Finger counting: The missing tool?

Do children need counting principle knowledge to count on their fingers?

It has been established that young children who use their fingers to solve arithmetic problems outperform those who do not. However, it remains unclear whether finger counting itself enhances arithmetic performance or if children with already advanced numerical abilities are more inclined to use this strategy. In the current study, to shed light on this matter, we observed the behavior of 189 4- and 5-year-old children in an addition task and a task assessing their knowledge of the three "how-to-count" principles (i.e., stable order, one-to-one correspondence, and cardinality principles). Of these children, 169 were reassessed 1 year later (the second testing point). At the first testing point, our results revealed that finger users better know the counting principles than non-finger users. Nevertheless, some children use their fingers without knowing the principles, but in this case they present low performance in the addition task. Moreover, we found that knowing the counting principles does not naturally prompt finger use. Finally, we did not find evidence supporting the idea that finger use has a specific role in the development of counting principles, which questions the idea that finger counting has a functional role in the construction of the number concept. All in all, our results tend to show that children need to know the counting principles to be efficient finger users. Therefore, finger counting seems to be a useful tool when used by children who already possess advanced numerical knowledge.

Combining virtual reality and tactile stimulation to investigate embodied finger-based numerical representations

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.

Influences of hand action on the processing of symbolic numbers: A special role of pointing?

Embodied and grounded cognition theories state that cognitive processing is built upon sensorimotor systems. In the context of numerical cognition, support to this framework comes from the interactions between numerical processing and the hand actions of reaching and grasping documented in skilled adults. Accordingly, mechanisms for the processing of object size and location during reach and grasp actions might scaffold the development of mental representations of numerical magnitude. The present study exploited motor adaptation to test the hypothesis of a functional overlap between neurocognitive mechanisms of hand action and numerical processing. Participants performed repetitive grasping of an object, repetitive pointing, repetitive tapping, or passive viewing. Subsequently, they performed a symbolic number comparison task. Importantly, hand action and number comparison were functionally and temporally dissociated, thereby minimizing context-based effects. Results showed that executing the action of pointing slowed down the responses in number comparison. Moreover, the typical distance effect (faster responses for numbers far from the reference as compared to close ones) was not observed for small numbers after pointing, while it was enhanced by grasping. These findings confirm the functional link between hand action and numerical processing, and suggest new hypotheses on the role of pointing as a meaningful gesture in the development and embodiment of numerical skills.

Impaired Arithmetic Fact Retrieval in an Adult with Developmental Dyscalculia: Evidence from Behavioral and Functional Brain Imaging Data

Developmental dyscalculia (DD) is a developmental disorder characterized by arithmetic difficulties. Recently, it has been suggested that the neural networks supporting procedure-based calculation (e.g., in subtraction) and left-hemispheric verbal arithmetic fact retrieval (e.g., in multiplication) are partially distinct. Here we compared the neurofunctional correlates of subtraction and multiplication in a 19-year-old student (RM) with DD to 18 age-matched controls. Behaviorally, RM performed significantly worse than controls in multiplication, while subtraction was unaffected. Neurofunctional differences were most pronounced regarding multiplication: RM showed significantly stronger activation than controls not only in left angular gyrus but also in a fronto-parietal network (including left intraparietal sulcus and inferior frontal gyrus) typically activated during procedure-based calculation. Region-of-interest analyses indicated group differences in multiplication only, which, however, did not survive correction for multiple comparisons. Our results are consistent with dissociable and processing-specific, but not operation-specific neurofunctional networks. Procedure-based calculation is not only associated with subtraction but also with (untrained) multiplication facts. Only after rote learning, facts can be retrieved quasi automatically from memory. We suggest that this learning process and the associated shift in activation patterns has not fully occurred in RM, as reflected in her need to resort to procedure-based strategies to solve multiplication facts.

