To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

Neural evidence of core foundations and conceptual change in preschool numeracy

Symbolic numeracy first emerges as children learn the meanings of number words and how to use them to precisely count sets of objects. This development starts before children enter school and forms a foundation for lifelong mathematics achievement. Despite its importance, exactly how children acquire this basic knowledge is unclear. Here we test competing theories of early number learning by measuring event-related brain potentials during a novel number word-quantity comparison task in 3-4-year-old preschool children (N = 128). We find several qualitative differences in neural processing of number by conceptual stage of development. Specifically, we find differences in early attention-related parietal electrophysiology (N1), suggesting that less conceptually advanced children process arrays as individual objects and more advanced children distribute attention over the entire set. Subsequently, we find that only more conceptually advanced children show later-going frontal (N2) sensitivity to the numerical-distance relationship between the number word and visual quantity. The nature of this response suggested that exact rather than approximate numerical meanings were being associated with number words over frontal sites. No evidence of numerical distance effects was observed over posterior scalp sites. Together these results suggest that children may engage parallel individuation of objects to learn the meanings of the first few number words, but, ultimately, create new exact cardinal value representations for number words that cannot be defined in terms of core, nonverbal number systems. More broadly, these results document an interaction between attentional and general cognitive mechanisms in cognitive development. RESEARCH HIGHLIGHTS: Conceptual development in numeracy is associated with a shift in attention from objects to sets. Children acquire meanings of the first few number words through associations with parallel attentional individuation of objects. Understanding of cardinality is associated with attentional processing of sets rather than individuals. Brain signatures suggest children attribute exact rather than approximate numerical meanings to the first few number words. Number-quantity relationship processing for the first few number words is evident in frontal but not parietal scalp electrophysiology of young children.

Recognition of small numbers in subset knowers Cardinal knowledge in early childhood

Previous research suggests that subset-knowers have an approximate understanding of small numbers. However, it is still unclear exactly what subset-knowers understand about small numbers. To investigate this further, we tested 133 participants, ages 2.6-4 years, on a newly developed eye-tracking task targeting cardinal recognition. Participants were presented with two sets differing in cardinality (1-4 items) and asked to find a specific cardinality. Our main finding showed that on a group level, subset-knowers could identify all presented targets at rates above chance, further supporting that subset-knowers understand several of the basic principles of small numbers. Exploratory analyses tentatively suggest that 1-knowers could identify the targets 1 and 2, but struggled when the target was 3 and 4, whereas 2-knowers and above could identify all targets at rates above chance. This might tentatively suggest that subset-knowers have an approximate understanding of numbers that is just (i.e. +1) above their current knower level. We discuss the implications of these results at length.

Everything is Infinite: Children's Beliefs About Endless Space, Time, and Number

How do children form beliefs about the infinity of space, time, and number? We asked whether children held similar beliefs about infinity across domains, and whether beliefs in infinity for domains like space and time might be scaffolded upon numerical knowledge (e.g., knowledge successors within the count list). To test these questions, 112 U.S. children (aged 4;0-7;11) completed an interview regarding their beliefs about infinite space, time, and number. We also measured their knowledge of counting, and other factors that might impact performance on linguistic assessments of infinity belief (e.g., working memory, ability to respond to hypothetical questions). We found that beliefs about infinity were very high across all three domains, suggesting that infinity beliefs may arise early in development for space, time, and number. Second, we found that-across all three domains-children were more likely to believe that it is always possible to add a unit than to believe that the domain is endless. Finally, we found that understanding the rules underlying counting predicted children's belief that it is always possible to add 1 to any number, but did not predict any of the other elements of infinity belief.

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.

