Emergent Morphology in Child Homesign: Evidence from Number Language

Human languages, signed and spoken, can be characterized by the structural patterns they use to associate communicative forms with meanings. One such pattern is paradigmatic morphology, where complex words are built from the systematic use and re-use of sub-lexical units. Here, we provide evidence of emergent paradigmatic morphology akin to number inflection in a communication system developed without input from a conventional language, homesign. We study the communication systems of four deaf child homesigners (mean age 8;02). Although these idiosyncratic systems vary from one another, we nevertheless find that all four children use handshape and movement devices productively to express cardinal and non-cardinal number information, and that their number expressions are consistent in both form and meaning. Our study shows, for the first time, that all four homesigners not only incorporate number devices into representational devices used as predicates , but also into gestures functioning as nominals, including deictic gestures. In other words, the homesigners express number by systematically combining and re-combining additive markers for number (qua inflectional morphemes) with representational and deictic gestures (qua bases). The creation of new, complex forms with predictable meanings across gesture types and linguistic functions constitutes evidence for an inflectional morphological paradigm in homesign and expands our understanding of the structural patterns of language that are, and are not, dependent on linguistic input.

The history of number words in the world's languages-what have we learnt so far?

For over 100 years, researchers from various disciplines have been enthralled and occupied by the study of number words. This article discusses implications for the study of deep history and human evolution that arise from this body of work. Phylogenetic modelling shows that low-limit number words are preserved across thousands of years, a pattern consistently observed in several language families. Cross-linguistic frequencies of use and experimental studies also point to widespread homogeneity in the use of number words. Yet linguistic typology and field documentation reports caution against positing a privileged linguistic category for number words, showing a wealth of variation in how number words are encoded across the world. In contrast with low-limit numbers, the higher numbers are characterized by a rapid and morphologically consistent pattern of expansion, and behave like grammatical phrasal units, following language-internal rules. Taken together, the evidence suggests that numbers are at the cross-roads of language history. For languages that do have productive and consistent number systems, numerals one to five are among the most reliable available linguistic fossils of deep history, defying change yet still bearing the marks of the past, while higher numbers emerge as innovative tools looking to the future, derived using language-internal patterns and created to meet the needs of modern speakers. This article is part of the theme issue 'Reconstructing prehistoric languages'.

Range and distribution effects on number line placement

People's placement of numbers on number lines sometimes shows linear and sometimes compressive scaling. We investigated whether people's placement of numbers was affected by their range and distribution, as indicated by Parducci's (Psychological Review, 72, 407-418, 1965) range-frequency theory. Experiment 1 found large compressive effects when the endpoints were 1 and 10¹⁶. Experiment 2 showed compression when 14 logarithmically distributed numbers were placed on a line marked 1-1,000 and close to linear scaling when the numbers were linearly distributed. Thus, we found both range and frequency effects on compression. Where compression arose, it was not as pronounced as that predicted by logarithmic scaling, but analyses of the results from Experiments 1 and 2 indicate this was not explained by participants switching between linear and logarithmic scaling.

The Challenge of Modeling the Acquisition of Mathematical Concepts

As a full-blown research topic, numerical cognition is investigated by a variety of disciplines including cognitive science, developmental and educational psychology, linguistics, anthropology and, more recently, biology and neuroscience. However, despite the great progress achieved by such a broad and diversified scientific inquiry, we are still lacking a comprehensive theory that could explain how numerical concepts are learned by the human brain. In this perspective, I argue that computer simulation should have a primary role in filling this gap because it allows identifying the finer-grained computational mechanisms underlying complex behavior and cognition. Modeling efforts will be most effective if carried out at cross-disciplinary intersections, as attested by the recent success in simulating human cognition using techniques developed in the fields of artificial intelligence and machine learning. In this respect, deep learning models have provided valuable insights into our most basic quantification abilities, showing how numerosity perception could emerge in multi-layered neural networks that learn the statistical structure of their visual environment. Nevertheless, this modeling approach has not yet scaled to more sophisticated cognitive skills that are foundational to higher-level mathematical thinking, such as those involving the use of symbolic numbers and arithmetic principles. I will discuss promising directions to push deep learning into this uncharted territory. If successful, such endeavor would allow simulating the acquisition of numerical concepts in its full complexity, guiding empirical investigation on the richest soil and possibly offering far-reaching implications for educational practice.

