Learning and development of embodied numerosity

Recent empirical evidence indicates that seemingly abstract numerical cognitions are rooted in sensory and bodily experiences. In particular in finger counting finger-based representations reflect a specific case of embodied cognition, we termed embodied numerosity. Furthermore, we suggest that finger-based representations should be considered a distinct representation of number (magnitude) and argue that this representation is activated automatically whenever we encounter a number. We discuss in what way such a theoretical framework can account for the associations of fingers and numbers observed so far. In the final part, we evaluate whether the concept of embodied numerosity should be generalized beyond finger-based representations with particular focus on whether bodily-sensory experiences (such as moving the whole body along the mental number line) may corroborate numerical capabilities. In a series of intervention studies, we consistently observed more pronounced training effects for our embodied numerosity trainings for different age groups, different digital media, different number ranges, and different control conditions. Taken together, we conclude that embodied representations of number (magnitude) exist, are not limited to finger-based representations, and influence number processing in a systematic and functional way that can be used to foster the efficiency of numerical trainings.     

Tactile Enumeration and Embodied Numerosity Among the Deaf

Representations of the fingers are embodied in our cognition and influence performance in enumeration tasks. Among deaf signers, the fingers also serve as a tool for communication in sign language. Previous studies in normal hearing (NH) participants showed effects of embodiment (i.e., embodied numerosity) on tactile enumeration using the fingers of one hand. In this research, we examined the influence of extensive visuo-manual use on tactile enumeration among the deaf. We carried out four enumeration task experiments, using 1–5 stimuli, on a profoundly deaf group (n = 16) and a matching NH group (n = 15): (a) tactile enumeration using one hand, (b) tactile enumeration using two hands, (c) visual enumeration of finger signs, and (d) visual enumeration of dots. In the tactile tasks, we found salient embodied effects in the deaf group compared to the NH group. In the visual enumeration of finger signs task, we controlled the meanings of the stimuli presentation type (e.g., finger-counting habit, fingerspelled letters, both or neither). Interestingly, when comparing fingerspelled letters to neutrals (i.e., not letters or numerical finger-counting signs), an inhibition pattern was observed among the deaf. The findings uncover the influence of rich visuo-manual experiences and language on embodied representations. In addition, we propose that these influences can partially account for the lag in mathematical competencies in the deaf compared to NH peers. Lastly, we further discuss how our findings support a contemporary model for mental numerical representations and finger-counting habits.     

手指的感知、运动以及数量表征对数字认知的促进作用

手指是儿童在习得数字符号之前最常使用的表征数量的工具,大量研究都表明手指在数字认知中具有促进作用。但是,目前仍不清楚手指在数字认知中的作用机制。综述从手指感知、手指运动以及手指数量表征三个方面总结了手指在数字认知中所起的作用。手指感知可能通过影响儿童的数量表征能力间接地影响其它数学能力;与表征量有关的手指运动可能促进了数量大小的加工。关于手指数量表征在数字认知中的作用存在两种有争议的观点：一种认为手指数量表征促进了儿童由非符号数量表征向符号数量表征的转化;另一种认为手指数量表征可能是一种数量语义表征方式。未来应该在发展、作用机制、性别差异等方向继续开展研究,进一步探讨手指在数字认知中所起的作用。     

Finger–digit compatibility in Arabic numeral processing

Finger–digit response compatibility was tested by asking participants to identify Arabic digits by pressing 1 of 10 keys with all 10 fingers. The direction of the finger–digit mapping was varied by manipulating the global direction of the hand–digit mapping as well as the direction of the finger–digit mapping within each hand (in each case, from small to large digits, or the reverse). The hypothesis of a left-to-right mental number line predicted that a complete left-to-right mapping should be easier whereas the hypothesis of a representation based on finger counting predicted that a counting-congruent mapping should be easier. The results show that when all 10 fingers are used to answer, a mapping congruent with the prototypical finger-counting strategy reported by the participants leads to better performance than does a mapping congruent with a left-to-right oriented mental number line, both in palm-down and palm-up postures of the hands, and they demonstrate that finger-counting strategies influence the way that numerical information is mentally represented and processed.     

Embodied numerosity: Implicit hand-based representations influence symbolic number processing across cultures

In recent years, a strong functional relationship between finger counting and number processing has been suggested. Developmental studies have shown specific effects of the structure of the individual finger counting system on arithmetic abilities. Moreover, the orientation of the mental quantity representation (“number line”) seems to be influenced by finger counting habits. However, it is unclear whether the structure of finger counting systems still influences symbolic number processing in educated adults.In the present transcultural study, we pursued this question by examining finger-based sub-base-five effects in an Arabic number comparison task with three different groups of participants (German deaf signers, German and Chinese hearing adults). We observed sub-base-five effects in all groups, but particularly so for both German groups who use an explicit sub-base-five system in their finger counting habits. It is concluded that bodily experiences – namely finger counting – influence the structure of the abstract mental number representations even in adults. Thus, the present findings support the general idea that even seemingly abstract cognition may at least partially be rooted in our bodily experiences.     