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

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

A million is more than a thousand: Children's acquisition of very large number words

Very large numbers words such as "hundred," "thousand," "million," "billion," and "trillion" pose a learning problem for children because they are sparse in everyday speech and children's experience with extremely large quantities is scarce. In this study, we examine when children acquire the relative ordering of very large number words as a first step toward understanding their acquisition. In Study 1, a hundred and twenty-five 5-8-year-olds participated in a verbal number comparison task involving very large number words. We found that children can judge which of two very large numbers is more as early as age 6, prior to entering first grade. In Study 2, we provided a descriptive analysis on the usage of very large number words using the CHILDES database. We found that the relative frequency of large number words does not change across the years, with "hundred" uttered more frequently than others by an order of magnitude. We also found that adults were more likely to use large number words to reference units of quantification for money, weight, and time, than for discrete, physical entities. Together, these results show that children construct a numerical scale for large number words prior to learning their precise cardinal meanings, and highlight how frequency and context may support their acquisition. Our results have pedagogical implications and highlight a need to investigate how children acquire meanings for number words that reference quantities beyond our everyday experience.

Acquisition of the counting principles during the subset-knower stages: Insights from children's errors

Studies on children's understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn's (1990) Give-N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal-principle-knowers) and those who cannot as lacking knowledge of it (subset-knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP-knowers on the Give-N Task. Specifically, we test the claim that subset-knowers lack cardinal principle knowledge by separating children's understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel's (1978) other how-to-count principles emerge in development. We analyzed how often children violated the three how-to-count principles in a secondary analysis of Give-N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP-knowers, and that understanding of the stable-order and word-object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children's acquisition of the counting principles and that learning to coordinate all three principles represents an additional step beyond learning them individually.

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.

Children's exploratory play tracks the discriminability of hypotheses

Effective curiosity-driven learning requires recognizing that the value of evidence for testing hypotheses depends on what other hypotheses are under consideration. Do we intuitively represent the discriminability of hypotheses? Here we show children alternative hypotheses for the contents of a box and then shake the box (or allow children to shake it themselves) so they can hear the sound of the contents. We find that children are able to compare the evidence they hear with imagined evidence they do not hear but might have heard under alternative hypotheses. Children (N = 160; mean: 5 years and 4 months) prefer easier discriminations (Experiments 1-3) and explore longer given harder ones (Experiments 4-7). Across 16 contrasts, children's exploration time quantitatively tracks the discriminability of heard evidence from an unheard alternative. The results are consistent with the idea that children have an "intuitive psychophysics": children represent their own perceptual abilities and explore longer when hypotheses are harder to distinguish.

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.

Starting small: exploring the origins of successor function knowledge

Although most U. S. children can accurately count sets by 4 years of age, many fail to understand the structural analogy between counting and number - that adding 1 to a set corresponds to counting up 1 word in the count list. While children are theorized to establish this Structure Mapping coincident with learning how counting is used to generate sets, they initially have an item-based understanding of this relationship, and can infer that, e.g, adding 1 to "five" is "six", while failing to infer that, e.g., adding 1 to "twenty-five" is "twenty-six" despite being able to recite these numbers when counting aloud. The item-specific nature of children's successes in reasoning about the relationship between changes in cardinality and the count list raises the possibility that such a Structure Mapping emerges later in development, and that this ability does not initially depend on learning to count. We test this hypothesis in two experiments and find evidence that children can perform item-based addition operations before they become competent counters. Even after children learn to count, we find that their ability to perform addition operations remains item-based and restricted to very small numbers, rather than drawing on generalized knowledge of how the count list represents number. We discuss how these early item-based associations between number words and sets might play a role in constructing a generalized Structure Mapping between counting and quantity.

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.