The Evolution of Finger Counting between Kindergarten and Grade 2

In this longitudinal study, we aimed at determining whether children who efficiently use finger counting are more likely to develop internalized arithmetic strategies than children who are less efficient. More precisely, we analyzed the behavior of 24 kindergarteners aged between 5 and 6 years who used their fingers to solve addition problems, and we were interested in determining the evolution of their finger counting strategies towards mental strategies after 2 years (Grade 2). Our results show that kindergarteners who were the most proficient in calculating on fingers were the more likely to have abandoned this strategy in Grade 2. This shows that the use of efficient finger counting strategies early during development optimizes the shift to mental strategies later on during school years. Moreover, children who still use their fingers to solve additions in Grade 2 present lower working memory capacities than children who had already abandoned this strategy.

Canonical representations of fingers and dots trigger an automatic activation of number semantics: an EEG study on 10-year-old children

Over the course of development, children must learn to map a non-symbolic representation of magnitude to a more precise symbolic system. There is solid evidence that finger and dot representations can facilitate or even predict the acquisition of this mapping skill. While several behavioral studies demonstrated that canonical representations of fingers and dots automatically activate number semantics, no study so far has investigated their cerebral basis. To examine these questions, 10-year-old children were presented a behavioral naming task and a Fast Periodic Visual Stimulation EEG paradigm. In the behavioral task, children had to name as fast and as accurately as possible the numbers of dots and fingers presented in canonical and non-canonical configurations. In the EEG experiment, one category of stimuli (e.g., canonical representation of fingers or dots) was periodically inserted (1/5) in streams of another category (e.g., non-canonical representation of fingers or dots) presented at a fast rate (4 Hz). Results demonstrated an automatic access to number semantics and bilateral categorical responses at 4 Hz/5 for canonical representations of fingers and dots. Some differences between finger and dot configuration's processing were nevertheless observed and are discussed in light of an effortful-automatic continuum hypothesis.

A robot that counts like a child: a developmental model of counting and pointing

In this paper, a novel neuro-robotics model capable of counting real items is introduced. The model allows us to investigate the interaction between embodiment and numerical cognition. This is composed of a deep neural network capable of image processing and sequential tasks performance, and a robotic platform providing the embodiment-the iCub humanoid robot. The network is trained using images from the robot's cameras and proprioceptive signals from its joints. The trained model is able to count a set of items and at the same time points to them. We investigate the influence of pointing on the counting process and compare our results with those from studies with children. Several training approaches are presented in this paper, all of them use pre-training routine allowing the network to gain the ability of pointing and number recitation (from 1 to 10) prior to counting training. The impact of the counted set size and distance to the objects are investigated. The obtained results on counting performance show similarities with those from human studies.

How visual is the « number sense »? Insights from the blind

Is vision a necessary building block for the foundations of mathematical cognition? A straightforward model to test the causal role visual experience plays in the development of numerical abilities is to study people born without sight. In this review we will demonstrate that congenitally blind people can develop numerical abilities that equal or even surpass those of sighted individuals, despite representing numbers using a qualitatively different representational format. We will also show that numerical thinking in blind people maps onto regions typically involved in visuo-spatial processing in the sighted, highlighting how intrinsic computational biases may constrain the reorganization of numerical networks in case of early visual deprivation. More generally, we will illustrate how the study of arithmetic abilities in congenitally blind people represents a compelling model to understand how sensory experience scaffolds the development of higher-level cognitive representations.

Putting a Finger on Numerical Development - Reviewing the Contributions of Kindergarten Finger Gnosis and Fine Motor Skills to Numerical Abilities

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.

Modal-independent Pattern Recognition Deficit in Developmental Dyscalculia Adults: Evidence from Tactile and Visual Enumeration

