When is four far more than three? Children's generalization of newly acquired number words

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.

Introducing Mr. Three: Attention, Perception, and Meaning Selection in the Acquisition of Number and Color Words

Children learn their first number words gradually over the course of many months, which is surprising given their ability to discriminate small numerosities. One potential explanation for this is that children are sensitive to the numerical features of stimuli, but don't consider exact cardinality as a primary hypothesis for novel word meanings. To test this, we trained 144 children on a number word they hadn't yet learned, and contrasted this with a condition in which they were merely required to attend to number to identify the word's referent, without encoding number as its meaning. In the first condition, children were trained to find a "giraffe with three spots." In the second condition, children were instead trained to find "Mr. Three", which also named a giraffe with three spots. In both conditions, children had to attend to number to identify the target giraffe, but, because proper nouns refer to individuals rather than their properties, the second condition did not require children to encode number as the meaning of the expression. We found that children were significantly better at identifying the giraffe when it had been labeled with the proper noun than with the number word. This finding contrasted with a second experiment involving color words, in which children (n = 56) were equally successful with a proper noun ("Mr. Purple") and an adjective ("the giraffe with purple spots"). Together, these findings suggest that, for number, but not for color, children's difficulty acquiring new words cannot be solely attributed to problems with attention or perception, but instead may be due to difficulty selecting the correct meaning from their hypothesis space for learning unknown words.

Early-emerging combinatorial thought: Human infants flexibly combine kind and quantity concepts

Combinatorial thought, or the ability to combine a finite set of concepts into a myriad of complex ideas and knowledge structures, is the key to the productivity of the human mind and underlies communication, science, technology, and art. Despite the importance of combinatorial thought for human cognition and culture, its developmental origins remain unknown. To address this, we tested whether 12-mo-old infants (N = 60), who cannot yet speak and only understand a handful of words, can combine quantity and kind concepts activated by verbal input. We proceeded in two steps: first, we taught infants two novel labels denoting quantity (e.g., "mize" for 1 item; "padu" for 2 items, Experiment 1). Then, we assessed whether they could combine quantity and kind concepts upon hearing complex expressions comprising their labels (e.g., "padu duck", Experiments 2-3). At test, infants viewed four different sets of objects (e.g., 1 duck, 2 ducks, 1 ball, 2 balls) while being presented with the target phrase (e.g., "padu duck") naming one of them (e.g., 2 ducks). They successfully retrieved and combined on-line the labeled concepts, as evidenced by increased looking to the named sets but not to distractor sets. Our results suggest that combinatorial processes for building complex representations are available by the end of the first year of life. The infant mind seems geared to integrate concepts in novel productive ways. This ability may be a precondition for deciphering the ambient language(s) and building abstract models of experience that enable fast and flexible learning.

Miscategorized subset-knowers: Five- and six-knowers can compare only the numbers they know

Initial acquisition of the first symbolic numbers is measured with the Give a Number (GaN) task. According to the classic method, it is assumed that children who know only 1, 2, 3, or 4 in the GaN task, (termed separately one-, two-, three-, and four-knowers, or collectively subset-knowers) have only a limited conceptual understanding of numbers. On the other hand, it is assumed that children who know larger numbers understand the fundamental properties of numbers (termed cardinality-principle-knowers), even if they do not know all the numbers as measured with the GaN task, that are in their counting list (e.g., five- or six-knowers). We argue that this practice may not be well-established. To validate this categorization method, here, the performances of groups with different GaN performances were measured separately in a symbolic comparison task. It was found that similar to one to four-knowers, five-, six-, and so forth, knowers can compare only the numbers that they know in the GaN task. We conclude that five-, six-, and so forth, knowers are subset-knowers because their conceptual understanding of numbers is fundamentally limited. We argue that knowledge of the cardinality principle should be identified with stricter criteria compared to the current practice in the literature. RESEARCH HIGHLIGHTS: Children who know numbers larger than 4 in the Give a Number (GaN) task are usually assumed to have a fundamental conceptual understanding of numbers. We tested children who know numbers larger than 4 but who do not know all the numbers in their counting list to see whether they compare numbers more similar to children who know only small numbers in the GaN task or to children who have more firm number knowledge. Five-, six-, and so forth, knowers can compare only the numbers they know in the GaN task, similar to the performance of the one, two, three, and four-knowers. We argue that these children have a limited conceptual understanding of numbers and that previous works may have miscategorized them.

