Development of Spatial-Numerical Associations

Eye movements reveal that young school children shift attention when solving additions and subtractions

Adults shift their attention to the right or to the left along a spatial continuum when solving additions and subtractions, respectively. Studies suggest that these shifts not only support the exact computation of the results but also anticipatively narrow down the range of plausible answers when processing the operands. However, little is known on when and how these attentional shifts arise in childhood during the acquisition of arithmetic. Here, an eye-tracker with high spatio-temporal resolution was used to measure spontaneous eye movements, used as a proxy for attentional shifts, while children of 2nd (8 y-o; N = 50) and 4th (10 y-o; N = 48) Grade solved simple additions (e.g., 4+3) and subtractions (e.g., 3-2). Gaze patterns revealed horizontal and vertical attentional shifts in both groups. Critically, horizontal eye movements were observed in 4th Graders as soon as the first operand and the operator were presented and thus before the beginning of the exact computation. In 2nd Graders, attentional shifts were only observed after the presentation of the second operand just before the response was made. This demonstrates that spatial attention is recruited when children solve arithmetic problems, even in the early stages of learning mathematics. The time course of these attentional shifts suggests that with practice in arithmetic children start to use spatial attention to anticipatively guide the search for the answer and facilitate the implementation of solving procedures. RESEARCH HIGHLIGHTS: Additions and subtractions are associated to right and left attentional shifts in adults, but it is unknown when these mechanisms arise in childhood. Children of 8-10 years old solved single-digit additions and subtractions while looking at a blank screen. Eye movements showed that children of 8 years old already show spatial biases possibly to represent the response when knowing both operands. Children of 10 years old shift attention before knowing the second operand to anticipatively guide the search for plausible answers.

Longitudinal relationship between number line estimation and other mathematical abilities in Chinese preschool children

Many cross-sectional studies have shown that number line estimation is associated with other mathematical skills, but there has been limited longitudinal research. To systematically examine such associations longitudinally at the earliest stage of mathematical learning, the current study tested 40 Chinese preschoolers (mean age = 4.97 years, SD = 0.18) and followed them up 8 months later. For both waves of data collection, children were administered six tasks: number line estimation, dot counting, comparison of two dot arrays, comparison of triple dot arrays, symbolic number comparison, and simple addition. Results of two-wave cross-lagged panel analysis showed that (a) dot counting and non-symbolic numerical comparison at Time 1 had significant longitudinal associations with number line estimation at Time 2, (b) number line estimation had bidirectional associations with symbolic number comparison, and (c) number line estimation at Time 1 had a marginally significant longitudinal association with simple addition at Time 2. These results extend the small but accumulating literature on the longitudinal relations between number line estimation and other mathematical skills and specify the important role of number line estimation in the early development of mathematical skills.

Fine motor skills and finger gnosia contribute to preschool children's numerical competencies

Facets of fine motor skills (FMS) and finger gnosia have been reported to predict young children's numerical competencies, possibly by affecting early finger counting experiences. Furthermore, neuronal connections between areas involved in finger motor movement, finger gnosia, and numerical processing have been posited. In this study, FMS and finger gnosia were investigated as predictors for preschool children's performance in numerical tasks. Preschool children (N = 153) completed FMS tasks measuring finger agility and finger dexterity as well as a non-motor finger gnosia task. Furthermore, children completed numerical tasks that involved finger use (i.e., finger counting and finger montring), and tasks that did not (i.e., picture-aided calculation and number line estimation). To control for possible confounding influences of domain general skills, we included measures of reasoning and spatial working memory. We found associations between FMS and both finger counting and calculation, but not finger montring. In contrast, finger gnosia was only associated with finger montring, but not finger counting and calculation. Surprisingly, there were no associations between FMS or finger gnosia with number line estimation. Findings highlight that the relationship between finger gnosia, FMS, and numerical skills is specific to task requirements. Possible implications are discussed.

Exploring Cognitive Processes of Knowledge Acquisition to Upgrade Academic Practices

The development of cognitive functions follows certain pathways through brain maturation. Concepts taught at school can be reinforced by understanding the related cognitive functions that enhance learning. The cultural and social diversities faced by the education system worldwide can be solved by understanding the unifying cognitive processes of learning. This knowledge can be effectively used to devise better curriculum and training for students. Cognition, conation, and emotional regulation are the main components that determine an individual’s efficiency to deal with various situations. How the brain receives input, perceives, and organizes these information lays the foundation for learning. The objectives of the study were (i) to explore age-group specific inputs for knowledge acquisition, (ii) to relate knowledge organization to the cognitive processes, and (iii) to identify factors that strengthen the knowledge ensemble through subject-domain allied training. The review focused on studies related to elementary school age (below 7 years), middle school age (7–12 years), and high school age (12 years and above). Published journal articles related to the objectives were randomly reviewed to establish a possible relationship. The findings of this review can help to advance student learning practices and instructional strategies. The findings are listed below. (i) Acquisition of knowledge during early childhood is based on sensory-motor integration on which attentional, perceptual, memory, language, and socialization systems develop. As brain development progresses toward adolescence, meta-awareness and social-emotional cognition influence the student learning process. (ii) Knowledge representations can be strengthened by domain-specific training inputs. (iii) Associational integration of the developmental, cognitive, and conative processes are indicators of curriculum strength. (iv) The strengthening of cognitive processes by rerouting through complementary neural circuitry, such as music, arts, real-life-based experiments, and physical exercises, is an effective way to improve child-friendly instructions.