The development of numerical estimation: evidence against a representational shift

How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations. Here we present a new theoretically motivated and empirically supported account of the development of numerical estimation, based on the idea that number‐line estimation tasks entail judgments of proportion. We extend existing models of perceptual proportion judgment to the case of abstract numerical magnitude. Two experiments provide support for these models; three likely sources of developmental change in children’s estimation performance are identified and discussed. This work demonstrates that proportion‐judgment models provide a unified account of estimation patterns that have previously been explained in terms of a developmental shift from logarithmic to linear representations of number.     

Arithmetic word problem solving: evidence for a magnitude-based mental representation

Previous findings have suggested that number processing involves a mental representation of numerical magnitude. Other research has shown that sensory experiences are part and parcel of the mental representation (or “simulation”) that individuals construct during reading. We aimed at exploring whether arithmetic word-problem solving entails the construction of a mental simulation based on a representation of numerical magnitude. Participants were required to solve word problems and to perform an intermediate figure discrimination task that matched or mismatched, in terms of magnitude comparison, the mental representations that individuals constructed during problem solving. Our results showed that participants were faster in the discrimination task and performed better in the solving task when the figures matched the mental representations. These findings provide evidence that an analog magnitude-based mental representation is routinely activated during word-problem solving, and they add to a growing body of literature that emphasizes the experiential view of language comprehension.     

Facing the mirror: A relativist account of immune nonconceptual self-representations

There is a consensus among philosophers that some “I”-thoughts are immune to error through misidentification. In some recent papers, this property has been formulated in the following deflationist way: an “I”-thought is immune to error through misidentification when it can misrepresent the mental or bodily property self-ascribed but cannot misrepresent the subject (if any) possessing that property. However, it has been put forward that the range of mental and bodily states that are immune in that limited sense cannot include nonconceptual forms of self-representation. In this paper, I claim the opposite. I argue in favor of a theoretical framework inspired by semantic relativism that solves the problem of immune nonconceptual self-representations. In order to do so, I refute an argument against the relativist account which is based on the existence of shared representations. This argument, I contend, rests on a confusion between two conditions to which a relativist may appeal when considering whether a certain mental content is relative to the self: a strong invariance condition and a weak invariance condition. I then argue that even if we acknowledge the existence of shared representations, the weak invariance condition is still satisfied, and consequently the relativist framework can make sense of INSRs. I argue that this weak invariance condition is satisfied by a representational function that self-relativizes certain representations. I then provide an empirical instance of such a function by discussing some of the recent literature on motor representations and the sense of agency. In the last part of the paper, I answer several potential objections. These potential objections lead me to distinguish two fundamental kinds of error relative to the self: error through misidentification and error through misapplication. This distinction allows me to answer a fundamental question raised by the very idea of de facto immunity to error through misidentification.     

Finger-digit compatibility in Arabic numeral processing

Finger-digit response compatibility was tested by asking participants to identify Arabic digits by pressing 1 of 10 keys with all 10 fingers. The direction of the finger-digit mapping was varied by manipulating the global direction of the hand-digit mapping as well as the direction of the finger-digit mapping within each hand (in each case, from small to large digits, or the reverse). The hypothesis of a left-to-right mental number line predicted that a complete left-to-right mapping should be easier whereas the hypothesis of a representation based on finger counting predicted that a counting-congruent mapping should be easier. The results show that when all 10 fingers are used to answer, a mapping congruent with the prototypical finger-counting strategy reported by the participants leads to better performance than does a mapping congruent with a left-to-right oriented mental number line, both in palm-down and palm-up postures of the hands, and they demonstrate that finger-counting strategies influence the way that numerical information is mentally represented and processed.     

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.     

Visual experience influences the interactions between fingers and numbers

Though a clear interaction between finger and number representations has been demonstrated, what drives the development of this intertwining remains unclear. Here we tested early blind, late blind and sighted control participants in two counting tasks, each performed under three different conditions: a resting condition, a condition requiring hands movements and a condition requiring feet movements. In the resting condition, every sighted and late blind spontaneously used their fingers, while the majority of early blind did not. Sighted controls and late blind were moreover selectively disrupted by the interfering hand condition, while the early blind who did not use the finger-counting strategy remained unaffected by the interference conditions. These results therefore demonstrate that visual experience plays an important role in implementing the sensori-motor habits that drive the development of finger–number interactions.     

Non-symbolic arithmetic in adults and young children

Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on non-symbolic numerosities. Previous research has demonstrated that human adults, human infants, and non-human animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these non-symbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on non-numerical continuous quantities, modality-specific quantity information, the adoption of alternative non-arithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.     