Sources of Directional Spatial Biases in Hemi-Image Drawing

We examined the performance of right- and left-handed brain-intact adult readers of English or Farsi on a hemi-image generation task in which participants were to imagine and then draw halves of objects using either their dominant or nondominant hand. Which half of the object was drawn was examined in relation to biomechanical, cerebral laterality, and cultural predictors. Findings showed a differential side bias as a function of reading/writing direction and hand used to draw. Specifically, when the dominant hand was used to draw (Experiment 1), English left-handers produced more left hemi-images while Farsi right-handers produced more right hemi-images. Body specificity associated with hand used, however, drove spatial preference when enlisting the nondominant hand (Experiment 2) with English right-handers now showing a significant left hemi-image bias and English left-handers showing a right hemi-image bias. Farsi right-handers using their nondominant hand did not show a significant bias in either direction. Taken together, the findings suggest a joint influence of handedness and reading/writing direction, aligned with an embodiment account of directional spatial biases.

Absence perception and the philosophy of zero

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating (1) an account of numbers as properties of collections, (2) work on the philosophy of absences, and (3) recent work in numerical cognition and ontogenetic studies.

The metaphoric nature of the ordinal position effect

Serial orders are thought to be spatially represented in working memory: The beginning items in the memorised sequence are associated with the left side of space and the ending items are associated with the right side of space. However, the origin of this ordinal position effect has remained unclear. It was suggested that the direction of serial order-space interaction is related to the reading/writing experience. An alternative hypothesis is that it originates from the "more is right"/"more is up" spatial metaphors we use in daily life. We can adjudicate between the two viewpoints in Chinese readers; they read left-to-right but also have a culturally ancient top-to-bottom reading/writing direction. Thus, the reading/writing viewpoint predicts no or a top-to-bottom effect in serial order-space interaction; whereas the spatial metaphor theory predicts a clear bottom-to-top effect. We designed four experiments to investigate this issue. First, we found a left-to-right ordinal position effect, replicating results obtained in Western populations. However, the vertical ordinal position effect was in the bottom-to-top direction; moreover, it was modulated by hand position (e.g., left hand bottom or up). We suggest that order-space interactions may originate from different sources and are driven by metaphoric comprehension, which itself may ground cognitive processing.

Closing the symbolic reference gap to support flexible reasoning about the passage of time

This commentary relates Hoerl & McCormack's dual systems perspective to models of cognitive development emphasizing representational redescription and the role of culturally constructed tools, including language, in providing flexible formats for thinking. We describe developmental processes that enable children to construct a mental time line, situate themselves in time, and overcome the primacy of the here and now.

Variability in the Alignment of Number and Space Across Languages and Tasks

While the domains of space and number appear to be linked in human brains and minds, their conceptualization still differs across languages and cultures. For instance, frames of reference for spatial descriptions vary according to task, context, and cultural background, and the features of the mental number line depend on formal education and writing direction. To shed more light on the influence of culture/language and task on such conceptualizations, we conducted a large-scale survey with speakers of five languages that differ in writing systems, preferences for spatial and temporal representations, and/or composition of number words. Here, we report data obtained from tasks on ordered arrangements, including numbers, letters, and written text. Comparing these data across tasks, domains, and languages indicates that, even within a single domain, representations may differ depending on task characteristics, and that the degree of cross-domain alignment varies with domains and culture.

The Use of Local and Global Ordering Strategies in Number Line Estimation in Early Childhood

A lot of research has been devoted to number line estimation in primary school. However, less is known about the early onset of number line estimation before children enter formal education. We propose that ordering strategies are building blocks of number line estimation in early childhood. In a longitudinal study, children completed a non-symbolic number line estimation task at age 3.5 and 5 years. Two ordering strategies were identified based on the children's estimation patterns: local and global ordering. Local ordering refers to the correct ordering of successive quantities, whereas global ordering refers to the correct ordering of all quantities across the number line. Results indicated a developmental trend for both strategies. The percentage of children applying local and global ordering strategies increased steeply from 3.5 to 5 years of age. Moreover, children used more advanced local and global ordering strategies at 5 years of age. Importantly, level of strategy use was related to more traditional number line estimation outcome measures, such as estimation accuracy and regression fit scores. These results provide evidence that children use dynamic ordering strategies when solving the number line estimation task in early stages of numerical development.

