Recent theories in numerical cognition suggest that humans are equipped with a mental system that supports the representation and processing of symbolic and nonsymbolic magnitudes, called the Approximate Number System (ANS). Prior research also suggests that the acuity of the ANS can predict individuals’ mathematical ability. However, results from research within the field has proven to be inconsistent with one another which raises questions about the reliability and validity of methods used to measure the ANS. The present study attempts to replicate the results found in studies suggesting that ANS acuity correlates with mathematical ability. The study also investigates the reliability and validity of different task that have been used to measure the ANS, and also presents a new method of measuring the ANS with an adaptive method. The results show that two tasks correlate significantly with mathematical ability, and multiple regression analyses show that ANS acuity can predict mathematical ability when controlling for general intelligence. Furthermore, the results also further highlight the issue of methodological flaws in previous studies.

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<p>The uniquely human mathematical mind sets us apart from all other animals. Although humans typically think about number symbolically, we also possess nonverbal representations of quantity that are present at birth and shared with many other animal species. These primitive numerical representations are thought to arise from an evolutionarily ancient system termed the Approximate Number System (ANS). The present dissertation aims to determine how these preverbal representations of quantity may serve as the foundation for more complex quantitative reasoning abilities. To this end, the five studies contained herein investigate the relations between representations of number, representations of other magnitude dimensions, and symbolic math proficiency in infants, children, and adults. The first empirical study, described in Chapter 2, investigated whether infants engage the ANS to represent the full range of natural numbers. The study presented in Chapter 3 compared infants' acuity for detecting changes in contour length to their acuity for detecting changes in number to assess whether representations of continuous quantities are primary to representations of number in infancy. The study presented in Chapter 4 compared individual differences in acuity for number, line length, and brightness in children and adults to determine how the relations between these magnitudes may change over development. Chapter 5 contains a longitudinal study investigating the relation between preverbal number sense in infancy and symbolic math abilities in preschool-aged children. Finally, the study presented in Chapter 6 investigated the mechanisms underlying the maturation of the number sense and determined which features of the number sense are predictive of symbolic math skill. Taken together, these findings confirm that number is a salient feature of the environment for infants and young children and suggest that approximate number representations are fundamental for the acquisition of symbolic math.</p> / Dissertation

Over the past twenty years, the Approximate Number System (ANS), a cognitive system for representing non-symbolic quantity information, has been the focus of much research attention. Psychologists seeking to understand how individuals learn and perform mathematics have investigated how this system might underlie symbolic mathematical skills. Dot comparison tasks are commonly used as measures of ANS acuity, however very little is known about the cognitive skills that are involved in completing these tasks. The aim of this thesis was to explore the factors that influence performance on dot comparison tasks and discuss the implications of these findings for future research and educational interventions. The first study investigated how the accuracy and reliability of magnitude judgements is influenced by the visual cue controls used to create dot array stimuli. This study found that participants performances on dot comparison tasks created with different visual cue controls were unrelated, and that stimuli generation methods have a substantial influence on test-retest reliability. The studies reported in the second part of this thesis (Studies 2, 3, 4 and 5) explored the role of inhibition in dot comparison task performance. The results of these studies provide evidence that individual differences in inhibition may, at least partially, explain individual differences in dot comparison task performance. Finally, a large multi-study re-analysis of dot comparison data investigated whether individuals take account of numerosity information over and above the visual cues of the stimuli when comparing dot arrays. This analysis revealed that dot comparison task performance may not reflect numerosity processing independently from visual cue processing for all participants, particularly children. This novel evidence may provide some clarification for conflicting results in the literature regarding the relationship between ANS acuity and mathematics achievement. The present findings call into question whether dot comparison tasks should continue to be used as valid measures of ANS acuity.

