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Origins of Number Sense Large-Number Discrimination in Human Infants
Abstract
Four experiments investigated infants' sensitivity to large, approximate numerosities in auditory sequences. Prior studies provided evidence that 6-month-old infants discriminate large numerosities that differ by a ratio of 2.0, but not 1.5, when presented with arrays of visual forms in which many continuous variables are controlled. The present studies used a head-turn preference procedure to test for infants' numerosity discrimination with auditory sequences designed to control for element duration, sequence duration, interelement interval, and amount of acoustic energy. Six-month-old infants discriminated 16 from 8 sounds but failed to discriminate 12 from 8 sounds, providing evidence that the same 2.0 ratio limits numerosity discrimination in auditory-temporal sequences and visual-spatial arrays. Nine-month-old infants, in contrast, successfully discriminated 12 from 8 sounds, but not 10 from 8 sounds, providing evidence that numerosity discrimination increases in precision over development, prior to the emergence of language or symbolic counting.
... Humans can rapidly estimate the approximate quantity of sets of items without counting or relying on symbolic notation. This ability is supported by the Approximate Number System (ANS; Dehaene, 1997;Feigenson et al., 2004), a core cognitive system that is operational in infants and young children before they begin receiving formal mathematics training (Gallistel and Gelman, 1992;Xu and Spelke, 2000;Lipton and Spelke, 2003;Barth et al., 2005Barth et al., , 2006. The ANS allows us to approximately quantify sets of objects by representing these quantities as noisy magnitudes (Meck and Church, 1983;Gallistel and Gelman, 1992), and the ability to discriminate these magnitudes depends both on the size of the quantities that they represent and the extent of the difference between the quantities (Lipton and Spelke, 2003;Roitman et al., 2007;Prather, 2012Prather, , 2014DeWind et al., 2015). ...
... This ability is supported by the Approximate Number System (ANS; Dehaene, 1997;Feigenson et al., 2004), a core cognitive system that is operational in infants and young children before they begin receiving formal mathematics training (Gallistel and Gelman, 1992;Xu and Spelke, 2000;Lipton and Spelke, 2003;Barth et al., 2005Barth et al., , 2006. The ANS allows us to approximately quantify sets of objects by representing these quantities as noisy magnitudes (Meck and Church, 1983;Gallistel and Gelman, 1992), and the ability to discriminate these magnitudes depends both on the size of the quantities that they represent and the extent of the difference between the quantities (Lipton and Spelke, 2003;Roitman et al., 2007;Prather, 2012Prather, , 2014DeWind et al., 2015). ...
... Previous work has shown that the representational precision of the ANS develops significantly across the lifespan, from infancy into late adulthood (Xu and Spelke, 2000;Lipton and Spelke, 2003;Halberda and Feigenson, 2008;Halberda et al., 2012; see Odic and Starr, 2018, for review) and precision can vary across individuals (Libertus et al., 2013;Thompson et al., 2016;Prather, 2019). In many of these studies, participants are asked to compare the relative magnitudes of visible arrays (e.g., participants are shown two sets of dots and are asked which set has more dots). ...
Children can represent the approximate quantity of sets of items using the Approximate Number System (ANS), and can perform arithmetic-like operations over ANS representations. Previous work has shown that the representational precision of the ANS develops substantially during childhood. However, less is known about the development of the operational precision of the ANS. We examined developmental change in the precision of the solutions to two non-symbolic arithmetic operations in 4-6-year-old U.S. children. We asked children to represent the quantity of an occluded set (Baseline condition), to compute the sum of two sequentially occluded arrays (Addition condition), or to infer the quantity of an addend after observing an initial array and then the array incremented by the unknown addend (Unknown-addend condition). We measured the precision of the solutions of these operations by asking children to compare their solutions to visible arrays, manipulating the ratio between the true quantity of the solution and the comparison array. We found that the precision of ANS representations that were not the result of operations (in the Baseline condition) was higher than the precision of solutions to ANS operations (in the Addition and Unknown-addend conditions). Further, we found that precision in the Baseline and Addition conditions improved significantly between 4 and 6 years, while precision in the Unknown-Addend condition did not. Our results suggest that ANS operations may inject “noise” into the representations they operate over, and that the development of the precision of different operations may follow different trajectories in childhood.
... ossèdent un sens de l'arithmétique. Agés de quelques mois seulement, les bébés manifestent une sensibilité aux quantités (Xu & Spelke, 2000); d'autre part, avant l'apprentissage du langage, les enfants sont capables d'opération arithémtiques simples comme l'addition, la soustraction, et les ratios (Brannon et al., 2004; J. S. Lipton & Spelke, 2004;J. Lipton & Spelke, 2003;Mccrink & Wynn, 2004;Wood & Spelke, 2005;Gilmore et al., 2007;Xu et al., 2005). Enfin, les performances de primates non humains dans des tâches d'addition et de soustraction montrent que ceux-ci possèdent également des compétences avoisinant celle des adultes humains, (Cantlon & Brannon, 2007), renforçant l'hypothèse d'un noyau de connai ...
... s that humans possess basic arithmetic knowledge, and that this system is evolutionarily ancient. First, number estimation skills, which support simple arithmetic operations, are present from the first months of life in infants (Xu & Spelke, 2000) as well as in children before language development (Brannon et al., 2004; J. S. Lipton & Spelke, 2004;J. Lipton & Spelke, 2003;Mccrink & Wynn, 2004;Wood & Spelke, 2005;Gilmore et al., 2007;Xu et al., 2005). These pre-verbal operations include multiplication and division, as well as ratio computation. Second, other animal species also exhibit these abilities, including macaques, which share most of the pre-verbal arithmetic skills of human adults (Cantlon & Brann ...
Although conceptual learning is at the heart of cognitive science, the ability to track signs of progress by studying the fine dynamics of learning has not been achieved, especially for a complex mathematical concept. This thesis aims to study the development of a geometric concept, geodesics, in the mind, and to describe the state of knowledge at three stages of learning: before, during, and after learning. In the first part of this thesis, I present results from a new paradigm that allows dynamic learning to be observed in a single session in the laboratory. This single-session paradigm allows for tracking the dynamics of learning, as well as measuring aspects of subjective and objective performance.Chapter 1 explores the content of intuitions about straight lines through a qualitative study of definitions provided by participants before, during, and after learning. Chapter 2 and Chapter 3 explore the dynamics of conceptual learning: Is learning made of discrete conceptual steps, or of progressive adjustments? To try to answer this question, in Chapter 2, we go through different measures of introspection. I present a study addressing alternative models where learning mechanisms are not necessarily conscious, and where consciousness has access only to discrete jumps. In Chapter 3, I present two studies that attempt to describe the learning curve in terms of periods of progress and recession and try to develop a measure of understanding to track progress, and thus estimate the shape of the learning curve. In the second part, I further explore the intuitive roots of the non-Euclidean concept of geodesics in non-mathematicians and mathematicians. In Chapter 4, I ask whether humans are shaped to think about straight trajectories in Euclidean terms. The study in Chapter 1 set up some intuitive properties that people think of when defining the notion of a straight line. Chapter 1 presents two systematic studies that test a more precise hypothesis about the content of the notion of a line: if people tend to identify planar sections (line resulting from the intersection of a plane with any surfaces) as geodesics (on the sphere, all circles are planar sections, however, only the great circles are geodesics). This study contributes to specifying which geometry would be the closest to the way the human mind conceives space.
... Using the same experimental logic -i.e., controlling for all non-numerical properties of the stimuli by equating some of them across groups and by equating the others across test stimuli -a different study (LIPTON;SPELKE, 2003) obtained evidence that six-month-olds can also compare sequences of sounds on the basis of number, even if the sounds are of different kinds (e.g., trumpets, phone rings, drums, or duck quacks). The fact that infants can compare both circles and sounds on the basis of number suggests that their representations of number are abstract -i.e., that the same representations apply to all collections of the same number regardless of the properties of the elements that comprise them. ...
... Nine-month-olds can make finer numerical discriminations than younger infants; they can discriminate eight from twelve. However, they cannot discriminate eight from ten (LIPTON;SPELKE, 2003). ...
