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Even though mathematics is considered one of the most abstract domains of human cognition, recent work on embodiment of mathematics has shown that we make sense of mathematical concepts by using insights and skills acquired through bodily activity. Fingers play a significant role in many of these bodily interactions. Finger-based interactions provide the preliminary access to foundational mathematical constructs, such as one-to-one correspondence and whole-part relations in early development. In addition, children across cultures use their fingers to count and do simple arithmetic. There is also some evidence for an association between children’s ability to individuate fingers (finger gnosis) and mathematics ability. Paralleling these behavioral findings, there is accumulating evidence for overlapping neural correlates and functional associations between fingers and number processing. In this paper, we synthesize mathematics education and neurocognitive research on the relevance of fingers for early mathematics development. We delve into issues such as how the early multimodal (tactile, motor, visuospatial) experiences with fingers might be the gateway for later numerical skills, how finger gnosis, finger counting habits, and numerical abilities are associated at the behavioral and neural levels, and implications for mathematics education. We argue that, taken together, the two bodies of research can better inform how different finger skills support the development of numerical competencies, and we provide a road map for future interdisciplinary research that can yield to development of diagnostic tools and interventions for preschool and primary grade classrooms.

Across cultures children and adults use their fingers to count, to communicate about numbers, and to do arithmetic. Body-based counting and arithmetic systems, all involving fingers and sometimes other body parts, emerged independently across human cultures and history, from New Guinea (

The foundations for mathematical abilities, like all cognition, can be traced to early development and have neurological bases that are linked to the active experiences of children. Children’s bodies, particularly hands and fingers, play a crucial role in grounding and in establishing the neural networks that underlie numerical abilities (

Our goal in this paper is to synthesize findings in mathematics education and cognitive science/neuroscience (neurocognitive) research on the relevance of fingers for numerical development, and reflect on gaps in our knowledge in an effort to lay out a road map for future research and practice. Mathematics education and neurocognitive research on the relevance of fingers for mathematical cognition differ in terms of theoretical orientations, goals and methodologies used.

Mathematics education research presented here follows a constructivist orientation and explores ways with which bodily interactions, including fingers, contribute to the active construction of number concepts. Methodologically, these studies follow traditions of genetic epistemology (

After covering how bodily interactions in early development contribute to the development of

There are several perspectives that have been taken over the years in regards to fingers and mathematics. According to one constructivist perspective fingers provide a physical and accessible representation for ordinal and cardinal representations in early development, and finger counting strategies facilitate arithmetic learning. These strategies evolve as a result of practice, automatization, and development of composite unit understanding (e.g., from count-all to count-in strategies, to count-on), and are gradually replaced by computational strategies supported by verbal, symbolic, and visuospatial representations (

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Through development the act of counting becomes progressively more independent from immediate perception. In addition to perceptual unit items,

When we consider infants’ pre-verbal, bodily experiences, finger-based interactions standout as, perhaps, the most significant of all bodily experiences, given that the movement of and tactile sensations with (both of which are related) fingers become progressively more distinct. Infants spend time looking at their hands and fingers and watch their fingers move in progressively more independent ways. The sensorimotor system for fingers involve tactile, motor and visual modalities, and early physical experiences serve for the integration of these modalities to develop visually guided fine motor movements and tactile perceptual abilities (

Even before fingers are used for explicitly representing numbers, they provide the preliminary and grounding sensorimotor experiences for perception of discrete units. During finger counting, fingers act as motor unit items. Shortcut finger counting strategies, like counting-on, represent aspects of abstract unit items, and are likely to pave the way for the symbolic representation of numbers. For example, while calculating “8 + 5” the counting-on strategy (starting to finger count from nine up to 13, instead of first counting from one to eight) includes three different counting types (i.e., figural, verbal, and abstract) (

Even though Steffe et al. framed the development of the number concept from a constructivist perspective, their approach is different from Piaget’s original formulation of how sensorimotor experiences shape numerical development.