Finding patterns in objects and numbers: Repeating patterning in pre-K predicts kindergarten mathematics knowledge

Both recent evidence and research-based early mathematics curricula indicate that repeating patterns-predictable sequences that follow a rule-are a topic of major importance for mathematics development. The purpose of the current study was to help build a theory for how early repeating patterning knowledge contributes to early math development, focusing on development in children aged 4-6 years. The current study examined the relation between 65 preschool children's repeating patterning knowledge (via a fast, teacher-friendly measure) and their end-of-kindergarten broad math and numeracy knowledge, controlling for verbal and visual-spatial working memory (WM) skills as well as end-of-pre-K (pre-kindergarten) broad math knowledge. Relations were also examined between repeating patterning and specific aspects of numeracy knowledge-knowledge of the count sequence to 100 and the successor principle. Children's repeating patterning knowledge was significantly predictive of their broad math and general numeracy knowledge, as well as one specific aspect of their numeracy knowledge (counting to 100), even after controlling for verbal and visual-spatial WM skills. Further, repeating patterning knowledge remained a unique predictor of general numeracy knowledge and counting to 100 after controlling for end-of-pre-K broad math knowledge. The relation between repeating patterning and mathematics may be explained by the central role that identifying predictable sequences based on underlying rules plays in both. Theories of math development and early math instruction standards should thus give even greater attention to the role of children's repeating patterning knowledge.

Counting to Infinity: Does Learning the Syntax of the Count List Predict Knowledge That Numbers Are Infinite?

By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled "five" should now be "six"). These findings suggest that children have implicit knowledge of the "successor function": Every natural number, n, has a successor, n + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language-specific counting routines (e.g., the rules in English that represent base-10 structure). We tested 4- and 5-year-old children's knowledge of counting with three tasks, which we then related to (a) children's belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end.

Is thirty-two three tens and two ones? The embedded structure of cardinal numbers

The acquisition and representation of natural numbers have been a central topic in cognitive science. However, a key question in this topic about how humans acquire the capacity to understand that numbers make 'infinite use of finite means' (or that numbers are generative) has been left unanswered. While previous theories rely on the idea of the successor principle, we propose an alternative hypothesis that children's understanding of the syntactic rules for building complex numerals-or numerical syntax-is a crucial foundation for the acquisition of number concepts. In two independent studies, we assessed children's understanding of numerical syntax by probing their knowledge about the embedded structure of cardinal numbers using a novel task called Give-a-number Base-10 (Give-N10). In Give-N10, children were asked to give a large number of items (e.g., 32 items) from a pool that is organized in sets of ten items. Children's knowledge about the embedded structure of numbers (e.g., knowing that thirty-two items are composed of three tens and two ones) was assessed from their ability to use those sets. Study 1 tested English-speaking 4- to 10-year-olds and revealed that children's understanding of the embedded structure of numbers emerges relatively late in development (several months into kindergarten), beyond when they are capable of making a semantic induction over a local sequence of numbers. Moreover, performance in Give-N10 was predicted by other task measures that assessed children's knowledge about the syntactic rules that govern numerals (such as counting fluency), demonstrating the validity of the measure. In Study 2, this association was tested again in monolingual Korean kindergarteners (5-6 years), as we aimed to test the same effect in a language with a highly regular numeral system. It replicated the association between Give-N10 performance and counting fluency, and it also demonstrated that Korean-speaking children understand the embedded structure of cardinal numbers earlier in the acquisition path than English-speaking peers, suggesting that regularity in numerical syntax facilitates the acquisition of generative properties of numbers. Based on these observations and our theoretical analysis of the literature, we propose that the syntax for building complex numerals, not the successor principle, represents a structural platform for numerical thinking in young children.

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

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

Do children use language structure to discover the recursive rules of counting?

We test the hypothesis that children acquire knowledge of the successor function - a foundational principle stating that every natural number n has a successor n + 1 - by learning the productive linguistic rules that govern verbal counting. Previous studies report that speakers of languages with less complex count list morphology have greater counting and mathematical knowledge at earlier ages in comparison to speakers of more complex languages (e.g., Miller & Stigler, 1987). Here, we tested whether differences in count list transparency affected children's acquisition of the successor function in three languages with relatively transparent count lists (Cantonese, Slovenian, and English) and two languages with relatively opaque count lists (Hindi and Gujarati). We measured 3.5- to 6.5-year-old children's mastery of their count list's recursive structure with two tasks assessing productive counting, which we then related to a measure of successor function knowledge. While the more opaque languages were associated with lower counting proficiency and successor function task performance in comparison to the more transparent languages, a unique within-language analytic approach revealed a robust relationship between measures of productive counting and successor knowledge in almost every language. We conclude that learning productive rules of counting is a critical step in acquiring knowledge of recursive successor function across languages, and that the timeline for this learning varies as a function of count list transparency.

