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Finger-based representation of numbers is a high-level cognitive strategy to assist numerical and arithmetic processing in children and adults. It is unclear whether this paradigm builds on simple perceptual features or comprises several attributes through embodiment. Here we describe the development and initial testing of an experimental setup to study embodiment during a finger-based numerical task using Virtual Reality (VR) and a low-cost tactile stimulator that is easy to build. Using VR allows us to create new ways to study finger-based numerical representation using a virtual hand that can be manipulated in ways our hand cannot, such as decoupling tactile and visual stimuli. The goal is to present a new methodology that can allow researchers to study embodiment through this new approach, maybe shedding new light on the cognitive strategy behind the finger-based representation of numbers. In this case, a critical methodological requirement is delivering precisely targeted sensory stimuli to specific effectors while simultaneously recording their behavior and engaging the participant in a simulated experience. We tested the device's capability by stimulating users in different experimental configurations. Results indicate that our device delivers reliable tactile stimulation to all fingers of a participant's hand without losing motion tracking quality during an ongoing task. This is reflected by an accuracy of over 95% in participants detecting stimulation of a single finger or multiple fingers in sequential stimulation as indicated by experiments with sixteen participants. We discuss possible application scenarios, explain how to apply our methodology to study the embodiment of finger-based numerical representations and other high-level cognitive functions, and discuss potential further developments of the device based on the data obtained in our testing., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2023 de Carvalho Souza, Barrocas, Fischer, Arnaud, Moeller and Rennó-Costa.)

Numerical cognition is a fundamental component of human intelligence that has not been fully understood yet. Indeed, it is a subject of research in many disciplines, e.g., neuroscience, education, cognitive and developmental psychology, philosophy of mathematics, linguistics. In Artificial Intelligence, aspects of numerical cognition have been modelled through neural networks to replicate and analytically study children behaviours. However, artificial models need to incorporate realistic sensory-motor information from the body to fully mimic the children's learning behaviours, e.g., the use of fingers to learn and manipulate numbers. To this end, this article presents a database of images, focused on number representation with fingers using both human and robot hands, which can constitute the base for building new realistic models of numerical cognition in humanoid robots, enabling a grounded learning approach in developmental autonomous agents. The article provides a benchmark analysis of the datasets in the database that are used to train, validate, and test five state-of-the art deep neural networks, which are compared for classification accuracy together with an analysis of the computational requirements of each network. The discussion highlights the trade-off between speed and precision in the detection, which is required for realistic applications in robotics., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2021 Davies, Lucas, Ricolfe-Viala and Di Nuovo.)

The study has two objectives: (1) to introduce grip force recording as a new technique for studying embodied numerical processing; and (2) to demonstrate how three competing accounts of numerical magnitude representation can be tested by using this new technique: the Mental Number Line (MNL), A Theory of Magnitude (ATOM) and Embodied Cognition (finger counting-based) account. While 26 healthy adults processed visually presented single digits in a go/no-go n-back paradigm, their passive holding forces for two small sensors were recorded in both hands. Spontaneous and unconscious grip force changes related to number magnitude occurred in the left hand already 100-140 ms after stimulus presentation and continued systematically. Our results support a two-step model of number processing where an initial stage is related to the automatic activation of all stimulus properties whereas a later stage consists of deeper conscious processing of the stimulus. This interpretation generalizes previous work with linguistic stimuli and elaborates the timeline of embodied cognition. We hope that the use of grip force recording will advance the field of numerical cognition research., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2021 Miklashevsky, Lindemann and Fischer.)