Os humanos nascem não com um, mas dois sistemas que criam representações com conteúdo numérico. Ao contrário da contagem verbal, nenhum deles define símbolos numéricos em termos de suas posições em uma lista ordenada. Juntamente com outras considerações, isso levou muitos a acreditar que as crianças, ao aprender a usar a contagem para definir o significado das palavras numéricas, resolvem o que parece ser um problema impossível: aprendem princípios numéricos que não podem ser definidos em termos das representações numéricas disponíveis para elas nesse período. Propomos que, diferentemente do que pesquisadores anteriores pensavam, pode haver mais continuidade entre o que as crianças aprendem, ao aprender a usar a contagem para definir os significados das palavras numéricas, e alguns dos princípios disponíveis para elas antes do aprendizado da linguagem. Especificamente, propomos que as crianças aprendem a usar a contagem para definir os significados das palavras numéricas, considerando que a contagem é um sistema de contagem, assim como um dos dois sistemas numéricos centrais – ou seja, individuação paralela (à qual nos referimos como “contagens mentais”). Ou seja, quando aprendem o sistema de contagem verbal, as crianças aprendem a definir o significado de expressões como “n X’s”, onde n é uma palavra numérica (por exemplo, “cinco”) e X é um sintagma nominal (por exemplo, “cadeiras velhas que vêm da Suécia”) como uma coleção de Xs que pareiam um a um com uma contagem que termina com “n”. Argumentamos que, comparativamente a qualquer outra proposta anterior, essa visão da aquisição do significado das palavras numéricas e da contagem fornece uma melhor explicação do modo como as representações não verbais do número são integradas à contagem verbal e do conhecimento das crianças sobre o significado das palavras numéricas antes e depois de aprenderem como a contagem verbal representa números.
... All these studies suggest that infants and children in the preverbal age already possess an ability to represent the numerosity, but also a highly inaccurate representation, requiring a minimum ratio between 2:3 and 1:2 (e.g., they successfully distinguish 8 vs 16 elements, but not 8 vs 12). However, developmental studies show that by age of 9/10 months, there is an improvement in accuracy and infants are able at discriminating even between 8 and 12 elements (Lipton & Spelke, 2003). Finally, the results obtained from all these developmental studies suggest the existence of two different numerosity representation systems: the Object-Tracking System (or Subitizing in the adult literature), which operates with numbers below 3 and 4 and the Approximate Number System, which represents large numbers. ...
The natural environment in which animals are forced to survive shapes their brain and the way in which they behave to adapt and overcome natural pressures. These selective pressures may have determined the emergence of an evolutionary ancient neural system suited to rapidly extract abstract information from collections, such as their numerosity, to take informed decisions pivotal for survivance and adaptation. The “Number Sense” theory represents the most influential neural model accounting for neuropsychological and psychophysical evidence in humans and animals. However, this model is still largely debated because of the methodological difficulties in isolating neural signals related to “discrete” (i.e., the real number of objects in a collection) abstract numerosity processing from those related to other features correlated or confounded with numerosity in the raw sensory input (e.g., visual area, density, spatial frequency, etc.). The present thesis aimed to investigate which mechanisms might be at the basis of visual numerosity representations, overcoming the difficulties in isolating discrete from continuous features. After reviewing the main theoretical models and findings from the literature (Chapter 1 and 2), in the Chapter 3 we presented a psychophysical paradigm in which Kanizsa-like illusory contours (ICs) lines were used to manipulate the connectedness (e.g., grouping strength) of the items in the set, controlling all the continuous features across connectedness levels. We showed that numerosity was underestimated when connections increased, suggesting that numerosity relies on segmented perceptual objects rather than on raw low-level features. In Chapter 4, we controlled for illusory brightness confounds accompanying ICs. Exploiting perceptual properties of the reverse-contrast Kanizsa illusion, we found that underestimation was insensitive to inducer contrast direction, suggesting that the effect was specifically induced by a sign invariant boundary grouping and not due to perceived brightness confounds. In Chapter 5, we concurrently manipulated grouping with ICs lines and the perceived size of the collections using classic size illusions (Ponzo Illusion). By using a combination of visual illusions, we showed that numerosity perception is not based on perceived continuous cues, despite continuous cue might affect numerical perception. In Chapter 6 we tackled the issue with a direct physical approach: using Fourier analysis to equalize spatial frequency (SF) in the stimuli, we showed that stimulus energy is not involved in numerosity representation. Rather segmentation of the items and perceptual organization explained our main findings. In Chapter 7 we also showed that the ratio effect, an important hallmark of Weber-like encoding of numerical perception, is not primarily explained by stimulus energy or SF. Finally, in Chapter 8, we also provided the first empirical evidence that non-symbolic numerosity is represented spatially regardless of the physical SF content of the stimuli. Overall, this thesis strongly supports the view that numerosity processing is not merely based on low-level features, and rather clearly suggests that discrete information is at the core of the Number Sense.
... and spatial length in as young as 8-month-old infants 9 . This view is further supported by similar degrees of discriminability observed for time, numerosity, and size (e.g., 2:1 ratio in 6-month-old and 3:2 ratio in 9-month-old infants [10][11][12][13] ) and cross-model mappings between time, numerosity and length in 8-and 9-month old infants 14,15 (also see 16 for a review). However, specificities in the development of number sense challenge this hypothesis 17,18 . ...
This study investigated the development of spatiotemporal perceptual interactions in 5-to-7 years old children. Participants reproduced the temporal and spatial interval between sequentially presented visual stimuli. The time and spacing between stimuli were experimentally manipulated. In addition, cognitive capacities were assessed using neuropsychological tests. Results revealed that starting at 5 years old, children exhibited spatial biases in their time estimations and temporal biases in their spatial estimations, pointing at space–time interference. In line with developmental improvement of temporal and spatial abilities, these spatiotemporal biases decreased with age. Importantly, short-term memory capacity was a predictor of space–time interference pointing to shared cognitive mechanisms between time and space processing. Our results support the symmetrical hypothesis that proposes a common neurocognitive mechanism for processing time and space.
... This capacity exists across ages, cultures and animal species, therefore suggesting that the ANS has a long phylogenetic history (Odic & Starr, 2018). A few months after birth, human babies are indeed already able to distinguish between arrays of dots (Xu & Spelke, 2000), sequences of sounds (Lipton & Spelke, 2003) and sequences of actions (Wood & Spelke, 2005). For instance, Xu and Spelke (2000) habituated 6-monthold infants with one constant number of dots, varying in size and location, until their looking time decreased. ...
As humans living in a numerate society, we must every day deal with numerical information presented under various codes and contexts. We can easily estimate the number of people sitting in a waiting room, write down the age of a child when he proudly raises four fingers, or get to square 5 in the goose game when our dice shows five points. All these activities seem automatic, effortless and yet, one major issue remains: how do we represent and process these various numerical codes? This thesis aimed to explore the interconnection between number representations using behavioral and electrophysiological approaches. We especially focused on the processing of canonical number configurations, which are assumed to play a fundamental role in human numerical development. Across studies, discrimination and integration mechanisms of numerical codes were highlighted. Findings reported here contribute to the comprehension of the specific and common processes underlying number representations.
... Roitman et al. [35] showed that PPC is a critical component in numerical cognition. Behavioral studies showed that numerosity judgement follow Weber's law [36] that differentiate the number presented in stimuli based on some features such as surface area or density [37]. However, for more precision and quantity that represents the magnitude and cardinal value, neurons within PPC are involved. ...
Visualization and visual analytic tools amplify one’s perception of data, facilitating deeper and faster insights that can improve decision making. For multidimensional data sets, one of the most common approaches of visualization methods is to map the data into lower dimensions. Scatterplot matrices (SPLOM) are often used to visualize bivariate relationships between combinations of variables in a multidimensional dataset. However, the number of scatterplots increases quadratically with respect to the number of variables. For high dimensional data, the corresponding enormous number of scatterplots makes data exploration overwhelmingly complex, thereby hindering the usefulness of SPLOM in human decision making processes. One approach to address this difficulty utilizes Graph-theoretic Scatterplot Diagnostic (Scagnostics) to automatically extract a subset of scatterplots with salient features and of manageable size with the hope that the data will be sufficient for improving human decisions. In this paper, we use Electroencephalogram (EEG) to observe brain activity while participants make decisions informed by scatterplots created using different visual measures. We focused on 4 categories of Scagnostics measures: Clumpy, Monotonic, Striated, and Stringy. Our findings demonstrate that by adjusting the level of difficulty in discriminating between data sets based on the Scagnostics measures, different parts of the brain are activated: easier visual discrimination choices involve brain activity mostly in visual sensory cortices located in the occipital lobe, while more difficult discrimination choices tend to recruit more parietal and frontal regions as they are known to be involved in resolving ambiguities. Our results imply that patterns of neural activity are predictive markers of which specific Scagnostics measures most assist human decision making based on visual stimuli such as ours.
