One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles

Dragging but not tapping promotes preschoolers' numerical estimating with touchscreens

When solving mathematical problems, young children will perform better when they can use gestures that match mental representations. However, despite their increasing prevalence in educational settings, few studies have explored this effect in touchscreen-based interactions. Thus, we investigated the impact on young children's performance of dragging (where a continuous gesture is performed that is congruent with the change in number) and tapping (involving a discrete gesture that is incongruent) on a touchscreen device when engaged in a continuous number line estimation task. By examining differences in the set size and position of the number line estimation, we were also able to explore the boundary conditions for the superiority effect of congruent gestures. We used a 2 (Gesture Type: drag or tap) × 2 (Set Size: Set 0-10 or Set 0-20) × 2 (Position: left of midpoint or right of midpoint) mixed design. A total of 70 children aged 5 and 6 years (33 girls) were recruited and randomly assigned to either the Drag or Tap group. We found that the congruent gesture (drag) generally facilitated better performance with the touchscreen but with boundary conditions. When completing difficult estimations (right side in the large set size), the Drag group was more accurate, responded to the stimulus faster, and spent more time manipulating than the Tap group. These findings suggest that when children require explicit scaffolding, congruent touchscreen gestures help to release mental resources for strategic adjustments, decrease the difficulty of numerical estimation, and support constructing mental representations.

The influence of language on the formation of number concepts: Evidence from preschool children who are bilingual in English and Arabic

We asked whether grammatical number marking has specific influence on the formation of early number concepts. In particular, does comprehension of dual case marking support young children's understanding of cardinality? We assessed number knowledge in 77 3-year-old Arabic-English bilingual children using the Give-a-Number task in both languages. Given recent concerns around the administration and scoring of the Give-a-Number task, we used two complementary approaches: one based on conceptual levels and the other based on overall test scores. We also tested comprehension of dual case marking in Arabic and number sequence knowledge in both languages. Regression analyses showed that dual case comprehension exerts a strong influence on cardinality tested in Arabic independent of age, general language skills, and number sequence knowledge. No such influence was found for cardinality tested in English, indicating a language-specific effect. Further analyses tested for transfer of cardinality knowledge between languages. These revealed, in addition to the findings outlined above, a powerful cross-linguistic transfer effect. Our findings are consistent with a model in which the direct effect of dual case marking is language specific, but concepts, once acquired, may be represented abstractly and transferred between languages.

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.

Mapping skills between symbols and quantities in preschoolers: The role of finger patterns

Mapping skills between different codes to represent numerical information, such as number symbols (i.e., verbal number words and written digits) and non-symbolic quantities, are important in the development of the concept of number. The aim of the current study is to investigate children's mapping skills by incorporating another numerical code that emerges at early stages in development, finger patterns. Specifically, the study investigates (i) the order in which mapping skills develop and the association with young children's understanding of cardinality; and (ii) whether finger patterns are processed similarly to symbolic codes or rather as non-symbolic quantities. Preschool children (3-year-olds, N = 113, Mage = 40.8 months, SDage = 3.6 months; 4-year-olds, N = 103, Mage = 52.9 months, SDage = 3.4 months) both cardinality knowers and subset-knowers, were presented with twelve tasks that assessed the mappings between number words, Arabic digits, finger patterns, and quantities. The results showed that children's ability to map symbolic numbers precedes the understanding that such symbols reflect quantities, and that children recognize finger patterns above their cardinality knowledge, suggesting that finger patterns are symbolic in essence. RESEARCH HIGHLIGHTS: Children are more accurate in mapping between finger patterns and symbols (number words and Arabic digits) than in mapping finger patterns and quantities, indicating that fingers are processed holistically as symbolic codes. Children can map finger patterns to symbols above their corresponding cardinality level even in subset-knowers. Finger patterns may play a role in the process by which children learn to map symbols to quantities. Fingers patterns' use in the classroom context may be an adequate instructional and diagnostic tool.

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.