The role of semantic similarity in verb learning events: Vocabulary-related changes across early development

Verb meaning is challenging for children to learn across varied events. This study examined how the taxonomic semantic similarity of the nouns in novel verb learning events in a progressive alignment learning condition differed from the taxonomic dissimilarity of nouns in a dissimilar learning condition in supporting near (similar) and far (dissimilar) verb generalization to novel objects in an eye-tracking task. A total of 48 children in two age groups (23 girls; younger: 21-24 months, Mage = 22.1 months; older: 27-30 months: Mage = 28.3 months) who differed in taxonomic vocabulary size were tested. There were no group or learning condition differences in near generalization. The younger group demonstrated better far generalization of verbs learned with semantically dissimilar nouns. The older group demonstrated the opposite pattern, with better far generalization of verbs learned with semantically similar nouns in the progressive alignment condition. These patterns were associated with children's in-category vocabulary knowledge more than other vocabulary measures, including verb vocabulary size. Taxonomic vocabulary knowledge differentially affects verb learning and generalization across development.

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.

A counting intervention promotes fair sharing in preschoolers

Recent work has probed the developmental mechanisms that promote fair sharing. This work investigated 2.5- to 5.5-year-olds' (N = 316; 52% female; 79% White; data collected 2016-2018) sharing behavior in relation to three cognitive correlates: number knowledge, working memory, and cognitive control. In contrast to working memory and cognitive control, number knowledge was uniquely associated with fair sharing even after controlling for the other correlates and for age. Results also showed a causal effect: After a 5-min counting intervention (vs. a control), children improved their fair sharing behavior from pre-test to post-test. Findings are discussed in light of how social, cognitive, and motivational factors impact sharing behavior.

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

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

Measuring Emerging Number Knowledge in Toddlers

Recent evidence suggests that infants and toddlers may recognize counting as numerically relevant long before they are able to count or understand the cardinal meaning of number words. The Give-N task, which asks children to produce sets of objects in different quantities, is commonly used to test children’s cardinal number knowledge and understanding of exact number words but does not capture children’s preliminary understanding of number words and is difficult to administer remotely. Here, we asked whether toddlers correctly map number words to the referred quantities in a two-alternative forced choice Point-to-X task (e.g., “Which has three?”). Two- to three-year-old toddlers (N = 100) completed a Give-N task and a Point-to-X task through in-person testing or online via videoconferencing software. Across number-word trials in Point-to-X, toddlers pointed to the correct image more often than predicted by chance, indicating that they had some understanding of the prompted number word that allowed them to rule out incorrect responses, despite limited understanding of exact cardinal values. No differences in Point-to-X performance were seen for children tested in-person versus remotely. Children with better understanding of exact number words as indicated on the Give-N task also answered more trials correctly in Point-to-X. Critically, in-depth analyses of Point-to-X performance for children who were identified as 1- or 2-knowers on Give-N showed that 1-knowers do not show a preliminary understanding of numbers above their knower-level, whereas 2-knowers do. As researchers move to administering assessments remotely, the Point-to-X task promises to be an easy-to-administer alternative to Give-N for measuring children’s emerging number knowledge and capturing nuances in children’s number-word knowledge that Give-N may miss.

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.

Causal Effects of Parent Number Talk on Preschoolers' Number Knowledge

Individual differences in children's number knowledge arise early and are associated with variation in parents' number talk. However, there exists little experimental evidence of a causal link between parent number talk and children's number knowledge. Parent number talk was manipulated by creating picture books which parents were asked to read with their children every day for 4 weeks. N = 100 two- to four-year olds and their parents were randomly assigned to read either Small Number (1-3), Large Number (4-6), or Control (non-numerical) books. Small Number books were particularly effective in promoting number knowledge relative to the Control books. However, children who began the study further along in their number development also benefited from reading the Large Number Books with their parents.

