Tactile enumeration of small quantities using one hand

Enhancing mathematics learning through finger-counting: A study investigating tactile strategies in 2 visually impaired cases

Finger-counting plays a crucial role in grounding and establishing mathematics, one of the most abstract domains of human cognition. While the combination of visual and proprioceptive information enables the coordination of finger movements, it was recently suggested that the emergence of finger-counting primarily relies on visual cues. In this study, we aimed to directly test this assumption by examining whether explicit finger-counting training (through tactile stimulation) may assist visually impaired children in overcoming their difficulties in learning mathematics. Two visually impaired participants (2 boys of 8.5 and 7.5 years) were therefore trained to use their fingers to calculate. Their pre- and post-training performance were compared to two control groups of sighted children who underwent either the same finger counting training (8 boys, 10 girls, Mage = 5.9 years; 10 kindergarteners and eight 1st graders) or another control vocabulary training (10 boys, 8 girls, Mage = 5.9 years; 11 kindergarteners and seven 1st graders). Results demonstrated that sighted children's arithmetic performance improved much more after the finger training than after the vocabulary training. Importantly, the positive impact of the finger training was also observed in both visually impaired participants (for addition and subtraction in one child; only for addition in the other child). These results are discussed in relation to the sensory compensation hypothesis and emphasize the importance of early and appropriate instruction of finger-based representations in both sighted and visually impaired children.

Encoding contact size using static and dynamic electrotactile finger stimulation: natural decoding vs. trained cues

Electrotactile stimulation through matrix electrodes is a promising technology to restore high-resolution tactile feedback in extended reality applications. One of the fundamental tactile effects that should be simulated is the change in the size of the contact between the finger and a virtual object. The present study investigated how participants perceive the increase of stimulation area when stimulating the index finger using static or dynamic (moving) stimuli produced by activating 1 to 6 electrode pads. To assess the ability to interpret the stimulation from the natural cues (natural decoding), without any prior training, the participants were instructed to draw the size of the stimulated area and identify the size difference when comparing two consecutive stimulations. To investigate if other "non-natural" cues can improve the size estimation, the participants were asked to enumerate the number of active pads following a training protocol. The results demonstrated that participants could perceive the change in size without prior training (e.g., the estimated area correlated with the stimulated area, p < 0.001; ≥ two-pad difference recognized with > 80% success rate). However, natural decoding was also challenging, as the response area changed gradually and sometimes in complex patterns when increasing the number of active pads (e.g., four extra pads needed for the statistically significant difference). Nevertheless, by training the participants to utilize additional cues the limitations of natural perception could be compensated. After the training, the mismatch in the activated and estimated number of pads was less than one pad regardless of the stimulus size. Finally, introducing the movement of the stimulus substantially improved discrimination (e.g., 100% median success rate to recognize ≥ one-pad difference). The present study, therefore, provides insights into stimulation size perception, and practical guidelines on how to modulate pad activation to change the perceived size in static and dynamic scenarios.

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.

Overlapping but separate number representations in the intraparietal sulcus – probing format- and modality-independence in sighted Braille readers

The Triple-Code Model stipulates that numerical information from different formats and modalities converges on a common magnitude representation in the Intraparietal Sulcus (IPS). To what extent the representations of all numerosity forms overlap remains unsolved. It has been postulated that the representation of symbolic numerosities (for example, Arabic digits) is sparser and grounded in an existing representation that codes for non-symbolic numerosity information (i.e., sets of objects). Other theories argue that numerical symbols represent a separate number category that emerges only during education. Here, we tested a unique group of sighted tactile Braille readers with numerosities 2, 4, 6 and 8 in three number notations: Arabic digits, sets of dots, tactile Braille numbers. Using univariate methods, we showed a consistent overlap in activations evoked by these three number notations. This result shows that all three used notations are represented in the IPS, which may suggest at least a partial overlap between the representations of the three notations used in this experiment. Using MVPA, we found that only non-automatized number information (Braille and sets of dots) allowed successful number classification. However, the numerosity of one notation could not be predicted above chance from the brain activation patterns evoked by another notation (no cross-classification). These results show that the IPS may host independent number codes in overlapping cortical circuits. In addition, they suggest that the level of training in encoding a given type of number information is an important factor that determines the amount of exploitable information and needs to be controlled for in order to identify the neural code underlying numerical information per se.