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Children who struggle to learn math are often identified by their poor performance on common math learning activities, such as number line estimations. While such behavioral assessments are useful in the classroom, naturalistic neuroimaging of children engaged in real-world math learning activities has the potential to identify concurrent behavioral and neurological correlates to poor math performance. Such correlates may help pinpoint effective teaching strategies for atypical learners, and may highlight instructional methods that elicit typical neurological response patterns to such activities. For example, multisensory stimulation that contains information about number enhances infants' and preschool children's behavioral performance on many numerical tasks and has been shown to elicit neural activation in areas related to number processing and decision-making. Thus, when applied to math teaching tools, multisensory stimulation may provide a platform through which both behavioral and neural math-related processes may be enhanced. Common approaches to neuroimaging of math processing lack ecological validity and are often not analogous to real-world learning activities. However, because of its liberal tolerance of movement, near-infrared spectroscopy (NIRS) provides an ideal platform for such studies. Here, NIRS is used to provide the first concurrent examination of neurological and behavioral data from number line estimation performance within children and adults. Moreover, in an effort to observe the behavioral and neurological benefits to number line estimations that may arise from multisensory stimulation, differential feedback (i.e., visual, auditory, or audiovisual) about estimation performance is provided throughout a portion of the task. Results suggest behavioral and neural performance is enhanced by feedback. Moreover, significant effects of age suggest young children show greater neurological response to feedback, and increase in task difficulty resulted in decreased behavioral performance and increased neurological activation associated with mathematical processing. Thus, typical math learners effectively recruit areas of the brain known to process number when math activities become increasingly difficult. Data inform understanding typical behavioral and neural responses to real-world math learning tasks, and may prove useful in triangulating signatures of atypical math learning. Moreover, results demonstrate the utility of NIRS as a platform to provide simultaneous neurological and behavioral data during naturalistic math learning activities.

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De nombreuses études ont montré qu'il existe deux systèmes indépendants permettant une représentation de la numérosité sans l'utilisation des noms des nombres. Ces deux systèmes ont été mis en évidence chez l'adulte, l'animal, ainsi que chez le nourrisson. Le premier système permet une représentation approximative de la numérosité: la capacité à discriminer entre deux numérosités dépend alors de leur ratio. Le second système a pour rôle initial le suivi spatio-temporel d'objets. L'individuation, en parallèle, de différents objets permet ainsi au système d'abstraire de façon indirecte la numérosité d'un ensemble. Contrairement au premier système, le second permet d'encoder de manière exacte la numérosité mais présente toutefois une limite quant au nombre d'items pouvant être suivis (4 pour l'adulte, 3 pour le nourrisson à partir de 5 mois). Si ces deux systèmes ont été mis en évidence chez le nourrisson avant l'acquisition des mots des nombres, la question de leur apparition et de leurs liens persiste toujours. Alors que le système approximatif a été mis en évidence dès la naissance, nous n'avons pas de preuves empiriques de l'existence du second système au même âge. C'est pourquoi, à travers deux groupes d'études, nous cherché à mettre en évidence l'existence d'un système spécifique des petits ensembles chez le nouveau-né, en utilisant des situations bimodales audio-visuelles. Le premier ensemble d'expériences a montré l'existence d'une dissociation entre les petites et les grandes numérosités. Les expériences suivantes ont permis de mettre en évidence l'existence du système pour les petits ensembles dès la naissance, mais qui présente toutefois une limite de suivi à deux items et non trois comme pour les nourrissons plus âgés. Afin d'étudier l'articulation entre les deux systèmes, une expérience, décrite dans un chapitre supplémentaire, a été réalisée chez le nourrisson de 5 mois testant la discrimination continue de la numérosité, à l'aide d'un paradigme de transfert intermodal entre le toucher et la vision. / Many studies showed that two systems are available to encode numerosities without the use of number words. These two systems have been shown to exist in adults, animals, and also in infants. The first system can represent approximate numerosities: the capacity to discriminate between two numerosities depends on their ratio difference. The second system has for primary role the spatiotemporal tracking of objects. The parallel individuation of several objects enables the system to encode implicitly the numerosity of a set. Contrary to the first system, the second can encode exact numerosities, however this system is limited by the number of objects that can be tracked (4 in adults, 3 in infants from 5 months of age). These two systems have been shown to exist in infants before the acquisition of number words, however, the question of their emergence and of their links remains. While a study showed that the approximate system exists from birth on, we do not know if the second system exists at the same age. In two series of studies, using audiovisual bimodal situations, we tested the existence of a system specific for small sets in newborn infants. The first set of experiments showed a dissociation between small and large numerosities. The next experiments revealed the existence of the system for small sets from birth on, however this system appears to be limited to 2 objects, while it has a limit of 3 in older infants. In order to study the link between the two systems for encoding numerosity, a third group of experiments was conducted with 5-month-old infants. These experiments, described in a supplementary chapter, tested the continuity between small and large numerosities using a crossmodal transfer paradigm between tactile and visual modalities.