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

Integration of symbolic and non-symbolic numerical information in children: Task dependence and its link to math abilities

From birth, children can access the approximate number system for noisy numerical estimates. With age, they acquire an exact number system for precise numerical information representation. The relations between these two systems and their correlations with math abilities in children remain unclear. In this study, 8- to 10-year-old children (N = 119) completed two tasks to test the integration of symbolic and non-symbolic numerical information (i.e., "symbolic integration") and how this integration relates to children's formal math abilities. For the number comparison task, involving dot arrays and Arabic numerals, children indicated which of two sequentially presented stimuli was larger. These stimuli were either in the same format (dot-dot or numeral-numeral) or in a mixed format (dot-numeral or numeral-dot). For the number-letter discrimination task, participants identified numerals or letter pairs co-occurring with dot arrays that either matched or mismatched the numeral's quantity. In the number comparison task, we found that children were significantly slower when comparing mixed-format stimuli versus same-format conditions, suggesting a lack of symbolic integration (i.e., "symbolic estrangement"). In contrast, in the number-letter discrimination task, children were significantly faster in tasks where the dot arrays and numerals matched, indicating symbolic integration. While we found correlations between number processing and math skills at the condition level for both tasks, neither of the derived measures of symbolic estrangement or symbolic integration correlated with children's performance on a standardized math assessment. Thus, we conclude that numerical integration or estrangement is task dependent and that symbolic integration has limited impact on 8- to 10-year-old children's math abilities.

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.

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.

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.

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.

A-B-3-Associations and dissociations of reading and arithmetic: Is domain-specific prediction outdated?

Reading and arithmetic are core domains of academic achievement with marked impact on career opportunities and socioeconomic status. While associations between reading and arithmetic are well established, evidence on underlying mechanisms is inconclusive. The main goal of this study was to reevaluate the domain-specificity of established predictors and to enhance our understanding of the (co-)development of reading and arithmetic. In a sample of 885 German-speaking children, standard domain-specific predictors of reading and arithmetic were assessed before and/or at the onset of formal schooling. Reading and arithmetic skills were measured at the beginning and end of second grade. Latent variables were extracted for all relevant constructs: Grapheme-phoneme processing (phonological awareness, letter identification), RAN (RAN-objects, RAN-digits), number system knowledge (number identification, successor knowledge), and magnitude processing (non-symbolic and symbolic magnitude comparison), as well as the criterion measures reading and arithmetic. Four structural equation models tested distinct research questions. Grapheme-phoneme processing was a specific predictor of reading, and magnitude processing explained variance specific to arithmetic. RAN explained variance in both domains, and it explained variance in reading even after controlling for arithmetic. RAN and number system knowledge further explained variance in skills shared between reading and arithmetic. Reading and arithmetic entail domain-specific cognitive components, and they both require tight networks of visual, verbal, and semantic information, as reflected by RAN. This perspective provides a useful background to explain associations and dissociations between reading and arithmetic performance.

Changing priorities in the development of cognitive competence and school learning: A general theory

This paper summarizes a theory of cognitive development and elaborates on its educational implications. The theory postulates that development occurs in cycles along multiple fronts. Cognitive competence in each cycle comprises a different profile of executive, inferential, and awareness processes, reflecting changes in developmental priorities in each cycle. Changes reflect varying needs in representing, understanding, and interacting with the world. Interaction control dominates episodic representation in infancy; attention control and perceptual awareness dominate in realistic representations in preschool; inferential control and awareness dominate rule-based representation in primary school; truth and validity control and precise self-evaluation dominate in principle-based thought in adolescence. We demonstrate that the best predictors of school learning in each cycle are the cycle's cognitive priorities. Also learning in different domains, e.g., language and mathematics, depends on an interaction between the general cognitive processes dominating in each cycle and the state of the representational systems associated with each domain. When a representational system is deficient, specific learning difficulties may emerge, e.g., dyslexia and dyscalculia. We also discuss the educational implications for evaluation and learning at school.

Development of information integration in the visual working memory of preschoolers

Visual working memory (WM) plays a pivotal role in integrating fragments into meaningful units, but no study has addressed how visual WM integration takes place in children. The current study examined whether WM integration emerges once preschoolers master Gestalt cue and can retain two representations in WM (automatic integration hypothesis), or still needs time to mature (maturation-of-integration hypothesis). Four experiments (N = 168, 81 females, 4- to 6-year-olds, Chinese, in Hangzhou, China, from 2016.10 to 2021.11) were conducted. Although 4-year-olds can retain two objects in WM and benefit from Gestalt cues in simultaneous display (Cohen's ds >1.00), they failed when memory arrays were presented sequentially. Meanwhile, 5- and 6-year-olds consistently demonstrated WM integration ability (all Cohen's ds >0.69), supporting the maturation-of-integration hypothesis.