Perceptual subitizing performance in 3- and 4-year-olds: The impact of visual features of sets

Perceptual subitizing is a pivotal skill in children's mathematical development. It is defined as the rapid identification of small numerosities. Previous studies pointed to the contribution of visual features of sets to perceptual subitizing performance in adults. Insights into the contribution of visual features to subitizing performance in the critical 3- to 4-year age range are scant. This study aimed to address this gap by investigating the impact of visual features on perceptual subitizing performance (accuracy and response time) in 3- and 4-year-olds. Participants (119 3- and 4-year-olds) were offered a subitizing task that incorporated pictures of sets of three to five objects. The pictures systematically varied across four visual features: (a) pictorial context (distractors present vs. absent), (b) set homogeneity (homogeneous vs. heterogeneous objects), (c) set arrangement (linearly vs. randomly arranged objects), and (d) set differentiation (distinct vs. overlapping objects). Pictures with distractors, heterogeneous objects, randomly arranged objects, or overlapping objects were associated with lower subitizing accuracy and longer response times compared with pictures without distractors, homogeneous objects, linearly arranged objects, or distinct objects, respectively. Pictures with randomly arranged or overlapping objects along with distractors were associated with even lower subitizing accuracy. Pictures featuring a simple visual design-without distractors and with homogeneous, linearly arranged, and distinct sets-yielded the best subitizing performance in terms of accuracy and response time. Our findings might be explained by the cognitive processes underlying 3- and 4-year-olds' subitizing performance. The findings offer building blocks for future research in the domain and preschool educational practice.

Thinking about numbers in different tongues: An overview of the influences of multilingualism on numerical and mathematical competencies

In an increasingly multilingual and multicultural world, understanding the interactions between language and mathematics is critical, especially when individuals must acquire and exercise their mathematical competencies in multiple languages. Indeed, research shows that, overall, L2 language learners are at an academic disadvantage compared to their L1 peers. The current article briefly overviews how multilingualism influences basic and advanced mathematical skills and interacts with mathematical learning difficulties. We first outline the traditional cognitive models of number learning and language processing. We then discuss the particularities of multilingualism and how it impacts numerical skills such as counting and building lexical-semantic associations, transcoding and arithmetic, mathematical word problems and mathematical performance tests, and dyscalculia diagnosis. We end this review by outlining challenges, recommendations, and solutions for multilingual educational settings. The article is intended as a guide for numerical cognition researchers who work with diverse populations and for mathematics educators and educational policy-makers facing the challenges of a multilingual classroom.

"Catastrophic" set size limits on infants' capacity to represent objects: A systematic review and Bayesian meta-analysis

Decades of research has revealed that humans can concurrently represent small quantities of three-dimensional objects as those objects move through space or into occlusion. For infants (but not older children or adults), this ability apparently comes with a significant limitation: when the number of occluded objects exceeds three, infants experience what has been characterized as a "catastrophic" set size limit, failing to represent even the approximate quantity of the hidden array. Infants' apparent catastrophic representational failures suggest a significant information processing limitation in the first years of life, and the evidence has been used as support for prominent theories of the development of object and numerical cognition. However, the evidence for catastrophic failure consists of individual small-n experiments that use null hypothesis significance testing to obtain null results (i.e., p > 0.05). Whether catastrophic representational failures are robust or reliable across studies, methods, and labs is not known. Here we report a systematic review and Bayesian meta-analysis to examine the strength of the evidence in favor of catastrophic representational failures in infancy. Our analysis of 22 experiments across 12 reports, with a combined total of n = 367 infants aged 10-20 months, revealed strong support for the evidence for catastrophic set size limits. A complementary analysis found moderate support for infants' success when representing fewer than four objects. We discuss the implications of our findings for theories of object and numerical cognitive development. RESEARCH HIGHLIGHTS: Previous work has suggested that infants are unable to concurrently represent four or more objects-a "catastrophic" set size limit. We reviewed this work and conducted a Bayesian meta-analysis to examine the robustness of this limit across individual small-n experiments. We found strong support for the evidence for catastrophic set size limits, and moderate support for infants' success when representing fewer than four objects.