Frequency of finger looking during finger counting is related to children's working memory capacities

Finger counting can be useful in solving arithmetic problems, noticeably because it reduces the working memory demand of mental calculations. However, proprioceptive information might not be sufficient to keep track of the number of fingers raised during problem solving, and visual input may play an important role in this process. The present study was designed to address this question and shows that 8-year-old children look at their fingers in 60% of the trials during finger counting when solving additive problems. Moreover, our results reveal that the frequency of finger looking is negatively correlated with working memory capacities and is higher for more difficult problems. These findings suggest that finger looking is recruited in managing the cognitive demand of the arithmetic task, probably by providing additional external cues to monitor the number of steps that have to be incremented during finger counting.     

Psychological foundations of number: numerical competence in human infants

An enduring question in philosophy and psychology is that of how we come to possess knowledge of number. Here I review research suggesting that the capacity to represent and reason about number is part of the inherent structure of the human mind. In the first few months of life, human infants can enumerate sets of entities and perform numerical computations. One proposal is that these abilities arise from general cognitive capacities not specific to number. I argue that the body of data supports a very different proposal: humans possess a specialized mental mechanism for number, one which we share with other species and which has evolved through natural selection. This mechanism is inherently restricted in the kinds of numerical knowledge it can support, leading to some striking limitations to early competence.     

Verbal interference suppresses exact numerical representation

Language for number is an important case study of the relationship between language and cognition because the mechanisms of non-verbal numerical cognition are well-understood. When the Pirahã (an Amazonian hunter-gatherer tribe who have no exact number words) are tested in non-verbal numerical tasks, they are able to perform one-to-one matching tasks but make errors in more difficult tasks. Their pattern of errors suggests that they are using analog magnitude estimation, an evolutionarily- and developmentally-conserved mechanism for estimating quantities. Here we show that English-speaking participants rely on the same mechanisms when verbal number representations are unavailable due to verbal interference. Followup experiments demonstrate that the effects of verbal interference are primarily manifest during encoding of quantity information, and—using a new procedure for matching difficulty of interference tasks for individual participants—that the effects are restricted to verbal interference. These results are consistent with the hypothesis that number words are used online to encode, store, and manipulate numerical information. This linguistic strategy complements, rather than altering or replacing, non-verbal representations.     

The effect of external representations on numeric tasks

This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported two-digit number comparison task, we obtained different results. Rather than holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: The comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect is discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations, and the behaviour in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical cognition should consider external representations as an essential component.     

Nonsymbolic, approximate arithmetic in children: abstract addition prior to instruction

Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the sum by 1 of 3 ratios. Children performed this task successfully with all the signatures of adults' nonsymbolic number representations: accuracy modulated by the ratio of the sum and the comparison quantity, equal performance for within- and cross-modality tasks and for addition and comparison tasks, and performance superior to that of a matched subtraction task. The findings provide clear evidence for nonsymbolic numerical operations on abstract numerical quantities in children who have not yet been taught formal arithmetic.     

反应-效应相容性范式下不同数字表征方式和身体经验对数字认知加工的影响

采用反应-效应相容性范式,探讨不同数字表征方式和身体经验对个体数字认知加工的影响。实验1首先采用木棒数字表征从知觉层面证明反应-效应相容性效应的存在;实验2采用累计手指数字表征考察手指数字表征在反应-效应相容性匹配中的优势;实验3则进一步排除了近似数量表征系统的影响,从语义层面探讨中国文化背景下语义概念手指数字表征的反应-效应相容性对数字认知加工的影响。结果发现,不同的数字表征系统中均存在反应-效应相容性效应,支持了观念运动理论;相比客体材料,手指数字表征在相容条件下具有明显的匹配优势,支持了具身数量的观点。     

Knowledge on the line: Manipulating beliefs about the magnitudes of symbolic numbers affects the linearity of line estimation tasks

It has been suggested that differences in performance on number-line estimation tasks are indicative of fundamental differences in people’s underlying representations of numerical magnitude. However, we were able to induce logarithmic-looking performance in adults for magnitude ranges over which they can typically perform linearly by manipulating their familiarity with the symbolic number formats that we used for the stimuli. This serves as an existence proof that individuals’ performances on number-line estimation tasks do not necessarily reflect the functional form of their underlying numerical magnitude representations. Rather, performance differences may result from symbolic difficulties (i.e., number-to-symbol mappings), independently of the underlying functional form. We demonstrated that number-line estimates that are well fit by logarithmic functions need not be produced by logarithmic functions. These findings led us to question the validity of considering logarithmic-looking performance on number-line estimation tasks as being indicative that magnitudes are being represented logarithmically, particularly when symbolic understanding is in question.