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One approach to understanding how the human cognitive system stores and operates with quantifiers such as "some," "many," and "all" is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link between quantifier processing and nonsymbolic quantity processing has been considered in the past, it has never been discussed extensively. Simultaneously, there is a long line of research within the field of numerical cognition on the relationship between processing exact number symbols (such as "3" or "three") and nonsymbolic quantity. This accumulated knowledge can potentially be harvested for research on quantifiers since quantifiers and number symbols are two different ways of referring to quantity information symbolically. The goal of the present review is to survey the research on the relationship between quantifiers and nonsymbolic quantity processing mechanisms and provide a set of research directions and specific questions for the investigation of quantifier processing. [ABSTRACT FROM AUTHOR]

This study focuses on adults who are neither preschool teachers nor professional caregivers and investigates their beliefs regarding the importance of engaging young children with numerical activities. It also examines the types of numerical activities adults report having observed children engaging with, as well as the types of activities they propose as a way for promoting counting, enumerating, recognizing number symbols, and number composition and decomposition. Findings showed that participants believed to a great extent that engaging young children with numerical activities is important. Most reported that they had observed children engaging with at least some numerical activity. In general, participants relayed more activities and more detailed activities when suggesting activities for each competency, than they did when reporting observed activities. Findings also suggested a need to enhance adults' knowledge regarding the necessity to promote verbal counting, separate from object counting, as well as to increase adults' awareness of number composition and decomposition. For mathematics educators wishing to plan workshops for adults, this study offers a method for investigating adults' knowledge of early numerical activities, as well as a starting point with which to plan appropriate workshops. [ABSTRACT FROM AUTHOR]

This study probed the cognitive mechanisms that underlie order processing for number symbols, specifically the extent to which the direction and format in which number symbols are presented influence the processing of numerical order, as well as the extent to which the relationship between numerical order processing and mathematical achievement is specific to Arabic numerals or generalisable to other notational formats. Seventy adults who were bilingual in English and Chinese completed a Numerical Ordinality Task, using number sequences of various directional conditions (i.e., ascending, descending, mixed) and notational formats (i.e., Arabic numerals, English number words, and Chinese number words). Order processing was found to occur for ascending and descending number sequences (i.e., ordered but not non-ordered trials), with the overall pattern of data supporting the theoretical perspective that the strength and closeness of associations between items in the number sequence could underlie numerical order processing. However, order processing was found to be independent of the notational format in which the numerical stimuli were presented, suggesting that the psychological representations and processes associated with numerical order are abstract across different formats of number symbols. In addition, a relationship between the processing speed for numerical order judgements and mathematical achievement was observed for Arabic numerals and Chinese number words, and to a weaker extent, English number words. Together, our findings have started to uncover the cognitive mechanisms that could underlie order processing for different formats of number symbols, and raise new questions about the generalisability of these findings to other notational formats. [ABSTRACT FROM AUTHOR]

Research has shown that two different, though related, ways of representing magnitude play foundational roles in the development of numerical and mathematical skills: a nonverbal approximate number system and an exact symbolic number system. While there have been numerous studies suggesting that the two systems are important predictors of math achievement, there has been substantial debate regarding whether and how these basic numerical competencies may be developmentally interrelated. Specifically, the causal direction of their relation has been the subject of debate: whether children's approximate number abilities predict later symbolic number abilities (the mapping account) or the other way around (the refinement account). Our sample included 622 kindergarten children (mean age = 62 months, SD = 3.5, 279 females, 75 born outside Canada), whose dot comparison, number comparison, and mixed comparison skills were assessed over three time-points and math achievement assessed over four time-points. We contrasted multiple theoretical predictions of the interrelations between the variables of interest posited by these two developmental accounts using longitudinal random intercept cross-lagged models. Results were most consistent with the refinement account, suggesting that earlier symbolic number ability is consistently the strongest predictor of approximate number ability, mixed-comparison ability, and arithmetic skills. Notably, our results demonstrated that, when individual models are examined in isolation, model fit was adequate or near adequate for all models tested. This highlights the need for future research to contrast competing accounts, as our results suggest the examination of any one account in isolation may not reveal the best fitting developmental model. [ABSTRACT FROM AUTHOR]

Children’s ability to map numbers into a spatial context has been shown to be a powerful predictor of math performance. Here, we investigate how three types of cognitive abilities – approximate number processing ability, symbolic number processing ability, and non-numerical cognitive abilities – predict 0–100 number-line estimation performance in first graders. While each type of measure predicts number-line performance when considered individually, when considered together, only symbolic number comparison and non-verbal reasoning predicted unique variance in number-line estimation. Moreover, the relation between symbolic number comparison and number-line ability was stronger for male students than for female students, suggesting potential gender differences in the way boys and girls accomplish mapping numbers into space. These results suggest that number-line estimation ability is largely reflective of the precision with which symbolic magnitudes are represented (at least among boys). Our findings therefore suggest that promoting children’s understanding of symbolic, rather than non-symbolic, numerical magnitudes may help children learn better from number-lines in the classroom.

