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.     

Small and large number discrimination in guppies

Non-verbal numerical behavior in human infants, human adults, and non-human primates appears to be rooted in two distinct mechanisms: a precise system for tracking and comparing small numbers of items simultaneously (up to 3 or 4 items) and an approximate system for estimating numerical magnitude of a group of objects. The most striking evidence that these two mechanisms are distinct comes from the apparent inability of young human infants and non-human primates to compare quantites across the small (<3 or 4)/large (>4) number boundary. We ask whether this distinction is present in lower animal species more distantly related to humans, guppies (Poecilia reticulata). We found that, like human infants and non-human primates, fish succeed at comparisons between large numbers only (5 vs. 10), succeed at comparisons between small numbers only (3 vs. 4), but systematically fail at comparisons that closely span the small/large boundary (3 vs. 5). Furthermore, increasing the distance between the small and large number resulted in successful discriminations (3 vs. 6, 3 vs. 7, and 3 vs. 9). This pattern of successes and failures is similar to those observed in human infants and non-human primates to suggest that the two systems are present and functionally distinct across a wide variety of animal species.     

核心知识系统及其对相关研究的启示

核心知识系统出现于人类个体发展和种系发展的早期,在人类复杂认知能力的发生发展中起着建构模块的作用。该文以表征物体和数量关系的两类核心知识系统的性质及二者在儿童的“数”概念形成和成人的数学思维中所起的作用为例,综述了以灵长类动物、婴儿、儿童和成人为被试的研究证据,并进而提出了核心知识系统给相关研究带来的若干启示。     

Large quantity discrimination by North Island robins (Petroica longipes)

While numerosity—representation and enumeration of different numbers of objects—and quantity discrimination in particular have been studied in a wide range of species, very little is known about the numerical abilities of animals in the wild. This study examined spontaneous relative quantity judgments (RQJs) by wild North Island robins (Petroica longipes) of New Zealand. In Experiment 1, robins were tested on a range of numerical values of up to 14 versus 16 items, which were sequentially presented and hidden. In Experiment 2, the same numerical contrasts were tested on a different group of subjects but quantities were presented as whole visible sets. Experiment 3 involved whole visible sets that comprised of exceedingly large quantities of up to 56 versus 64 items. While robins shared with other species a ratio-based representation system for representing very large values, they also appeared to have developed an object indexing system with an extended upper limit (well beyond 4) that may be an evolutionary response to ecological challenges faced by scatter-hoarding birds. These results suggest that cognitive mechanism influencing an understanding of physical quantity may be deployed more flexibly in some contexts than previously thought, and are discussed in light of findings across other mammalian and avian species.     

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.     

人类数能力的演化基础——数能力比较研究的启示

人类的数能力可以发展到很高的抽象水平,但大量研究表明,人类婴儿和非人灵长类也具有基本的数表征和数推理能力。文章从表征内容和表征形式两个方面系统地比较了人类婴儿和非人灵长类的数能力,并对比了成人和非人灵长类数表征的生理基础。人类和非人灵长类在这三方面存在的相似性揭示了他们可能享有相同的数表征系统。在今后的研究中,进一步探讨这两者相似的数表征核心系统,可以加深我们对人类数能力起源和本质的理解     

Spontaneous analog number representations in 3‐year‐old children

When enumerating small sets of elements nonverbally, human infants often show a set‐size limitation whereby they are unable to represent sets larger than three elements. This finding has been interpreted as evidence that infants spontaneously represent small numbers with an object‐file system instead of an analog magnitude system ( Feigenson, Dehaene & Spelke, 2004 ). In contrast, non‐human animals and adult humans have been shown to rely on analog magnitudes for representing both small and large numbers ( Brannon & Terrace, 1998 ; Cantlon & Brannon, 2007 ; Cordes, Gelman, Gallistel & Whalen, 2001). Here we demonstrate that, like adults and non‐human animals, children as young as 3 years of age spontaneously employ analog magnitude representations to enumerate both small and large sets. Moreover, we show that children spontaneously attend to numerical value in lieu of cumulative surface area. These findings provide evidence of young children’s greater sensitivity to number relative to other quantities and demonstrate continuity in the process they spontaneously recruit to judge small and large values.     