Number-related Brain Potentials Are Differentially Affected by Mapping Novel Symbols on Small versus Large Quantities in a Number Learning Task

The nature of the mapping process that imbues number symbols with their numerical meaning-known as the "symbol-grounding process"-remains poorly understood and the topic of much debate. The aim of this study was to enhance insight into how the nonsymbolic-symbolic number mapping process and its neurocognitive correlates might differ between small (1-4; subitizing range) and larger (6-9) numerical ranges. Hereto, 22 young adults performed a learning task in which novel symbols acquired numerical meaning by mapping them onto nonsymbolic magnitudes presented as dot arrays (range 1-9). Learning-dependent changes in accuracy and RT provided evidence for successful novel symbol quantity mapping in the subitizing (1-4) range only. Corroborating these behavioral results, the number processing related P2p component was only modulated by the learning/mapping of symbols representing small numbers 1-4. The symbolic N1 amplitude increased with learning independent of symbolic numerical range but dependent on the set size of the preceding dot array; it only occurred when mapping on one to four item dot arrays that allow for quick retrieval of a numeric value, on the basis of which, with learning, one could predict the upcoming symbol causing perceptual expectancy violation when observing a different symbol. These combined results suggest that exact nonsymbolic-symbolic mapping is only successful for small quantities 1-4 from which one can readily extract cardinality. Furthermore, we suggest that the P2p reflects the processing stage of first access to or retrieval of numeric codes and might in future studies be used as a neural correlate of nonsymbolic-symbolic mapping/symbol learning.

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.

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.

Number gestures predict learning of number words

When asked to explain their solutions to a problem, children often gesture and, at times, these gestures convey information that is different from the information conveyed in speech. Children who produce these gesture-speech "mismatches" on a particular task have been found to profit from instruction on that task. We have recently found that some children produce gesture-speech mismatches when identifying numbers at the cusp of their knowledge, for example, a child incorrectly labels a set of two objects with the word "three" and simultaneously holds up two fingers. These mismatches differ from previously studied mismatches (where the information conveyed in gesture has the potential to be integrated with the information conveyed in speech) in that the gestured response contradicts the spoken response. Here, we ask whether these contradictory number mismatches predict which learners will profit from number-word instruction. We used the Give-a-Number task to measure number knowledge in 47 children (Mage  = 4.1 years, SD = 0.58), and used the What's on this Card task to assess whether children produced gesture-speech mismatches above their knower level. Children who were early in their number learning trajectories ("one-knowers" and "two-knowers") were then randomly assigned, within knower level, to one of two training conditions: a Counting condition in which children practiced counting objects; or an Enriched Number Talk condition containing counting, labeling set sizes, spatial alignment of neighboring sets, and comparison of these sets. Controlling for counting ability, we found that children were more likely to learn the meaning of new number words in the Enriched Number Talk condition than in the Counting condition, but only if they had produced gesture-speech mismatches at pretest. The findings suggest that numerical gesture-speech mismatches are a reliable signal that a child is ready to profit from rich number instruction and provide evidence, for the first time, that cardinal number gestures have a role to play in number-learning.

Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves

Numerous studies from developmental psychology have suggested that human symbolic representation of numbers is built upon the evolutionally old capacity for representing quantities that is shared with other species. Substantial research from mathematics education also supports the idea that mathematical concepts are best learned through their corresponding physical representations. We argue for an independent pathway to learning "big" multi-digit symbolic numbers that focuses on the symbol system itself. Across five experiments using both between- and within-subject designs, we asked preschoolers to identify written multi-digit numbers with their spoken names in a two-alternative-choice-test or to indicate the larger quantity between two written numbers. Results showed that preschoolers could reliably map spoken number names to written forms and compare the magnitudes of two written multi-digit numbers. Importantly, these abilities were not related to their non-symbolic representation of quantities. These findings have important implications for numerical cognition, symbolic development, teaching, and education.