Pitch–Luminance Crossmodal Correspondence in the Baby Chick: An Investigation on Predisposed and Learned Processes

Our senses are constantly reached by a multitude of stimuli from all different sensory modalities. To create a coherent representation of the environment, we must integrate the various unimodal inputs that refer to the same object into a single multimodal representation. In some cases, however, we tend to bind certain properties of the stimuli without any apparent reason, which is a phenomenon named crossmodal correspondence. For instance, we match a spiky or a rounded shape with the sound “Kiki” or “Bouba”, respectively. Similarly, we associate the left hemispace with low luminance and the right one with high luminance. Instances of crossmodal correspondences were described also in other mammals, and recently, a case of space-luminance crossmodal correspondence was reported in birds (i.e., domestic chicks). Here, we investigate the presence of pitch–luminance crossmodal correspondence in three-day-old chicks, employing experimental methods that exploit either predisposed or learned processes. While failing to report evidence for this phenomenon, we discuss the difference between statistical and structural crossmodal correspondences and the possible role of environmental factors in determining their emergence. Moreover, we discuss the importance of the different experimental methodologies to investigate distinct aspects of this perceptual phenomenon to reach a deeper understanding and unveil the role of innate vs. learned mechanisms.

The link between number and action in human infants

Humans' inborn ability to represent and manipulate numerical quantities is supported by the parietal cortex, which is also involved in a variety of spatial and motor abilities. While the behavioral links between numerical and spatial information have been extensively studied, little is known about the connection between number and action. Some studies in adults have shown a series of interference effects when simultaneously processing numerical and action information. We investigated the origins of this link by testing forty infants (7- to 9-month-old) in one of two experimental conditions: one group was habituated to congruent number-hand pairings, where the larger the number, the more open the hand-shape associated; the second group was habituated to incongruent number-hand pairings, where the larger the number, the more close the hand-shape associated. In test trials, both groups of infants were presented with congruent and incongruent pairings. We found that only infants habituated to congruency showed a significantly higher looking time to the test trial depicting incongruent pairings. These findings show for the first time that infants spontaneously associate magnitude-related changes across the dimensions of number and action-related information, thus offering support to the existence of an early, preverbal number-action link in the human mind.

Uncertainty and Prior Assumptions, Rather Than Innate Logarithmic Encoding, Explain Nonlinear Number-to-Space Mapping

Mapping number to space is natural and spontaneous but often nonveridical, showing a clear compressive nonlinearity that is thought to reflect intrinsic logarithmic encoding of numerical values. We asked 78 adult participants to map dot arrays onto a number line across nine trials. Combining participant data, we confirmed that on the first trial, mapping was heavily compressed along the number line, but it became more linear across trials. Responses were well described by logarithmic compression but also by a parameter-free Bayesian model of central tendency, which quantitatively predicted the relationship between nonlinearity and number acuity. To experimentally test the Bayesian hypothesis, we asked 90 new participants to complete a color-line task in which they mapped noise-perturbed color patches to a "color line." When there was more noise at the high end of the color line, the mapping was logarithmic, but it became exponential with noise at the low end. We conclude that the nonlinearity of both number and color mapping reflects contextual Bayesian inference processes rather than intrinsic logarithmic encoding.

Spatial concepts of number, size, and time in an indigenous culture

In industrialized groups, adults implicitly map numbers, time, and size onto space according to cultural practices like reading and counting (e.g., from left to right). Here, we tested the mental mappings of the Tsimane', an indigenous population with few such cultural practices. Tsimane' adults spatially arranged number, size, and time stimuli according to their relative magnitudes but showed no directional bias for any domain on any spatial axis; different mappings went in different directions, even in the same participant. These findings challenge claims that people have an innate left-to-right mapping of numbers and that these mappings arise from a domain-general magnitude system. Rather, the direction-specific mappings found in industrialized cultures may originate from direction-agnostic mappings that reflect the correlational structure of the natural world.

Space-luminance crossmodal correspondences in domestic chicks

Crossmodal correspondences are spontaneous associations of non-redundant information across different modalities. Infants and some non-human animals (i.e., chimpanzees, rhesus monkeys, and dogs) showed crossmodal correspondences like adult humans, suggesting a shared origin (at least among mammals) of such a phenomenon. Here we investigate visual-spatial crossmodal correspondences in a precocial avian species, i.e., the domestic chicken. Three-day-old chicks (n = 40) were presented with two (one in the left and one in the right hemispace of an arena) identical panels, either dyed black (low luminance) or white (high luminance). Chicks could circumnavigate either panel to obtain a food reward. Akin to humans, they preferentially chose the left side when presented with black panels and the right side when presented with white panels. The control group (n = 39), tested with grey panels, showed no spatial preference. In light of our results, we discuss crossmodal correspondences in terms of an early available mechanism widespread across different species.