Making Sense of Number Words and Arabic Digits: Does Order Count More?

The ability to choose the larger between two numbers reflects a mature understanding of the magnitude associated with numerical symbols. The present study explores how the knowledge of the number sequence and memory capacity (verbal and visuospatial) relate to number comparison skills while controlling for cardinal knowledge. Preschool children's (N = 140, M_{age-in-months}  = 58.9, range = 41-75) knowledge of the directional property of the counting list as well as the spatial mapping of digits on the visual line were assessed. The ability to order digits on the visual line mediated the relation between memory capacity and number comparison skills while controlling for cardinal knowledge. Beyond cardinality, the knowledge of the (spatial) order of numbers marks the understanding of the magnitude associated with numbers.

The knowledge of the preceding number reveals a mature understanding of the number sequence

There is an ongoing debate concerning how numbers acquire numerical meaning. On the one hand, it has been argued that symbols acquire meaning via a mapping to external numerosities as represented by the approximate number system (ANS). On the other hand, it has been proposed that the initial mapping of small numerosities to the corresponding number words and the knowledge of the properties of counting list, especially the order relation between symbols, lead to the understanding of the exact numerical magnitude associated with numerical symbols. In the present study, we directly compared these two hypotheses in a group of preschool children who could proficiently count (most of the children were cardinal principle knowers). We used a numerosity estimation task to assess whether children have created a mapping between the ANS and the counting list (i.e., ANS-to-word mapping). Children also completed a direction task to assess their knowledge of the directional property of the counting list. That is, adding one item to a set leads to he next number word in the sequence (i.e., successor knowledge) whereas removing one item leads to the preceding number word (i.e., predecessor knowledge). Similarly, we used a visual order task to assess the knowledge that successive and preceding numbers occupy specific spatial positions on the visual number line (i.e., preceding: [?], [13], [14]; successive: [12], [13], [?]). Finally, children's performance in comparing the magnitude of number words and Arabic numbers indexed the knowledge of exact symbolic numerical magnitude. Approximately half of the children in our sample have created a mapping between the ANS and the counting list. Most of the children mastered the successor knowledge whereas few of them could master the predecessor knowledge. Children revealed a strong tendency to respond with the successive number in the counting list even when an item was removed from a set or the name of the preceding number on the number line was asked. Crucially, we found evidence that both the mastering of the predecessor knowledge and the ability to name the preceding number in the number line relate to the performance in number comparison tasks. Conversely, there was moderate/anecdotal evidence for a relation between the ANS-to-word mapping and number comparison skills. Non-rote access to the number sequence relates to knowledge of the exact magnitude associated with numerical symbols, beyond the mastering of the cardinality principle and domain-general factors.

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.

Preschool deficits in cardinal knowledge and executive function contribute to longer-term mathematical learning disability

In a preschool through first grade longitudinal study, we identified groups of children with persistently low mathematics achievement (n = 14) and children with low achievement in preschool but average achievement in first grade (n = 23). The preschool quantitative developments of these respective groups of children with mathematical learning disability (MLD) and recovered children and a group of typically achieving peers (n = 35) were contrasted, as were their intelligence, executive function, and parental education levels. The core characteristics of the children with MLD were poor executive function and delayed understanding of the cardinal value of number words throughout preschool. These compounded into even more substantive deficits in number and arithmetic at the beginning of first grade. The recovered group had poor executive function and cardinal knowledge during the first year of preschool but showed significant gains during the second year. Despite these gains and average mathematics achievement, the recovered children had subtle deficits with accessing magnitudes associated with numerals and addition combinations (e.g., 5 + 6 = ?) in first grade. The study provides unique insight into domain-general and quantitative deficits in preschool that increase risk for long-term mathematical difficulties.