... Numerous studies indicate that humans and animals are capable of processing quantitative information without symbols, specifically, they can compare arrays of objects and detect the largest one, detecting changes in numerosities, or establishing similarities between quantities without [1][2][3]. This ability is usually referred to as Approximate Number Sense (ANS) [4,5]. ...
The nonsymbolic comparison task is used to investigate the precision of the Approximate Number Sense, the ability to process discrete numerosity without counting and symbols. There is an ongoing debate regarding the extent to which the ANS is influenced by the processing of non-numerical visual cues. To address this question, we assessed the congruency effect in a nonsymbolic comparison task, examining its variability across different stimulus presentation formats and numerical proportions. Additionally, we examined the variability of the numerical ratio effect with the format and congruency. Utilizing generalized linear mixed-effects models with a sample of 290 students (89% female, mean age 19.33 years), we estimated the congruency effect and numerical ratio effect for separated and intermixed formats of stimulus presentation, and for small and large numerical proportions. The findings indicated that the congruency effect increased in large numerical proportion conditions, but this pattern was observed only in the separated format. In the intermixed format, the congruency effect was insignificant for both types of numerical proportion. Notably, the numerical ratio effect varied for congruent and incongruent trials in different formats. The results may suggest that the processing of visual non-numerical parameters may be crucial when numerosity processing becomes noisier, specifically when numerical proportion becomes larger. The implications of these findings for refining the ANS theory are discussed.
... Previous research has indicated that ANS is based on intrinsic intuition and innate cognitive abilities that are common in both adults and newborns, as well as animals (Dehaene, 1997;Feigenson et al., 2004). As ANS-related number identification relies on Weber's ratio -which is based on the ratio between numbers rather than their absolute size -the identification process becomes faster and more accurate as the difference between two or more numbers increases (Xu and Spelke, 2000;Lipton and Spelke, 2003;Xu and Arriaga, 2007;Cho, 2013;Libertus et al., 2013). Previous studies have suggested that the ability to identify ratios develops gradually as children age. ...
Recently, it has become evident that cognitive abilities such as the approximate number system (ANS), number knowledge, and intelligence affect individuals’ fundamental mathematical ability. However, it is unclear which of these cognitive abilities have the greatest impact on the non-symbolic division ability in preschoolers. Therefore, in the present study, we included 4- to 6-year-old Korean preschoolers without prior formal education of division in order to test their ability to solve non-symbolic division problems, ANS acuity, and intelligence, and to determine the interrelationships among those functions (N = 38). We used the Panamath Dot Comparison Paradigm to measure the ANS acuity, employed non-symbolic division tasks to measure the ability to solve non-symbolic division problems, and measured the intelligence using the Korean version of the WPPSI-IV (Wechsler Preschool Primary Scale of Intelligence-IV). Our results showed that, in all conditions of the non-symbolic division tasks, the 4- to 6-years old children were able to perform better than chance level. Additionally, in a relatively easy condition, the children’s performance showed a significant positive correlation with full-scale intelligence quotient (FSIQ) and ANS acuity; however, in a more complex condition, only FSIQ was significantly correlated with their performance. Overall, we found significant relationships between the children’s performance in the non-symbolic division tasks and verbal comprehension, fluid reasoning, and processing speed index. Taken together, our findings demonstrate that preschoolers without formal education on the arithmetic problem solving can solve non-symbolic division problems. Moreover, we suggest that both FSIQ and ANS ability play essential roles in children’s ability to solve non-symbolic division problems, highlighting the significance of intelligence on children’s fundamental mathematical ability.
... The ANS is a basic and universal cognitive system that enables us to quantify sets of items without language or formal symbols 1 (Dehaene, 1997;Feigenson, Dehaene, & Spelke, 2004). The ANS represents sets of individual items as a single magnitude that is noisy and imprecise (Dehaene, 1997;Gallistel & Gelman, 1992;Meck & Church, 1983;Wynn, 1995), and the discriminability of ANS representations depends on the ratio between the quantities they represent (Gallistel & Gelman, 2000;Lipton & Spelke, 2003;Pica, Lemer, Izard, & Dehaene, 2004;Xu, 2003;Xu & Spelke, 2000). The ANS is what allows us to get a rough sense of the size of a crowd, or to tell whether our sibling got more cereal than we did, without having to count. ...
Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n = 74 4- to -8-year-olds in Experiment 1; n = 52 7- to 8-year-olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them "about the same." We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge.
... The ANS is a basic and universal cognitive system that enables us to quantify sets of items without language or formal symbols 1 (Dehaene, 1997;Feigenson, Dehaene, & Spelke, 2004). The ANS represents sets of individual items as a single magnitude that is noisy and imprecise (Dehaene, 1997;Gallistel & Gelman, 1992;Meck & Church, 1983;Wynn, 1995), and the discriminability of ANS representations depends on the ratio between the quantities they represent (Gallistel & Gelman, 2000;Lipton & Spelke, 2003;Pica, Lemer, Izard, & Dehaene, 2004;Xu, 2003;Xu & Spelke, 2000). The ANS is what allows us to get a rough sense of the size of a crowd, or to tell whether our sibling got more cereal than we did, without having to count. ...
Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over non-symbolic, approximate representations of quantity. However, the algorithmic rules that guide such non-symbolic operations are not entirely clear. We asked whether non-symbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n=74 4-8-year-olds in Experiment 1; n=52 7-8-year-olds in Experiment 2) first solved two non-symbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them “about the same”. We hypothesized that, if non- symbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of non-symbolic computations as inputs into another non-symbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other non-symbolic computations. These results suggest that non-symbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage non-symbolic arithmetic intuitions to acquire formal mathematics knowledge.
... Particularly, it has been demonstrated that the accuracy of nonsymbolic comparison or discrimination depends on numerical proportion or the numerical distance between numerosities. Children and adults are less accurate and spend more time in the comparison of numerosities with a larger numerical ratio or closer numerical distance [16][17][18][19]. Reducing the precision of numerosities' comparison/matching associated with the increase in the ratio between two numerosities is known as the numerical ratio effect (NRE) or numerical distance effect (NDE). ...
The study used a large sample of elementary schoolchildren in Russia (N = 3,448, 51.6% were girls, with a mean age of 8.70 years, ranging 6–11 years) to investigate the congruency, format and heterogeneity effects in a nonsymbolic comparison test and between-individual differences in these effects with generalized linear mixed effects models (GLMMs). The participants were asked to compare two arrays of figures of different colours in spatially separated or spatially intermixed formats. In addition, the figures could be similar or different for the two arrays. The results revealed that congruency (difference between congruent and incongruent items), format (difference between mixed and separated formats) and heterogeneity (difference between homogeneous and heterogeneous conditions) interacted. The heterogeneity effect was higher in the separated format, while the format effect was higher for the homogeneous condition. The separated format produced a greater congruency effect than the mixed format. In addition, the congruency effect was lower in the heterogeneous condition than in the homogeneous condition. Analysis of between-individual differences revealed that there was significant between-individual variance in the format and congruency effects. Analysis of between-grade differences revealed that accuracy improved from grade 1 to grade 4 only for congruent trials in separated formats. Consequently, the congruency effect increased in separated/homogeneous and separated/heterogeneous conditions. In general, the study demonstrated that the test format and heterogeneity affected accuracy and that this effect varied for congruent and incongruent items.