Children dynamically update and extend the interface between number words and perceptual magnitudes

As adults, we represent and think about number, space, and time in at least two ways: our intuitive-but imprecise-perceptual representations, and the slowly learned-but precise-number words. With development, these representational formats interface, allowing us to use precise number words to estimate imprecise perceptual experiences. We test two accounts of this developmental milestone. Either slowly learned associations are required for the interface to form, predicting that deviations from typical experiences (e.g., presentation of a novel unit or unpracticed dimension) will disrupt children's ability to map number words to their perceptual experiences or children's understanding of the logical similarity between number words and perceptual representations allows them to flexibly extend this interface to novel experiences (e.g., units and dimensions they have not yet learned how to formally measure). 5-11-year-olds completed verbal estimation and perceptual sensitivity tasks across three dimensions: Number, Length, and Area. For verbal estimation, they were given novel units (i.e., a three-dot unit called one "toma" for Number, a 44 px long line called one "blicket" for Length, a 111 px2 blob called one "modi" for Area) and asked to estimate how many tomas/blickets/modies they saw when shown a larger set of dots, lines, and blobs. Children could flexibly link number words to novel units across dimensions, demonstrating positive estimation slopes, even for Length and Area, which younger children had limited experience with. This suggests that the logic of structure mapping can be dynamically utilized across perceptual dimensions, even without extensive experience.

The longitudinal contribution of mapping to arithmetic: Do numeral knowledge, inhibition or analogical reasoning matter?

BACKGROUND

Recent studies have revealed the association between mapping and arithmetic (Ferres-Forga et al., J. Numer. Cogn., 8, 2022, 123; LeFevre et al., J. Numer. Cogn., 8, 2022, 1).

AIM

The underlying mechanism remains unclear.

MATERIALS & METHODS

The current study recruited 118 kindergarten children and followed up on them three times at 6-month intervals. They completed measures to assess mapping skills (T1), non-verbal IQ (T1), numeral knowledge (T2), inhibitory control (T2), analogical reasoning (T2) and arithmetic (T3).

RESULTS

The results showed that mapping accounted for significant variance in arithmetic ability over and above age, gender and non-verbal IQ. Furthermore, analogical reasoning played an important role in the relationship between mapping and mathematics ability.

DISCUSSION

The findings suggest the association between mapping and mathematics ability prior to formal schooling.

CONCLUSION

Analogical reasoning, rather than numeral knowledge or inhibitory control, may drive that association in young children.

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.

Early access to language supports number mapping skills in deaf children

The ability to associate different types of number representations referring to the same quantity (symbolic Arabic numerals, signed/spoken number words, and nonsymbolic quantities), is an important predictor of overall mathematical success. This foundational skill-mapping-has not been examined in deaf and hard-of-hearing (DHH) children. To address this gap, we studied 188 4 1/2 to 9-year-old DHH and hearing children and systematically examined the relationship between their language experiences and mapping skills. We asked whether the timing of children's language exposure (early vs. later), the modality of their language (signed vs. spoken), and their rote counting abilities related to mapping performance. We found that language modality did not significantly relate to mapping performance, but timing of language exposure and counting skills did. These findings suggest that early access to language, whether spoken or signed, supports the development of age-typical mapping skills and that knowledge of number words is critical for this development.

Conceptual foundations of early numeracy: Evidence from infant brain data

Understanding the conceptual resources that children bring to mathematics learning is crucial for developing effective instruction and interventions. Despite the considerable number of studies examining the neural underpinnings of number representations in adults and the growing number of reports in children, very few studies have examined the neural correlates of the link between foundational resources related to numerical information and symbolic number representations in infants. There is currently an active debate about which foundational resources are critical for symbolic mathematics. Is early numerical discrimination best explained by a holistic and generalized sense of magnitude rather than a number sense? Does early number sense provide the conceptual basis for mapping numerical symbols to their meaning? Are foundational number systems marginal while children learn to count and perform symbolic arithmetic, and only later children map non symbolic representations of numerical magnitudes onto symbols? After describing the mainstream theories of numerical cognition and the sources of controversy, we review recent studies of the neural bases of human infants' numerical performance with the aim of clarifying the link between early conceptual resources and symbolic number systems as children's mathematical minds develop.

