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Theoretical Contributions
You Can Count on Your Fingers: The Role of Fingers in Early
Mathematical Development
Firat Soylu*a, Frank K. Lester Jr.b, Sharlene D. Newmanb
[a] The University of Alabama, Tuscaloosa, AL, USA. [b] Indiana University, Bloomington, IN, USA.
Abstract
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.
Keywords: cognitive development, numerical cognition, finger counting, finger gnosis, embodied cognition, neuroscience, mathematics
education
Journal of Numerical Cognition, 2018, Vol. 4(1), 107–135, doi:10.5964/jnc.v4i1.85
Received: 2016-10-31. Accepted: 2017-05-21. Published (VoR): 2018-06-07.
Handling Editors: Anderson Norton, Department of Mathematics, Virginia Tech, Blacksburg, VA, USA; Julie Nurnberger-Haag, School of Teaching,
Learning, and Curriculum Studies, Kent State University, Kent, OH, USA
*Corresponding author at: College of Education, Box 870231, Tuscaloosa, AL 35487, USA. Phone: (205) 348 6267. E-mail: fsoylu@ua.edu
This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License, CC BY 4.0
(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited.
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 (Saxe, 1981) to Mayans, ancient
Babylon, and Romans (Richardson, 1916), and show a wide range of variability (Bender & Beller, 2012). Given
the deep relation between fingers and numbers, it is crucial to understand how and why finger counting and
other finger-based interactions with numbers relate to numerical development, and how we can develop
approaches in mathematics education that can harness the relevance of fingers for numerical development. In
addition, the finger and mathematics relation constitutes an important aspect of embodiment of mathematics
(Domahs, Moeller, Huber, Willmes, & Nuerk, 2010; Lakoff & Nunez, 2000). Understanding how the finger
Journal of Numerical Cognition
jnc.psychopen.eu | 2363-8761
sensorimotor system grounds numerical processes and scaffolds mathematical development can also inform
the wider questions on embodiment of cognition.
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 (Butterworth, 1999; Moeller et al., 2012). The bodily activities that we engage in have a direct impact on
brain development and future cognitive processing (Greenough, Black, & Wallace, 1987; Posner & Rothbart,
2007). This is particularly true of children due to the rapid neural development that occurs in early life.
Therefore, choosing activities that will provide foundational experiences to ground and support the development
of more advanced cognitive skills is very important. In the case of fingers, there is no consensus on what form
these activities should take or how fingers and hands play a role in numerical development. For example, a
recent article in The Atlantic, titled “Why kids should use their fingers in math class,” reports on a tendency in
schools towards discouraging finger use in math classrooms, despite research that shows an intricate relation
between fingers and early numerical skills, and mathematics educators advocating for use of finger-based
strategies early on (Boaler & Chen, 2016). As Cowan (2013) noted, “Although an earlier generation considered
all methods apart from retrieval as crutches and teachers were likely to forbid the use of fingers, we
[mathematics educators] now see these as a necessary part of children’s development” (p. 56).
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. Moeller et al. (2011) have previously argued that neurocognition
and mathematics education present opposing views on effects of finger counting on numerical development.
According to their characterization, while “mathematics education research recommends the reliance on
external representations, including finger-based ones, only as an aid in the transition to mental representations
of numbers” (p. 4), neurocognitive evidence points to a deep and sustained role of embodied representations,
including finger-based ones, in numerical cognition. Our synthesis here presents a slightly different picture. We
argue that constructivist approaches in mathematics education and neurocognitive research, from an embodied
cognition theoretical framing, can be complementary in understanding the finger and mathematics relation, in
spite of theoretical and methodological differences.
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 (Piaget, 1970) and present support for
theoretical claims based on clinical interviews and classroom observations. Neurocognitive research focuses
primarily on the cognitive and neural mechanisms that support number processing, and the developmental and
evolutionary origins of these mechanisms. The theoretical claims under this body of research are supported by
behavioral and neural (e.g., fMRI, EEG/ERP) data from lab experiments and intervention studies. Taken
together, these two bodies of research can provide new insights into the relation between finger and number
processing, and the forms of bodily interactions that can support numerical development during development.