Alternatively, the finger and mathematics relation can be considered from an embodied cognition perspective. Embodied cognition is not a single and unified theory of cognition, but rather a transdisciplinary research program with a multitude of claims and theories, with the shared notion that cognition is grounded in bodily systems (

Even though

For the purposes of this article, we distinguish between two forms of finger processing: finger numeral representations and finger gnosis. Finger numeral representations involve both counting and montring. Finger counting is ordinal and is used as an aid for the purpose of counting quantities or for doing arithmetic. A related concept is finger montring, which refers to finger configurations to represent and communicate cardinal number information (

Previous research distinguished between two forms of finger numeral representations; montring and counting. Montring gestures communicate cardinal number information (

Finger counting is universal and ubiquitous across cultures, and show a high-level of cultural variability (

Compared to cultural differences, we know very little of how sociocultural factors, like socioeconomic status, affect early finger-based interactions, and finger counting habits and skills. Previously it was shown that kindergarteners from low-income households use their fingers less often to count and add than children from middle-income households (

When children first start learning to solve arithmetic problems, they use their knowledge of counting, which is often executed with the help of fingers. The use of finger counting strategies eventually results with memorization of basic arithmetic facts, which then leads to a shift from finger counting strategies to memory-based strategies (

Children progressively develop more efficient finger counting strategies during early arithmetic learning. According to

Children with mathematics learning difficulties show differences in reliance on and habits of finger counting. First, second, and third grade children with mathematics difficulties are more reliant on finger counting strategies, and they have a harder time transitioning from finger counting to verbal counting and retrieval strategies, which typically occurs towards the end of first grade and early second grade with children who are not identified with any impairments (

The fact that children with MD/RD show lower performance with both finger and verbal counting strategies compared to the group with MD-only was proposed to hint at a weakness with counting procedures for children with MD/RD and with mental computation (e.g., fact retrieval) for children with MD-only (

Unimpaired children transition from finger counting strategies to arithmetic fact retrieval strategies from first grade to second grade, while children with math-learning and math-learning/reading disabilities do not show this shift and continue to rely on finger counting even after first grade (

One way to approach the finger and numerical development is that fingers provide a “natural scaffold for calculation” (^{th} grader is a negative indicator of age-appropriate literacy (

What is it about finger counting that could be so important? Steffe and colleagues’ theory about fingers constituting a level of abstraction between specific things and abstract concepts of number is one possibility. There is growing evidence indicating that fingers play a significant role in the development of a mature counting system, inasmuch as early counting consists of touching in sequence the thing counted, as one says the number name (

Starting at a very young age, even before number symbols are learned, a link is created between magnitude and fingers, with children being able to represent numbers with their fingers as early as three years of age (

Thus far we have focused primarily on the role of finger counting in numerical and mathematical development. However, it may be that finger gnosis is the building block upon which finger counting and numerical competence rests. There is a growing body of research focused on the association between finger gnosis and numerical and mathematical competency. For example, in a study examining whether finger gnosis was foundational to number representation,

The relationship between finger gnosis and finger counting has not been well understood. Finger gnosis has been found to be correlated with number knowledge (

Improvements in finger gnosis may positively impact the use of finger counting in young children via two mechanisms. The first possible route is via motoric processing. For example, speculatively, better finger gnosis is associated with better fine motor skills, which may be necessary for both finger counting and for counting small entities (like rows of counters). A recent study showing an association between fine motor abilities, finger counting skills, and conceptual counting knowledge in pre-school children provides some preliminary evidence in this direction (

Critically, the construct of finger gnosis is not exclusively or originally about number concepts. The ability to localize the stimulation of fingers (finger gnosis) is, in part, a measure of the preciseness of discriminating regions of sensory stimulation (but as we will discuss, it also involves more than that). Poor finger gnosis has been used as an indicator of brain dysfunction and learning disability for several decades (