Number sense at 12 months predicts 4-year-olds' maths skills

Preverbal infants spontaneously represent the number of objects in collections. Is this 'sense of number' (also referred to as Approximate Number System, ANS) part of the cognitive foundations of mathematical skills? Multiple studies reported a correlation between the ANS and mathematical achievement in children. However, some have suggested that such correlation might be mediated by general-purpose inhibitory skills. We addressed the question using a longitudinal approach: we tested the ANS of 60 12 months old infants and, when they were 4 years old (final N = 40), their symbolic math achievement as well as general intelligence and inhibitory skills. Results showed that the ANS at 12 months is a specific predictor of later maths skills independent from general intelligence or inhibitory skills. The correlation between ANS and maths persists when both abilities are measured at four years. These results confirm that the ANS has an early, specific and longstanding relation with mathematical abilities in childhood. RESEARCH HIGHLIGHTS: In the literature there is a lively debate about the correlation between the ANS and maths skills. We longitudinally tested a sample of 60 preverbal infants at 12 months and rested them at 4 years (final sample of 40 infants). The ANS tested at 12 months predicted later symbolic mathematical skills at 4 years, even when controlling for inhibition, general intelligence and perceptual skills. The ANS tested at 4 years remained linked with symbolic maths skills, confirming this early and longstanding relation in childhood.

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.

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.

What aspects of counting help infants attend to numerosity?

Recent work shows that 18-month old infants understand that counting is numerically relevant-infants who see objects counted are more likely to represent the approximate number of objects in the array than infants who see the objects labeled but not counted. Which aspects of counting signal infants to attend to numerosity in this way? Here we asked whether infants rely on familiarity with the count words in their native language, or on procedures instantiated by the counting routine, independent of specific tokens. In three experiments (N = 48), we found that 18-month old infants from English-speaking households successfully distinguished four hidden objects from two when the objects were counted correctly, regardless of their familiarity with the count words (i.e., when objects were counted in familiar English and in unfamiliar German). However, when the objects were counted using familiar English count words in ways that violated basic counting principles, infants no longer represented the arrays, failing to distinguish four hidden objects from two. Together with previous findings, these results suggest that children may link the procedure of counting with numerosity years before they learn the meanings of the count words.

The plural counts: Inconsistent grammatical number hinders numerical development in preschoolers - A cross-linguistic study

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiting differences between Polish, which has a complex grammatical number paradigm, leading to a partially inconsistent mapping between numerical quantities and grammatical number, and German, which has a comparatively easy verbal paradigm: 151 Polish-speaking and 123 German-speaking kindergarten children were tested using a symbolic numerical comparison task. Additionally, counting skills (Give-a-Number and count-list), and mapping between non-symbolic (dot sets) and symbolic representations of numbers, as well as working memory (Corsi blocks and Digit span) were assessed. Based on the Give-a-Number and mapping tasks, the children were divided into subset-knowers, CP-knowers-non-mappers, and CP-knowers-mappers. Linguistic background was related to performance in several ways: Polish-speaking children expectedly progressed to the CP-knowers stage later than German children, despite comparable non-numerical capabilities, and even after this stage was achieved, they fared worse in the numerical comparison task. There were also meaningful differences in spatial-numerical mapping between the Polish and German groups. Our findings are in line with the theory that grammatical number paradigms influence. the development of representations and processing of numbers, not only at the stage of acquiring the meaning of the first number-words but at later stages as well, when dealing with symbolic numbers.