Children's spatial-numerical associations on horizontal, vertical, and sagittal axes

There is substantial evidence linking numerical magnitude to the physical properties of space. The most influential support for this connection comes from the SNARC effect (spatial-numerical association of response codes), in which responses to small/large numbers are faster on the left/right side of space, respectively. The SNARC effect has been extensively replicated, and is understood as horizontal mapping of numerical magnitude. However, much less is known about how numbers are represented on the vertical and sagittal axes, and whether spatial-numerical associations on different axes emerge during childhood. To that end, we tested two groups of children, aged 5-7 years and 8 and 9 years, on a single-digit magnitude comparison task with response buttons positioned either upper/lower (vertical), left/right (horizontal) or near/far (sagittal). Our results provide evidence of spatial-numerical mapping on all three axes for both age groups that are similar in strength. This indicates that, even at an early stage of formal education, children can flexibly assign numerical magnitude to any spatial dimension. To examine the contribution of extracorporeal space and spatio-anatomical mapping to the SNARC effect across axes, these sources were pitted against each other by swapping the position of the response hands in Experiment 1b. Switching hand position did not reveal convincing evidence for SNARC effects on any axis. Results are discussed with respect to the utility of three-dimensional mental number lines, and potential avenues for future research are outlined.

Limited Internal Comparability of General Intelligence Composites: Impact on External Validity, Possible Predictors, and Practical Remedies

Research on comparability of general intelligence composites (GICs) is scarce and has focused exclusively on comparing GICs from different test batteries, revealing limited individual-level comparability. We add to these findings, investigating the group- and individual-level comparability of different GICs within test batteries (i.e., internal score comparability), thereby minimizing transient error and ruling out between-battery variance completely. We (a) determined the magnitude of intraindividual IQ differences, (b) investigated their impact on external validity, (c) explored possible predictors for these differences, and (d) examined ways to deal with incomparability. Results are based on the standardization samples of three intelligence test batteries, spanning from early childhood to late adulthood. Despite high group-level comparability, individual-level comparability was often unsatisfactory, especially toward the tails of the IQ distribution. This limited comparability has consequences for external validity, as GICs were differentially related to and often less predictive for school grades for individuals with high IQ differences. Of several predictors, only IQ level and age were systematically related to comparability. Consequently, findings challenge the use of overall internal consistencies for confidence intervals and suggest using confidence intervals based on test–retest reliabilities or age- and IQ-specific internal consistencies for clinical interpretation. Implications for test construction and application are discussed.

The development of working memory spatialization revealed by using the cave paradigm in a two-alternative spatial choice

When Western participants are asked to keep in mind a sequence of verbal items, they tend to associate the first items to the left and the last items to the right. This phenomenon, known as the spatial-positional association response codes effect, has been interpreted as showing that individuals spatialize the memoranda by creating a left-to-right mental line with them. One important gap in our knowledge concerns the development of this phenomenon: when do Western individuals start organizing their thought from left to right? To answer this question, 274 participants in seven age groups were tested (kindergarten, Grades 1, 2, 3, 4, and 5, and adults). We used a new protocol meant to be child-friendly, which involves associating two caves with two animals using a two-alternative spatial forced choice. Participants had to guess in which cave a specific animal could be hidden. Results showed that it is from Grade 3 on that participants spatialize information in working memory in a left-to-right fashion like adults.

A biological foundation for spatial-numerical associations: the brain's asymmetric frequency tuning

"Left" and "right" coordinates control our spatial behavior and even influence abstract thoughts. For number concepts, horizontal spatial-numerical associations (SNAs) have been widely documented: we associate few with left and many with right. Importantly, increments are universally coded on the right side even in preverbal humans and nonhuman animals, thus questioning the fundamental role of directional cultural habits, such as reading or finger counting. Here, we propose a biological, nonnumerical mechanism for the origin of SNAs on the basis of asymmetric tuning of animal brains for different spatial frequencies (SFs). The resulting selective visual processing predicts both universal SNAs and their context-dependence. We support our proposal by analyzing the stimuli used to document SNAs in newborns for their SF content. As predicted, the SFs contained in visual patterns with few versus many elements preferentially engage right versus left brain hemispheres, respectively, thus predicting left-versus rightward behavioral biases. Our "brain's asymmetric frequency tuning" hypothesis explains the perceptual origin of horizontal SNAs for nonsymbolic visual numerosities and might be extensible to the auditory domain.

Spatial biases in mental arithmetic are independent of reading/writing habits: Evidence from French and Arabic speakers

The representation of numbers in human adults is linked to space. In Western cultures, small and large numbers are associated respectively with the left and right sides of space. An influential framework attributes the emergence of these spatial-numerical associations (SNAs) to cultural factors such as the direction of reading and writing, because SNAs were found to be reduced or inverted in right-to-left readers/writers (e.g., Arabic, Farsi, or Hebrew speakers). However, recent cross-cultural and animal studies cast doubt on the determining role of reading and writing directions on SNAs. In this study, we assessed this role in mental arithmetic, which requires explicit number manipulations and has revealed robust leftward or rightward biases in Western participants. We used a temporal order judgement task in French and Arabic speakers, two languages that have opposite reading/writing directions. Participants had to solve subtraction and addition problems presented auditorily while at the same time determining which of a left or right visual target appeared first on a screen. The results showed that the right target was favoured more often when solving additions than when solving subtractions both in the French- (n = 31) and Arabic-speaking (n = 25) groups. This was true even in Arabic-speaking participants whose preference for ordering of various series of numerical and non-numerical stimuli went from right to left (n = 10). These results indicate that SNAs in mental arithmetic cannot be explained by the direction of reading/writing habits and call for a reconsideration of current models to acknowledge the pervasive role of biological factors in SNAs in adults.

