gelman and Gallistel, 1978; Dehaene, 1997). Thus, the origin of this numerical capacity and its relation to language have been of much interdisciplinary interest in developmental and behavioral psychology, cognitive neuroscience, and linguistics (Dehaene, 1997; Hauser et al., 2002; Pica et al., 2004). Hauser et al. (2002) and Chomsky (2008) hypothesize that a recursive generative operation that is central to the computational system of language (called Merge) can give rise to the successor function in a set-theoretic fashion, from which capacities for discretely infinite natural numbers may be derived. However, a careful look at two domains in language, grammatical number and numerals, reveals no trace of the successor function. Following behavioral and neuropsychological evidence that there are two core systems of number cognition innately available, a core system of representation of large, approximate numerical magnitudes and a core system of precise representation of distinct small numbers (Feigenson et al., 2004), I argue that grammatical number reflects the core system of precise representation of distinct small numbers alone. In contrast, numeral systems arise from integrating the pre-existing two core systems of number and the human language faculty. To the extent that my arguments are correct, linguistic representations of number, grammatical number, and numerals do not incorporate anything like the successor function.

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.

Linguistic explanation and domain specialization: a case study in bound variable anaphora

The core question behind this Frontiers research topic is whether explaining linguistic phenomena requires appeal to properties of human cognition that are specialized to language. We argue here that investigating this issue requires taking linguistic research results seriously, and evaluating these for domain-specificity. We present a particular empirical phenomenon, bound variable interpretations of pronouns dependent on a quantifier phrase, and argue for a particular theory of this empirical domain that is couched at a level of theoretical depth which allows its principles to be evaluated for domain-specialization. We argue that the relevant principles are specialized when they apply in the domain of language, even if analogs of them are plausibly at work elsewhere in cognition or the natural world more generally. So certain principles may be specialized to language, though not, ultimately, unique to it. Such specialization is underpinned by ultimately biological factors, hence part of UG.

Representations of numerical sequences and the concept of middle in preschoolers

The present study concerns preschoolers' understanding of the middle concept as it applies to numerical sequences. Previous research using implicit psychophysical assessment suggests that the numerical midpoint is embedded within numerical representations by 4 years of age. Here, we examined 3- to 5-year-olds' ability to identify the midpoint value in triplets of non-symbolic numbers when explicitly probed to do so. We found that whereas 4- and 5-year-olds were capable of explicit access to numerical midpoint values and showed ratio-dependent performance, a signature of the approximate number system (ANS), 3-year-olds performed at chance. Children's difficulty in identifying numerical midpoint values was not due to comparing multiple arrays, nor was it entirely due to a spatial association with the word "middle" used in the task. We speculate that explicit access to numerical midpoint values may be jointly supported by endogenous control of attentional mechanisms and the development of a mental number line.

The origins of counting algorithms

Humans' ability to count by verbally labeling discrete quantities is unique in animal cognition. The evolutionary origins of counting algorithms are not understood. We report that nonhuman primates exhibit a cognitive ability that is algorithmically and logically similar to human counting. Monkeys were given the task of choosing between two food caches. First, they saw one cache baited with some number of food items, one item at a time. Then, a second cache was baited with food items, one at a time. At the point when the second set was approximately equal to the first set, the monkeys spontaneously moved to choose the second set even before that cache was completely baited. Using a novel Bayesian analysis, we show that the monkeys used an approximate counting algorithm for comparing quantities in sequence that is incremental, iterative, and condition controlled. This proto-counting algorithm is structurally similar to formal counting in humans and thus may have been an important evolutionary precursor to human counting.

Core knowledge and the emergence of symbols: The case of maps

Map reading is unique to humans but present in people of diverse cultures, at ages as young as 4 years. Here we explore the nature and sources of this ability, asking both what geometric information young children use in maps and what non-symbolic systems are associated with their map-reading performance. Four-year-old children were given two tests of map-based navigation (placing an object within a small 3D surface layout at a position indicated on a 2D map), one focused on distance relations and the other on angle relations. Children also were given two non-symbolic tasks, testing their use of geometry for navigation (a reorientation task) and for visual form analysis (a deviant-detection task). Although children successfully performed both map tasks, their performance on the two map tasks was uncorrelated, providing evidence for distinct abilities to represent distance and angle on 2D maps of 3D surface layouts. In contrast, performance on each map task was associated with performance on one of the two non-symbolic tasks: map-based navigation by distance correlated with sensitivity to the shape of the environment in the reorientation task, whereas map-based navigation by angle correlated with sensitivity to the shapes of 2D forms and patterns in the deviant detection task. These findings suggest links between one uniquely human, emerging symbolic ability, geometric map use, and two core systems of geometry.

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.

Pre-school children’s bridge to symbolic knowledge: first literature framework for a learning and cognition lab at a South African university

The authors present the theoretical groundwork for a research project on learning and cognitive development of number concepts in the early years of childhood. Giving a background sketch of the genesis of a learning and cognition lab at a university in the metropolitan heartland of South Africa, they present their initial literature framework for inquiries into children’s symbolic learning of number in the pre-school years. They argue that conceptual development of young children is a neglected area in childhood cognition research in South Africa. The study of some of the literature for the first project of the new lab is then introduced with a view of identifying a few of the main components of a conceptual framework for what will become a multiple-year study. The authors propose that this literature can serve as foundation for examining a linguistically diverse group of children’s responses on experimental tasks and in clinical interviews in four or more languages. The designs of these inquiries are imminent. They suggest that the views of leading authors such as Elizabeth Spelke, Susan Carey, and Stanislas Dehaene can shed much light on their understanding of early number concept development of South African children.