In the first part of the paper we review research that connects development of early perceptual abilities with
understanding whole-part relations and learning how to count. It is argued that the concept of “number” is
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grounded in our ability first to perceive distinct objects, then to categorize and group them based on a measure
of “sameness” (Steffe, Cobb, & von Glasersfeld, 1988; von Glasersfeld, 1981). Our abilities to recognize distinct
objects based on their perceptual features, and to categorize them in groups of plural objects relies on our
bodily interactions with the world. From this perspective, numbers are related to how we perceive and interact
with the world, and our readiness to understand numbers very much relates to our early bodily interactions.
After covering how bodily interactions in early development contribute to the development of number sense, we
explore how finger counting strategies emerge and evolve, and the relevance of finger counting to mathematics
learning difficulties. Then we focus on neurocognitive studies on the relation between finger counting, finger
gnosis and mathematics learning and development. Finally, we discuss ways of bridging neurocognitive and
mathematics education perspectives, and explore crucial questions for future interdisciplinary efforts.
Different Approaches to the Role of Fingers in Early
Numerical Development
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 (Baroody, 1987;
Dehaene & Cohen, 1995; Steffe et al., 1988). This transition model paints a picture of early numerical
development characterized by representational transitions from concrete to more abstract forms of processing,
and provides us with a description of how the concept of numerosity emerges as a result of interacting with the
physical world. An alternative approach, which is partially complementary with the constructivist perspective,
interprets the finger and number relation from an embodied cognition perspective (e.g., Domahs et al., 2010;
Moeller et al., 2012). Here we review and compare these approaches, and discuss ways with which these two
main approaches can together present a more comprehensive account for the role of fingers in development of
numerical skills.
Constructivist Approaches: From Sensorimotor to Figural Unit Items
Built on Piaget’s work (1954) on how children develop an understanding of whole-part relations, Steffe, von
Glasersfeld and colleagues (Steffe et al., 1988, 1983; Steffe & von Glasersfeld, 1985) traced children’s ability to
count to early perceptual experiences. Before counting and understanding numerosity, infants develop the
ability to perceive discrete objects –perceptual unit items– by abstracting aspects of their ongoing sensorimotor
experience. Steffe et al. (1988) proposed that “… isolating something from the experiential continuum is an
indispensable prerequisite for any conception of number” (p. 3). Von Glasersfeld (1981) pointed out that
perceiving an entity as an “object” requires coordination of sensory material from multiple sources. Most often,
this would be a combination of visual and tactile information. Von Glasersfeld proposed a pulse like theory of
attention, where the perception of a sensorimotor unit (an object) is bounded by states of unfocused attention,
and includes pulses of attention in between states of unfocused attention. Each pulse of attention focuses on a
different aspect of the sensorimotor experience (e.g., in perceiving a cup these might include features of color,
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texture, shape). There are two prerequisites for a sensorimotor item to be recognizable: First, it needs to
contain more than one focused impulse, which “make the thing qualitatively discriminable and give it location in
the experiential field” (p. 88). The second aspect is “boundedness.” The sensorimotor item needs to be
distinguishable from its background. The pre-verbal ability to perceive distinct objects is followed by the ability
to label objects. By about two years of age most children can not only perceive objects but label them, and use
the plural form of the labels. The use of plural requires not only extraction of certain aspects of the fuzzy
sensorimotor experience as a distinct “thing”, but also recognizing the recurrent nature of this unit across
different instantiations.
Steffe et al. (1988) distinguished between pluralities and collections. Pluraties refer to more than one
instantiation of a previously extracted sensorimotor experience; however, it is not bounded in any specific way.
The plural refers to more than one, but not to a collection, because what it refers to has no beginning or end.
Collections are bounded, and they refer to pluralities defined with a beginning and end. For example “the cups
on the table” is a collection; the items in the collection can be considered as recurrences of a perceived unit
due to similarities in the sensorimotor experiences they elicit. They are bounded by the perceptual field
presented by the table, and therefore they are countable.