Large-number addition and subtraction by 9-month-old infants

Do genuinely numerical computational abilities exist in infancy? It has recently been argued that previous studies putatively illustrating infants' ability to add and subtract tapped into specialized object-tracking processes that apply only with small numbers. This argument contrasts with the original interpretation that successful performance was achieved via a numerical system for estimating and calculating magnitudes. Here, we report that when continuous variables (such as area and contour length) are controlled, 9-month-old infants successfully add and subtract over numbers of items that exceed object-tracking limits. These results support the theory that infants possess a magnitude-based estimation system for representing numerosities that also supports procedures for numerical computation.     

Core knowledge

Human cognition is founded, in part, on four systems for representing objects, actions, number, and space. It may be based, as well, on a fifth system for representing social partners. Each system has deep roots in human phylogeny and ontogeny, and it guides and shapes the mental lives of adults. Converging research on human infants, non-human primates, children and adults in diverse cultures can aid both understanding of these systems and attempts to overcome their limits.     

A sensorimotor perspective on numerical cognition

Numbers are present in every part of modern society and the human capacity to use numbers is unparalleled in other species. Understanding the mental and neural representations supporting this capacity is of central interest to cognitive psychology, neuroscience, and education. Embodied numerical cognition theory suggests that beyond the seemingly abstract symbols used to refer to numbers, their underlying meaning is deeply grounded in sensorimotor experiences, and that our specific understanding of numerical information is shaped by actions related to our fingers, egocentric space, and experiences with magnitudes in everyday life. We propose a sensorimotor perspective on numerical cognition in which number comprehension and numerical proficiency emerge from grounding three distinct numerical core concepts: magnitude, ordinality, and cardinality.     

Multiple spatially overlapping sets can be enumerated in parallel

A system for nonverbally representing the approximate number of items in visual and auditory arrays has been documented in multiple species, including humans. Although many aspects of this approximate number system are well characterized, fundamental questions remain unanswered: how does attention select which items in a scene to enumerate, and how many enumerations can be computed simultaneously? Here we show that when presented an array containing different numbers of spatially overlapping dots of many colors, human adults can select and enumerate items on the basis of shared color and can enumerate approximately three color subsets from a single glance. This three-set limit converges with previously observed three-item limits of parallel attention and visual short-term memory. This suggests that participants can select a subset of items from a complex array as a single individual set, which then serves as the input to the approximate number system.     

Varieties of numerical abilities.

This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.     

Neurocognitive start-up tools for symbolic number representations

Attaching meaning to arbitrary symbols (i.e. words) is a complex and lengthy process. In the case of numbers, it was previously suggested that this process is grounded on two early pre-verbal systems for numerical quantification: the approximate number system (ANS or 'analogue magnitude'), and the object tracking system (OTS or 'parallel individuation'), which children are equipped with before symbolic learning. Each system is based on dedicated neural circuits, characterized by specific computational limits, and each undergoes a separate developmental trajectory. Here, I review the available cognitive and neuroscientific data and argue that the available evidence is more consistent with a crucial role for the ANS, rather than for the OTS, in the acquisition of abstract numerical concepts that are uniquely human.     

Better elementary number processing in higher skill arithmetic problem solvers: evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

Better elementary number processing in higher skill arithmetic problem solvers: Evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

从数量表征到数表征：具身认知视角下的人类数能力获得

数能力是数学认知的基本成分。与动物所具有的基本数能力不同,人类不仅具备数量表征能力,更重要的是还拥有对数概念进行表征的数表征能力。虽然具身认知与离身认知都对数概念的表征问题进行了解释,但二者却存在明显理论分歧。具身认知观点主要从具身数量表征和数能力发展的具身认知机制两方面为人类独特数能力的获得提供了理论支撑及实证证据。这启示人们需要重视具身学习在数能力形成实践中的关键作用,重视具身数量表征在数学教学中的作用,仍需进一步揭示其内在的心理和神经基础。     

Relationships between number and space processing in adults with and without dyscalculia