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

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

Distinct Representations of Magnitude and Spatial Position within Parietal Cortex during Number-Space Mapping

Mapping numbers onto space is foundational to mathematical cognition. These cognitive operations are often conceptualized in the context of a "mental number line" and involve multiple brain regions in or near the intraparietal sulcus (IPS) that have been implicated both in numeral and spatial cognition. Here we examine possible differentiation of function within these brain areas in relating numbers to spatial positions. By isolating the planning phase of a number line task and introducing spatiotopic mapping tools from fMRI into mental number line task research, we are able to focus our analysis on the neural activity of areas in anterior IPS (aIPS) previously associated with number processing and on spatiotopically organized areas in and around posterior IPS (pIPS), while participants prepare to place a number on a number line. Our results support the view that the nonpositional magnitude of a numerical symbol is coded in aIPS, whereas the position of a number in space is coded in posterior areas of IPS. By focusing on the planning phase, we are able to isolate activation related to the cognitive, rather than the sensory-motor, aspects of the task. Also, to allow the separation of spatial position from magnitude, we tested both a standard positive number line (0 to 100) and a zero-centered mixed number line (-100 to 100). We found evidence of a functional dissociation between aIPS and pIPS: Activity in aIPS was associated with a landmark distance effect not modulated by spatial position, whereas activity in pIPS revealed a contralateral preference effect.

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

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

Current Perspectives on Cognitive Diversity

To what extent is cognition influenced by a person’s cultural background? This question has remained controversial in large fields of the cognitive sciences, including cognitive psychology, and is also underexplored in anthropology. In this perspective article, findings from a recent wave of cross-cultural studies will be outlined with respect to three aspects of cognition: perception and categorization, number representation and counting, and explanatory frameworks and beliefs. Identifying similarities and differences between these domains allows for general conclusions regarding cognitive diversity and helps to highlight the importance of culturally shaped content for a comprehensive understanding of cognition.

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

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

Development of Spatial-Numerical Associations

Links between spatial and numerical thinking are well established in studies of adult cognition. Here, we review recent research on the origins and development of these links, with an emphasis on the formative role of experience in typical development and on the theoretical insights to be gained from infant cognition. This research points to three important influences on the development of spatial-numerical associations: innate mechanisms linking space and nonsymbolic number, gross and fine motor activity that couples spatial location to both symbolic and nonsymbolic number, and culturally bound activities (e.g., reading, writing, and counting) that shape the relationship between spatial direction and symbolic number.

Cognitive enhancement kept within contexts: neuroethics and informed public policy

Neurothics has far greater responsibilities than merely noting potential human enhancements arriving from novel brain-centered biotechnologies and tracking their implications for ethics and civic life. Neuroethics must utilize the best cognitive and neuroscientific knowledge to shape incisive discussions about what could possibly count as enhancement in the first place, and what should count as genuinely "cognitive" enhancement. Where cognitive processing and the mental life is concerned, the lived context of psychological performance is paramount. Starting with an enhancement to the mental abilities of an individual, only performances on real-world exercises can determine what has actually been cognitively improved. And what can concretely counts as some specific sort of cognitive improvement is largely determined by the classificatory frameworks of cultures, not brain scans or laboratory experiments. Additionally, where the public must ultimately evaluate and judge the worthiness of individual performance enhancements, we mustn't presume that public approval towards enhancers will somehow automatically arrive without due regard to civic ideals such as the common good or social justice. In the absence of any nuanced appreciation for the control which performance contexts and public contexts exert over what "cognitive" enhancements could actually be, enthusiastic promoters of cognitive enhancement can all too easily depict safe and effective brain modifications as surely good for us and for society. These enthusiasts are not unaware of oft-heard observations about serious hurdles for reliable enhancement from neurophysiological modifications. Yet those observations are far more common than penetrating investigations into the implications to those hurdles for a sound public understanding of cognitive enhancement, and a wise policy review over cognitive enhancement. We offer some crucial recommendations for undertaking such investigations, so that cognitive enhancers that truly deserve public approval can be better identified.

Spatial Associations in Numerical Cognition—From Single Digits to Arithmetic

The literature on spatial associations during number processing is dominated by the SNARC (spatial–numerical association of response codes) effect. We describe spatial biases found for single digits and pairs of numbers, first in the “original” speeded parity task and then extending the scope to encompass different tasks, a range of measures, and various populations. Then we review theoretical accounts before surveying the emerging evidence for similar spatial associations during mental arithmetic. We conclude that the mental number line hypothesis and an embodied approach are useful frameworks for further studies.

Doing Arithmetic by Hand: Hand Movements during Exact Arithmetic Reveal Systematic, Dynamic Spatial Processing