Understanding number magnitude is an important prerequisite for children's mathematical development. One early experience that contributes to this understanding is the common practice of finger counting. Recent research suggested that through repeated finger counting, children internalize their fingers as representations of number magnitude. Furthermore, finger counting habits have been proposed to predict concurrent and future mathematical performance. However, little is known about how finger-based number representations are formed and by which processes they could influence mathematical development. Regarding the emergence of finger-based number representations, it is likely that they result from repeated practice of finger counting. Accordingly, children need sufficient fine motor skills (FMS) to successfully count on their fingers. However, the role that different types of FMS (such as dexterity and graphomotor skills) might play in the development of finger-based number representations is still unknown. In the current study, we investigated (a) whether children's FMS (dexterity and graphomotor skills) are associated with their emerging finger-based number representations (ordinal and cardinal), (b) whether FMS explain variance in children's finger-based number representations beyond the influence of general cognitive skills, and (c) whether the association between FMS and numerical skills is mediated by finger-based representations. We tested associations between preschool children's ( N = 80) FMS (dexterity and graphomotor skills), finger-based number representations, and numerical skills. Furthermore, visuo-spatial working memory and nonverbal intelligence were controlled for. Dexterity was related to children's finger-based number representations as well as numerical skills after controlling for chronological age, but not after also controlling for cognitive skills. Moreover, the relationship between dexterity and numerical skills was mediated by finger-based number representations. No such associations were observed for graphomotor skills. These results suggest that dexterity plays a role in children's development of finger-based number representations, which in turn contribute to their numerical skills. Possible explanations are discussed., (Copyright © 2020 Fischer, Suggate and Stoeger.)

The well-documented association between fingers and numbers is not only based on the observation that most children use their fingers for counting and initial calculation, but also on extensive behavioral and neuro-functional evidence. In this article, we critically review developmental studies evaluating the association between finger sensorimotor skills (i.e., finger gnosis and fine motor skills) and numerical abilities. In sum, reviewed studies were found to provide evidential value and indicated that both finger gnosis and fine motor skills predict measures of counting, number system knowledge, number magnitude processing, and calculation ability. Therefore, specific and unique contributions of both finger gnosis and fine motor skills to the development of numerical skills seem to be substantiated. Through critical consideration of the reviewed evidence, we suggest that the association of finger gnosis and fine motor skills with numerical abilities may emerge from a combination of functional and redeployment mechanisms, in which the early use of finger-based numerical strategies during childhood might be the developmental process by which number representations become intertwined with the finger sensorimotor system, which carries an innate predisposition for said association to unfold. Further research is nonetheless necessary to clarify the causal mechanisms underlying this association., (Copyright © 2020 Barrocas, Roesch, Gawrilow and Moeller.)

Numerical cognition has long been considered the perfect example of abstract information processing. Nevertheless, there is accumulating evidence in recent years suggesting that the representation of number magnitude may not be entirely abstract but may present a specific case of embodied cognition rooted in the sensory and bodily experiences of early finger counting and calculating. However, so far none of the existing models of numerical development considers the influence of finger-based representations. Therefore, we make first suggestions on (i) how finger-based representations may be integrated into a current model of numerical development; and (ii) how they might corroborate the acquisition of basic numerical competencies at different development levels.

Evidence from developmental as well as neuroscientific studies suggest that finger counting activity plays an important role in the acquisition of numerical skills in children. It has been claimed that this skill helps in building motor-based representations of number that continue to influence number processing well into adulthood, facilitating the emergence of number concepts from sensorimotor experience through a bottom-up process. The act of counting also involves the acquisition and use of a verbal number system of which number words are the basic building blocks. Using a Cognitive Developmental Robotics paradigm we present results of a modeling experiment on whether finger counting and the association of number words (or tags) to fingers, could serve to bootstrap the representation of number in a cognitive robot, enabling it to perform basic numerical operations such as addition. The cognitive architecture of the robot is based on artificial neural networks, which enable the robot to learn both sensorimotor skills (finger counting) and linguistic skills (using number words). The results obtained in our experiments show that learning the number words in sequence along with finger configurations helps the fast building of the initial representation of number in the robot. Number knowledge, is instead, not as efficiently developed when number words are learned out of sequence without finger counting. Furthermore, the internal representations of the finger configurations themselves, developed by the robot as a result of the experiments, sustain the execution of basic arithmetic operations, something consistent with evidence coming from developmental research with children. The model and experiments demonstrate the importance of sensorimotor skill learning in robots for the acquisition of abstract knowledge such as numbers.