Recursive sequence generation in crows

Recursion, the process of embedding structures within similar structures, is often considered a foundation of symbolic competence and a uniquely human capability. To understand its evolution, we can study the recursive aptitudes of nonhuman animals. We adopted the behavioral protocol of a recent study demonstrating that humans and nonhuman primates grasp recursion. We presented sequences of bracket pair stimuli (e.g., [ ] and { }) to crows who were instructed to peck at training lists. They were then tested on their ability to transfer center-embedded structure to never-before-seen pairings of brackets. We reveal that crows have recursive capacities; they perform on par with children and even outperform macaques. The crows continued to produce recursive sequences after extending to longer and thus deeper embeddings. These results demonstrate that recursive capabilities are not limited to the primate genealogy and may have occurred separately from or before human symbolic competence in different animal taxa.

The cultural origins of symbolic number

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

More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3–5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

Preschoolers prior formal mathematics education engage numerical magnitude representation rather than counting principles in symbolic +/-1 arithmetic: Evidence from the Operational Momentum effect

In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This study is the first to demonstrate OM in symbolic arithmetic in preschoolers. It was modeled on Haman and Lipowska's (2021) non-symbolic arithmetic task, using Arabic numerals instead of visual sets. Seventy-seven children (4-7 years old) who know Arabic numerals and counting principles, but without prior school math education, solved addition and subtraction problems presented as videos with 1 as the second operand. In principle, such problems may be difficult when involving a non-symbolic approximate number processing system, whereas in symbolic format they can be solved based solely on the successor/predecessor functions and knowledge of numerical orders, without reference to representation of numerical magnitudes. Nevertheless, participants made systematic errors, in particular, overestimating results of addition in line with the typical OM tendency. Moreover, subtraction and addition induced longer response times when primed with left- and right-directed movement, respectively, which corresponds to the reversed spatial form of OM. These results largely replicate those of non-symbolic task and show that children at early stages of mastering symbolic arithmetic may rely on numerical magnitude processing and spatial-numerical associations rather than newly-mastered counting principles and the concept of an exact number. Adding and subtracting 1 in a symbolic format formally requires only knowledge of numerical orders and the predecessor/successor function, but not numerical magnitude processing Preschoolers knowing the counting principles and Arabic numerals, but without prior mathematics education, demonstrated operational momentum by overestimating results of symbolic addition of 1 In the same arithmetic task children showed faster reactions for addition primed with an object moving leftward and in subtraction primed with rightward motion These effects replicate findings from non-symbolic ±1 arithmetic, indicating that preschoolers use magnitude representation and spatial-numerical associations for symbolic calculation This article is protected by copyright. All rights reserved.

What children's number naming errors tell us about early understanding of multidigit numbers

This investigation examined the nature and development of a foundational symbolic numeric skill-number identification-focusing on children's emerging knowledge of multidigit numbers. Two studies were conducted with Russian preschoolers. Study 1 (N = 350; 51-77 months of age) investigated age-related changes in the accuracy of number naming and in the types of errors children produced. The errors fell into distinct categories: syntactic (structural errors such as naming each digit separately without using place-value markers) and lexical (nonstructural errors such as replacing the name of a digit with the name of another digit). Number reading accuracy improved with age, primarily due to a decreased frequency of syntactic errors. Boys made fewer syntactic errors than girls. Study 2 (N = 110; 61-74 months of age) showed that accuracy of naming double-digit numbers was related to conceptual understanding of the base-10 numeric structure. The frequency of syntactic errors in number naming was negatively associated with the use of base-10 representations, whereas lexical errors were not related to children's ability to represent base-10 number structure. Implications for understanding children's mathematics trajectories are discussed.

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.

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.

Can training in the approximate number system improve the informal mathematics ability of preschoolers?

Recent studies show that comparison or arithmetic training in the approximate number system (ANS) can improve the early mathematics ability of preschool children. However, no studies have compared the training effects of ANS comparison training with those of ANS arithmetic training on the early mathematics ability of preschool children. The current study pseudorandomly assigned 87 children aged 4-5 years to one of three training groups (the ANS comparison, ANS arithmetic, and control groups) for 4 weeks of training. The results showed that compared with the control group, the ANS comparison training and ANS arithmetic training equally improved the ANS acuity and informal mathematics ability of preschool children. In addition, the study found that there may be a bidirectional causal relationship between ANS and mathematics in preschoolers, but this relationship needs to be further investigated using longitudinal studies. Taken together, these findings emphasize the importance of ANS-based training in improving preschoolers' ANS acuity and informal mathematics ability before formal school enrollment.