... Preverbal and verbal number concepts develop predictably and tractably in children. Children have robust and abstract nonverbal representations of quantity (numerosities) beginning in infancy (Izard et al., 2009;Lipton & Spelke, 2003), but they do not begin to learn formal verbal counting until around 2.5 years. A central focus of research in numerical cognition concerns what role nonverbal numerosity representations play in the acquisition of symbolic number (Carey et al., 2017;Odic et al., 2015;vanMarle et al., 2018). ...
Learning to map number words onto their ordinal and quantitative meanings is a key step in the acquisition of formal mathematics. Previous neuroimaging work suggests that the intraparietal sulcus (IPS), the inferior frontal gyrus (IFG), and the fronto-temporal language network may be involved in representing number words. However, the contribution of early-developing numerosity representations to the acquisition of counting has not been tested in children. If regions that support numerosity processing are important for the acquisition of counting, then there should be functional overlap between numerosity representations and number word representations in the brain, before children have mastered counting. Using functional magnetic resonance imaging (fMRI), we identified numerosity processing regions in 3- to 5-year-old children during a numerosity comparison task. To identify neural representations of number words, we measured changes in neural amplitudes while those same children listened to number words and color words and while they listened to counting and alphabet sequences. Across multiple whole-brain analyses, we found that the bilateral IPS consistently supported representations of numerosities, number words, and counting sequences. Functional overlap between numerosities and unknown counting sequences was also evident in the left IFG, and in some cases number word representations emerged in the left hemisphere fronto-temporal language network. These results provide new evidence from children that primitive numerosity processing regions of the brain interface with the language network to ground the acquisition of verbal counting.
Highlights
fMRI data revealed the neural basis of counting acquisition in 3- to 5-year-olds.
Overlap between neural responses to count words and numerosity emerged in the IPS.
Sensitivity to number words emerged in the IPS across two different tasks.
Number word stimuli also engaged regions of the language network in children.
The IPS and language network may ground number words during counting acquisition.
... The Weber fraction derived from such a numerosity comparison task provides a quantitative estimate of ANS precision, with smaller Weber fractions (w) indicating more precise discrimination acuity. Weber fractions decrease dramatically over the first year of life and reach peak sensitivity around 30 years of age (e.g., [2][3][4][5]). ANS acuity throughout the lifespan predicts formal math ability substantiating the importance of studying ANS acuity in general, and ultimately its potential contribution toward math education [6][7][8]. ...
Background
A hallmark of the approximate number system (ANS) is ratio dependence. Previous work identified specific event-related potentials (ERPs) that are modulated by numerical ratio throughout the lifespan. In adults, ERP ratio dependence was correlated with the precision of the numerical judgments with individuals who make more precise judgments showing larger ratio-dependent ERP effects. The current study evaluated if this relationship generalizes to preschoolers.
Method
ERPs were recorded from 56 4.5 to 5.5-year-olds while they compared the numerosity of two sequentially presented dot arrays. Nonverbal numerical precision, often called ANS acuity, was assessed using a similar behavioral task.
Results
Only children with high ANS acuity exhibited a P2p ratio-dependent effect onsetting ∼250 ms after the presentation of the comparison dot array. Furthermore, P2p amplitude positively correlated with ANS acuity across tasks.
Conclusion
Results demonstrate developmental continuity between preschool years and adulthood in the neural basis of the ANS.
... All humans possess a "number sense": a capacity to perceive and manipulate the approximate number of objects in collections without counting (Dehaene, 2011). This capacity is present in children before formal education and can be traced back to infancy (e.g., Feigenson et al., 2004;Izard et al., 2009;Libertus & Brannon, 2010;Lipton & Spelke, 2003;Xu & Spelke, 2000;Xu et al., 2005). Acuity for perceiving numerosity improves over the course of development. ...
Inter‐individual differences in infants' numerosity processing have been assessed using a change detection paradigm, where participants were presented with two concurrent streams of images, one alternating between two numerosities and the other showing one constant numerosity. While most infants look longer at the changing stream in this paradigm, the reasons underlying these preferences have remained unclear. We suggest that, besides being attracted by numerosity changes, infants perhaps also respond to the alternating pattern of the changing stream. We conducted two experiments (N = 32) with 6‐month‐old infants to assess this hypothesis. In the first experiment, infants responded to changes in numerosity even when the changing stream showed numerosities in an unpredictable random order. In the second experiment, infants did not display any preference when an alternating stream was pitted against a random stream. These findings do not provide evidence that the alternating pattern of the changing stream contributes to drive infants' preferences. Instead, around the age of 6 months, infants' responses in the numerosity change detection paradigm appear to be mainly driven by changes in numerosity, with different levels of preference reflecting inter‐individual difference in the acuity of numerosity perception.
... It has been shown that humans, like other species, can perceive and process quantitative information with some level of imprecision. In particular, the precision of discrimination between two numerosities is reduced when comparing numerosities that have a small numerical distance or a large proportion between them (Dietrich et al., 2016;Lipton & Spelke, 2003;Lyons et al., 2015;Xu & Spelke, 2000). This dependence of accuracy regarding the ratio between two numerosities is known as the numerical ratio effect (NRE). ...
The extent to which the approximate number sense is based on the estimation of continuous visual properties has been widely discussed. Some investigators have hypothesized that humans are able to estimate numerosity directly and independently of visual cues. Other investigators have posited that numerosity can be processed only via the estimation of non-numeric visual properties. The latter theory is confirmed by the existence of the congruency effect, that is, greater accuracy in congruent trials where visual properties were positively correlated with numerosity compared with that in incongruent trials. In this study, we tested the assumption that the congruency effect, reflecting the bias in numerosity estimation due to the estimation of visual cues, varies depending on the format of the stimulus presentation and object heterogeneity. The study involved a sample of pupils in Grades 5–9 from Kyrgyzstan (N = 764; 48% girls; mean age = 13.06 years), whereby participants performed a nonsymbolic comparison test in four formats of stimulus presentation: paired/homogeneous, paired/heterogeneous, mixed/homogeneous, and mixed/heterogeneous. Compared arrays of figures might be congruent or incongruent for one visual parameter (convex hull or cumulative area), whereas another visual parameter was held constant for two arrays. The results of generalized linear mixed-effect models demonstrated that the largest congruency effect occurred in a paired format with homogeneous figures. The congruency effect was insignificant in the mixed/heterogeneous format. The results also revealed that the effects of the convex hull and cumulative area varied in different formats of stimulus presentations.
Development is a lifelong process. It begins with conception and ends with death. Developmental psychology reflects this idea by examining the changes in mind and behavior over the lifespan. Development is also the result of an interaction of various processes on the biological, cognitive, and socioemotional level. Lifespan psychological development is, therefore, complex, and this course book seeks to help you examine it more closely.
The course book Developmental Psychology will present you with an overview of the field, the methods of research used in this field, and the key debates that govern it. You will then progress through the lifespan by examining core psychological components. These domains include physical and motoric development, perceptual development, cognitive development, language development, emotional development, and social and moral development. In each case, you will encounter the relevant developmental progressions from the prenatal period into childhood, including atypical developmental pathways.
Because developmental changes are more dominant in early stages of the lifespan, warranting a separate examination of each domain, later stages are more compact. In the subsequent units, you will, therefore, engage with specific life stages. You will learn about the significant changes in adolescence, particularly on the brain and social identity levels. From here, you will engage with the psychosocial challenges of early and middle adulthood. Finally, the course book will end with the final stage of life. Here, you will come across key issues in late adulthood, including the closure of the lifespan.
Functional lateralization was previously established for various cognitive domains—but not for number processing. Although numbers are considered to be bilaterally represented in the intraparietal sulcus (IPS), there are some indications of different functional roles of the left vs. right IPS in processing number pairs with small vs. large distance, respectively. This raises the question whether number size plays a distinct role in the lateralization within the IPS. In our preregistered study, we applied anodal transcranial direct current stimulation (tDCS) over the left vs. right IPS to investigate the effect of stimulation as compared to sham on small vs. large distance, in both single-digit and two-digit number comparison. We expected that anodal tDCS over the left IPS facilitates number comparison with small distance, while anodal tDCS over the right IPS facilitates number comparison with large distance. Results indicated no effect of stimulation; however, exploratory analyses revealed that tDCS over the right IPS slowed down single-digit number processing after controlling for the training effect. In conclusion, number magnitude processing might be bilaterally represented in the IPS, however, our exploratory analyses emphasise the need for further investigation on functional lateralization of number processing.