Automatic integration of numerical formats examined with frequency-tagged EEG

How humans integrate and abstract numerical information across different formats is one of the most debated questions in human cognition. We addressed the neuronal signatures of the numerical integration using an EEG technique tagged at the frequency of visual stimulation. In an oddball design, participants were stimulated with standard sequences of numbers (< 5) depicted in single (digits, dots, number words) or mixed notation (dots-digits, number words-dots, digits-number words), presented at 10 Hz. Periodically, a deviant stimulus (> 5) was inserted at 1.25 Hz. We observed significant oddball amplitudes for all single notations, showing for the first time using this EEG technique, that the magnitude information is spontaneously and unintentionally abstracted, irrespectively of the numerical format. Significant amplitudes were also observed for digits-number words and number words-dots, but not for digits-dots, suggesting an automatic integration across some numerical formats. These results imply that direct and indirect neuro-cognitive links exist across the different numerical formats.

Measuring spontaneous and automatic processing of magnitude and parity information of Arabic digits by frequency-tagging EEG

Arabic digits (1–9) are everywhere in our daily lives. These symbols convey various semantic information, and numerate adults can easily extract from them several numerical features such as magnitude and parity. Nonetheless, since most studies used active processing tasks to assess these properties, it remains unclear whether and to what degree the access to magnitude and especially to parity is automatic. Here we investigated with EEG whether spontaneous processing of magnitude or parity can be recorded in a frequency-tagging approach, in which participants are passively stimulated by fast visual sequences of Arabic digits. We assessed automatic magnitude processing by presenting a stream of frequent small digit numbers mixed with deviant large digits (and the reverse) with a sinusoidal contrast modulation at the frequency of 10 Hz. We used the same paradigm to investigate numerical parity processing, contrasting odd digits to even digits. We found significant brain responses at the frequency of the fluctuating change and its harmonics, recorded on electrodes encompassing right occipitoparietal regions, in both conditions. Our findings indicate that both magnitude and parity are spontaneously and unintentionally extracted from Arabic digits, which supports that they are salient semantic features deeply associated to digit symbols in long-term memory.

When do preschoolers learn specific mathematics skills? Mapping the development of early numeracy knowledge

Children's mathematics skills undergo extensive development during the preschool years. Opportunities for engaging in mathematics in the preschool classroom, however, are limited, and activities and instruction are often targeted below children's skill levels. Although researchers have developed general learning trajectories of children's mathematics skills, no fine-grained trajectories across a broad range of mathematics skills exist. Such a fine-grained trajectory of when specific numeracy skills develop would allow preschool and kindergarten instruction to more appropriately match and scaffold children's mathematics capabilities. The current study examined preschool children's item-level performance on eight numeracy subtests at half-year age points throughout the preschool period. Data were compiled across six studies, and participants comprised 801 preschool children (age range = 3.12-5.99 years, M = 4.63, SD = 0.68). Children were grouped into six age categories (3, 3.5, 4, 4.5, 5, and 5.5 years). Linear regression analyses were used to investigate the number of children who correctly answered each item of a specific subtest within a particular age group. Findings provide clear trajectories of children's early mathematics skills that can be used to inform preschool classroom practices and facilitate the design of intervention studies.

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.

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

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

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

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

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

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

Spatial and Verbal Routes to Number Comparison in Young Children

The ability to compare the numerical magnitude of symbolic numbers represents a milestone in the development of numerical skills. However, it remains unclear how basic numerical abilities contribute to the understanding of symbolic magnitude and whether the impact of these abilities may vary when symbolic numbers are presented as number words (e.g., "six vs. eight") vs. Arabic numbers (e.g., 6 vs. 8). In the present study on preschool children, we show that comparison of number words is related to cardinality knowledge whereas the comparison of Arabic digits is related to both cardinality knowledge and the ability to spatially map numbers. We conclude that comparison of symbolic numbers in preschool children relies on multiple numerical skills and representations, which can be differentially weighted depending on the presentation format. In particular, the spatial arrangement of digits on the number line seems to scaffold the development of a "spatial route" to understanding the exact magnitude of numerals.