Developmental dyscalculia (DD) is characterized by lower numerical and finger-related skills. Studies of enumeration among those DD that suggested core deficiency in pattern recognition, working memory or/and attention were mostly carried out in the visual modality. In our study, we examined visual (dots) enumeration of 1-10 stimuli and tactile (vibration) enumeration of 1-10 fingers among DD and matched-control adults. We used 800-ms stimuli exposure time of either random/non-neighboring or canonical/neighboring stimuli arrangements (visual/tactile). Compared to controls, those with DD responded faster in visual random enumeration and did not differ in reaction time (RT) of canonical stimuli arrangements. However, while the control group had near perfect accuracy in random stimuli arrangements of up to five stimuli, DD participants performed accurately for only up to four stimuli, and they were less accurate in the canonical stimuli arrangements in the counting range. In the tactile task, DD participants showed less accurate tactile enumeration only for neighboring arrangements, more profoundly for finger counting (FC) patterns. The longer exposure time in the visual task enabled us to explore pattern recognition effects when working memory and attention loads were low. We discuss possible modal-independent deficits in pattern recognition and working memory on enumeration performance among those with DD and the unique role of fingers in ordinal and cardinal representation of numbers.

Numerical cognition: A meta-analysis of neuroimaging, transcranial magnetic stimulation and brain-damaged patients studies

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We review neuroimaging, TMS, and patients studies on numerical cognition.

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We focused on the predictions derived from the Triple Code Model (TCM).

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Our findings generally agree with TCM predictions.

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Our results open avenues for the study of the neural bases of numerical cognition.

This article offers the first comprehensive review examining the neurocognitive bases of numerical cognition from neuroimaging, Transcranial Magnetic Stimulation (TMS) and brain-damaged patients studies. We focused on the predictions derived from the Triple Code Model (TCM), particularly the assumption that the representation of numerical quantities rests on a single format-independent representation (i.e., the analogical code) involving both intraparietal sulci (IPS). To do so, we conducted a meta-analysis based on 28 neuroimaging, 12 TMS and 12 brain-damaged patients studies, including arithmetic and magnitude tasks in symbolic and non-symbolic formats. Our findings generally agree with the TCM predictions indicating that both IPS are engaged in all tasks. Nonetheless, the results of brain-damaged patients studies conflicted with neuroimaging and TMS studies, suggesting a right hemisphere lateralization for non-symbolic formats. Our findings also led us to discuss the involvement of brain regions other than IPS in the processing of the analogical code as well as the neural substrate of other codes underlying numerical cognition (i.e., the auditory-verbal code).

A large-scale survey on finger counting routines, their temporal stability and flexibility in educated adults

A strong link between bodily activity and number processing has been established in recent years. Although numerous observations indicate that adults use finger counting (FC) in various contexts of everyday life for different purposes, existing knowledge of FC routines and their use is still limited. In particular, it remains unknown how stable the (default) FC habits are over time and how flexible they can be. To investigate these questions, 380 Polish participants completed a questionnaire on their FC routines, the stability of these routines, and the context of FC usage, preceded by the request to count on their fingers from 1 to 10. Next, the test-retest stability of FC habits was examined in 84 participants 2 months following the first session. To the best of our knowledge, such a study design has been adopted for the first time. The results indicate that default FC routines of the majority of participants (75%) are relatively stable over time. At the same time, FC routines can flexibly adapt according to the situation (e.g., when holding an object). As regards prevalence, almost all participants, in line with previous findings on Western individuals, declared starting from the closed palm and extending consecutive fingers. Furthermore, we observed relations between FC preferences and handedness (more left-handers start from the left hand) and that actual finger use is still widespread in healthy adults for a variety of activities (the most prevalent uses of FC are listing elements, presenting arguments and plans, and calendar calculations). In sum, the results show the practical relevance of FC in adulthood, the relative stability of preferences over time along with flexible adaptation to a current situation, as well as an association of FC routines with handedness. Taken together our results suggest that FC is the phenomenon, which is moderated or mediated by multiple embodied factors.