I argue that analogue mental representations possess a canonical decomposition into privileged constituents from which they compose. I motivate this suggestion, and rebut arguments to the contrary, through reflection on the approximate number system , whose representations are widely expected to have an analogue format. I then argue that arguments for the compositionality and constituent structure of these analogue representations generalize to other analogue mental representations posited in the human mind, such as those in early vision and visual imagery.
Children, who are introduced to technology as soon as they are born, have access to internet-based videos as young as one year old. These colorful, animated cartoon/animation content supported by songs have turned into informal learning tools for children. Some of these videos, which families do not hesitate to present as educational content, are math videos. These videos reach millions of views. These videos with mathematical content prepared for children ,mostly shared with the label of children's songs, are the subject of this study. In this study, we focused on Turkish number and counting videos from the open access YOUTUBE content provider. Document analysis method was used in this study. As a result of the analysis, it was observed that only a few videos were adequate in terms of mathematical language, content and number teaching, and almost all of the other videos included incorrect or inaccurate mathematical representations. While counting, it was observed that number symbols were used as "ordinal numbers". In addition, it was observed that there were scenes where the amount counted and the number did not match. It was thought that these situations could lead to false learning. Based on all these findings, it can be concluded that the mathematical language used in YOUTUBE video content is weak.
Abilitism is an approach to the metaphysics of concepts, according to which each concept consists of a managing cognitive ability coordinating other abilities (cognitive and non-cognitive) and a set of subordinate abilities associated with this managing ability. As I argue here, if we accept the abilitist approach, we can efficiently solve such puzzles in the metaphysics of concepts as the partial possession problem, the concept pluralism problem, etc. However, there are some possible objections to abilitism, concerning the abilitist explanation of compositional properties of concepts, knowledge-that, an extension/intension of concepts, and the idea that concepts are constituents of thought. Nonetheless, as I demonstrate here, they can be answered.
A longitudinal study was conducted to identify unique sources of individual differences in later understanding of the equal sign as a relational symbol of equivalence (i.e., formal understanding of mathematical equivalence). The sample included 141 children from a mid-sized city in the Midwestern United States (Mage = 6 years, 2 months in kindergarten; 88 boys, 53 girls; 71% white, 8% Hispanic or Latine, 7% Black, 3% Asian, 11% multiracial or other race/ethnicity; 42% qualified for free/reduced lunch). Children were assessed on three categories of skills in kindergarten including number knowledge, relational thinking, and executive functioning. These skills were hypothesized to provide a foundation for a formal understanding of mathematical equivalence (assessed in second grade) by preventing a specific, narrow misunderstanding of the equal sign that hinders learning. Results showed that kindergarten relational thinking, particularly tasks assessing nonsymbolic equivalence understanding, uniquely and positively predicted formal understanding of mathematical equivalence and negatively predicted the specific misunderstanding of the equal sign in second grade, controlling for IQ, gender, and free/reduced lunch status. Exploratory analyses unpacking the categories of skills into individual tasks also indicated specific areas of kindergarten instructional focus that may help children construct understanding of mathematical equivalence in future years.
Over development, children acquire symbols to represent abstract concepts such as time and number. Despite the importance of quantity symbols, it is unknown how acquiring these symbols impacts one's ability to perceive quantities (i.e., nonsymbolic representations). While it has been proposed that learning symbols shapes nonsymbolic quantitative abilities (i.e., the refinement hypothesis), this hypothesis has been understudied, especially in the domain of time. Moreover, the majority of research in support of this hypothesis has been correlational in nature, and thus, experimental manipulations are critical for determining whether this relation is causal. In the present study, kindergarteners and first graders (N = 154) who have yet to learn about temporal symbols in school completed a temporal estimation task during which they were either (1) trained on temporal symbols and effective timing strategies ("2 s" and counting on the beat), (2) trained on temporal symbols only ("2 s"), or (3) participated in a control training. Children's nonsymbolic and symbolic timing abilities were assessed before and after training. Results revealed a correlation between children's nonsymbolic and symbolic timing abilities at pre-test (when controlling for age), indicating this relation exists prior to formal classroom instruction on temporal symbols. Notably, we found no support for the refinement hypothesis, as learning temporal symbols did not impact children's nonsymbolic timing abilities. Implications and future directions are discussed.
Although attempts to create evidence-based television content for infants from birth to 2 years of age are notable, it has not been empirically verified to what extent infants understand such content. Our study evaluated whether Japanese 11- to 20-month-olds (N = 97; 52 boys and 45 girls) understand evidence-based television content using a looking-time method. When presented with content based on number themes, infants demonstrated an understanding of addition. When presented with content related to moral cognition, infants preferentially looked at a helper more than at a non-helper. Results reveal that infants understand educational television content based on scientific findings, demonstrating robustness and ecological validity. We discuss the possibility that broadcasting such content promotes infants’ sensitivity to numbers and morals and provides learning opportunities through television.
Numerosity effects have been investigated in the psychology and marketing literatures. While the effects are documented in outcomes including money and temperature judgments, the potential application and effects of numerosity for nutrition labeling remain unexplored. In this work, we propose that vigilance offers one circumstance when individuals might succumb to numerosity effects. Within the context of nutrition labeling, we propose that the increased vigilance that people with diet‐sensitive illnesses have for specific nutrients on nutrition labels, counter‐intuitively, exacerbates the numerosity effect. We demonstrate that those with diabetes and those with hypertension, for example, are more vigilant for information on nutrition labels relevant to their condition, sugar, and salt, and this greater vigilance counterintuitively leads them to exhibit greater numerosity effects for those nutrients, influencing their food perceptions. As an illustration, we find that a person with hypertension would consider a food product with, say, 3 g of sodium to have less sodium content and be more healthful than one with 3000 mg, although the quantities are equivalent. Our research highlights to policymakers that a “one‐size‐fits‐all” solution for nutrition labeling is not appropriate.
Children with below average school entry mathematics knowledge are likely to remain behind their peers throughout schooling, and go on to face poor employment prospects and low wage-earning potential as adults. Previous research has identified key skills (number system knowledge) in the first-grade that predict math achievement ten years later. Recent studies have tried to identify early-emerging skills in infancy and during the preschool years that may provide a foundation upon which the formal skills they learn in school can be built. One candidate skill is the non-verbal ability to estimate and reason about numerosities that depends on an evolutionary ancient, and phylogenetically common, mechanism for representing quantities – the approximate number system (ANS). While there is evidence that ANS acuity (precision) predicts math achievement concurrently and prospectively, several studies have failed to find a link, or identified other quantitative skills (e.g., ordinal knowledge) or domain-general skills (e.g., intelligence, executive function, working memory) that are better predictors. Evidence for and against the claim that the ANS is a foundational skill that is critical for the development of formal mathematical knowledge is reviewed, followed by a synthesis of work by Geary and colleagues in which they examined the growth of many different domain-specific (quantitative) and domain-general skills longitudinally over four years. The findings suggest that the link between ANS acuity and math achievement is mediated by other skills, especially cardinal knowledge. The ANS still remains a significant factor, however, because our findings also suggest that it plays an important role in the acquisition of cardinal knowledge, and the conceptual insight children appear to have when they first come to understand counting and the natural numbers.
Reunindo contribuições de importantes pesquisadores, o livro Métodos Experimentais em Psicolinguística apresenta uma introdução a técnicas comumente utilizadas no Brasil em estudos sobre aquisição, processamento e produção da linguagem. Ideal para estudantes
com interesse nessas áreas, o livro conta com capítulos que não apenas descrevem e exemplificam cada método, mas também discutem suas possíveis vantagens e desvantagens.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
‘Measuring the World and Beyond’ was the official title of the conference that led to this book. In my response to the papers that became the chapters of this book and the discussion, I would like to focus on the phrase ‘and Beyond’ as a point of entry into the broader issue explored by this particular symposium and the project as a whole – the roots of spirituality. The archaeological insights gathered from data analysis around the globe shed new light on the extent to which the construction of modes of measurement in early cultures functioned as a new means of recognizing and engaging with the material world. How is this related to that which we experience as ‘beyond’ the world, ‘beyond’ measurement?