Through development the act of counting becomes progressively more independent from immediate
perception. In addition to perceptual unit items, Steffe et al. (1988) proposed four counting types to characterize
the gradual independence of counting from immediate sensorimotor interaction. Figural unit items are
visualized substitutes for objects that are not visible (e.g., when some units in a collection are partially blocked,
children assume that they are there by imagining or visualizing them as part of the collection). Motor unit items
are motor movements (e.g., pointing with the index finger) that substitute for the perceptual units. Verbal unit
items are utterances of a number word, without the need to present the physical perceptual unit item. And
finally, abstract unit items represent complete independence from the sensorimotor experience in the counting
task. For example, when the child realizes that the utterance of the number word “eight” both represents the
number word sequence “one, two, …, eight” and a collection of eight discrete items, the child is said to have
“an abstract conception of number” (p. 6). When asked how many checkers are hidden, in a situation where the
total is 12 and eight is visible, if the child counts “9 as 1, 10 as 2, 11 as 3, 12 as 4”, then she is referring to each
verbal item as a countable unit. At this stage the abstract, verbal item replaces the sensorimotor item.
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 (Barrett, Traupman,
& Needham, 2008). It could be said that fingers are discrete sensorimotor units that are attached to our body. If
extracting “distinct things” from our sensorimotor experience is a prerequisite to understanding plurality and
collections, and eventually to understanding numerosity (Steffe et al., 1988, 1983), then experiences with
fingers should be the gateway for these competencies. Fingers are distinctly presented across multiple
modalities (i.e., tactile, modal, visual), they are attached to our body as two separate collections of five units,
and they can be moved independently to create different configurations (e.g., the simplest way being an open
or closed state, which is useful in counting and representing collections of different number of units).
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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) (Steffe & von Glasersfeld, 1985).
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. Von Glasersfeld (1981) pointed out that even though Piaget had referred to children’s
understanding of whole-part relations, he did not provide an account for how this understanding develops: “In
all his work on the development of number, Piaget focuses on a conceptual complex that involves class
inclusion and order, and, like most of his predecessors, he takes the construction of units for granted” (p. 84).
Steffe, von Glasersfeld and colleagues (Steffe et al., 1988, 1983; Steffe & von Glasersfeld, 1985) filled this gap
by explicating how perception of sensorimotor units eventually lead to abstract representations for numbers
through a series of representational shifts, including motor, figural, and verbal unit items.
Embodied Approaches
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 (Clark, 1999; Gallese & Lakoff, 2005; Smith & Gasser, 2005). According to the embodied account of
mathematics, bodily interactions do not aid development of abstract, and eventually disembodied,
representations and logical principles, but rather help structure the sensorimotor systems in a way to provide
semantic content for number processing (de Freitas & Sinclair, 2013; Gallese & Lakoff, 2005; Núñez, 2012;
Sfard, 1994). The use of symbolic representations (e.g., Arabic numerals, algebraic notations) does not mean
that cognitive processing becomes progressively more disembodied, and takes a purely abstract and symbolic
processing form. Instead, symbolic processing itself is grounded in the sensorimotor systems. For example,
Barsalou (1999) argued that during perceptual experiences, association areas in the brain capture bottom-up
sensorimotor patterns. Later, during the use of perceptual symbols, association areas activate some of the
same sensorimotor areas in a top-down manner; meaning that certain perceptual experiences can trigger
associated sensorimotor states. From this perspective, the meaning of symbols (semantics) emerge from the
sensorimotor simulation of relevant systems. The early finger-based interactions and finger counting
experiences not only aid numerical development during childhood, but also ground and shape the number
processing system, extending its effects to how adults process numbers (see Berteletti & Booth, 2016 for a
review). From the embodied cognition perspective, number processing and bodily systems do not gradually
become independent. According to the embodiment thesis (Lakoff & Nunez, 2000; Soylu, 2011) the number
processing system continues to be embodied through development and adulthood, and is grounded in
sensorimotor systems. In other words, the sensorimotor system is part of the number processing network,
instead of just being a precursor to or constituting the foundation for it.