Mathematics requires precise inferences about abstract objects inaccessible to perception. How is this possible? One proposal is that mathematical reasoning, while concerned with entirely abstract objects, nevertheless relies on neural resources specialized for interacting with the world—in other words, mathematics may be grounded in spatial or sensorimotor systems. Mental arithmetic, for instance, could involve shifts in spatial attention along a mental “number-line”, the product of cultural artefacts and practices that systematically spatialize number and arithmetic. Here, we investigate this hypothesized spatial processing during exact, symbolic arithmetic (e.g., 4 + 3 = 7). Participants added and subtracted single-digit numbers and selected the exact solution from responses in the top corners of a computer monitor. While they made their selections using a computer mouse, we recorded the movement of their hand as indexed by the streaming x, y coordinates of the computer mouse cursor. As predicted, hand movements during addition and subtraction were systematically deflected toward the right and the left, respectively, as if calculation involved simultaneously simulating motion along a left-to-right mental number-line. This spatial–arithmetical bias, moreover, was distinct from—but correlated with—individuals’ spatial–numerical biases (i.e., spatial–numerical association of response codes, SNARC, effect). These results are the first evidence that exact, symbolic arithmetic prompts systematic spatial processing associated with mental calculation. We discuss the possibility that mathematical calculation relies, in part, on an integrated system of spatial processes.

Mapping spatial frames of reference onto time: a review of theoretical accounts and empirical findings

When speaking and reasoning about time, people around the world tend to do so with vocabulary and concepts borrowed from the domain of space. This raises the question of whether the cross-linguistic variability found for spatial representations, and the principles on which these are based, may also carry over to the domain of time. Real progress in addressing this question presupposes a taxonomy for the possible conceptualizations in one domain and its consistent and comprehensive mapping onto the other-a challenge that has been taken up only recently and is far from reaching consensus. This article aims at systematizing the theoretical and empirical advances in this field, with a focus on accounts that deal with frames of reference (FoRs). It reviews eight such accounts by identifying their conceptual ingredients and principles for space-time mapping, and it explores the potential for their integration. To evaluate their feasibility, data from some thirty empirical studies, conducted with speakers of sixteen different languages, are then scrutinized. This includes a critical assessment of the methods employed, a summary of the findings for each language group, and a (re-)analysis of the data in view of the theoretical questions. The discussion relates these findings to research on the mental time line, and explores the psychological reality of temporal FoRs, the degree of cross-domain consistency in FoR adoption, the role of deixis, and the sources and extent of space-time mapping more generally.

Nature and culture of finger counting: diversity and representational effects of an embodied cognitive tool

Studies like the one conducted by Domahs et al. (2010, in Cognition) corroborate that finger counting habits affect how numbers are processed, and legitimize the assumption that this effect is culturally modulated. The degree of cultural diversity in finger counting, however, has been grossly underestimated in the field at large, which, in turn, has restricted research questions and designs. In this paper, we demonstrate that fingers as a tool for counting are not only naturally available, but are also-and crucially so-culturally encoded. To substantiate this, we outline the variability in finger counting and illustrate each of its types with instances from the literature. We argue that the different types of finger counting all constitute distinct representational systems, and we use their properties-dimensionality, dimensional representation, base and sub-base values, extendibility and extent, sign count, and regularity-to devise a typology of such systems. This allows us to explore representational effects, that is, the cognitive implications these properties may have, for instance, for the efficiency of information encoding and representation, ease of learning and mastering the system, or memory retrieval and cognitive load. We then highlight the ambivalent consequences arising from structural inconsistencies between finger counting and other modes of number representation like verbal or notational systems, and we discuss how this informs questions on the evolution and development of counting systems. Based on these analyses, we suggest some directions for future research in the field of embodied cognition that would profit substantially from taking into account the cultural diversity in finger counting.

Number concepts without number lines in an indigenous group of Papua New Guinea

BACKGROUND

The generic concept of number line, which maps numbers to unidimensional space, is a fundamental concept in mathematics, but its cognitive origins are uncertain. Two defining criteria of the number line are that (i) there is a mapping of each individual number (or numerosity) under consideration onto a specific location on the line, and (ii) that the mapping defines a unidimensional space representing numbers with a metric--a distance function. It has been proposed that the number line is based on a spontaneous universal human intuition, rooted directly in brain evolution, that maps number magnitude to linear space with a metric. To date, no culture lacking this intuition has been documented.

METHODOLOGY/PRINCIPAL FINDINGS

By means of a number line task, we investigated the universality proposal with the Yupno of Papua New Guinea. Unschooled adults did exhibit a number-to-space mapping (criterion i) but, strikingly, despite having precise cardinal number concepts, they located numbers only on the endpoints, thus failing to use the extent of the line. The produced mapping was bi-categorical and metric-free, in violation of criterion ii. In contrast, Yupnos with scholastic experience used the extent of the segment according to known standards, but they did so not as evenly as western controls, exhibiting a bias towards the endpoints.

CONCLUSIONS/SIGNIFICANCE

Results suggest that cardinal number concepts can exist independently from number line representations. They also suggest that the number line mapping, although ubiquitous in the modern world, is not universally spontaneous, but rather seems to be learned through--and continually reinforced by--specific cultural practices.