From a philosophical and theological point of view, it is not simply the emergence of the capacity for mensuration that makes early human cultures interesting but also, and even especially, the growing self-awareness among human beings of their lack of capacity in this regard. That is, the human construction of measurement may be a manifestation both of an evolutionary and an adaptive skill for controlling the environment and of an awakening to the recognition of the limits of adaptation mechanisms for manipulating the cosmos. Alongside the discovery of the susceptibility of the world to measurement arose the discovery of the concept of the immeasurable, which invites questions about spirituality and religious awareness.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
The construction of formal measurement systems underlies the development of science, technology, economy and new ways of understanding and explaining the world. Human societies have developed such systems in different ways, in different places and at different times, and recent archaeological investigations highlight the importance of these activities for fundamental aspects of human life. Measurement systems have provided the structure for addressing key concerns of cosmological belief systems, as well as the means for articulating relationships between the human form, human action, and the world. The Archaeology of Measurement explores the archaeological evidence for the development of measuring activities in numerous ancient societies, as well as the implications of these discoveries for an understanding of their worlds and beliefs. Featuring contributions from a cast of internationally renowned scholars, it analyses the relationships between measurement, economy, architecture, symbolism, time, cosmology, ritual, and religion among prehistoric and early historic societies.
Humans can estimate the number of visually displayed items without counting. This capacity of numerosity perception has often been attributed to a dedicated system to estimate numerosity, or alternatively to the exploitation of various stimulus features, such as density, convex hull, the size of items, and occupancy area. The distribution of the presented items is usually not varied with eccentricity in the visual field. However, our visual fields are highly asymmetric. To date, it is unclear how inhomogeneities of the visual field impact numerosity perception. Besides eccentricity, a pronounced asymmetry is the radial-tangential anisotropy. For example, in crowding, radially placed flankers interfere more strongly with target perception than tangentially placed flankers. Similarly, in redundancy masking, the number of perceived items in repeating patterns is reduced when the items are arranged radially but not when they are arranged tangentially. Here, we investigated whether numerosity perception is subject to the radial-tangential anisotropy of spatial vision to shed light on the underlying topology of numerosity perception. In Experiment 1, observers were presented with varying numbers of discs, predominantly arranged radially or tangentially, and asked to report their perceived number. In Experiment 2, observers were presented with the same displays as in Experiment 1, and were asked to encircle items that were perceived as a group. We found that numerosity estimation depended on the arrangement of discs, suggesting a radial-tangential anisotropy of numerosity perception. Grouping among discs did not seem to explain our results. We suggest that the topology of spatial vision modulates numerosity estimation and that asymmetries of visual space should be taken into account when investigating numerosity estimation.
Learners use certainty to guide learning. They maintain existing beliefs when certain, but seek further information when they feel uninformed. Here, we review developmental evidence that this metacognitive strategy does not require reportable processing. Uncertainty prompts nonverbal human infants and nonhuman animals to engage in strategies like seeking help, searching for additional information, or opting out. Certainty directs children’s attention and active learning strategies and provides a common metric for comparing and integrating conflicting beliefs across people. We conclude that certainty is a continuous, domain-general signal of belief quality even early in life.
Written for pre-service and in-service educators, as well as parents of children in preschool through grade five, this book connects research in cognitive development and math education to offer an accessibly written and practical introduction to the science of elementary math learning. Structured according to children's mathematical development, How Children Learn Math systematically reviews and synthesizes the latest developmental research on mathematical cognition into accessible sections that explain both the scientific evidence available and its practical classroom application. Written by an author team with decades of collective experience in cognitive learning research, clinical learning evaluations, and classroom experience working with both teachers and children, this amply illustrated text offers a powerful resource for understanding children's mathematical development, from quantitative intuition to word problems, and helps readers understand and identify math learning difficulties that may emerge in later grades. Aimed at pre-service and in-service teachers and educators with little background in cognitive development, the book distills important findings in cognitive development into clear, accessible language and practical suggestions. The book therefore serves as an ideal text for pre-service early childhood, elementary, and special education teachers, as well as early career researchers, or as a professional development resource for in-service teachers, supervisors and administrators, school psychologists, homeschool parents, and other educators.
This study examined infants' use of contour length in num- ber discrimination tasks. We systematically varied number and con- tour length in a visual habituation experiment in order to separate these two variables. Sixteen 6- to 8-month-old infants were habituated to displays of either two or three black squares on a page. They were then tested with alternating displays of either a familiar number of squares with a novel contour length or a novel number of squares with a familiar contour length. Infants dishabituated to the display that changed in contour length, but not to the display that changed in num- ber. We conclude that infants base their discriminations on contour length or some other continuous variable that correlates with it, rather than on number.
Previous research has reported that infants use amount rather than number to discriminate small sets (Clearfield & Mix, 1999). This study sought to replicate and extend this finding. Experiment 1 confirmed that infants respond to changes in contour length but not to changes in number when contour length is controlled. However, contour length and area were confounded in this experiment and the original study. To determine what specific measure of spatial extent infants use to discriminate small sets, Experiment 2 included 2 conditions that varied either area or contour length, but not both. As before, infants responded to the changes in spatial extent-either contour length alone or area alone-but not to the changes in number.
Three experiments were conducted to assess the impact of adaptation to dense visual texture on the perceived numerosity and spatial distribution of texture. Study participants compared fields of dots presented in 2 locations, 1 of which was adapted to dense arrays of dots. The effect of adaptation was assessed by measuring points of subjective equality between the adapted and unadapted regions for the perceived density, numerosity, and distribution (cluster) of texture with staircase procedures for textures containing 20–320 dots. Perceived texture density was reduced at all numerosities. For high numerosities, density adaptation markedly diminished the perceived numerosity but not the apparent cluster of the dots. At low numerosities, the opposite pattern of results emerged, suggesting that density is more influential in the perception of high numerosities and that perceptual distortions of number and cluster may be traded off with one another. A simulation study of Allik and Tuulmets's (1991) occupancy model of perceived numerosity is also presented, and suggestions are made for modifying the model based on the patterns of results found. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Three experiments, using the high-amplitude sucking procedure, tested whether 4-day-old infants discriminate multisyllabic utterances on the basis of number of syllables or number of phonemes. Exp 1 showed that infants discriminate 2 large sets of phonetically variable utterances composed of 2- vs 3-CV (consonant–vowel) syllables. Exp 2 was run to assess whether infants discriminated the 2 sets on the basis of duration differences between the 2- and 3-CV stimuli. Results indicate that reducing the duration differences does not affect infants' discrimination. Finally, Exp 3 investigated whether infants discriminate 4- vs 6-phoneme bisyllabic utterances. The results provide no evidence that infants are sensitive to such a change in number of phonemic constituents. Although not decisive, these results appear to be congruent with the hypothesis that infants perceptually structure complex speech inputs. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Humans appear to share with animals a nonverbal counting process. In a nonverbal counting condition, subjects pressed a key
a numeral-specified number of times, while saying “the” at every press. The mean number of presses increased as a power function
of the target number, with a constant coefficient of variation (c.v.), both within and beyond the proposed subitizing range
(1–4 or 5), suggesting small numbers are represented on the same continuum as larger numbers and subject to the same noise
process (scalar variability). By contrast, when subjects counted their presses out loud as fast as they could, the c.v. decreased
as the inverse square root of the target value (binomial variability instead of scalar variability). The unexpected power-law
relation between target value and mean number of presses in nonverbal counting suggests a new hypothesis about the development
of the function relating number symbols to mental magnitudes.
The area of cognitive arithmetic is concerned with the mental representation of number and arithmetic, and the processes and procedures that access and use this knowledge. In this article, I provide a tutorial review of the area, first discussing the four basic empirical effects that characterize the evidence on cognitive arithmetic: the effects of problem size or difficulty, errors, relatedness, and strategies of processing. I then review three current models of simple arithmetic processing and the empirical reports that support or challenge their explanations. The third section of the review discusses the relationship between basic fact retrieval and a rule-based component or system, and considers current evidence and proposals on the overall architecture of the cognitive arithmetic system. The review concludes with a final set of speculations about the all-pervasive problem difficulty effect, still a central puzzle in the field.