Even though Steffe et al. (1988) have not extensively theorized about what role the sensorimotor system plays
once abstract representations (abstract unit items) are constructed, they hinted at some form of sensorimotor
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simulation: “The transition from being a counter of perceptual unit items to a counter of verbal unit items
involves the internalization of sensory-motor activity” (p. 6). Separately, in their explanation of what they
referred to as the “uniting” operation (distinguishing discrete units from collections) Steffe and von Glasersfeld
(1985) added, “A mental operation is an interiorized action, an action that can be carried out in thought. What
may be overlooked is that these mental acts lead to a result and must have material to operate on. The uniting
operation can have a collection as material and a unit of units - a whole number - as a result.” (p. 272).
Nevertheless, they fell short of describing what is meant by “internalization of sensory-motor activity” or the
nature of an “interiorized action.” Embodied accounts of mathematics, especially sensorimotor simulation
theories (Soylu, 2011, 2016; Svensson, 2007), address this missing component. The embodied account is
corroborated by neurocognitive studies showing that the finger sensorimotor network continues to play a role in
adults’ number processing (Andres, Davare, Pesenti, Olivier, & Seron, 2004; Badets, Andres, Di Luca, &
Pesenti, 2007; Di Luca & Pesenti, 2008; Rusconi, Walsh, & Butterworth, 2005; Soylu & Newman, 2016), and
that early finger counting experiences leave a lasting effect on adults’ performance and neural correlates for
number processing (Newman & Soylu, 2014; Tschentscher, Hauk, Fischer, & Pulvermuller, 2012). The
grounded model argues that number processing does not become disembodied with age and instead early
bodily interactions help establish the number network and leave a lasting trace on numerical cognition.
Finger Numeral Representations and Finger Gnosis
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 (Di Luca & Pesenti, 2008). Finger gnosis (also referred to as finger-localization; Benton, 1955;
finger gnosia; Noël, 2005; and finger sense) is the ability to localize the stimulation of fingers and is, in part, a
measure of the preciseness of discriminating regions of sensory stimulation. Various tests have been used to
measure finger gnosis (Benton, 1955; Fayol, Barrouillet, & Marinthe, 1998; Noël, 2005); all involving stimulation
of one or multiple fingers, through the touch of a physical object, while the hands are not visible (either because
the eyes are closed, or the hands are covered), and the participants being asked to report the fingers touched
either verbally, or by moving the matching fingers on the other hand. There are a multitude of studies, which we
will cover in detail, that showed that finger gnosis scores correlate with or predict mathematical skills (e.g.,
Fayol et al., 1998; Newman, 2016; Noël, 2005).
Finger Numeral Representations (Counting and Montring)
Previous research distinguished between two forms of finger numeral representations; montring and counting.
Montring gestures communicate cardinal number information (Di Luca & Pesenti, 2008). For example, when
asked how many strawberries a child wants for her lunch, the child may spontaneously open her index and
middle fingers, indicating “two.” Finger counting is ordinal and is used as an external aid for the purpose of
counting quantities or for doing arithmetic. Wasner et al. (2015) reported the ordinal and cardinal finger
representations to be different in 44 % of an adult sample, showing that different finger-based representations
coexist and are recruited based on the task context. In experimental studies montring and counting gestures
were found to automatically activate number semantics, when compared non-canonical ones (e.g., index, ring,
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and little fingers for “3”) (Di Luca, Lefèvre, & Pesenti, 2010; Di Luca & Pesenti, 2008), showing that montring
and counting gestures acquire a symbolic status and are processed similar to Arabic numerals, while non-
canonical gestures are processed similar to any other collection of objects, requiring subitizing, counting, or
estimation processes. In an ERP study processing of montring gestures were found to show differential
modulation of attentional resources in early perceptual processing compared to counting and non-canonical
gestures, where later semantic processing related to access to numerosity information overlapped between
montring and counting gestures, compared to non-canonical (Soylu, 2018). The distinction between montring
and counting representations was made relatively recently (Di Luca & Pesenti, 2008) in numerical cognition
research, and therefore most research on finger numeral representations refer to finger counting, even when
cardinal finger representations are studied.