Exact Number Concepts Are Limited to the Verbal Count Range

Previous findings suggest that mentally representing exact numbers larger than four depends on a verbal count routine (e.g., "one, two, three . . ."). However, these findings are controversial because they rely on comparisons across radically different languages and cultures. We tested the role of language in number concepts within a single population-the Tsimane' of Bolivia-in which knowledge of number words varies across individual adults. We used a novel data-analysis model to quantify the point at which participants (N = 30) switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switch points were bounded by participants' verbal count ranges; their representations of exact cardinalities were limited to the number words they knew. Beyond that range, they resorted to numerical approximation. These results resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.

Assessing the knower-level framework: How reliable is the Give-a-Number task?

The Give-a-Number task has become a gold standard of children's number word comprehension in developmental psychology. Recently, researchers have begun to use the task as a predictor of other developmental milestones. This raises the question of how reliable the task is, since test-retest reliability of any measure places an upper bound on the size of reliable correlations that can be found between it and other measures. In Experiment 1, we presented 81 2- to 5-year-old children with Wynn (1992) titrated version of the Give-a-Number task twice within a single session. We found that the reliability of this version of the task was high overall, but varied importantly across different assigned knower levels, and was very low for some knower levels. In Experiment 2, we assessed the test-retest reliability of the non-titrated version of the Give-a-Number task with another group of 81 children and found a similar pattern of results. Finally, in Experiment 3, we asked whether the two versions of Give-a-Number generated different knower levels within-subjects, by testing 75 children with both tasks. Also, we asked how both tasks relate to another commonly used test of number knowledge, the "What's-On-This-Card" task. We found that overall, the titrated and non-titrated versions of Give-a-Number yielded similar knower levels, though the non-titrated version was slightly more conservative than the titrated version, which produced modestly higher knower levels. Neither was more closely related to "What's-On-This-Card" than the other. We discuss the theoretical and practical implications of these results.

Numerical Training Videos and Early Numerical Achievement: A Study on 3-Year-Old Preschoolers

Early numerical abilities predict later math achievement and could be improved in children by using various training methods. As the literature on the use of training videos to develop numerical abilities is still surprisingly scant, the aim of the present study was to test the efficacy of a numerical training video on the development of counting and number line knowledge in 3-year-old preschoolers. Far transfer effects to cardinality and working memory were also examined. The study involved 86 children randomly assigned to two intervention groups: a numerical training group exposed to videos on counting and number lines; and a control group exposed to videos on colors and animal names in a foreign language. After the video training, there was an improvement in the numerical training group's counting skills, but not in their number line knowledge, and this improvement persisted six months later. The numerical training group also showed a far-transfer enhancement of cardinality six months after the intervention. Based on our results, numerical training videos could be effective in helping to enhance early numeracy skills in very young preschoolers.

Constructing rationals through conjoint measurement of numerator and denominator as approximate integer magnitudes in tradeoff relations

To investigate mechanisms of rational representation, I consider (1) construction of an ordered continuum of psychophysical scale of magnitude of sensation; (2) counting mechanism leading to an approximate numerosity scale for integers; and (3) conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

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

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

The neural basis of number word processing in children and adults

The ability to map number words to their corresponding quantity representations is a gatekeeper for children's future math success (Spaepen et al., 2018). Without number word knowledge at school entry, children are at greater risk for developing math learning difficulties (Chu et al., 2019). In the present study, we used functional magnetic resonance imaging (fMRI) to examine the neural basis for processing the meaning of spoken number words and its developmental trajectory in 4- to 10-year-old children, and in adults. In a number word-quantity mapping paradigm, participants listened to number words while simultaneously viewing quantities that were congruent or incongruent to the number word they heard. Whole brain analyses revealed that adults showed a neural congruity effect with greater neural activation for incongruent relative to congruent trials in anterior cingulate cortex (ACC) and left intraparietal sulcus (LIPS). In contrast, children did not show a significant neural congruity effect. However, a region of interest analysis in the child sample demonstrated age-related increases in the neural congruity effect, specifically in the LIPS. The positive correlation between neural congruity in LIPS and age was stronger in children who were already attending school, suggesting that developmental changes in LIPS function are experience-dependent.