Executive Functions and Visual-Spatial Skills Predict Mathematical Achievement: Asymmetrical Associations Across Age

Children's mathematical achievement depends on their domain-specific abilities and their domain-general skills such as executive functions (EFs) and visual-spatial skills (VSS). Research indicates that these two domain-general skills predict mathematical achievement. However, it is unclear whether these skills are differently associated with mathematical achievement across a large age range. The current cross-sectional study answered this question using a large, representative sample aged 5-20 years (N = 1754). EFs, VSS, and mathematical achievement were assessed using the Intelligence and Development Scales-2. Hierarchical regression analyses were computed with EFs and VSS as predictor variables and mathematical achievement as dependent variable. We examined (non-) linear effects and interactions of EFs and VSS with age. Results indicated that EFs and VSS were distinctly associated with mathematical achievement above and beyond effects of age, sex, maternal education, and verbal reasoning. Effects of EFs were linear and age-invariant. Effects of VSS were curvilinear and stronger in adolescents than in children. Our results indicated that EFs and VSS related differently to mathematical proficiency across age, suggesting a varying impact on mathematics across age.

The Relational SNARC: Spatial Representation of Nonsymbolic Ratios

Recent research in numerical cognition has begun to systematically detail the ability of humans and nonhuman animals to perceive the magnitudes of nonsymbolic ratios. These relationally defined analogs to rational numbers offer new potential insights into the nature of human numerical processing. However, research into their similarities with and connections to symbolic numbers remains in its infancy. The current research aims to further explore these similarities by investigating whether the magnitudes of nonsymbolic ratios are associated with space just as symbolic numbers are. In two experiments, we found that responses were faster on the left for smaller nonsymbolic ratio magnitudes and faster on the right for larger nonsymbolic ratio magnitudes. These results further elucidate the nature of nonsymbolic ratio processing, extending the literature of spatial-numerical associations to nonsymbolic relative magnitudes. We discuss potential implications of these findings for theories of human magnitude processing in general and how this general processing relates to numerical processing.

Number line unidimensionality is a critical feature for promoting fraction magnitude concepts

Children's ability to estimate fractions on a number line is strongly related to algebra and overall high school math achievement, and number line training leads to better fraction magnitude comparisons compared with area model training. Here, we asked whether unidimensionality is necessary for the number line to promote fraction magnitude concepts and whether left-to-right orientation and labeled endpoints are sufficient. We randomly assigned second- and third-graders (N = 148) to one of four 15-min one-on-one, experimenter-led trainings. Three number line trainings had identical scripts, where the experimenter taught children to segment and shade the number line along the horizontal dimension. The number line conditions varied only in the vertical dimension of the training number line: pure unidimensional number line (17.5 cm horizontal line), hybrid unidimensional number line (17.5 × 0.6 cm rectangle), and square number line (17.5 × 17.5 cm). In the area model condition, children were taught to segment and shade a square (17.5 × 17.5 cm) along both dimensions. The conditions significantly differed in posttest fraction magnitude comparison accuracy (a transfer task), controlling for pretest accuracy, reading achievement, and age. In preregistered analyses, the hybrid unidimensional number line condition significantly outperformed the square area model condition and the square number line condition. In exploratory analyses accounting for training protocol fidelity, these results held and the pure unidimensional number line also outperformed the area model condition on fraction magnitude comparisons. We argue that unidimensionality is a critical feature of the number line for promoting fraction magnitude concepts because it aligns with a key concept-that real numbers, including fractions, can be ordered along a single dimension.

Spatial-numerical associations: Shared symbolic and non-symbolic numerical representations

During the last decades, there have been a large number of studies into the number-related abilities of humans. As a result, we know that humans and non-human animals have a system known as the approximate number system that allows them to distinguish between collections based on their number of items, separately from any counting procedures. Dehaene and others have argued for a model on which this system uses representations for numbers that are spatial in nature and are shared by our symbolic and non-symbolic processing of numbers. However, there is a conflicting theoretical perspective in which there are no representations of numbers underlying the approximate number system, but only quantity-related representations. This perspective would then suggest that there are no shared representations between symbolic and non-symbolic processing. We review the evidence on spatial biases resulting from the activation of numerical representations, for both non-symbolic and symbolic tests. These biases may help decide between the theoretical differences; shared representations are expected to lead to similar biases regardless of the format, whereas different representations more naturally explain differences in biases, and thus behaviour. The evidence is not yet decisive, as the behavioural evidence is split: we expect bisection tasks to eventually favour shared representations, whereas studies on the spatial-numerical association of response codes (SNARC) effect currently favour different representations. We discuss how this impasse may be resolved, in particular, by combining these behavioural studies with relevant neuroimaging data. If this approach is carried forward, then it may help decide which of these two theoretical perspectives on number representations is correct.