Finger counting is universal and ubiquitous across cultures, and show a high-level of cultural variability (Bender
& Beller, 2012; Butterworth, 1999). In some cultures body-based counting systems are used, which include
other body parts like feet, arms, the head, in addition to fingers, and in others, higher-order counting
configurations are used to represent numbers bigger than 10, for example by using one hand to count numbers
from one to five, and using the other one to keep track of how many times counting on the other hand was
completed (see Bender & Beller, 2012 for a review of the wide range of finger counting systems used across
cultures). There are also some theories arguing that finger-based number representations go back in human
history as early as upper Paleolithic, based on hand stencils at the Cosquer Cave (Overmann, 2014). The most
prevalent modern form of finger counting involves a one-to-one correspondence with numbers from one to 10,
with variations on some aspects of it, for example which hand one starts to count with, or which finger is first
used on each hand (e.g., thumb or index). One notable exception to this is the Chinese finger counting system,
which uses fingers only on one hand, and uses symbolic gestures for numbers six to 10 (no one-to-one match
with fingers) (Domahs et al., 2010).
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 (Jordan, Huttenlocher, & Levine, 1992; Jordan, Kaplan, Ramineni, & Locuniak,
2008). Finger use, as well as finger gnosis, have been shown to positively predict mathematical achievement in
children (Fayol et al., 1998; Noël, 2005; Penner-Wilger & Anderson, 2013). In addition, several studies (Crollen
& Noël, 2015; Fuson, 1988; Soylu & Newman, 2016) suggested that finger processing may play a role in setting
up the neural networks on which more advanced mathematical computations are built. If the relation between
fingers and numbers goes beyond use of finger counting strategies between ages four to eight, and if the finger
sensorimotor system grounds and scaffolds the development of numerical skills, then we need to reevaluate
approaches to finger counting and finger skills (particularly finger gnosis) in mathematics education.
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 (Jordan et al., 2008). The transition from finger and verbal counting strategies to automatic
fact retrieval allows children to focus the limited processing resources to more complex aspects of arithmetic
like multi-digit subtraction/addition, long division or complex multiplication (Gersten & Chard, 1999).
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Children progressively develop more efficient finger counting strategies during early arithmetic learning.
According to Baroody (1987), counting-all is the most basic strategy, which involves counting fingers out one by
one to represent the first, then the second addend, consecutively until all fingers put out are counted to
determine the sum (e.g., for “3 + 2”, thumb, index, and middle fingers are opened, then ring and little fingers; all
of them are counted together to find the sum). If the addends are both smaller than five, then each addend can
be represented separately on each hand, which may not require counting each addend if the finger
configuration for numbers from one to five is automatized, and the child has to count only once. A more
developed strategy is to bypass the sum count by automatically recognizing the sum either visually (i.e.,
subitizing) or kinesthetically. A more efficient alternative to the counting-all strategy is the counting-on strategy,
where the child counts not from one but starts with the cardinal designation of the first number. This strategy
can be made even more efficient by always starting with the larger number, which reduces the counting length
(e.g., 8 + 3 instead of 3 + 8). Children’s use of strategies that imply knowledge of commutativity (e.g., always
starting with the larger number) does not warrant knowledge of commutativity (Baroody, 1987). For example, a
child who prefers to start counting from the larger number in an arithmetic task might still claim that 3 + 5 is not
the same as 5 + 3.
Finger Counting and Mathematics Learning Difficulties
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 (Bryant, 2005; Geary, 2004; Jordan & Hanich, 2000; Jordan et al., 2008). In
addition, mathematics learning disability seems to interact with reading disability; children with both types of
disabilities (comorbid) show further problems in memorizing and retrieving arithmetic facts than children with
only mathematics learning disability. At first grade, children with MD/RD (mathematics and reading disability
together) or MD (mathematics disability only) show more counting-procedure and retrieval errors, and use more
efficient finger counting strategies less frequently compared to children who are not identified with any
impairments (Gersten & Chard, 1999). At second grade, children with MD develop more efficient finger
counting strategies compared to their peers with MD/RD (Gersten & Chard, 1999; Jordan & Montani, 1997).