Numerosity Identification Used to Assess Tactile Stimulation Methods for Communication

Finger-Braille is a tactile communication method used by people who are Deafblind. Individuals communicate Finger-Braille messages with combinations of taps on three fingers of each of the hands of the person receiving the communication. Devices have been developed to produce Finger-Braille symbols using different tactile stimulation methods. Before engaging in communication studies based on technologically-mediated Finger-Braille, we evaluated the relative efficacy of these methods by comparing two devices similarly constructed; the first based on widely employed eccentric rotating-mass vibrating motors and the other using specifically designed tapping actuators. We asked volunteers to identify the numerosity of presented items and for each device we measured (1) error-rate, (2) reaction time, (3) confidence ratings, and (4) a comparison of confidence ratings to actual performance. The four measures obtained for each device showed a net advantage of the tapping stimulation method over the method of vibrations. In this article, we conclude that the tapping stimulation method is recommended for use in the design of tactile communication devices based on Finger-Braille and fingerspelling methods reliant on finger tapping actions. The results did not demonstrate clear evidence for tactile subitising with passively experienced stimulation on the fingers.

Raised Dot Enumeration Via Haptic Exploration

In two experiments we investigated blindfolded, sighted participants' capacity to extract the number of raised dots from arrays of braille cells that they scanned once via active touch. The arrays could contain between one and 12 raised dots and estimates were based on scanning with one or more fingers on one or both hands (Experiment 1), or when the dots were as maximally or minimally spaced as the braille code permits (Experiment 2). We sought evidence of discontinuities in performance that reflect more than one mode of enumeration. We found that participants' estimates of numerosity increased in a linear fashion with actual numerosity, but were increasingly underestimated beyond numerosity of six, and confidence in the judgment declined linearly with increasing numerosity. Finger combinations made no difference to accuracy, errors, or confidence. Increasing dot density had the effect of diminishing perceptual accuracy, exaggerating underestimation and reducing confidence. While perceptual accuracy was generally high up to six raised dots, patterns of confusions and scaling analyses suggest that numerosities of four or less are perceptually unique. In this article, we discuss these data in terms of enumeration in touch and other modalities, and consider whether this discontinuity in enumeration signifies a subitize-to-count or a count-to-estimate transition.

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.

A shared code for Braille and Arabic digits revealed by cross-modal priming in sighted Braille readers

Quantities can be represented by different formats (e.g. symbolic or non-symbolic) and conveyed via different modalities (e.g. tactile or visual). Despite different priming curves: V-shape and step-shape for place and summation coded representation, respectively, the occurrence of priming effect supports the notion of different format overlap on the same mental number line. However, little is known about tactile-visual overlap of symbolic numerosities i.e. Braille numbers to Arabic digits on the magnitude number representation. Here, in a priming experiment, we tested a unique group of sighted Braille readers to investigate whether tactile Braille digits would activate a place-coding type of mental number representation (V-shape), analogous to other symbolic formats. The primes were either tactile Braille digits presented on a Braille display or number words presented on a computer screen. The targets were visually presented Arabic digits, and subjects performed a naming task. Our results reveal a V-shape priming function for both prime formats: tactile Braille and written words representing numbers, with strongest priming for primes of identical value (e.g. "four" and "4"), and a symmetrical decrease of priming strength for neighboring numbers, which indicates that the observed priming is due to identity priming. We thus argue that the magnitude information is processed according to a shared phonological code, independent of the input modality.

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.

If it looks, sounds, or feels like subitizing, is it subitizing? A modulated definition of subitizing

Research in cognitive psychology has focused mainly on the visual modality as the input interface for mental processes. We suggest that integrating studies from different modalities can aid in resolving theoretical controversies. We demonstrate this in the case of subitizing. Subitizing, the quick and accurate enumeration of small quantities, has been studied since the 19th century. Nevertheless, to date, the underlying mechanism is still debated. Two mechanisms have been suggested: a domain-general mechanism-attention, and a domain-specific mechanism-pattern recognition. Here, we review pivotal studies in the visual, tactile, and auditory modalities. The accumulative findings shed light on the theoretical debate. Accordingly, we suggest that subitizing is a subprocess of counting that occurs in the presence of facilitating factors, such as attentional resources and familiar patterns.

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.

Stimulating numbers: signatures of finger counting in numerosity processing

Finger counting is one of the first steps in the development of mature number concepts. With a one-to-one correspondence of fingers to numbers in Western finger counting, fingers hold two numerical meanings: one is based on the number of fingers raised and the second is based on their ordinal position within the habitual finger counting sequence. This study investigated how these two numerical meanings of fingers are intertwined with numerical cognition in adults. Participants received tactile stimulation on their fingertips of one hand and named either the number of fingers stimulated (2, 3, or 4 fingers; Experiment 1) or the number of stimulations on one fingertip (2, 3, or 4 stimulations; Experiment 2). Responses were faster and more accurate when the set of stimulated fingers corresponded to finger counting habits (Experiment 1) and when the number of stimulations matched the ordinal position of the stimulated finger (Experiment 2). These results show that tactile numerosity perception is affected by individual finger counting habits and that those habits give numerical meaning to single fingers.