About why there is a shift from cardinal to ordinal processing in the association with arithmetic between first and second grade

Digit comparison is strongly related to individual differences in children's arithmetic ability. Why this is the case, however, remains unclear to date. Therefore, we investigated the relative contribution of three possible cognitive mechanisms in first and second graders' digit comparison performance: digit identification, digit-number word matching and digit ordering ability. Furthermore, we examined whether these components could account for the well-established relation between digit comparison performance and arithmetic. As expected, all candidate predictors were related to digit comparison in both age groups. Moreover, in first graders, digit ordering and in second graders both digit identification and digit ordering explained unique variance in digit comparison performance. However, when entering these unique predictors of digit comparison into a mediation model with digit comparison as predictor and arithmetic as outcome, we observed that whereas in second graders digit ordering was a full mediator, in first graders this was not the case. For them, the reverse was true and digit comparison fully mediated the relation between digit ordering and arithmetic. These results suggest that between first and second grade, there is a shift in the predictive value for arithmetic from cardinal processing and procedural knowledge to ordinal processing and retrieving declarative knowledge from memory; a process which is possibly due to a change in arithmetic strategies at that age. A video abstract of this article can be viewed at: https://youtu.be/dDB0IGi2Hf8.

Number prompts left-to-right spatial mapping in toddlerhood

Toddlers performed a spatial mapping task in which they were required to learn the location of a hidden object in a vertical array and then transpose this location information 90° to a horizontal array. During the vertical training, they were given (a) no labels, (b) alphabetical labels, or (c) numerical labels for each potential spatial location. After the array was transposed to become a horizontal continuum, the children who were provided with numerical labels during training and those who heard alphabetical labels and formed a strong memory for the vertical location, selectively chose the location corresponding to a left-to-right mapping bias. Children who received no concurrent ordinal labels during training were not able to transpose the array, and did not exhibit any spatial directionality bias after transposition. These results indicate that children exhibit more flexible spatial mapping than other animals, and this mapping is modulated depending on the type of concurrent ordinal information the child receives. (PsycINFO Database Record

The integration of symbolic and non-symbolic representations of exact quantity in preschool children

Preschoolers (n=62) completed tasks that tapped their knowledge of symbolic and non-symbolic exact quantities, their ability to translate among different representations of exact quantity (i.e., digits, number words, and non-symbolic quantities), and their non-symbolic, digit, and spoken number comparison skills (e.g., which is larger, 2 or 4?). As hypothesized, children's knowledge about non-symbolic exact quantities, spoken number words, and digits predicted their ability to map between symbolic and non-symbolic exact quantities. Further, their knowledge of the mappings between digits and non-symbolic quantities predicted symbolic number comparison (i.e., of spoken number words or written digits). Mappings between written digits and non-symbolic exact quantities developed later than the other mappings. These results support a model of early number knowledge in which integration across symbolic and non-symbolic representations of exact quantity underlies the development of children's number comparison skills.

The Symbol Grounding Problem Revisited: A Thorough Evaluation of the ANS Mapping Account and the Proposal of an Alternative Account Based on Symbol-Symbol Associations

Recently, a lot of studies in the domain of numerical cognition have been published demonstrating a robust association between numerical symbol processing and individual differences in mathematics achievement. Because numerical symbols are so important for mathematics achievement, many researchers want to provide an answer on the ‘symbol grounding problem,’ i.e., how does a symbol acquires its numerical meaning? The most popular account, the approximate number system (ANS) mapping account, assumes that a symbol acquires its numerical meaning by being mapped on a non-verbal and ANS. Here, we critically evaluate four arguments that are supposed to support this account, i.e., (1) there is an evolutionary system for approximate number processing, (2) non-symbolic and symbolic number processing show the same behavioral effects, (3) non-symbolic and symbolic numbers activate the same brain regions which are also involved in more advanced calculation and (4) non-symbolic comparison is related to the performance on symbolic mathematics achievement tasks. Based on this evaluation, we conclude that all of these arguments and consequently also the mapping account are questionable. Next we explored less popular alternative, where small numerical symbols are initially mapped on a precise representation and then, in combination with increasing knowledge of the counting list result in an independent and exact symbolic system based on order relations between symbols. We evaluate this account by reviewing evidence on order judgment tasks following the same four arguments. Although further research is necessary, the available evidence so far suggests that this symbol–symbol association account should be considered as a worthy alternative of how symbols acquire their meaning.