To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

Recent accounts of number word learning posit that when children learn to accurately count sets (i.e., become “cardinal principle” or “CP” knowers), they have a conceptual insight about how the count list implements the successor function – i.e., that every natural number n has a successor defined as n + 1 (Carey, 2004, 2009; Sarnecka & Carey, 2008). However, recent studies suggest that knowledge of the successor function emerges sometime after children learn to accurately count, though it remains unknown when this occurs, and what causes this developmental transition. We tested knowledge of the successor function in 100 children aged 4 through 7 and asked how age and counting ability are related to: (1) children’s ability to infer the successors of all numbers in their count list and (2) knowledge that all numbers have a successor. We found that children do not acquire these two facets of the successor function until they are about 5½ or 6 years of age – roughly 2 years after they learn to accurately count sets and become CP-knowers. These findings show that acquisition of the successor function is highly protracted, providing the strongest evidence yet that it cannot drive the cardinal principle induction. We suggest that counting experience, as well as knowledge of recursive counting structures, may instead drive the learning of the successor function.     

Attaching meaning to the number words: contributions of the object tracking and approximate number systems

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems – the approximate number system (ANS) and the object tracking system (OTS) – but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single‐mechanism (ANS‐only and OTS‐only) and two dual‐mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single‐mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.     

多元表征假设：概念表征机制的新观点

传统的认知主义认为概念表征是与主体的感知系统无关的抽象符号。而具身理论则认为,概念表征以主体的感觉、知觉运动系统为基础的,感知系统在概念表征中具有中心作用。然而,具身性假设无法恰当的解释抽象概念表征这一问题。这种局限性说明主体的概念系统可能具有多元表征机制：既包括感知表征以加工与身体经验相关的具体知识,也包括抽象符号表征以加工与身体经验无关的抽象知识。来自病理学、认知神经科学和行为实验的实证研究证明了不同类型的概念会涉及不同的表征机制,证实了多元表征存在的合理性。今后的研究应探讨各种表征机制之间的关系等问题。     

Influences of problem format and SES on preschoolers’ understanding of approximate addition

Recent studies suggest that 5-year-olds can add and compare large numerical quantities through approximate representations of number. However, the nature of this understanding and its susceptibility to environmental influences remain unclear. We examined whether children's early competence depends on the canonical problem format (i.e., arithmetic operations presented on the left side). Sixty children from middle-to-high-SES backgrounds (Experiment 1) and 47 children from low-SES backgrounds (Experiment 2) viewed events that required them to add and compare large numbers. Events were shown in a canonical or noncanonical format. Children from both SES backgrounds performed above chance on the approximate addition tasks, but children from middle-to-high-SES backgrounds performed significantly better. Moreover, children from middle-to-high SES backgrounds performed better when problems were presented in the canonical format, whereas children from low-SES backgrounds did not. These results suggest that children's understanding of approximate number is affected by some of the same environmental factors that affect performance on exact arithmetic tasks.     

The development of numerosity estimation: Evidence for a linear number representation early in life

Several studies investigating the development of approximate number representations used the number-to-position task and reported evidence for a shift from a logarithmic to a linear representation of numerical magnitude with increasing age. However, this interpretation as well as the number-to-position method itself has been questioned recently. The current study tested 5- and 8-year-old children on a newly established numerosity production task to examine developmental changes in number representations and to test the idea of a representational shift. Modelling of the children's numerical estimations revealed that responses of the 8-year-old children approximate a simple positive linear relation between estimated and actual numbers. Interestingly, however, the estimations of the 5-year-old children were best described by a bilinear model reflecting a relatively accurate linear representation of small numbers and no apparent magnitude knowledge for large numbers. Taken together, our findings provide no support for a shift of mental representations from a logarithmic to a linear metric but rather suggest that the range of number words which are appropriately conceptualised and represented by linear analogue magnitude codes expands during development.     

Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.     

两种数量表征系统

数量表征是人类数学能力的基础,数量表征研究中的一个争论焦点在于是否存在两种不同的数量表征系统：对小数的精确表征系统和对大数的近似表征系统。通过综述不同研究领域对数量表征的研究,总结了支持两种表征系统分离的证据：对1~3范围内小数的表征受数量大小的限制,基于指向物体本身的注意,更依赖于物体的知觉特征,对物体及其数量进行精确表征;而对4以上的数量的近似表征系统则受韦伯定律的限制,基于指向数量的模拟幅度的表征,而不依赖单个物体的知觉特征,是对数量的近似的、心理的表征。fMRI、PET和ERP的脑成像研究结果迄今尚无定论,但认知神经科学研究的深入开展将最终阐明数量表征的机制     