Children's ability to map numbers into a spatial context has been shown to be a powerful predictor of math performance. Here, we investigate how three types of cognitive abilities – approximate number processing ability, symbolic number processing ability, and non-numerical cognitive abilities – predict 0–100 number-line estimation performance in first graders. While each type of measure predicts number-line performance when considered individually, when considered together, only symbolic number comparison and non-verbal reasoning predicted unique variance in number-line estimation. Moreover, the relation between symbolic number comparison and number-line ability was stronger for male students than for female students, suggesting potential gender differences in the way boys and girls accomplish mapping numbers into space. These results suggest that number-line estimation ability is largely reflective of the precision with which symbolic magnitudes are represented (at least among boys). Our findings therefore suggest that promoting children's understanding of symbolic, rather than non-symbolic, numerical magnitudes may help children learn better from number-lines in the classroom. [ABSTRACT FROM AUTHOR]

How the brain encodes abstract numerical symbols is a fundamental question in philosophy and cognitive neuroscience alike. Here we probe the nature of symbolic number representation in the brain by characterizing the neural similarity space for symbolic quantities in regions sensitive to their semantic content. In parietal and occipital regions, the similarity space of number symbols was positively predicted by the lexical frequency of numerals in parietal and occipital areas, and was unrelated to numerical ratio. These results are more consistent with a categorical, frequency-based account of symbolic quantity encoding. In contrast, the similarity space of analog quantities was positively predicted by ratio in prefrontal, parietal and occipital regions. We thus provide an explanation for why previous work has indicated that symbolic and analog quantities are distinct: number symbols operate primarily like discrete categories sensitive to input frequency, while analog quantities operate more like approximate perceptual magnitudes. In addition, we find substantial evidence for related patterns of activity across formats in prefrontal, parietal and occipital regions. Crucially however, between-format relations were not specific to individual quantities, indicating common processing as opposed to common representation. Moreover, evidence for between-format processing was strongest for quantities that could be represented as exact, discrete values in both systems (quantities in the 'subitizing' range: 1–4). In sum, converging evidence presented here indicates that symbolic quantities are coded in the brain as discrete categories sensitive to input frequency and largely independent of approximate, analog quantities. [ABSTRACT FROM AUTHOR]

There is currently considerable discussion about the relative influences of evolutionary and cultural factors in the development of early numerical skills. In particular, there has been substantial debate and study of the relationship between approximate, nonverbal (approximate magnitude system [AMS]) and exact, symbolic (symbolic number system [SNS]) representations of number. Here we examined several hypotheses concerning whether, in the earliest stages of formal education, AMS abilities predict growth in SNS abilities, or the other way around. In addition to tasks involving symbolic (Arabic numerals) and nonsymbolic (dot arrays) number comparisons, we also tested children's ability to translate between the 2 systems (i.e.. mixed-format comparison). Our data included a sample of 539 kindergarten children (M - 5.17 years, SD = .29), with AMS, SNS, and mixed-comparison skills assessed at the beginning and end of the academic year. In this way, we provide, to the best of our knowledge, the most comprehensive test to date of the direction of influence between the AMS and SNS in early formal schooling. Results were more consistent with the view that SNS abilities at the beginning of kindergarten lay the foundation for improvement in both AMS abilities and the ability to translate between the 2 systems. It is important to note that we found no evidence to support the reverse. We conclude that, once one acquires a basic grasp of exact number symbols, it is this understanding of exact number (and perhaps repeated practice therewith) that facilitates growth in the AMS. Though the precise mechanism remains to be understood, these data challenge the widely held view that the AMS scaffolds the acquisition of the SNS. [ABSTRACT FROM AUTHOR]

A key question in the field of numerical cognition is how the human brain represents numerical symbols (e.g., Arabic digits). A large body of research has implicated left parietal regions in symbolic number processing. One possible explanation for this lateralization of neural activity is the handedness of participants. Specifically, participants in neuroimaging research are almost exclusively right-handed. The current study sought to probe whether number representation in the brain is associated with hand preference for handwriting. To address this question, we used functional Magnetic Resonance Imaging (fMRI) to compare brain activation of a group of right-handed participants with a group of left-handed participants during the passive viewing of symbolic numbers. At the whole-brain level, the right-handers demonstrated a previously obtained left-lateralized effect within the intraparietal sulcus (IPS). The left-handers showed some evidence of reverse lateralization of this effect in the IPS. However, when the groups were statistically compared, we found no regions that demonstrated group-level differences. In a follow-up region of interest (ROI) analysis within the left and right parietal lobes, we calculated laterality indices for each participant. Results did not provide support for group differences in laterality within the right or left IPS ROI. These findings do not support the hypothesis that handwriting plays a role in the lateralization of symbolic number processing in the brain. Further research is needed to better understand the factors that lead to left-lateralization for symbolic number processing., (Copyright © 2019 Elsevier Ltd. All rights reserved.)

Numerical and mathematical skills are critical predictors of academic success. The last three decades have seen a substantial growth in our understanding of how the human mind and brain represent and process numbers. In particular, research has shown that we share with animals the ability to represent numerical magnitude (the total number of items in a set) and that preverbal infants can process numerical magnitude. Further research has shown that similar processing signatures characterize numerical magnitude processing across species and developmental time. These findings suggest that an approximate system for nonsymbolic (e.g., dot arrays) numerical magnitude representation serves as the basis for the acquisition of cultural, symbolic (e.g., Arabic numerals) representations of numerical magnitude. This chapter explores this hypothesis by reviewing studies that have examined the relation between individual differences in nonsymbolic numerical magnitude processing and symbolic math abilities (e.g., arithmetic). Furthermore, we examine the extent to which the available literature provides strong evidence for a link between symbolic and nonsymbolic representations of numerical magnitude at the behavioral and neural levels of analysis. We conclude that claims that symbolic number abilities are grounded in the approximate system for the nonsymbolic representation of numerical magnitude are not strongly supported by the available evidence. Alternative models and future research directions are discussed., (© 2015 Elsevier Inc. All rights reserved.)