Operational momentum for magnitude ordering in preschool children and adults

When adding or subtracting quantities, adults tend to overestimate addition outcomes and underestimate subtraction outcomes. They also shift visuospatial attention to the right when adding and to the left when subtracting. These operational momentum phenomena are thought to reflect an underlying representation in which small magnitudes are associated with the left side of space and large magnitudes with the right side of space. Currently, there is limited research on operational momentum in early childhood or for operations other than addition and subtraction. The current study tested whether English-speaking 3- and 4-year-old children and college-aged adults exhibit operational momentum when ordering quantities. Participants were presented with two experimental blocks. In one block of trials, they were tasked with choosing the same quantity they had previously seen three times; in the other block, they were asked to generate the next quantity in a doubling sequence composed of three ascending quantities. A bias to shift attention to the right after an ascending operation was found in both age groups, and a bias to overestimate the next sequential quantity during an ascending ordering operation was found in adults under conditions of uncertainty. These data suggest that, for children, the spatial biases during operating are more pronounced than the mis-estimation biases. These findings highlight the spatial underpinnings of operational momentum and suggest that both very young children and adults conceptualize quantity along a horizontal continuum during ordering operations, even before formal schooling.

Materials count: Linear-spatial materials improve young children's addition strategies and accuracy, irregular arrays don't

Children who use advanced arithmetic strategies, such as count-on and decomposition, are more accurate when solving arithmetic problems and are more likely to later have higher levels of math achievement. The present study tested the hypothesis that instruction using linear-spatial representations would activate children's knowledge necessary for use of mental addition strategies and, thus, lead to greater accuracy on addition problems, than instruction using irregular representations of magnitude. As predicted, low-income kindergartners (n = 29) randomly assigned to practice sums up to 10 using materials that instantiated the linear-spatial features of a mental number line (i.e., discrete squares arranged in rows) demonstrated substantially more improvement in solving unpracticed addition problems than children who practiced with irregular materials (i.e., pictures of stars arranged in random arrays). This was particularly true for children with better initial numerical knowledge, which provided support for the idea that existing knowledge was activated. The use of count-on more than doubled from pretest to posttest among children in the linear-spatial condition and this mediated the difference in improvement between conditions. The importance of aligning instructional materials to relevant mental representations-consistent with the Cognitive Alignment Framework for instructional design-is discussed.

Dynamics and development in number-to-space mapping

Young children's estimates of numerical magnitude increase approximately logarithmically with actual magnitude. The conventional interpretation of this finding is that children's estimates reflect an innate logarithmic encoding of number. A recent set of findings, however, suggests that logarithmic number-line estimates emerge via a dynamic encoding mechanism that is sensitive to previously encountered stimuli. Here we examine trial-to-trial changes in logarithmicity of numerosity estimates to test an alternative dynamic model (D-MLLM) with both a strong logarithmic component and a weak response to previous stimuli. In support of D-MLLM, first-trial numerosity estimates in both adults (Study 1, 2, 3, and 4) and children (Study 4) were strongly logarithmic, despite zero previous stimuli. Additionally, although numerosity of a previous trial affected adults' estimates, the influence of previous numbers always accompanied the logarithmic-to-linear shift predicted by D-MLLM. We conclude that a dynamic encoding mechanism is not necessary for compressive mapping, but sequential effects on response scaling are a possible source of linearity in adults' numerosity estimation.

Differential Development of Children's Understanding of the Cardinality of Small Numbers and Zero

Counting and the understanding of cardinality are important steps in children’s numerical development. Recent studies have indicated that language and visuospatial abilities play an important role in the development of children’s cardinal knowledge of small numbers. However, predictors for the knowledge about zero were usually not considered in these studies. Therefore, the present study investigated whether the acquisition of cardinality knowledge on small numbers and the concept of zero share cross-domain and domain-specific numerical predictors. Particular interest was paid to the question whether visuospatial abilities – in addition to language abilities – were associated with children’s understanding of small numbers and zero. Accordingly, we assessed kindergarteners aged 4 to 5 years in terms of their understanding of small numbers and zero as well as their visuospatial, general language, counting, Arabic number identification abilities, and their finger number knowledge. We observed significant zero-order correlations of vocabulary, number identification, finger knowledge, and counting abilities with children’s knowledge about zero as well as understanding of the cardinality of small numbers. Subsequent regression analyses substantiated the influences of counting abilities on knowledge about zero and the influences of both counting abilities and finger knowledge on children’s understanding of the cardinality of small numbers. No significant influences of cross-domain predictors were observed. In sum, these results indicate that domain-specific numerical precursor skills seem to be more important for children’s development of an understanding of the cardinality of small numbers as well as of the concept of zero than the more proximal cross-domain abilities such as language and visuospatial abilities.

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.

On the genesis of spatial-numerical associations: Evolutionary and cultural factors co-construct the mental number line

Mapping numbers onto space is a common cognitive representation that has been explored in both behavioral and neuroimaging contexts. Empirical work probing the diverse nature of these spatial-numerical associations (SNAs) has led researchers to question 1) how the human brain links numbers with space, and 2) whether this link is biologically vs. culturally determined. We review the existing literature on the development of SNAs and situate that empirical work within cognitive and neuroscientific theoretical frameworks. We propose that an evolutionarily-ancient frontal-parietal circuit broadly tuned to multiple magnitude dimensions provides the phylogenetic substrate for SNAs, while enculturation and sensorimotor experience shape their specific profiles. We then use this perspective to discuss educational implications and highlight promising avenues for future research.