A large body of evidence indicates clear relationships between number and space processing in healthy and brain-damaged adults, as well as in children. The present paper addressed this issue regarding atypical math development. Adults with a diagnosis of dyscalculia (DYS) during childhood were compared to adults with average or high abilities in mathematics across two bisection tasks. Participants were presented with Arabic number triplets and had to judge either the number magnitude or the spatial location of the middle number relative to the two outer numbers. For the numerical judgment, adults with DYS were slower than both groups of control peers. They were also more strongly affected by the factors related to number magnitude such as the range of the triplets or the distance between the middle number and the real arithmetical mean. By contrast, adults with DYS were as accurate and fast as adults who never experienced math disability when they had to make a spatial judgment. Moreover, number–space congruency affected performance similarly in the three experimental groups. These findings support the hypothesis of a deficit of number magnitude representation in DYS with a relative preservation of some spatial mechanisms in DYS. Results are discussed in terms of direct and indirect number–space interactions.     

Quantity discrimination in Tenebrio molitor: evidence of numerosity discrimination in an invertebrate?

Numerosity discrimination, the ability to distinguish between sets with more and less items, is recognised as the foundation for higher numerical abilities. Understanding numerosity discrimination from a comparative perspective is hence pivotal in tracing the evolution of numerical representation systems. However, numerosity discrimination has been well studied only in vertebrates, where two innate systems of number representation have been described: an ‘analog magnitude system’ used to discriminate among numerosities by representing them as cardinal magnitudes and a ‘parallel individualisation system’ that allows precise discrimination among small arrays of items (≤4) by representing objects individually. We investigated the existence of quantity discrimination in an insect species (Tenebrio molitor) by using a spontaneous two-choice procedure in which males were exposed to substrates bearing odours from different numbers of females (≤4) in increasing numerosity ratios (1:4, 1:3 and 1:2). We show that males can discriminate sources of odours reflecting 1 versus 4 and 1 versus 3 females, but not 2 versus 4 or 1 versus 2, indicating that T. molitor males exhibit a marked preference for sources reflecting more female donors only when numerosity ratios are below 1:2. We discuss the functional significance of this finding and whether our pattern of results could be best explained by summation of a non-numerical continuous variable or by the existence of a numerosity discrimination mechanism with an operational signature ratio of 1:2.     

The own-species face bias: A review of developmental and comparative data

Face recognition is characterized in part by biases resulting in enhanced perception and memory for individuals within frequently encountered groups and impaired recognition for individuals within uncommonly encountered groups. These biases are found across multiple categories and levels, including species, race, age, and gender (Scherf & Scott, 2012). At the highest level of categorization, human adults and nonhuman primates exhibit improved recognition (Dufour, Pascalis, & Petit, 2006; Pascalis & Bachevalier, 1998) and discrimination abilities (Pascalis, de Haan, & Nelson, 2002) for individuals within their own species, reflecting what has been called an “own-species” or “species-specific” bias in face processing. The own-species bias is a model system for delineating the role of experience and the developmental trajectory of face-processing biases, for further understanding the malleability of face biases in adults, and for examining and comparing face-processing abilities across species. Here, we will review findings from studies investigating the perception of other-species faces during development, into adulthood and across species.     

Quantity discrimination in female mosquitofish

The ability in animals to count and represent different numbers of objects has received a great deal of attention in the past few decades. Cumulative evidence from comparative studies on number discriminations report obvious analogies among human babies, non-human primates and birds and are consistent with the hypothesis of two distinct and widespread mechanisms, one for counting small numbers (<4) precisely, and one for quantifying large numbers approximately. We investigated the ability to discriminate among different numerosities, in a distantly related species, the mosquitofish, by using the spontaneous choice of a gravid female to join large groups of females as protection from a sexually harassing male. In one experiment, we found that females were able to discriminate between two shoals with a 1:2 numerosity ratio (2 vs. 4, 4 vs. 8 and 8 vs. 16 fish) but failed to discriminate a 2:3 ratio (8 vs. 12 fish). In the second experiment, we studied the ability to discriminate between shoals that differed by one element; females were able to select the larger shoal when the paired numbers were 2 vs. 3 or 3 vs. 4 but not 4 vs. 5 or 5 vs. 6. Our study indicates that numerical abilities in fish are comparable with those of other non-verbal creatures studied; results are in agreement with the hypothesis of the existence of two distinct systems for quantity discrimination in vertebrates.