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

Exact integer concepts are fundamental to a wide array of human activities, but their origins are obscure. Some have proposed that children are endowed with a system of natural number concepts, whereas others have argued that children construct these concepts by mastering verbal counting or other numeric symbols. This debate remains unresolved, because it is difficult to test children's mastery of the logic of integer concepts without using symbols to enumerate large sets, and the symbols themselves could be a source of difficulty for children. Here, we introduce a new method, focusing on large quantities and avoiding the use of words or other symbols for numbers, to study children's understanding of an essential property underlying integer concepts: the relation of exact numerical equality. Children aged 32-36 months, who possessed no symbols for exact numbers beyond 4, were given one-to-one correspondence cues to help them track a set of puppets, and their enumeration of the set was assessed by a non-verbal manual search task. Children used one-to-one correspondence relations to reconstruct exact quantities in sets of 5 or 6 objects, as long as the elements forming the sets remained the same individuals. In contrast, they failed to track exact quantities when one element was added, removed, or substituted for another. These results suggest an alternative to both nativist and symbol-based constructivist theories of the development of natural number concepts: Before learning symbols for exact numbers, children have a partial understanding of the properties of exact numbers.

Why do we differ in number sense? Evidence from a genetically sensitive investigation

Basic intellectual abilities of quantity and numerosity estimation have been detected across animal species. Such abilities are referred to as 'number sense'. For human species, individual differences in number sense are detectable early in life, persist in later development, and relate to general intelligence. The origins of these individual differences are unknown. To address this question, we conducted the first large-scale genetically sensitive investigation of number sense, assessing numerosity discrimination abilities in 837 pairs of monozygotic and 1422 pairs of dizygotic 16-year-old twin pairs. Univariate genetic analysis of the twin data revealed that number sense is modestly heritable (32%), with individual differences being largely explained by non-shared environmental influences (68%) and no contribution from shared environmental factors. Sex-Limitation model fitting revealed no differences between males and females in the etiology of individual differences in number sense abilities. We also carried out Genome-wide Complex Trait Analysis (GCTA) that estimates the population variance explained by additive effects of DNA differences among unrelated individuals. For 1118 unrelated individuals in our sample with genotyping information on 1.7 million DNA markers, GCTA estimated zero heritability for number sense, unlike other cognitive abilities in the same twin study where the GCTA heritability estimates were about 25%. The low heritability of number sense, observed in this study, is consistent with the directional selection explanation whereby additive genetic variance for evolutionary important traits is reduced.

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.

A number of options: rationalist, constructivist, and Bayesian insights into the development of exact-number concepts

The question of how human beings acquire exact-number concepts has interested cognitive developmentalists since the time of Piaget. The answer will owe something to both the rationalist and constructivist traditions. On the one hand, some aspects of numerical cognition (e.g. approximate number estimation and the ability to track small sets of one to four individuals) are innate or early-developing and are shared widely among species. On the other hand, only humans create representations of exact, large numbers such as 42, as distinct from both 41 and 43. These representations seem to be constructed slowly, over a period of months or years during early childhood. The task for researchers is to distinguish the innate representational resources from those that are constructed, and to characterize the construction process. Bayesian approaches can be useful to this project in at least three ways: (1) As a way to analyze data, which may have distinct advantages over more traditional methods (e.g. making it possible to find support for a nuli hypothesis); (2) as a way of modeling children's performance on specific tasks: Peculiarities of the task are captured as a prior; the child's knowledge is captured in the way the prior is updated; and behavior is captured as a posterior distribution; and (3) as a way of modeling learning itself, by providing a formal account of how learners might choose among alternative hypotheses.

Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.

Flexible intuitions of Euclidean geometry in an Amazonian indigene group

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ~180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics.

Spatial and numerical abilities without a complete natural language

We studied the cognitive abilities of a 13-year-old deaf child, deprived of most linguistic input from late infancy, in a battery of tests designed to reveal the nature of numerical and geometrical abilities in the absence of a full linguistic system. Tests revealed widespread proficiency in basic symbolic and non-symbolic numerical computations involving the use of both exact and approximate numbers. Tests of spatial and geometrical abilities revealed an interesting patchwork of age-typical strengths and localized deficits. In particular, the child performed extremely well on navigation tasks involving geometrical or landmark information presented in isolation, but very poorly on otherwise similar tasks that required the combination of the two types of spatial information. Tests of number- and space-specific language revealed proficiency in the use of number words and deficits in the use of spatial terms. This case suggests that a full linguistic system is not necessary to reap the benefits of linguistic vocabulary on basic numerical tasks. Furthermore, it suggests that language plays an important role in the combination of mental representations of space.

How Humans Count: Numerosity and the Parietal Cortex

Numerosity (the number of objects in a set), like color or movement, is a basic property of the environment. Animal and human brains have been endowed by evolution by mechanisms based on parietal circuitry for representing numerosity in an highly abstract, although approximate fashion. These mechanisms are functional at a very early age in humans and spontaneously deployed in the wild by animals of different species. The recent years have witnessed terrific advances in unveiling the neural code(s) underlying numerosity representations and showing similarities as well as differences across species. In humans, during development, with the introduction of symbols for numbers and the implementation of the counting routines, the parietal system undergoes profound (yet still largely mysterious) modifications, such that the neural machinery previously evolved to represent approximate numerosity gets partially “recycled” to support the representation of exact number.