We describe the preverbal system of counting and arithmetic reasoning revealed by experiments on numerical representations in animals. In this system, numerosities are represented by magnitudes, which are rapidly but inaccurately generated by the Meck and Church (1983) preverbal counting mechanism. We suggest the following. (1) The preverbal counting mechanism is the source of the implicit principles that guide the acquisition of verbal counting. (2) The preverbal system of arithmetic computation provides the framework for the assimilation of the verbal system. (3) Learning to count involves, in part, learning a mapping from the preverbal numerical magnitudes to the verbal and written number symbols and the inverse mappings from these symbols to the preverbal magnitudes. (4) Subitizing is the use of the preverbal counting process and the mapping from the resulting magnitudes to number words in order to generate rapidly the number words for small numerosities. (5) The retrieval of the number facts, which plays a central role in verbal computation, is mediated via the inverse mappings from verbal and written numbers to the preverbal magnitudes and the use of these magnitudes to find the appropriate cells in tabular arrangements of the answers. (6) This model of the fact retrieval process accounts for the salient features of the reaction time differences and error patterns revealed by experiments on mental arithmetic. (7) The application of verbal and written computational algorithms goes on in parallel with, and is to some extent guided by, preverbal computations, both in the child and in the adult.
It is widely believed that humans are endowed with a specialized numerical process, called subitizing, which enables them to apprehend rapidly and accurately the numerosity of small sets of objects. A major part of the evidence for this process is a purported discontinuity in the mean response time (RT) versus numerosity curves at about 4 elements, when subjects enumerate up to 7 or more elements in a visual display. In this article, RT data collected in a speeded enumeration experiment are subjected to a variety of statistical analyses, including several tests on the RT distributions. None of these tests reveals a significant discontinuity as numerosity increases. The data do suggest a strong stochastic dominance in RT by display numerosity, indicating that the mental effort required to enumerate does increase with each additional element in the display, both within and beyond the putative subitizing range.
Do Ss compare multidigit numbers digit by digit (symbolic model) or do they compute the whole magnitude of the numbers before comparing them (holistic model)? In 4 experiments of timed 2-digit number comparisons with a fixed standard, the findings of Hinrichs, Yurko, and Hu (1981) were extended with French Ss. Reaction times (RTs) decreased with target-standard distance, with discontinuities at the boundaries of the standard's decade appearing only with standards 55 and 66 but not with 65. The data are compatible with the holistic model. A symbolic interference model that posits the simultaneous comparison of decades and units can also account for the results. To separate the 2 models, the decades and units digits of target numbers were presented asynchronously in Experiment 4. Contrary to the prediction of the interference model, presenting the units before the decades did not change the influence of units on RTs. Pros and cons of the holistic model are discussed.
This paper proposes a probabilistic model of how humans identify the number of dots within a briefly presented visual display.
The model is an application of Thurstone’s law of comparative judgment, and it is assumed that the internal representation
of numerosity consists of log-spaced random variables. The discrimination between any two different numerosities is consequently
described as a function of max/rain, where max and rain are the larger and smaller numbers, respectively. The model was tested
in two experiments in which the Weber fraction for numerosity, corresponding with the critical ratio of max and min, was found
to have the value of 162. It was concluded that the classical span of subitizing numerosity is but a special case of the span
of discrimination.
"Subitizing," the process of enumeration when there are fewer than 4 items, is rapid (40-100 ms/item), effortless, and accurate. "Counting," the process of enumeration when there are more than 4 items, is slow (250-350 ms/item), effortful, and error-prone. Why is there a difference in the way the small and large numbers of items are enumerated? A theory of enumeration is proposed that emerges from a general theory of vision, yet explains the numeric abilities of preverbal infants, children, and adults. We argue that subitizing exploits a limited-capacity parallel mechanism for item individuation, the FINST mechanism, associated with the multiple target tracking task (Pylyshyn, 1989; Pylyshyn & Storm, 1988). Two kinds of evidence support the claim that subitizing relies on preattentive information, whereas counting requires spatial attention. First, whenever spatial attention is needed to compute a spatial relation (cf. Ullman, 1984) or to perform feature integration (cf. Treisman & Gelade, 1980), subitizing does not occur (Trick & Pylyshyn, 1993a). Second, the position of the attentional focus, as manipulated by cue validity, has a greater effect on counting than subitizing latencies (Trick & Pylyshyn, 1993b).
Seven studies explored the empirical basis for claims that infants represent cardinal values of small sets of objects. Many studies investigating numerical ability did not properly control for continuous stimulus properties such as surface area, volume, contour length, or dimensions that correlate with these properties. Experiment 1 extended the standard habituation/dishabituation paradigm to a 1 vs 2 comparison with three-dimensional objects and confirmed that when number and total front surface area are confounded, infants discriminate the arrays. Experiment 2 revealed that infants dishabituated to a change in front surface area but not to a change in number when the two variables were pitted against each other. Experiments 3 through 5 revealed no sensitivity to number when front surface area was controlled, and Experiments 6 and 7 extended this pattern of findings to the Wynn (1992) transformation task. Infants' lack of a response to number, combined with their demonstrated sensitivity to one or more dimensions of continuous extent, supports the hypothesis that the representations subserving object-based attention, rather than those subserving enumeration, underlie performance in the above tasks.
Three experiments were conducted to assess the impact of adaptation to dense visual texture on the perceived numerosity and spatial distribution of texture. Study participants compared fields of dots presented in 2 locations, 1 of which was adapted to dense arrays of dots. The effect of adaptation was assessed by measuring points of subjective equality between the adapted and unadapted regions for the perceived density, numerosity, and distribution (cluster) of texture with staircase procedures for textures containing 20-320 dots. Perceived texture density was reduced at all numerosities. For high numerosities, density adaptation markedly diminished the perceived numerosity but not the apparent cluster of the dots. At low numerosities, the opposite pattern of results emerged, suggesting that density is more influential in the perception of high numerosities and that perceptual distortions of number and cluster may be traded off with one another. A simulation study of Allik and Tuulmets's (1991) occupancy model of perceived numerosity is also presented, and suggestions are made for modifying the model based on the patterns of results found.
''Subitizing,'' the process of enumeration when there are fewer than 4 items, is rapid (40-100 ms/item), effortless, and accurate. ''Counting' the process of enumeration when there are more than 4 items, is slow (250-350 ms/item), effortful, and error-prone. Why is there a difference in the way the small and large numbers of items are enumerated? A theory of enumeration is proposed that emerges from a general theory of vision, yet explains the numeric abilities of preverbal infants, children, and adults. We argue that subitizing exploits a limited-capacity parallel mechanism for item individuation, the FINST mechanism, associated with the multiple target tracking task (Pylyshyn, 1989; Pylyshyn & Storm, 1988). Two kinds of evidence support the claim that subitizing relies on preattentive information, whereas counting requires spatial attention. First, whenever spatial attention is needed to compute a spatial relation (cf. Ullman, 1984) or to perform feature integration (cf. Treisman & Gelade, 1980), subitizing does not occur (Trick & Pylyshyn, 1993a). Second, the position of the attentional focus, as manipulated by cue validity, has a greater effect on counting than subitizing latencies (Trick & Pylyshyn, 1993b).
Two experiments examined 6-month-old infants' ability to individuate and enumerate physical actions—the sequential jumps of a puppet In both experiments, which employed a habituation paradigm, infants successfully discriminated two-jump from three-jump sequences The sequences of activity in the two experiments provided for an initial exploration of the cues infants use to individuate actions Results show that (a) infants can individuate and enumerate actions in a sequence, indicating that their enumeration mechanism is quite general in the kinds of entities over which it will operate, (b) actions whose temporal boundaries are characterized by a contrast between motion and absence of motion are especially easy to individuate and enumerate, but nonetheless (c) infants can individuate and enumerate actions embedded in a sequence of continuous motion, indicating that infants possess procedures for parsing an ongoing motionful scene into distinct portions of activity
Can animals do math? In her Perspective, Carey discusses the abilities of primates to mentally represent and manipulate numbers--reported on p. [746][1] by Brannon and Terrance--and compares these to the cultural and ontogenic development of numerical representation in humans.