Ontogenetic Origins of Human Integer Representations

Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.

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

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

Predicting Age of Becoming a Cardinal Principle Knower

Children's first mathematics concept is their understanding of the quantities represented by number words (cardinal value), and the age at which they achieve this insight predicts their readiness for mathematics learning in school. We provide the first exploration of the factors that influence the age of becoming a cardinal principle knower (CPK), with a longitudinal study of 197 (94 boys) children from the beginning to the end of two years of preschool. Core symbolic and non-symbolic quantitative competencies at the beginning of preschool, as well as measures of intelligence, executive function, preliteracy skills, and parental education were used to predict timing of CPK status. Children who achieved early CPK status had higher IQ scores, knew more count words and numerals, and had a better intuitive understanding of relative quantity than their peers. Children who were delayed CPKs, in contrast, had deficits in executive function and poor preliteracy skills. The results add to our understanding of children's conceptual development in mathematics and have implications for the identification of at-risk children and design of interventions for them.

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.

Do children's number words begin noisy?

How do children acquire exact meanings for number words like three or forty-seven? In recent years, a lively debate has probed the cognitive systems that support learning, with some arguing that an evolutionarily ancient "approximate number system" drives early number word meanings, and others arguing that learning is supported chiefly by representations of small sets of discrete individuals. This debate has centered around the findings generated by Wynn's (, ) Give-a-Number task, which she used to categorize children into discrete "knower level" stages. Early reports confirmed Wynn's analysis, and took these stages to support the "small sets" hypothesis. However, more recent studies have disputed this analysis, and have argued that Give-a-Number data reveal a strong role for approximate number representations. In the present study, we use previously collected Give-a-Number data to replicate the analyses of these past studies, and to show that differences between past studies are due to assumptions made in analyses, rather than to differences in data themselves. We also show how Give-a-Number data violate the assumptions of parametric tests used in past studies. Based on simple non-parametric tests and model simulations, we conclude that (a) before children learn exact meanings for words like one, two, three, and four, they first acquire noisy preliminary meanings for these words, (b) there is no reliable evidence of preliminary meanings for larger meanings, and (c) Give-a-Number cannot be used to readily identify signatures of the approximate number system.

Spatial order relates to the exact numerical magnitude of digits in young children

Spatial representation of numbers has been repeatedly associated with the development of numerical and mathematical skills. However, few studies have explored the contribution of spatial mapping to exact number representation in young children. Here we designed a novel task that allows a detailed analysis of direction, ordinality, and accuracy of spatial mapping. Preschool children, who were classified as competent counters (cardinal principle knowers), placed triplets of sequentially presented digits on the visual line. The ability to correctly order triplets tended to decrease with the larger digits. When triplets were correctly ordered, the direction of spatial mapping was predominantly oriented from left to right and the positioning of the target digits was characterized by a pattern of underestimation with no evidence of logarithmic compression. Crucially, only ordinality was associated with performance in a digit comparison task. Our results suggest that the spatial (ordinal) arrangement of digits is a powerful source of information that young children can use to construct the representation of exact numbers. Therefore, digits may acquire numerical meaning based on their spatial order on the number line.

Meaning before order: Cardinal principle knowledge predicts improvement in understanding the successor principle and exact ordering