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 (Jordan, Hanich, &
Kaplan, 2003). There is evidence showing that, similar to dyslexia (which results from a deficit in the
phonological system; see Schlaggar & McCandliss, 2007), mathematics disability is not due to general low
intellectual performance, but is more likely to be due to a more specific deficit that affects number processing
and other closely related processes (see Rousselle & Noel, 2007). There is some controversy, however, as to
which aspect of the number processing system is affected in mathematics disability (Rubinsten & Henik, 2009;
Shalev & Gross-Tsur, 2001). There is no single cause that explains all cases of MD, therefore it is likely that MD
represents a spectrum of conditions involving one or more deficits in various number-related processes.
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 (Gersten & Chard, 1999). Since fingers
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can function as a working-memory aid in keeping track of addends, difficulties with representing numerical
magnitudes in the working memory during addition is one of the explanations proposed for why children with
MD rely on finger counting for a more extended amount of time (Geary, 2004). Working memory deficiency is
also proposed to explain why children with MD undercount or overcount during finger counting, since
miscounting might be due to losing track of what has already been counted and what remains to be counted
(Geary, 1990; Hanich, Jordan, Kaplan, & Dick, 2001). Given the further severity of mathematics disability of
children with both MD and RD, the working memory deficiency might also be related to language-based
information representation. If phonetic representation of number words cannot be retained in working memory
or misrepresented due to a problem in the phonetic-articulatory system, children might resort to relying on
finger representations, which bypass the phonetic representation (Geary, 1990).
Finger Counting and Number Skills
One way to approach the finger and numerical development is that fingers provide a “natural scaffold for
calculation” (Jordan et al., 2008, p. 662). As with all manipulatives, however, there is controversy as to whether
finger use during problem-solving is helpful or detrimental. This may depend on the timing of finger use.
Evidence suggests that finger use early is helpful. For example, Jordan and colleagues (1992, 1994) found that
finger use was linked to higher accuracy on number combinations (includes responding to questions like ‘How
much is 9 take away 2?’) in kindergarten and first grade students. Those students who rarely used finger
counting spontaneously had poorer performance. By second grade, however, there was a shift in that better
mathematical performance was associated with reduced finger use and a greater reliance on retrieval
strategies (Jordan et al., 2008). These findings may reflect a developmental trajectory in which finger counting
sets the stage for more advanced skills, but once those skills are acquired finger counting is no longer needed;
possibly because finger counting is supposed to become, what Steffe and von Glasersfeld (1985) referred to
as, an “interiorized action” (or alternatively an embodied simulation) by then. It follows then that children who
are still finger counting are the children who have not acquired those more advanced grade level skills. By
analogy, a kindergartener who uses invented spelling (writing “kitten” as KTTN) shows signs of precocity and
readiness to learn to read; however, the same behavior in a 4th grader is a negative indicator of age-appropriate
literacy (Treiman & Zukowski, 1991).
Steffe and colleagues’ (1988, 1983) interpretations of their own findings on numerical development in
elementary children may be helpful in understanding the developmental question. They proposed a stage
theory of counting suggesting that children’s learning of number concepts follows levels of increasing
sophistication according to their ability to conceptualize and interiorize individual units. Fingers (which are used
to count all kinds of different things) stand in between the concrete things that are counted and the abstract
concept of number itself, and therefore help children develop an initial level of numerical representation that is
more abstract than touchable items and through which addition and subtraction may be understood and
generalized. As children become more competent in their numerical skills, however, they will discard the use of
fingers or other physical representations, as the conceptualization of numbers becomes more abstract. In this
regard, finger counting and other finger activity can be seen as indicators of the students’ level of numerical
development, and preference for using a tool that provides increased numerical power, until they advance to
the point where the use of fingers is no longer necessary.
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.
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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 (Butterworth, 1999, 2005; Fuson, 1982, 1988). Others have suggested that fingers serve as a
memory aid during counting (Geary & Wiley, 1991), and that fingers aid in understanding cardinality (Fayol &
Seron, 2005) and the development of the one-to-one correspondence principle (Alibali & DiRusso, 1999).
Indeed, Rusconi et al. (2005) suggested that “counting and combining quantities on one’s own fingers seem to
represent an obligatory passage to the mastery of number concepts and of arithmetical operations” (p. 1610).