Much ado about nothing? Why going non-semantic is not merely semantics

This paper argues that deciding on whether the cognitive sciences need a Representational Theory of Mind matters. Far from being merely semantic or inconsequential, the answer we give to the RTM-question makes a difference to how we conceive of minds. How we answer determines which theoretical framework the sciences of mind ought to embrace. The structure of this paper is as follows. Section 1 outlines Rowlands’s (2017) argument that the RTM-question is a bad question and that attempts to answer it, one way or another, have neither practical nor theoretical import. Rowlands concludes this because, on his analysis, there is no non-arbitrary fact of the matter about which properties something must possess in order to qualify as a mental representation. By way of reply, we admit that Rowlands’s analysis succeeds in revealing why attempts to answer the RTM-question simpliciter are pointless. Nevertheless, we show that if specific formulations of the RTM-question are stipulated, then it is possible, conduct substantive RTM debates that do not collapse into merely verbal disagreements. Combined, Sections 2 and 3 demonstrate how, by employing specifying stipulations, we can get around Rowlands’s arbitrariness challenge. Section 2 reveals why RTM, as canonically construed in terms of mental states exhibiting intensional (with-an-s) properties, has been deemed a valuable explanatory hypothesis in the cognitive sciences. Targeting the canonical notion of mental representations, Section 3 articulates a rival nonrepresentational hypothesis that, we propose, can do all the relevant explanatory work at much lower theoretical cost. Taken together, Sections 2 and 3 show what can be at stake in the RTM debate when it is framed by appeal to the canonical notion of mental representation and why engaging in it matters. Section 4 extends the argument for thinking that RTM debates matter. It provides reasons for thinking that, far from making no practical or theoretical difference to the sciences of the mind, deciding to abandon RTM would constitute a revolutionary conceptual shift in those sciences.     

The spatial representation of numerical and non-numerical ordered sequences: Insights from a random generation task

It is widely believed that numbers are spatially represented from left to right on the mental number line. Whether this spatial format of representation is specific to numbers or is shared by non-numerical ordered sequences remains controversial. When healthy participants are asked to randomly generate digits they show a systematic small-number bias that has been interpreted in terms of “pseudoneglect in number space”. Here we used a random generation task to compare numerical and non-numerical order. Participants performed the task at three different pacing rates and with three types of stimuli (numbers, letters, and months). In addition to a small-number bias for numbers, we observed a bias towards “early” items for letters and no bias for months. The spatial biases for numbers and letters were rate independent and similar in size, but they did not correlate across participants. Moreover, letter generation was qualified by a systematic forward direction along the sequence, suggesting that the ordinal dimension was more salient for letters than for numbers in a task that did not require its explicit processing. The dissociation between numerical and non-numerical orders is consistent with electrophysiological and neuroimaging studies and suggests that they rely on at least partially different mechanisms.     

Neural signatures of number processing in human infants: evidence for two core systems underlying numerical cognition

Behavioral research suggests two cognitive systems are at the foundations of numerical thinking: one for representing 1-3 objects in parallel and one for representing and comparing large, approximate numerical magnitudes. We tested for dissociable neural signatures of these systems in preverbal infants, by recording event-related potentials (ERPs) as 6-7.5 month-old infants (n = 32) viewed dot arrays containing either small (1-3) or large (8-32) sets of objects in a number alternation paradigm. If small and large numbers are represented by the same neural system, then the brain response to the arrays should scale with ratio for both number ranges, a behavioral and brain signature of the approximate numerical magnitude system obtained in animals and in human adults. Contrary to this prediction, a mid-latency positivity (P500) over parietal scalp sites was modulated by the ratio between successive large, but not small, numbers. Conversely, an earlier peaking positivity (P400) over occipital-temporal sites was modulated by the absolute cardinal value of small, but not large, numbers. These results provide evidence for two early developing systems of non-verbal numerical cognition: one that responds to small quantities as individual objects and a second that responds to large quantities as approximate numerical values. These brain signatures are functionally similar to those observed in previous studies of non-symbolic number with adults, suggesting that this dissociation may persist over vast differences in experience and formal training in mathematics.     

Measuring the approximate number system

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.     

Calibrating the mental number line

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.     

THE TROUBLE WITH MARY

Abstract: Two arguments are famously held to support the conclusion that consciousness cannot be explained in purely physical or functional terms – hence, that physicalism is false: the modal argument and the knowledge argument. While anti‐physicalists appeal to both arguments, this paper argues there is a methodological incoherence in jointly maintaining them: the modal argument supports the possibility of zombies; but the possibility of zombies undercuts the knowledge argument. At best, this leaves anti‐physicalists in a considerably weakened rhetorical position. At worst, it shows that commonsense intuitions on which anti‐physicalists rely mislead us about the true nature of conscious experience.     