Numerical Affordance Influences Action Execution: A Kinematic Study of Finger Movement

Humans represent symbolic numbers as oriented from left to right: the mental number line (MNL). Up to now, scientific studies have mainly investigated the MNL by means of response times. However, the existing knowledge on the MNL can be advantaged by studies on motor patterns while responding to a number. Cognitive representations, in fact, cannot be fully understood without considering their impact on actions. Here we investigated whether a motor response can be influenced by number processing. Participants seated in front of a little soccer goal. On each trial they were visually presented with a numerical (2, 5, 8) or a non-numerical ($) stimulus. They were instructed to kick a small ball with their right index toward a frontal soccer goal as soon as a stimulus appeared on a screen. However, they had to refrain from kicking when number five was presented (no-go signal). Our main finding is that performing a kicking action after observation of the larger digit proved to be more efficient: the trajectory path was shorter and lower on the surface, velocity peak was anticipated. The smaller number, instead, specifically altered the temporal and spatial aspects of trajectories, leading to more prolonged left deviations. This is the first experimental demonstration that the reaching component of a movement is influenced by number magnitude. Since this paradigm does not require any verbal skill and non-symbolic stimuli (array of dots) can be used, it could be fruitfully adopted to evaluate number abilities in children and even preschoolers. Notably, this is a self-motivating and engaging task, which might help children to get involved and to reduce potential arousal connected to institutional paper-and-pencil examinations.

How not to develop a sense of number

The authors rightly point to the theoretical importance of interactions of space and number through the life span, yet propose a theory with several weaknesses. In addition to proclaiming itself unfalsifiable, its stage-like format and emphasis on the role of selective attention are at odds with what is known about the development of spatial-numerical associations in infancy.

Are Books Like Number Lines? Children Spontaneously Encode Spatial-Numeric Relationships in a Novel Spatial Estimation Task

Do children spontaneously represent spatial-numeric features of a task, even when it does not include printed numbers (Mix et al., 2016)? Sixty first grade students completed a novel spatial estimation task by seeking and finding pages in a 100-page book without printed page numbers. Children were shown pages 1 through 6 and 100, and then were asked, “Can you find page X?” Children’s precision of estimates on the page finder task and a 0-100 number line estimation task was calculated with the Percent Absolute Error (PAE) formula (Siegler and Booth, 2004), in which lower PAE indicated more precise estimates. Children’s numerical knowledge was further assessed with: (1) numeral identification (e.g., What number is this: 57?), (2) magnitude comparison (e.g., Which is larger: 54 or 57?), and (3) counting on (e.g., Start counting from 84 and count up 5 more). Children’s accuracy on these tasks was correlated with their number line PAE. Children’s number line estimation PAE predicted their page finder PAE, even after controlling for age and accuracy on the other numerical tasks. Children’s estimates on the page finder and number line tasks appear to tap a general magnitude representation. However, the page finder task did not correlate with numeral identification and counting-on performance, likely because these tasks do not measure children’s magnitude knowledge. Our results suggest that the novel page finder task is a useful measure of children’s magnitude knowledge, and that books have similar spatial-numeric affordances as number lines and numeric board games.

Observation of directional storybook reading influences young children's counting direction

Even before formal schooling, children map numbers onto space in a directional manner. The origin of this preliterate spatial-numerical association is still debated. We investigated the role of enculturation for shaping the directionality of the association between numbers and space, focusing on counting behavior in 3- to 5-year-old preliterate children. Two studies provide evidence that, after observing reading from storybooks (left-to-right or right-to-left reading) children change their counting direction in line with the direction of observed reading. Just observing visuospatial directional movements had no such effect on counting direction. Complementarily, we document that book illustrations, prevalent in children's cultures, exhibit directionality that conforms to the direction of a culture's written language. We propose that shared book reading activates spatiotemporal representations of order in young children, which in turn affect their spatial representation of numbers.

Act on Numbers: Numerical Magnitude Influences Selection and Kinematics of Finger Movement

In the past decade hand kinematics has been reliably adopted for investigating cognitive processes and disentangling debated topics. One of the most controversial issues in numerical cognition literature regards the origin - cultural vs. genetically driven - of the mental number line (MNL), oriented from left (small numbers) to right (large numbers). To date, the majority of studies have investigated this effect by means of response times, whereas studies considering more culturally unbiased measures such as kinematic parameters are rare. Here, we present a new paradigm that combines a "free response" task with the kinematic analysis of movement. Participants were seated in front of two little soccer goals placed on a table, one on the left and one on the right side. They were presented with left- or right-directed arrows and they were instructed to kick a small ball with their right index toward the goal indicated by the arrow. In a few test trials participants were presented also with a small (2) or a large (8) number, and they were allowed to choose the kicking direction. Participants performed more left responses with the small number and more right responses with the large number. The whole kicking movement was segmented in two temporal phases in order to make a hand kinematics' fine-grained analysis. The Kick Preparation and Kick Finalization phases were selected on the basis of peak trajectory deviation from the virtual midline between the two goals. Results show an effect of both small and large numbers on action execution timing. Participants were faster to finalize the action when responding to small numbers toward the left and to large number toward the right. Here, we provide the first experimental demonstration which highlights how numerical processing affects action execution in a new and not-overlearned context. The employment of this innovative and unbiased paradigm will permit to disentangle the role of nature and culture in shaping the direction of MNL and the role of finger in the acquisition of numerical skills. Last but not least, similar paradigms will allow to determine how cognition can influence action execution.