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
(Fuson, 1988). This may simply be because magnitude is first mapped onto a body-based system and then
translated into a more abstract numeral system, where numbers and fingers are connected. This is essentially
the hypothesis proposed by Butterworth (1999), who suggested that fingers and numbers link because fingers
are used to represent numerosity. If this is the case, then individual experiences in finger counting should
matter. There is evidence from studies on the relation between finger counting habits and neural correlates of
number processing that is consistent with this proposal. Tschentscher et al. (2012) presented number words to
adults and found that the fMRI-measured brain activation observed was systematically related to finger
counting habits: those who began counting with the left hand, “left-starters,” showed increased activation in the
right motor cortex, while “right-starters” showed more activation in the left motor cortex (no motor response was
required during the task). Because finger counting habits have been found to be stable across development
(Sato & Lalain, 2008), this finding suggests that while adults may not use their fingers in the service of
mathematical calculation very often, those early experiences relating fingers with number (and magnitude)
create a lasting neural impression that was activated on seeing a number word, even when no motor response
was required.
Finger Gnosis
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, Noël (2005) examined first graders and found that finger gnosis in first
grade was correlated with the ability to map between numerals and their associated magnitudes in second
grade.
The relationship between finger gnosis and finger counting has not been well understood. Finger gnosis has
been found to be correlated with number knowledge (Newman, 2016; Noël, 2005; Penner-Wilger & Anderson,
2013), which in turn is essential to mathematical performance, and finger counting appears to be an important
and possibly necessary part of early mathematical calculation skill development (Moeller et al., 2012). But how
finger gnosis and finger counting relate to each other is not clearly understood. One possibility is that finger
counting depends on finger gnosis. Alternatively, finger gnosis itself may be tuned by activities such as
counting. Reeve and Humberstone (2011) demonstrated that finger gnosis ability and use of finger counting for
arithmetic co-develop between ages 5 and 7, and that together, but to a lesser extent, with visuo-spatial
working memory, finger gnosis ability predicted both use of finger counting strategies for arithmetic and
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calculation ability (i.e., high finger gnosis ability group used more finger counting and had higher calculation
ability). While this result does not weight in towards either of the casual directions suggested, it does show that
finger gnosis, finger counting, and calculation abilities co-develop. The co-development of finger gnosis,
counting, and arithmetic skills might explain some of the conflicting findings on the relation between finger
gnosis and arithmetic skills in 5 to 8 year old children; some reporting strong associations (Fayol et al., 1998;
Noël, 2005), while others showing no behavioral associations (Long et al., 2016; Newman, 2016; Soylu,
Raymond, Gutierrez, & Newman, 2017) for roughly the same age groups. Newman (2016) reported that 5 to 8
year old children failed to show a relationship between addition performance and finger gnosis, while children 9
to 12 year old did show such a relationship; finger gnosis better predicted performance in older children than
younger children. The explanation provided for the discrepancy, between the two age groups on how finger
gnosis relates to addition performance, was that both addition skills and finger gnosis are still developing in the
younger group. However, both skills should be developed within the older group (9-12 yo), making the
relationship between these two factors more evident. In a recent fMRI study with 7 and 8 year old children
Soylu et al. (2017) reported that even when no behavioral correlations between finger gnosis and arithmetic
(single-digit addition and subtraction) performance were found, activations in particular visuospatial brain areas
related to number processing (i.e., left fusiform and lingual gyri, and bilateral precuneus) were found to
negatively correlate with finger gnosis scores –high finger gnosis children showing reduced activations in these
areas during both addition and subtraction. They also found a negative correlation between finger gnosis
scores and activation in the left inferior parietal lobule only during addition, but not during subtraction. They
proposed that arithmetic fact retrieval is related to finger gnosis at the neural level, both for addition and
subtraction, even when behavioral correlations are not observed in younger children. Given the differential
correlation in the left IPL for addition, the nature of this relation was proposed to be different for addition
compared to subtraction, possibly due to differences in finger-counting strategies used for these two
operations.