[1]: http://www.sciencemag.org/cgi/content/short/282/5389/746
Three studies explored how infants parse a stream of motion into distinct actions. Results show that infants (a) can perceptually discriminate different actions performed by a puppet and (b) can individuate and enumerate heterogeneous sequences of such actions (e.g., jump-fall-jump) when the actions are separated by brief motionless pauses, but (c) are not able to individuate such actions when embedded within a continuous stream of motion. Combined with previous research showing that infants can individuate homogeneous actions from an ongoing stream of motion, these findings suggest that infants can use repeating patterns of motion in the perceptual input to define action boundaries. Results have implications as well for infants' conceptual structure for actions.
This paper presents a new conceptualization of the origins of numerical competence in humans. I first examine the existing claim that infants are innately provided with a system of specifically numerical knowledge, consisting of both cardinal and ordinal concepts. I suggest instead that the observed behaviors require only simple perceptual discriminations based on domain-independent competencies. At most, these involve the formal equivalent of cardinal information. Finally, I present a “non-numerical” account that characterizes infants competencies with regard to numerosity as emerging primarily from some general characteristics of the human perception and attention system.
The Head-Turn Preference Procedure (HPP) is valuable for testing perception of sustained auditory materials, particularly speech. This article presents a detailed description of the current version of HPP, new evidence of the objectivity of measurements within it, and an account of recent modifications.
An enduring question in philosophy and psychology is that of how we come to possess knowledge of number. Here I review research suggesting that the capacity to represent and reason about number is part of the inherent structure of the human mind. In the first few months of life, human infants can enumerate sets of entities and perform numerical computations. One proposal is that these abilities arise from general cognitive capacities not specific to number. I argue that the body of data supports a very different proposal: humans possess a specialized mental mechanism for number, one which we share with other species and which has evolved through natural selection. This mechanism is inherently restricted in the kinds of numerical knowledge it can support, leading to some striking limitations to early competence.
The present study investigated the ability of 3- and 4-year-old children to perform tasks which require matching sets of sounds to numerically equivalent visual displays. We found that 3-year-olds performed at chance on the auditory-visual matching task, while 4-year-olds performed significantly above chance. There is evidence that mastery of the linguistic counting system is related to success on this task. These findings are unexpected given previous research reporting that 6-8-month-olds can detect the numerical equivalence between a set of sounds and items in a visual display.
We describe two acalculic patients, one with a left subcortical lesion and the other with a right inferior parietal lesion and Gerstmann's syndrome. Both suffered from "pure anarithmetia": they could read arabic numerals and write them to dictation, but experienced a pronounced calculation deficit. On closer analysis, however, distinct deficits were found. The subcortical case suffered from a selective deficit of rote verbal knowledge, including but not limited to arithmetic tables, while her semantic knowledge of numerical quantities was intact. Conversely the inferior parietal case suffered from a category-specific impairment of quantitative numerical knowledge, particularly salient in subtraction and number bissection tasks, with preserved knowledge of rote arithmetic facts. This double dissociation suggests that numerical knowledge is processed in different formats within distinct cerebral pathways. We suggest that a left subcortical network contributes to the storage and retrieval of rote verbal arithmetic facts, while a bilateral inferior parietal network is dedicated to the mental manipulation of numerical quantities.
Does the human capacity for mathematical intuition depend on linguistic competence or on visuo-spatial representations? A
series of behavioral and brain-imaging experiments provides evidence for both sources. Exact arithmetic is acquired in a language-specific
format, transfers poorly to a different language or to novel facts, and recruits networks involved in word-association processes.
In contrast, approximate arithmetic shows language independence, relies on a sense of numerical magnitudes, and recruits bilateral
areas of the parietal lobes involved in visuo-spatial processing. Mathematical intuition may emerge from the interplay of
these brain systems.
Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio (8 vs. 16 but not 8 vs. 12). The capacities to represent approximate numerosity found in adult animals and humans evidently develop in human infants prior to language and symbolic counting.
Two independent research communities have produced large bodies of data concerning object representations: the community concerned with the infant's object concept and the community concerned with adult object-based attention. We marshal evidence in support of the hypothesis that both communities have been studying the same natural kind. The discovery that the object representations of young infants are the same as the object files of mid-level visual cognition has implications for both fields.
The time to compare two numbers shows additive effects of number notation and of semantic distance, suggesting that the comparison task can be decomposed into distinct stages of identification and semantic processing. Using event-related fMRI and high-density ERPs, we isolated cerebral areas where activation was influenced by input notation (verbal or Arabic notation). The bilateral extrastriate cortices and a left precentral region were more activated during verbal than during Arabic stimulation, while the right fusiform gyrus and a set of bilateral inferoparietal and frontal regions were more activated during Arabic than during verbal stimulation. We also identified areas that were influenced solely by the semantic content of the stimuli (numerical distance between numbers to be compared) independent of the input notation. Activation tightly correlated with numerical distance was observed mainly in a group of parietal areas distributed bilaterally along the intraparietal sulci and in the precuneus, as well as in the left middle temporal gyrus and posterior cingulate. Our results support the assumption of a central semantic representation of numerical quantity that relies on a common parietal network shared among notations.
Recent findings suggest that infants are capable of distinguishing between different numbers of objects, and of performing simple arithmetical operations. But there is debate over whether these abilities result from capacities dedicated to numerical cognition, or whether infants succeed in such experiments through more general, non-numerical capacities, such as sensitivity to perceptual features or mechanisms of object tracking. We report here a study showing that 5-month-olds can determine the number of collective entities -- moving groups of items -- when non-numerical perceptual factors such as contour length, area, density, and others are strictly controlled. This suggests both that infants can represent number per se, and that their grasp of number is not limited to the domain of objects.
Is numerical comparison digital: Analogi-cal and symbolic effects in two-digit number comparison Sources of mathe-matical thinking: Behavioral and brain-imaging evidence
- S Dehaene
- E Dupoux
- J Mehler
Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital: Analogi-cal and symbolic effects in two-digit number comparison. Psychology: Human Perception and Performance Dehaene, S., Spelke, E.S., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathe-matical thinking: Behavioral and brain-imaging evidence
Infants’ discrimination of number vs. con-tinuous extent). to vision by 5-month-old infants . Poster presented at the Biennial International Con-ference on Infant Studies
- L Feigenson
- S Carey
- E S Spelke
- J Feron
- A Streri
- E Gentaz
Feigenson, L., Carey, S., & Spelke, E.S. (2002). Infants’ discrimination of number vs. con-tinuous extent. Cognitive Psychology , 44 Feron, J., Streri, A., & Gentaz, E. (2002, April). to vision by 5-month-old infants . Poster presented at the Biennial International Con-ference on Infant Studies, Toronto, Ontario, Canada
Numerical intermodal transfer from touch to vision by 5-month-old infants
- J Feron
- A Streri
- E Gentaz
Feron, J., Streri, A., & Gentaz, E. (2002, April). Numerical intermodal transfer from touch to vision by 5-month-old infants. Poster presented at the Biennial International Con-ference on Infant Studies, Toronto, Ontario, Canada.
Intermodal numerical correspon-dences in 6-month-old infants . Poster presented at the Biennial International Con-ference on Infant Studies Do preschool children recognize audi-tory-visual numerical correspondences? Child Development
- T Kobayashi
- K Hiraki
- T Hasegawa
- S C Levine
Kobayashi, T., Hiraki, K., & Hasegawa, T. (2002, April). Intermodal numerical correspon-dences in 6-month-old infants. Poster presented at the Biennial International Con-ference on Infant Studies, Toronto, Ontario, Canada. Mix, K.S., Huttenlocher, J., & Levine, S.C. (1996). Do preschool children recognize audi-tory-visual numerical correspondences? Child Development, 67, 1592–1608.
Intermodal numerical correspondences in 6-month-old infants
- T Kobayashi
- K Hiraki
- T Hasegawa
Kobayashi, T., Hiraki, K., & Hasegawa, T. (2002, April). Intermodal numerical correspondences in 6-month-old infants. Poster presented at the Biennial International Conference on Infant Studies, Toronto, Ontario, Canada.