Infants' enumeration of actions: numerical discrimination and its signature limits

Are abstract representations of number – representations that are independent of the particular type of entities that are enumerated – a product of human language or culture, or do they trace back to human infancy? To address this question, four experiments investigated whether human infants discriminate between sequences of actions (jumps of a puppet) on the basis of numerosity. At 6 months, infants successfully discriminated four‐ versus eight‐jump sequences, when the continuous variables of sequence duration, jump duration, jump rate, jump interval and duration, and extent of motion were controlled, and rhythm was eliminated. In contrast, infants failed to discriminate two‐ versus four‐jump sequences, suggesting that infants fail to form cardinal number representations of small numbers of actions. Infants also failed to discriminate between sequences of four versus six jumps at 6 months, and succeeded at 9 months, suggesting that infants’ number representations are imprecise and increase in precision with age. All of these findings agree with those of studies using visual–spatial arrays and auditory sequences, providing evidence that a single, abstract system of number representation is present and functional in infancy.     

Talking our way to systematicity

Do we think in a language-like format? Taking the marker of language-like formats to be the property of unconstrained systematicity, this paper considers the following master argument for the claim that we do: (1) language is unconstrainedly systematic, (2) if language is unconstrainedly systematic then so is thought, (3) so thought is unconstrainedly systematic. It is easy to feel that there is something right about this argument, that there will be some way of filling in its details that will vindicate the idea that our thought must be unconstrainedly systematic given that the language in which we express it is. Clearly, however, the second premise needs support—we need a principled reason for moving from the unconstrained systematicity of language to the unconstrained systematicity of thought. This paper gives three passes at formulating such a principle. This turns out to be much harder than it might seem. We should, I conclude, resist falling too easily for the lure of this master argument for the language-like format of thought.

     

The Idea of an Exact Number: Children's Understanding of Cardinality and Equinumerosity

Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one‐to‐one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood.     

数形联觉——数字空间表征研究“新”取向

数字空间表征是人类对数字进行表征的重要方式。数形联觉（number-form synesthesia）是一种数字可以有意识地引起空间知觉的独特现象,与此类似的是非联觉者中广泛存在的无意识的心理数字线（mental number line）现象。两者在行为和脑机制上存在着很多重叠,也存在着值得思考的差异。数形联觉的研究能够提供实质性的行为和脑机制数据,用以解决数字空间表征研究中出现的问题,加强对于数字空间表征的理解;也为更加全面深入地开展进一步研究提供了新的启示,成为数字空间表征研究中值得推崇的新取向。     

Auditory spatial concepts in blind football experts

ObjectivesWe compared the spatial concepts given to sounds' directions by blind football players with both blind non-athletes and sighted individuals.MethodParticipants verbally described the directions of sounds around them by using predefined spatial concept labels, under two blocked conditions: 1) facing front, 2) pointing with the hand towards the stimulus.ResultsBlind football players categorized the directions more precisely (i.e., they used simple labels for describing the cardinal directions and combined labels for the intermediate ones) than the other groups, and their categorization was less sensitive to the response conditions than blind non-athletes. Sighted participants' categorization was similar to previous studies, in which the front and back regions were generally more precisely described than the sides, where simple labels were often used for describing directions around the absolute left and right.ConclusionsThe differences in conceptual categorization of sound directions are a) in sighted individuals, influenced by the representation of the visual space b) in blind individuals, influenced by the level of expertise in action and locomotion based on non-visual information, which can be increased by auditive stimulation provided by blind football training.     

A Dissociation between Symbolic Number Knowledge and Analogue Magnitude Information

Semantic understanding of numbers and related concepts can be dissociated from rote knowledge of arithmetic facts. However, distinctions among different kinds of semantic representations related to numbers have not been fully explored. Working with numbers and arithmetic requires representing semantic information that is both analogue (e.g., the approximate magnitude of a number) and symbolic (e.g., what / means). In this article, the authors describe a patient (MC) who exhibits a dissociation between tasks that require symbolic number knowledge (e.g., knowledge of arithmetic symbols including numbers, knowledge of concepts related to numbers such as rounding) and tasks that require an analogue magnitude representation (e.g., comparing size or frequency). MC is impaired on a variety of tasks that require symbolic number knowledge, but her ability to represent and process analogue magnitude information is intact. Her deficit in symbolic number knowledge extends to a variety of concepts related to numbers (e.g., decimal points, Roman numerals, what a quartet is) but not to any other semantic categories that we have tested. These findings suggest that symbolic number knowledge is a functionally independent component of the number processing system, that it is category specific, and that it is anatomically and functionally distinct from magnitude representations.     

Propositions,Numbers, And The Problem Of Arbitrary Identification

Those inclined to believe in the existence of propositions as traditionally conceived might seek to reduce them to some other type of entity. However, parsimonious propositionalists of this type are confronted with a choice of competing candidates – for example, sets of possible worlds, and various neo-Russellian and neo-Fregean constructions. It is argued that this choice is an arbitrary one, and that it closely resembles the type of problematic choice that, as Benacerraf pointed out, bedevils the attempt to reduce numbers to sets – should the number 2 be identified with the set Ø or with the set Ø, Ø? An “argument from arbitrary identification” is formulated with the conclusion that propositions (and perhaps numbers) cannot be reduced away. Various responses to this argument are considered, but ultimately rejected. The paper concludes that the argument is sound: propositions, at least, are sui generis entities.