Learning the cardinal principle (the last word reached when counting a set represents the size of the whole set) is a major milestone in early mathematics. But researchers disagree about the relationship between cardinal principle knowledge and other concepts, including how counting implements the successor function (for each number word N representing a cardinal value, the next word in the count list represents the cardinal value N + 1) and exact ordering (cardinal values can be ordered such that each is one more than the value before it and one less than the value after it). No studies have investigated acquisition of the successor principle and exact ordering over time, and in relation to cardinal principle knowledge. An open question thus remains: Is the cardinal principle a "gatekeeper" concept children must acquire before learning about succession and exact ordering, or can these concepts develop separately? Preschoolers (N = 127) who knew the cardinal principle (CP-knowers) or who knew the cardinal meanings of number words up to "three" or "four" (3-4-knowers) completed succession and exact ordering tasks at pretest and posttest. In between, children completed one of two trainings: counting only versus counting, cardinal labeling, and comparison. CP-knowers started out better than 3-4-knowers on succession and exact ordering. Controlling for this disparity, we found that CP-knowers improved over time on succession and exact ordering; 3-4-knowers did not. Improvement did not differ between the two training conditions. We conclude that children can learn the cardinal principle without understanding succession or exact ordering and hypothesize that children must understand the cardinal principle before learning these concepts.

Children's Understanding of the Natural Numbers' Structure

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between "5" and "10" is larger than the distance between "75" and "80." This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, ; Siegler & Opfer, ). However, several investigators have questioned this argument (e.g., Barth & Paladino, ; Cantlon, Cordes, Libertus, & Brannon, ; Cohen & Blanc-Goldhammer, ). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as "5" or "80"). In Experiments 1 and 2, when 5- and 6-year-olds choose between number lines in a forced-choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1-100 are arranged. The bead placement of 4- and 5-year-olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number-line task may depend on strategies specific to the task.

Is There Really an Evolved Capacity for Number?

Humans and other species have biologically endowed abilities for discriminating quantities. A widely accepted view sees such abilities as an evolved capacity specific for number and arithmetic. This view, however, is based on an implicit teleological rationale, builds on inaccurate conceptions of biological evolution, downplays human data from non-industrialized cultures, overinterprets results from trained animals, and is enabled by loose terminology that facilitates teleological argumentation. A distinction between quantical (e.g., quantity discrimination) and numerical (exact, symbolic) cognition is needed: quantical cognition provides biologically evolved preconditions for numerical cognition but it does not scale up to number and arithmetic, which require cultural mediation. The argument has implications for debates about the origins of other special capacities - geometry, music, art, and language.

Early Conceptual Understanding of Cardinality Predicts Superior School-Entry Number-System Knowledge

We demonstrate a link between preschoolers' quantitative competencies and their school-entry knowledge of the relations among numbers (number-system knowledge). The quantitative competencies of 141 children (69 boys) were assessed at the beginning of preschool and throughout the next 2 years of preschool, as was their mathematics and reading achievement at the end of kindergarten and their number-system knowledge at the beginning of first grade. A combination of Bayes analyses and standard regressions revealed that the age at which the children had the conceptual insight that number words represent specific quantities (cardinal value) was strongly related to their later number-system knowledge and was more consistently related to broader mathematics than to reading achievement, controlling for intelligence, executive function, and parental education levels. The key implication is that it is not simply knowledge of cardinal value but the age of acquisition of this principle that is central to later mathematical development and school readiness.

What counts in preschool number knowledge? A Bayes factor analytic approach toward theoretical model development

Preschool children vary tremendously in their numerical knowledge, and these individual differences strongly predict later mathematics achievement. To better understand the sources of these individual differences, we measured a variety of cognitive and linguistic abilities motivated by previous literature to be important and then analyzed which combination of these variables best explained individual differences in actual number knowledge. Through various data-driven Bayesian model comparison and selection strategies on competing multiple regression models, our analyses identified five variables of unique importance to explaining individual differences in preschool children's symbolic number knowledge: knowledge of the count list, nonverbal approximate numerical ability, working memory, executive conflict processing, and knowledge of letters and words. Furthermore, our analyses revealed that knowledge of the count list, likely a proxy for explicit practice or experience with numbers, and nonverbal approximate numerical ability were much more important to explaining individual differences in number knowledge than general cognitive and language abilities. These findings suggest that children use a diverse set of number-specific, general cognitive, and language abilities to learn about symbolic numbers, but the contribution of number-specific abilities may overshadow that of more general cognitive abilities in the learning process.

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.