During a finger gnosis test,

Finger gnosis tasks are measuring more than just the ability to discriminate the finger touched, they also measure the ability to activate an internal representation and then map it onto another representation, which requires a host of processes required for response generation, including working memory and spatial processes (this might decrease validity and could be considered a weakness of the task). The ability to generate internal representations of physical objects is, in general, an important cognitive process. But fingers, as part of the body, are special. There is a direct mapping of fingers in somatosensory and motor cortices (e.g., touching the thumb activates a particular part of somatosensory cortex). Learning to generate precise and finely tuned representations of one’s own body and to map the spatial relationships of body parts may scaffold learning about external objects. Some support for this idea can be seen in the study of children with motor processing deficits. In a study by

The finger gnosis task can also be considered one of perceptual discrimination (i.e., requires distinguishing between sets of stimuli based on perceptual features). Recent research on the relation between perceptual discrimination, in this case visual, and mathematical skills suggests another potentially related pathway. In particular, the ability to distinguish small differences in the set sizes of arrays of objects (a set of 100 stars versus a set of 108 stars) as a preschooler has been shown to predict later success in mathematics (

The different theories that explain what underlies the finger and number relation can generally be categorized into three as, localizationist, developmental (functionalist) and evolutionary (neural reuse) approaches (

According to the evolutionary account, the finger sensorimotor system is involved in number processing because of a general evolutionary phenomenon named “neural reuse” (

The number processing network in the brain is distributed across many areas. According to the Triple Code Model (TCM) –a model that has significantly influenced discussions on the neural correlates of number processing in the last two decades– three parietal areas, each matching a distinct form of numerical representation, constitute the core neural correlates for number processing (

While these parietal areas are crucial for number processing, overlapping areas in the parietal cortex were also found to be related to fingers. The study of the overlap between number and finger processing in the parietal cortex goes back almost a century. In 1924 Josef Gerstmann diagnosed an adult patient who was not able to name her own fingers or point to them on request. Tests on this patient also revealed that she had difficulty differentiating between her right and left hand, or another person’s right and left hands. In addition, she performed poorly on calculation tests and had impairments in spontaneous writing. The source of the symptoms was a lesion located in the left AG (

Gerstmann’s syndrome is controversial and the existence of such a conditions is questioned, both because it is hard to find a pure case of Gerstmann’s syndrome, and because the four symptoms do not have an obvious shared sub-function that can be affiliated with the angular gyrus (

Contrary to the localizationist approach, there is accumulating evidence for a functional relation between finger and number representations. In a study where excitability in hand muscles was measured during a visual parity judgment task, involving numbers between one and nine, modulation of right hand muscles, but not the left hand, was found with right-handed subjects (14 out of 16 started finger counting with their right hands). The effect was stronger for numbers between 1 and 4 (

The research reviewed so far shows that children’s initial finger counting experiences, as well as different forms of both number related and non-number related hand and finger experiences, might be crucial in establishing the number processing network and in paving the way for learning of more advanced mathematics topics. To explore the implications of this body of research we should shift our focus to which forms of finger-based activities can help support mathematics learning, and how finger-based indicators (e.g., counting, fine motor skills, finger gnosis) can help diagnose potential problems and predict future performance. Here we pose a set of research questions to provide a roadmap for future research.

Earlier in this paper we detailed how Steffe, von Glasersfeld and colleagues (

“Laurent looks at his hands most attentively, as if he did not know them. He is alone in his bassinet, his hands motionless, but he constantly moves his fingers and examines them. After this he moves his hands slowly, looking at them with the same interested expression. Then he joins them and separates them more slowly while continuing to study the phenomenon; he ends by scratching his covers; striking them, etc., but watching his hands the whole time.” (p. 232)

If constructing an understanding of sensorimotor units, pluralities, and collections is a prerequisite for counting and number sense, and finger-based interactions are gateways to building these competencies, then hereditary and developmental differences in in the functioning of the finger sensorimotor system might constrain development of the prerequisite competencies for numerical development.

While there is accumulating research on how finger gnosis (

In several lab studies finger counting habits (the way one counts on her fingers from one to ten) showed significant effects in performance measures.

In addition to lateralization effects, the chunking of numbers in groups of five (due to having five fingers on each hand) during finger counting seems to affect number processing.