The effects of training on tactile enumeration

Subitizing is a fast and accurate process of enumerating small quantities. Whether subitizing carried out in the tactile modality is under debate. We previously found a moderately increasing RT slope from one to four stimuli and a large decrease in RT for five stimuli when using one hand. Yet, a high error rate was observed, making it difficult to determine if the RT pattern found was indeed subitizing. To increase accuracy, we carried out training of the tactile enumeration task using one hand for 6 days. We compared performance in the trained and additional non-trained tasks between two groups-the 6-day training group (6DT) and the non-trained controls (C)-after three periods (1 week, 1 and 6 months after the training of the 6DT group ended). Results showed an increase in accuracy rates for both groups but a decrease in RT for the 6DT group only for the trained task. This RT improvement was present even after 6 months. Importantly, the RT slope of one-hand enumeration did not change after training, showing a moderately increased slope up to four stimuli and a decrease for five stimuli. Our study shows the training long-term effect on tactile enumeration and emphasizes the embodiment of finger counting on enumeration. Two possible enumeration processes are discussed-accelerated counting and subitizing-both based on spatial cues and pattern recognition of familiarized finger-counting patterns.

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.

Comparison of Tactile Temporal Numerosity Judgments Between Unimanual and Bimanual Presentations

In recent years a growing interest has emerged in numerosity perception in touch. Most of the studies on tactile numerosity perception have investigated its spatial aspect by testing the ability to count the number of items presented simultaneously. On the other hand, only a small number of studies have examined its temporal aspect, and the underlying mechanisms of tactile temporal numerosity judgments (TTNJs) remain elusive. In this study, we presented a rapid sequence of vibrations, each of which was presented to one of two bodily locations, and then compared the performance of the TTNJ between two stimulus-location conditions. In one condition, each of the vibration trains was presented to one of two fingers of the left hand (unimanual condition). In the other condition, each of the vibration trains was presented to the index finger of either the right or left hand (bimanual condition). With these conditions, we aimed to examine how the differences in stimulus locations and in types of tasks affect TTNJ performance. Our results showed that when the participants were asked to count the total number of vibrations presented at two locations, the performance (proportion of correct answers) was not so much different between the two conditions. In contrast, when the participants had to report the two numbers of vibrations presented at each location or to focus on the number of vibrations at a single location, the TTNJ performance in the bimanual condition was drastically better than in the unimanual condition. These results suggest that the underlying mechanism for tactile temporal numerosity perception can segregate the interhemispheric information (bimanual condition) more precisely than the within-hemispheric information (unimanual condition), when spatiotemporal tasks are performed.

Effects of Numerosity Range on Tactile and Visual Enumeration

Our study explores tactile enumeration using both hands and investigates the effects of numerosity range's (NR) on general enumeration. In Experiment 1, using custom-made vibro-tactile apparatus, we replicated results of Cohen, Naparstek, and Henik (2014, Acta Psychologica, 150C, 26-34) and again found a moderate increase in RT up to four stimuli and then a decrease for five stimuli. In Experiment 2, we used a within participants design and compared NR 1 to 5 and 1 to 10 in tactile and visual enumeration. The results showed that enumeration for NR 5 to 1 was faster than for NR 1 to 10, especially for numerosities four and five. Within NR 1 to 10, in the visual modality the subitizing range was 4, the counting range was from 5 to 9, and there was an end effect of 10 dots. In the tactile modality, when excluding one-hand arrangements, the subitizing range was 2, the counting range was from 3 to 5, there was an acceleration of counting from 5 and on, and there was an end effect for 10 stimuli that was stronger than for 10 visual stimuli. We suggest that NR influences enumeration and that number-hand association (i.e. resulting from finger counting) influences enumeration, resulting in faster counting.

Perceptual Grouping Affects Haptic Enumeration Over the Fingers

Spatial arrangement is known to influence enumeration times in vision. In haptic enumeration, it has been shown that dividing the total number of items over the two hands can speed up enumeration. Here we investigated how spatial arrangement of items and non-items presented to the individual fingers impacts enumeration times. More specifically, we tested whether grouping by proximity facilitates haptic serial enumeration (counting). Participants were asked to report the number of tangible items, amongst non-items, presented to the finger pads of both hands. In the first experiment, we divided the tangible items in one, two, or three groups that were defined by proximity (i.e., one nonitem in between two groups) and found that number of groups and not number of items were the critical factor in enumeration times. In a second experiment, we found that this grouping even takes place when groups extend across fingers of both hands. These results suggest that grouping by proximity affects haptic serial enumeration and that this grouping takes place on a spatial level possibly in addition to the somatotopic level. Our results support the idea that grouping by proximity, a principle introduced in vision, also greatly affects haptic processing of spatial information.