Response patterns of young children from two contrasting SES contexts to different numerical tasks with numbers 1-5

This study contributes to a multifaceted picture of young children's emergent number knowledge by focusing on the variety of ways in which children express and use quantitative information. The authors' aims are to (a) explore the extent to which quantifying collections 1-5 and using that information pose different levels of difficulty to children from 33 to 47 months old, (b) identify intra- and intertask response patterns, and (c) analyze the influence of socioeconomic status and age on these response patterns. Sixty-six children from two contrasting socioeconomic status groups (very low and middle) were asked to solve tasks with 1-5 elements in the context of a game. Using quantitative information turned out to be more complex than quantification. Intra- and intertask response patterns showed that children gradually come to understand the first five numerical values according to the numerical sequence in a much less strict way than that proposed by the cardinal-knowers model that posits that children progress in an orderly way in their number knowledge. Children in different ages and socioeconomic status groups were found to be more similar to each other when the whole arc of responses provided was considered than when solely correct performance was measured.

The effect of multimodal information on children's numerical judgments

Although much research suggests that adults, infants, and nonhuman primates process number (among other properties) across distinct modalities, limited studies have explored children's abilities to integrate multisensory information when making judgments about number. In the current study, 3- to 6-year-old children performed numerical matching or numerical discrimination tasks in which numerical information was presented either unimodally (visual only), cross-modally (comparing audio with visual), or bimodally (simultaneously presenting audio and visual input). In three experiments, we investigated children's multimodal numerical processing across distinct task demands and difficulty levels. In contrast to previous work, results indicate that even the youngest children (3 and 4 years) performed above chance across all three modality presentations. In addition, the current study contributes two other novel findings, namely that (a) children exhibit a cross-modal disadvantage when numerical comparisons are easy and that (b) accuracy on bimodal trial types led to even more accurate numerical judgments under more difficult circumstances, particularly for the youngest participants and when precise numerical matching was required. Importantly, findings from this study extend the literature on children's numerical cross-modal abilities to reveal that, like their adult counterparts, children readily track and compare visual and auditory numerical information, although their abilities to do so are not perfect and are affected by task demands and trial difficulty.

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.

Do analog number representations underlie the meanings of young children's verbal numerals?

Children learn to count, and even learn the cardinal meanings of the first three or four verbal numerals ("one" through "three" or "four"), before they master the numerical significance of counting. If so, it follows that the cardinal meanings of those first few numerals cannot be derived, initially, from their place in the count list and the counting routine. What non-verbal representations, then, support the cardinal meanings of verbal numerals before children have mastered how counting does so? Four experiments addressed the commonly adopted assumption that in the earliest period of learning the meanings of number words, children map verbal numerals to regions of the analog number system (ANS), a system of representation with numerical content that is widely attested in animals and in human infants. Experiment 1 confirmed that children who know what "three" means, but who do not yet know what "four" means, and do not yet know how counting represents number, can be easily taught the meaning of "four," if they are trained to indicate sets of four when they are paired with a series of sets that contrast numerically with four. If children learn "four" by mapping the word to an ANS representation of sets of four, and if such ANS value-to-word mappings underlie the meanings of other known numerals early in development, then analogous teaching should enable young children to establish a ANS value-to-word mapping for between "ten" and sets of 10 as specified by the ANS. Furthermore, the ease of learning should be a function of the ratio of the number of individuals in the comparison set to 10. Three further experiments tested these hypotheses by attempting to teach young Cardinal Principle-knowers the meaning of the word "ten," under the same training conditions "three-"knowers are easily taught the meaning of "four". The children learned which picture in each training pair had "ten." However, test trials with novel animals and spatial configurations showed that they had failed to learn what set sizes should be labeled "ten", even when, after training, they were asked to indicate a set of 10 vs. a set of 20 or 30 (well within the ratio sensitivity of the ANS even early in infancy). Furthermore, there was no effect of ratio on success during test trials. These data provide new evidence that ANS value-to-word mappings do not underlie the meanings of number words early in development. We discuss what other non-verbal representations might do so, and discuss other ways the ANS may support learning how counting represents number.

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

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

Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.