Tactile enumeration: A case study of acalculia

Enumeration is one of the building blocks of arithmetic and fingers are used as a counting tool in early steps. Subitizing-fast and accurate enumeration of small quantities-has been vastly studied in the visual modality, but less in the tactile modality. We explored tactile enumeration using fingers, and gray matter (GM) changes using voxel-based morphometry (VBM), in acalculia. We examined JD, a 22-year-old female with acalculia following a stroke to the left inferior parietal cortex. JD and a neurologically healthy normal comparison (NC) group reported how many fingers were stimulated. JD was tested at several time points, including at acute and chronic phases. Using the sensory intact hand for tactile enumeration, JD showed deficit in the acute phase, compared to the NC group, and improvement in the chronic phase of (1) the RT slope of enumerating up to four stimuli, (2) enumerating neighboring fingers, and (3) arithmetic fluency performance. Moreover, VBM analysis showed a larger GM volume for JD relative to the NC group in the right middle occipital cortex, most profoundly in the chronic phase. JD's performance serves as a first glance of tactile enumeration in acalculia. Pattern-recognition-based results support the suggestion of subitizing being the enumeration process when using one hand. Moreover, the increase in GM in the occipital cortex lays the groundwork for studying the innate and primitive ability to perceive and evaluate sizes or amounts-"sense of magnitude"- as a multisensory magnitude area and as part of a recovery path for deficits in basic numerical abilities.

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.

The role of fingers in the development of counting and arithmetic skills

Interactions between fingers and numbers have been reported in the existing literature on numerical cognition. The aim of the present research was to test whether hand interference movements might have an impact on children performance in counting and basic arithmetic problem solving. In Experiment 1, 5-year-old children had to perform both a one-target and a two-target counting task in three different conditions: with no constraints, while making interfering hand movements or while making interfering foot movements. In Experiment 2, first and fourth graders were required to perform addition problems under the same control and sensori-motor interfering conditions. In both tasks, the hand movements caused more disruption than the foot movements, suggesting that finger-counting plays a functional role in the development of counting and arithmetic.

Tactile enumeration of small quantities using one hand

Our study explores various aspects of enumerating small quantities in the tactile modality. Fingertips of one hand were stimulated by a vibro-tactile apparatus (for 100/800 ms). Between 1 and 5 stimuli were presented to the right or the left hand and applied to neighboring (e.g., thumb-index-middle) or non-neighboring (e.g., thumb-middle-pinkie) fingers. The results showed a moderate increase in RT up to 4 stimuli and then a decrease for 5 stimuli. Right hand stimulation evoked more accurate performance than left hand stimulation only under short exposures (100 ms). Importantly, when the stimuli were presented to neighboring fingers, the accuracy rate was higher and the RT was faster than when presented to non-neighboring fingers. We discuss the results and suggest that when the stimuli are presented to one hand the subitizing range is 4 rather than 3. Furthermore, the right hand advantage and the efficiency for neighboring fingers are further support for the association between number and spatial arrangement of the fingers.

Accelerating the early numeracy development of kindergartners with limited working memory skills through remedial education

BACKGROUND

Young children with limited working memory skills are a special interest group among all children that score below average on early numeracy tests. This study examines the effect of accelerating the early numeracy development of these children through remedial education, by comparing them with children with typically working memory skills and early numeracy abilities below average.

METHOD

Selected from a sample of 933 children, children with early numeracy ability below average are assigned into four groups: two intervention groups with limited working memory skills (IL-group) or typical working memory skills (IT-group), and two control groups with limited working memory skills (CL-group) or typical working memory skills (CT-group). All four groups were followed for a period of 1.5 years. Four measurements were carried out.

CONCLUSION

The remedial program proved to be similarly effective for the IL-group and the IT-group. The findings are discussed in the light of several limitations and implications.

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.

You can count on the motor cortex: finger counting habits modulate motor cortex activation evoked by numbers

The embodied cognition framework suggests that neural systems for perception and action are engaged during higher cognitive processes. In an event-related fMRI study, we tested this claim for the abstract domain of numerical symbol processing: is the human cortical motor system part of the representation of numbers, and is organization of numerical knowledge influenced by individual finger counting habits? Developmental studies suggest a link between numerals and finger counting habits due to the acquisition of numerical skills through finger counting in childhood. In the present study, digits 1 to 9 and the corresponding number words were presented visually to adults with different finger counting habits, i.e. left- and right-starters who reported that they usually start counting small numbers with their left and right hand, respectively. Despite the absence of overt hand movements, the hemisphere contralateral to the hand used for counting small numbers was activated when small numbers were presented. The correspondence between finger counting habits and hemispheric motor activation is consistent with an intrinsic functional link between finger counting and number processing.