What can we learn from finger counting habits for populations with mathematics learning disabilities and other cognitive disorders

For example, patients diagnosed with autism (

Findings from disparate studies show that, in addition to finger counting, fine motor skills (

The early bodily experiences of children today differ significantly from previous generations due to availability of and early exposure to technology. How does early interaction with technology affect finger skills and what are the indirect effects on numerical development? If early bodily experiences are crucial for setting the stage for later numerical development, there are reasons to be concerned about how children’s extensive interaction with technology can affect development of bodily -and in particular finger- skills, which then can have an effect on numerical development. In the U.S., preschool children are exposed to an average of 4.1 hours of screen time per day (

The research on the relation between fingers and numerical development can also guide development of new learning technologies. For example,

The home environment can be another factor that impacts the development of both finger and numerical skills. There has been extensive research regarding the impact of the home environment on reading (e.g.,

In the reading literature there are several studies that examine the impact of phonological training on reading skill (

There are a few studies examining the impact of finger training on arithmetic performance.

While number processing is often conceived as abstract, the systems that support number processing are related to systems that allow us to engage in seemingly mundane bodily tasks. An evolutionary perspective can help us understand why and how mathematical abilities are grounded in bodily systems. Symbolic mathematics being a relatively recent cultural invention, there was not enough time in our evolutionary past to develop brain systems that are explicitly dedicated to mathematical cognition. Like other recent steps in human evolution, such as verbal abilities and writing, mathematical abilities had to rely on and reuse existing systems and abilities. Evolution of hands plays a special role in human evolution. We use (and have used in the past) our hands to build tools, to manipulate our environment, and to communicate. Hands are represented disproportionately in both somatosensory and primary motor areas of the brain. The sensorimotor system that supports functioning of the hands mainly involves the sensory (tactile), motor, and visual modalities. These three modalities coordinate to allow for visually guided movement of the hands to engage in a wide range of tasks, from manipulating objects (

Neurocognitive studies provide insights about how the finger sensorimotor systems supports and scaffolds number processing; how behavioral indicators for fingers (e.g., finger gnosis, finger counting habits) correlate with and predict numerical indicators (e.g., subitizing, counting, arithmetic), and how neural correlates of finger (e.g., finger gnosis, finger motor movements) and number (e.g., magnitude processing, arithmetic) skills interact and overlap. Most neurocognitive studies on the relevance of fingers for number skills are theoretically framed by embodied approaches to cognition, and consider involvement of the finger sensorimotor system in number processing as an aspect of the embodiment of mathematics (

The mathematics education research presented here share a constructivist orientation, according to which number concepts are constructed through bodily interactions with the world. According to this approach, early interactions with the world set the foundation for understanding number concepts, and through development, number skills progressively become more independent of the physical interactions and representations used early on; eventually to be replaced by more abstract and formal constructs. This body of research gives us a sense of how pre-verbal abilities that lay the foundation for later number concepts develop through ordinary bodily interactions. It also sheds light on how number concepts are constructed from a first-person perspective, by capturing children’s ways of thinking through clinical interviews.

The two bodies of research differ in terms of methodologies and levels of analysis, as well as the theoretical framing of why and how bodily interactions, in particular with fingers, are relevant (if not central) to numerical development. Constructivist research, exemplified by the works of von Glasersfeld, Steffe and colleagues (

Grouping a wide range of work on the relation between fingers and mathematics under two main categories; constructivist mathematics education and embodied neurocognitive research, reduces the complexity within each body of research. Nevertheless, this categorization provides us with ways of comparing the two bodies of research, and looking at how the two can be bridged in synergistic ways to provide a more complete picture of how our body scaffolds and grounds number skills, how this grounding takes place across different levels of analysis (e.g., first-person, behavioral, neural), and how the finger-based measures can help with understanding developmental patterns and skills in mathematics.

Instead of characterizing fingers as physical manipulatives that are relevant to numerical cognition during only a limited window of development, we argue that features of numerical cognition may be grounded in the finger sensorimotor system. This grounding may be similar to the relationship between phonological awareness and reading. Phonological awareness (awareness of the speech sounds in the language and how they map onto letters or syllables) has been shown to be a predictor of success in learning to read (

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