Evidence for an approximate analog system of numbers has been provided by the finding that the comparison of two numerals takes longer and is more error-prone if the semantic distance between the numbers becomes smaller (so-called numerical distance effect). Recent embodied theories suggest that analog number representations are based on previous sensory experiences and constitute therefore a common magnitude metric shared by multiple domains. Here we demonstrate the existence of a cross-modal semantic distance effect between symbolic and tactile numerosities. Participants received tactile stimulations of different amounts of fingers while reading Arabic digits and indicated verbally whether the amount of stimulated fingers was different from the simultaneously presented digit or not. The larger the semantic distance was between the two numerosities, the faster and more accurate participants made their judgments. This cross-modal numerosity distance effect suggests a direct connection between tactile sensations and the concept of numerical magnitude. A second experiment replicated the interaction between symbolic and tactile numerosities and showed that this effect is not modulated by the participants' finger counting habits. Taken together, our data provide novel evidence for a shared metric for symbolic and tactile numerosities as an instance of an embodied representation of numbers.

The spontaneous use of finger counting has been for long recognized as critical to the acquisition of number skills. Recently, the great interest on space-number associations shifted attention to the practice of finger counting itself, and specifically, to its spatial components. Besides general cross-cultural differences in mapping numbers onto fingers, contrasting results have been reported with regard to the directional features of this mapping. The key issue we address is to what extent directionality is culturally mediated, i.e., linked to the conventional reading-writing system direction, and/or biologically determined, i.e., linked to hand dominance. Although the preferred starting-hand for counting seems to depend on the surveyed population, even within the same population high inter-individual variability minimizes the role of cultural factors. Even if so far largely overlooked, handedness represents a sound candidate for shaping finger counting direction. Here we discuss adults and developmental evidence in support of this view and we reconsider the plausibility of multiple and coexistent number-space mapping in physical and representational space.

Children typically learn basic numerical and arithmetic principles using finger-based representations. However, whether or not reliance on finger-based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education. From the neurocognitive perspective, finger counting provides multisensory input, which conveys both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis, or "finger sense," enhances mathematical skills. Therefore neurocognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting. More precisely, mathematics education recommends first using finger counting, then concrete structured representations and, finally, mental representations of numbers to perform numerical operations. Taken together, these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be discussed.

Spatial-numerical associations (SNAs) are prevalent yet their origin is poorly understood. We first consider the possible prime role of reading habits in shaping SNAs and list three observations that argue against a prominent influence of this role: (1) directional reading habits for numbers may conflict with those for non-numerical symbols, (2) short-term experimental manipulations can overrule the impact of decades of reading experience, (3) SNAs predate the acquisition of reading. As a promising alternative, we discuss behavioral, neuroscientific, and neuropsychological evidence in support of finger counting as the most likely initial determinant of SNAs. Implications of this "manumerical cognition" stance for the distinction between grounded, embodied, and situated cognition are discussed.

Recently, a strong functional relationship between finger counting and number processing has been suggested. It has been argued that bodily experiences such as finger counting may influence the structure of the basic mental representations of numbers even in adults. However, to date it remains unclear whether the structure of finger counting systems also influences educated adults' performance in mental arithmetic. In the present study, we pursued this question by examining finger-based sub-base-five effects in an addition production task. With the standard effect of a carry operation (i.e., base-10 crossing) being replicated, we observed an additional sub-base-five effect such that crossing a sub-base-five boundary led to a relative response time increase. For the case of mental arithmetic sub-base-five effects have previously been reported only in children. However, it remains unclear whether finger-based numerical effects in mental arithmetic reflect an important but transitory step in the development of arithmetical skills. The current findings suggest that even in adults embodied representations such as finger counting patterns modulate arithmetic performance. Thus, they support the general idea that even seemingly abstract cognition in adults may at least partly be rooted in our bodily experiences.

In the present study, we experimentally tested the role of hand motor circuits in simple-arithmetic strategies. Educated adults solved simple additions (e.g., 8 + 3) or simple subtractions (e.g., 11 - 3) while they were required to retrieve the answer from long-term memory (e.g., knowing that 8 + 3 = 11), to transform the problem by making an intermediate step (e.g., 8 + 3 = 8 + 2 + 1 = 10 + 1 = 11) or to count one-by-one (e.g., 8 + 3 = 8…9…10…11). During the process of solving the arithmetic problems, the experimenter did or did not move the participants' hand on a four-point matrix. The results show that passive hand movements disrupted the counting strategy while leaving the other strategies unaffected. This pattern of results is in agreement with a procedural account, showing that the involvement of hand motor circuits in adults' mathematical abilities is reminiscent of finger counting during childhood.