Cognitive Mechanisms Underlying Directional and Non-directional Spatial-Numerical Associations across the Lifespan

There is accumulating evidence suggesting an association of numbers with physical space. However, the origin of such spatial-numerical associations (SNAs) is still debated. In the present study we investigated the development of two SNAs in a cross-sectional study involving children, young and middle-aged adults as well as the elderly: (1) the SNARC (spatial-numerical association of response codes) effect, reflecting a directional SNA; and (2) the numerical bisection bias in a line bisection task with numerical flankers. Results revealed a consistent SNARC effect in all age groups that continuously increased with age. In contrast, a numerical bisection bias was only observed for children and elderly participants, implying an U-shaped distribution of this bias across age groups. Additionally, individual SNARC effects and numerical bisection biases did not correlate significantly. We argue that the SNARC effect seems to be influenced by longer-lasting experiences of cultural constraints such as reading and writing direction and may thus reflect embodied representations. Contrarily, the numerical bisection bias may originate from insufficient inhibition of the semantic influence of irrelevant numerical flankers, which should be more pronounced in children and elderly people due to development and decline of cognitive control, respectively. As there is an ongoing debate on the origins of SNAs in general and the SNARC effect in particular, the present results are discussed in light of these differing accounts in an integrative approach. However, taken together, the present pattern of results suggests that different cognitive mechanisms underlie the SNARC effect and the numerical bisection bias.

Spatial representation of magnitude in gorillas and orangutans

Humans mentally represent magnitudes spatially; we respond faster to one side of space when processing small quantities and to the other side of space when processing large quantities. We determined whether spatial representation of magnitude is a fundamental feature of primate cognition by testing for such space-magnitude correspondence in gorillas and orangutans. Subjects picked the larger quantity in a pair of dot arrays in one condition, and the smaller in another. Response latencies to the left and right sides of the screen were compared across the magnitude range. Apes showed evidence of spatial representation of magnitude. While all subjects did not adopt the same orientation, apes showed consistent tendencies for spatial representations within individuals and systematically reversed these orientations in response to reversal of the task instruction. Results suggest that spatial representation of magnitude is phylogenetically ancient and that consistency in the orientation of these representations in humans is likely culturally mediated.

An Embodiment Perspective on Number-Space Mapping in 3.5-Year-Old Dutch Children

Previous research suggests that block adding, subtracting and counting direction are early forms of number–space mapping. In this study, an embodiment perspective on these skills was taken. Embodiment theory assumes that cognition emerges through sensory–motor interaction with the environment. In line with this assumption, it was investigated if counting and adding/subtracting direction in young children is related to the hand they use during task performance. Forty‐eight 3.5‐year‐old children completed a block adding, subtracting and counting task. They had to add and remove a block from a row of three blocks and count a row of five blocks. Adding, subtracting and counting direction were related to the hand the children used for task performance. Most children who used their right hand added, removed and started counting the blocks at the right side of the row. Most children who used their left hand added, removed and started counting the blocks at the left side of the row. It can be concluded that number–space mapping, as measured by direction of adding, subtracting and counting blocks, in young children is embodied: It is not fixed, but is related to the situation. © 2016 The Authors Infant and Child Development Published by John Wiley & Sons, Ltd.

Number-space associations without language: Evidence from preverbal human infants and non-human animal species

It is well known that humans describe and think of numbers as being represented in a spatial configuration, known as the 'mental number line'. The orientation of this representation appears to depend on the direction of writing and reading habits present in a given culture (e.g., left-to-right oriented in Western cultures), which makes this factor an ideal candidate to account for the origins of the spatial representation of numbers. However, a growing number of studies have demonstrated that non-verbal subjects (preverbal infants and non-human animals) spontaneously associate numbers and space. In this review, we discuss evidence showing that pre-verbal infants and non-human animals associate small numerical magnitudes with short spatial extents and left-sided space, and large numerical magnitudes with long spatial extents and right-sided space. Together this evidence supports the idea that a more biologically oriented view can account for the origins of the 'mental number line'. In this paper, we discuss this alternative view and elaborate on how culture can shape a core, fundamental, number-space association.

The Early Construction of Spatial Attention: Culture, Space, and Gesture in Parent-Child Interactions

American and Israeli toddler-caregiver dyads (mean age of toddler = 26 months) were presented with naturalistic tasks in which they must watch a short video (N = 97) or concoct a visual story together (N = 66). English-speaking American caregivers were more likely to use left to right spatial structuring than right to left, especially for well-ordered letters and numbers. Hebrew-speaking Israeli parents were more likely than Americans to use right to left spatial structuring, especially for letters. When constructing a pictorial narrative for their children, Americans were more likely to place pictures from left to right than Israelis. These spatial structure biases exhibited by caregivers are a potential route for the development of spatial biases in early childhood, before children have developed automatic reading and writing habits.

Where Does the Ordered Line Come From? Evidence From a Culture of Papua New Guinea

Number lines, calendars, and measuring sticks all represent order along some dimension (e.g., magnitude) as position on a line. In high-literacy, industrialized societies, this principle of spatial organization- linear order-is a fixture of visual culture and everyday cognition. But what are the principle's origins, and how did it become such a fixture? Three studies investigated intuitions about linear order in the Yupno, members of a culture of Papua New Guinea that lacks conventional representations involving ordered lines, and in U.S. undergraduates. Presented with cards representing differing sizes and numerosities, both groups arranged them using linear order or sometimes spatial grouping, a competing principle. But whereas the U.S. participants produced ordered lines in all tasks, strongly favoring a left-to-right format, the Yupno produced them less consistently, and with variable orientations. Conventional linear representations are thus not necessary to spark the intuition of linear order-which may have other experiential sources-but they nonetheless regiment when and how the principle is used.