A common right fronto-parietal network for numerosity and duration processing: an fMRI study

Numerosity and duration processing have been modeled by a functional mechanism taking the form of an accumulator working under two different operative modes. Separate investigations of their cerebral substrates have revealed partly similar patterns of activation, mainly in parietal and frontal areas. However, the precise cerebral implementation of the accumulator model within these areas has not yet been directly assessed. In this study, we asked participants to categorize the numerosity of flashed dot sequences or the duration of single dot displays, and we used functional magnetic resonance imaging (fMRI) to examine the common neural correlates of these processes. The results reveal a large right-lateralized fronto-parietal network, including the intraparietal sulcus (IPS) and areas in the precentral, middle and superior frontal gyri, which is activated by both numerosity and duration processing. Complementary psychophysiological interaction (PPI) analyses show a functional connectivity between the right IPS and the frontal areas in both tasks, whereas the right IPS was functionally connected to the left IPS and the right precentral area in the numerosity categorization task only. We propose that the right IPS underlies a common magnitude processing system for both numerosity and duration, possibly corresponding to the encoding and accumulation stages of the accumulator model, whereas the frontal areas are involved in subsequent working-memory storage and decision-making processes.

The role of vision in the development of finger-number interactions: Finger-counting and finger-montring in blind children

Previous research has suggested that the use of the fingers may play a functional role in the development of a mature counting system. However, the role of developmental vision in the elaboration of a finger numeral representation remains unexplored. In the current study, 14 congenitally blind children and 14 matched sighted controls undertook three different test batteries that examined (a) general cognitive abilities, (b) the spontaneous use of finger-counting and finger-montring strategies (where "finger-montring" is a term used to characterize the way people raise their fingers to show numerosities to other people), and (c) the canonicity level of the finger-counting and finger-montring habits. Compared with sighted controls, blind children used their fingers less spontaneously to count and in a less canonical way to count and show quantities. These results demonstrate that the absence of vision precludes the development of a typical finger numeral representation and suggest that the use of canonical finger-counting and finger-montring strategies relies on the visual recognition of particular hand shapes.

Let us redeploy attention to sensorimotor experience

With his massive redeployment hypothesis (MRH), Anderson claims that novel cognitive functions are likely to rely on pre-existing circuits already possessing suitable resources. Here, we put forward recent findings from studies in numerical cognition in order to show that the role of sensorimotor experience in the ontogenetical development of a new function has been largely underestimated in Anderson's proposal.

Absence of low-level visual difference between canonical and noncanonical finger-numeral configurations

Canonical finger numeral configurations are named faster than less familiar finger configurations and activate a semantic place-coding representation as symbolic stimuli. However, this does not exclude categorically the possibility that mere visuo-perceptual differences between canonical and noncanonical finger configurations may induce differences in processing speed. This study capitalizes on the fact that, in typical visual-detection tasks, participants focus on low-level visuo-perceptual features to detect a target among distractors sharing the same high-level semantic features, producing the so-called pop-out effect. Participants had to decide whether a canonical finger configuration was present among a set of distractors expressing the same numerosity in a noncanonical way. The results showed that the time needed to detect the presence of the target grew linearly with the number of distractors. This indicates that the canonical target enjoyed no perceptual saliency among the noncanonical configurations (i.e., no pop-out effect) excluding visuo-perceptual differences as the source of the better identification of and semantic access of canonical configurations.

Response–effect compatibility of finger–numeral configurations in arithmetical context

The present study aimed at testing, by means of a response–effect compatibility paradigm, whether finger–numeral representations derived from finger counting may underlie simple arithmetic problem solving in adults. Participants were asked to provide a verbal response to simple additions, which triggered the presentation of the correct (Experiment 1) or an incorrect (Experiment 2) response, displayed either as a configuration of fingers or as a series of rods. Answers were faster with finger configurations than with rods, and only when the finger configuration showed the correct result. These findings support the idea that, even in adults, simple arithmetic operations are still unconsciously underlain by finger–numeral representations.