Of Huge Mice and Tiny Elephants: Exploring the Relationship Between Inhibitory Processes and Preschool Math Skills

The cognitive mechanisms underpinning the well-established relationship between inhibitory control and early maths skills remain unclear. We hypothesized that a specific aspect of inhibitory control drives its association with distinct math skills in very young children: the ability to ignore stimulus dimensions that are in conflict with task-relevant representations. We used an Animal Size Stroop task in which 3- to 6-year-olds were required to ignore the physical size of animal pictures to compare their real-life dimensions. In Experiment 1 (N = 58), performance on this task correlated with standardized early mathematics achievement. In Experiment 2 (N = 48), performance on the Animal Size Stroop task related to the accuracy of magnitude comparison, specifically for trials on which the physical size of dot arrays was incongruent with their numerosity. This highlights a process-oriented relationship between interference control and resolving conflict between discrete and continuous quantity, and in turn calls for further detailed empirical investigations of whether, how and why inhibitory processes matter to emerging numerical cognition.

Thinking about quantity: the intertwined development of spatial and numerical cognition

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological, and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time and mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, and density) and number can be conceptualized discretely or continuously (i.e., natural vs real numbers). Variation on these quantitative dimensions is typically correlated, e.g., larger objects often weigh more than smaller ones. Number is a distinctive continuous dimension because the natural numbers (i.e., positive integers) are used to quantify collections of discrete objects. This aspect of number is emphasized by teaching of the count word sequence and arithmetic during the early school years. We review research on spatial and numerical estimation, and argue that a generalized magnitude system is the starting point for development in both domains. Development occurs along several lines: (1) changes in capacity, durability, and precision, (2) differentiation of the generalized magnitude system into separable dimensions, (3) formation of a discrete number system, i.e., the positive integers, (4) mapping the positive integers onto the continuous number line, and (5) acquiring abstract knowledge of the relations between pairs of systems. We discuss implications of this approach for teaching various topics in mathematics, including scaling, measurement, proportional reasoning, and fractions.

Gesture as a window onto children's number knowledge

Before learning the cardinal principle (knowing that the last word reached when counting a set represents the size of the whole set), children do not use number words accurately to label most set sizes. However, it remains unclear whether this difficulty reflects a general inability to conceptualize and communicate about number, or a specific problem with number words. We hypothesized that children's gestures might reflect knowledge of number concepts that they cannot yet express in speech, particularly for numbers they do not use accurately in speech (numbers above their knower-level). Number gestures are iconic in the sense that they are item-based (i.e., each finger maps onto one item in a set) and therefore may be easier to map onto sets of objects than number words, whose forms do not map transparently onto the number of items in a set and, in this sense, are arbitrary. In addition, learners in transition with respect to a concept often produce gestures that convey different information than the accompanying speech. We examined the number words and gestures 3- to 5-year-olds used to label small set sizes exactly (1-4) and larger set sizes approximately (5-10). Children who had not yet learned the cardinal principle were more than twice as accurate when labeling sets of 2 and 3 items with gestures than with words, particularly if the values were above their knower-level. They were also better at approximating set sizes 5-10 with gestures than with words. Further, gesture was more accurate when it differed from the accompanying speech (i.e., a gesture-speech mismatch). These results show that children convey numerical information in gesture that they cannot yet convey in speech, and raise the possibility that number gestures play a functional role in children's development of number concepts.

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.

Young children's mapping between arrays, number words, and digits

This study investigates when young children develop the ability to map between three numerical representations: arrays, spoken number words, and digits. Children (3, 4, and 5 years old) had to map between the two directions (e.g., array-to-digit vs. digit-to-array) of each of these three representation pairs, with small (1-3) and large numbers (4-6). Five-year-olds were at ceiling in all tasks. Three-year-olds succeeded when mapping between arrays and number words for small numbers (but not large numbers), and failed when mapping between arrays and digits and between number words and digits. The main finding was that four-year-olds performed equally well when mapping between arrays and number words and when mapping between arrays and digits. However, they performed more poorly when mapping between number words and digits. Taken together, these results suggest that children first learn to map number words to arrays, then learn to map digits to arrays and finally map number words to digits. These findings highlight the importance of directly exploring when children acquire digits rather than assuming that they acquire digits directly from number words.

What exactly do numbers mean?