How Parents Read Counting Books and Non-numerical Books to Their Preverbal Infants: An Observational Study

Studies have stressed the importance of counting with children to promote formal numeracy abilities; however, little work has investigated when parents begin to engage in this behavior with their young children. In the current study, we investigated whether parents elaborated on numerical information when reading a counting book to their preverbal infants and whether developmental differences in numerical input exist even in the 1st year of life. Parents and their 5–10 months old infants were asked to read, as they would at home, two books to their infants: a counting book and another book that did not have numerical content. Parents’ spontaneous statements rarely focused on number and those that did consisted primarily of counting, with little emphasis on labeling the cardinality of the set. However, developmental differences were observed even in this age range, such that parents were more likely to make numerical utterances when reading to older infants. Together, results are the first to characterize naturalistic reading behaviors between parents and their preverbal infants in the context of counting books, suggesting that although counting books promote numerical language in parents, infants still receive very little in the way of numerical input before the end of the 1st year of life. While little is known regarding the impact of number talk on the cognitive development of young infants, the current results may guide future work in this area by providing the first assessment of the characteristics of parental numerical input to preverbal infants.

Mapping number to space in the two hemispheres of the avian brain

Pre-verbal infants and non-human animals associate small numbers with the left space and large numbers with the right space. Birds and primates, trained to identify a given position in a sagittal series of identical positions, whenever required to respond on a left/right oriented series, referred the given position starting from the left end. Here, we extended this evidence by selectively investigating the role of either cerebral hemisphere, using the temporary monocular occlusion technique. In birds, lacking the corpus callosum, visual input is fed mainly to the contralateral hemisphere. We trained 4-day-old chicks to identify the 4th element in a sagittal series of 10 identical elements. At test, the series was identical but left/right oriented. Test was conducted in right monocular, left monocular or binocular condition of vision. Right monocular chicks pecked at the 4th right element; left monocular and binocular chicks pecked at the 4th left element. Data on monocular chicks demonstrate that both hemispheres deal with an ordinal (sequential) task. Data on binocular chicks indicate that the left bias is linked to a right hemisphere dominance, that allocates the attention toward the left hemispace. This constitutes a first step towards understanding the neural basis of number space mapping.

Bewegtes Lernen numerischer Kompetenzen

Zusammenfassung. Nicht nur in Konzepten wie der Bewegten Schule ist körperliche Bewegung zur Unterstützung des Lernens von großer Bedeutung. Inzwischen liegen erste empirische Befunde zum positiven Einfluss spezifischer körperlicher Bewegungen vor, wie zum Beispiel dem Einsatz der Finger beim Erstrechnen oder dem Laufen entlang eines Zahlenstrahls. Diese aktuellen Studien deuten darauf hin, dass Bewegung den Erwerb numerischer Konzepte unterstützen kann. Neue bewegungssensitive Eingabemedien (z. B. Tanzmatte, Kinect Sensor) ermöglichen nicht nur solche Bewegungen in der Interaktion mit einer Lernumgebung, sondern machen diese mess- und damit spezifisch nutzbar. Dadurch können Trainings realisiert werden, die gezielt den Zusammenhang von Zahlen und Raum und damit für die Ausprägung des mentalen Zahlenstrahls relevante Prozesse trainieren. Die Entwicklung solcher Trainings ist von besonderer Bedeutung, weil der mentale Zahlenstrahl wichtig für eine erfolgreiche numerisch-mathematische Entwicklung zu sein scheint. In diesem Artikel stellen wir neben den theoretischen Grundlagen eine Zusammenfassung der Ergebnisse verschiedener eigener Arbeiten zu verkörperlichten numerischen Trainings vor.

A helping hand putting in order: Visuomotor routines organize numerical and non-numerical sequences in space

Theories of embodied cognition emphasize the importance of sensorimotor schemas linked to external world experience for representing conceptual knowledge. Accordingly, some researchers have proposed that the spatial representation of numerical and non-numerical sequences relies on visuomotor routines, like reading habit and finger counting. There is a growing interest in how these two routines contribute to the spatial representation of ordinal sequences, although no investigation has so far directly compared them. The present study aims to investigate how these routines contribute to represent ordinal information in space. To address this issue, bilingual participants reading either from left-to-right or right-to-left were required to map ordinal information to all fingers of their right dominant hand. Critically, we manipulated both the direction of the mapping and the language of the verbal information. More specifically, a finger-mapping compatibility task was adopted in three experiments to explore the spatial representation of numerical (digit numbers and number words) and non-numerical (days of the week, presented in Hebrew and in English) sequences. Results showed that numerical information was preferentially mapped according to participants' finger counting habits, regardless of hand posture (prone and supine), number notation and reading habit. However, for non-numerical ordinal sequences, reading and finger counting directions both contributed to determine a preferential spatial mapping. These findings indicate that abstract knowledge representation relies on multiple over-trained visuomotor routines. More generally, these results highlight the capacity of our cognitive system to flexibly represent abstract ordered information, by relying on different directional experiences (finger counting, reading direction) depending on the stimuli and on the task at hand.