Preschoolers' mastery of advanced counting: The best predictor of addition skills 2 years later

The current study addressed the following question: Among preschoolers' basic numerical abilities, what are the best predictors for the later addition skills? We measured numerical abilities at preschool age and used dominance analysis to determine the dominant predictor for addition skills 2 years later. We tested seven numerical specific predictors (counting, advanced counting, enumeration, Give-N, collection comparison, number-word comparison, and approximate addition). Both quantitative and qualitative aspects (accuracy, strategy choice, and fluency) of addition skills were measured. The results show that the predictor weights for addition skills were 39% (counting), 37% (advanced counting), and 25% (collection comparison). We concluded that counting ability and especially advanced counting measured in early preschool is the most robust predictor of addition skills 2 years later (even after controlling for global cognitive abilities). This study generalized the previous findings found for Western children to Vietnamese preschoolers (N = 157, M_age = 4.8 years); extended and highlighted the role of advanced counting (count from a number other than 1) to later addition performance, mature strategy, and calculation fluency; and suggested further implications.

Development of Preschoolers' Understanding of Zero

While knowledge on the development of understanding positive integers is rapidly growing, the development of understanding zero remains not well-understood. Here, we test several components of preschoolers’ understanding of zero: Whether they can use empty sets in numerical tasks (as measured with comparison, addition, and subtraction tasks); whether they can use empty sets soon after they understand the cardinality principle (cardinality-principle knowledge is measured with the give-N task); whether they know what the word “zero” refers to (tested in all tasks in this study); and whether they categorize zero as a number (as measured with the smallest-number and is-it-a-number tasks). The results show that preschoolers can handle empty sets in numerical tasks as soon as they can handle positive numbers and as soon as, or even earlier than, they understand the cardinality principle. Some also know that these sets are labeled as “zero.” However, preschoolers are unsure whether zero is a number. These results identify three components of knowledge about zero: operational knowledge, linguistic knowledge, and meta-knowledge. To account for these results, we propose that preschoolers may understand numbers as the properties of items or objects in a set. In this view, zero is not regarded as a number because an empty set does not include any items, and missing items cannot have any properties, therefore, they cannot have the number property either. This model can explain why zero is handled correctly in numerical tasks even though it is not regarded as a number.

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.

Electrophysiological evidence for internalized representations of canonical finger-number gestures and their facilitating effects on adults' math verification performance

Fingers facilitate number learning and arithmetic processing in early childhood. The current study investigated whether images of early-learned, culturally-typical (canonical), finger montring patterns presenting smaller (2,3,4) or larger (7,8,9) quantities still facilitate adults' performance and neural processing in a math verification task. Twenty-eight adults verified solutions to simple addition problems that were shown in the form of canonical or non-canonical finger-number montring patterns while measuring Event Related Potentials (ERPs). Results showed more accurate and faster sum verification when sum solutions were shown by canonical (versus non-canonical) finger patterns. Canonical finger montring patterns 2-4 led to faster responses independent of whether they presented correct or incorrect sum solutions and elicited an enhanced early right-parietal P2p response, whereas canonical configurations 7-9 only facilitated performance in correct sum solution trials without evoking P2p effects. The later central-parietal P3 was enhanced to all canonical finger patterns irrespective of numerical range. These combined results provide behavioral and brain evidence for canonical cardinal finger patterns still having facilitating effects on adults' number processing. They further suggest that finger montring configurations of numbers 2-4 have stronger internalized associations with other magnitude representations, possibly established through their mediating role in the developmental phase in which children acquire the numerical meaning of the first four number symbols.