Number words are generally used to refer to the exact cardinal value of a set, but cognitive scientists disagree about their meanings. Although most psychological analyses presuppose that numbers have exact semantics (two means EXACTLY TWO), many linguistic accounts propose that numbers have lower-bounded semantics (AT LEAST TWO), and that speakers restrict their reference through a pragmatic inference (scalar implicature). We address this debate through studies of children who are in the process of acquiring the meanings of numbers. Adults and 2- and 3-year-olds were tested in a novel paradigm that teases apart semantic and pragmatic aspects of interpretation (the covered box task). Our findings establish that when scalar implicatures are cancelled in the critical trials of this task, both adults and children consistently give exact interpretations for number words. These results, in concert with recent work on real-time processing, provide the first unambiguous evidence that number words have exact semantics.

Cognitive processes of numerical estimation in children

We tested children in Grades 1 to 5, as well as college students, on a number line estimation task and examined latencies and errors to explore the cognitive processes involved in estimation. The developmental trends in estimation were more consistent with the hypothesized shift from logarithmic to linear representation than with an account based on a proportional judgment application of a power function model; increased linear responding across ages, as predicted by the log-to-lin shift position, yielded reasonable developmental patterns, whereas values derived from the cyclical power model were difficult to reconcile with expected developmental patterns. Neither theoretical position predicted the marked "M-shaped" pattern that was observed, beginning in third graders' errors and fourth graders' latencies. This pattern suggests that estimation comes to rely on a midpoint strategy based on children's growing number knowledge (i.e., knowledge that 50 is half of 100). As found elsewhere, strength of linear responding correlated significantly with children's performance on standardized math tests.

The enigma of number: why children find the meanings of even small number words hard to learn and how we can help them do better

Although number words are common in everyday speech, learning their meanings is an arduous, drawn-out process for most children, and the source of this delay has long been the subject of inquiry. Children begin by identifying the few small numerosities that can be named without counting, and this has prompted further debate over whether there is a specific, capacity-limited system for representing these small sets, or whether smaller and larger sets are both represented by the same system. Here we present a formal, computational analysis of number learning that offers a possible solution to both puzzles. This analysis indicates that once the environment and the representational demands of the task of learning to identify sets are taken into consideration, a continuous system for learning, representing and discriminating set-sizes can give rise to effective discontinuities in processing. At the same time, our simulations illustrate how typical prenominal linguistic constructions ("there are three balls") structure information in a way that is largely unhelpful for discrimination learning, while suggesting that postnominal constructions ("balls, there are three") will facilitate such learning. A training-experiment with three-year olds confirms these predictions, demonstrating that rapid, significant gains in numerical understanding and competence are possible given appropriately structured postnominal input. Our simulations and results reveal how discrimination learning tunes children's systems for representing small sets, and how its capacity-limits result naturally out of a mixture of the learning environment and the increasingly complex task of discriminating and representing ever-larger number sets. They also explain why children benefit so little from the training that parents and educators usually provide. Given the efficacy of our intervention, the ease with which it can be implemented, and the large body of research showing how early numerical ability predicts later educational outcomes, this simple discovery may have far-reaching consequences.

Quinian bootstrapping or Fodorian combination? Core and constructed knowledge of number

According to Carey (2009), humans construct new concepts by abstracting structural relations among sets of partly unspecified symbols, and then analogically mapping those symbol structures onto the target domain. Using the development of integer concepts as an example, I give reasons to doubt this account and to consider other ways in which language and symbol learning foster conceptual development.

Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling

Children take years to learn symbolic arithmetic. Nevertheless, non-human animals, human adults with no formal education, and human infants represent approximate number in arrays of objects and sequences of events, and they use these capacities to perform approximate addition and subtraction. Do children harness these abilities when they begin to learn school mathematics? In two experiments in different schools, kindergarten children from diverse backgrounds were tested on their non-symbolic arithmetic abilities during the school year, as well as on their mastery of number words and symbols. Performance of non-symbolic arithmetic predicted children's mathematics achievement at the end of the school year, independent of achievement in reading or general intelligence. Non-symbolic arithmetic performance was also related to children's mastery of number words and symbols, which figured prominently in the assessments of mathematics achievement in both schools. Thus, non-symbolic and symbolic numerical abilities are specifically related, in children of diverse socio-economic backgrounds, near the start of mathematics instruction.