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 (Fischer, Suggate, Schmirl, & Stoeger, 2017). Finger-based sensorimotor activities may foster
an increased ability to individuate the fingers, which in turn may lead to better finger counting. A second
possibility is that finger gnosis has a direct impact on number sense, perhaps through the discrimination of
numerical quantities (Halberda & Feigenson, 2008; Mazzocco & Thompson, 2005), which in turn leads to better
finger counting.
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 (Benton, 1979;
Critchleey, 1953; Gerstmann, 1940). More, current research shows that finger gnosis predicts numerical
performance in children (Newman, 2016; Noël, 2005). To understand how finger gnosis may relate to number
and mathematics, it is useful to explore a process model of the typical finger gnosis test.
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During a finger gnosis test, Noël (2005) had participants sit with both hands palm down on a table in front of
them with their hands obscured so that they could not see them. The experimenter, with a pointer, touched the
fingers (either just one or a combination of two) in a pre-determined order. After each touch (or combination of
touches) the participant was expected to indicate which finger(s) was touched either by moving the
corresponding finger of the other hand or pointing to the finger that was touched on a displayed picture. In order
to correctly respond, not only did the participants had to “sense” the touch, but they also had to generate an
internal representation of their hands and fingers. This internal representation is an internal information model
(either an image or propositional representation) of the hand and fingers that contains spatial information
regarding the location of the individual fingers. In order to respond correctly during the finger gnosis task
participants had to map this internal representation of their hand and fingers onto another representation, either
their opposite hand or a picture of a hand. This mapping process is the determination of a one-to-one
correspondence of the representation of the hand being touched and the representation through which the
participant indicates their responses.
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 O’Brien et al. (2002), children with developmental dyspraxia, an inability to
choose, plan, sequence and execute movements, were found to show deficits in global spatial processing,
which may implicate a link between finger gnosis (as defined by the task) and spatial processing in general. In
fact, Newman (2016) found a strong relationship between finger gnosis and matrix reasoning (a non-verbal
reasoning test, which involves a series of figures representing a pattern with one figure left blank) scores;
children (5-12 years of age) with high finger gnosis also having higher matrix reasoning scores. The negative
correlations found between finger gnosis scores and activations in three visuospatial processing areas (i.e., left
fusiform, and bilateral precuneus) during addition and subtraction, reported by Soylu et al. (2017), provide
further neural evidence for the partially visuospatial nature of the link between finger gnosis and number
processing.
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 (Halberda
& Feigenson, 2008; Mazzocco, Feigenson, & Halberda, 2011). Discrimination of discrete quantity was found to
be trainable in adults, although it is not clear how well training transfers to mathematical problem solving
(DeWind & Brannon, 2012), a limitation that might be due to the fact the participants were adults (which might
make it harder to modulate the established problem solving system). In a study of children from low-income
families (ranging in age from 44 to 71 months), Fuhs and McNeil (2013) found that the approximate number
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system (ANS) acuity (i.e., the ability to correctly compare which of the two collection of dots represent a larger
numerosity based on an approximate judgment, without counting) did not predict mathematics ability (including
measures like counting skills, number facts, calculations skills) when controlled for inhibitory control. This result
is in contrast to previous studies where ANS acuity was found to correlate with mathematics ability in
comparable age groups (Libertus, Feigenson, & Halberda, 2011; Mazzocco et al., 2011). Fuhs and McNeil
(2013) explained this discrepancy based on lack of opportunities for low-income children to connect their ANS
with symbolic mathematics skills. A separate body of research (e.g., Jordan, Kaplan, Ola, & Locuniak, 2006;
Ramani & Siegler, 2008) showing that children from low-income homes have less exposure to early number
concepts supports this argument. Even though multiple studies have shown that finger gnosis ability predicts
mathematics ability in children (Newman, 2016; Noël, 2005; Penner-Wilger & Anderson, 2013; Wasner, Nuerk,
Martignon, Roesch, & Moeller, 2016), no previous study has explored if the finger gnosis and mathematics
ability relation holds true for children from low-income homes. It is possible that early association of finger
gnosis ability with number skills depends on exposure to number concepts in the home environment before
formal education.
The Relation Between Finger and Number Representations in
the Brain
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 (Penner-
Wilger & Anderson, 2013). According to the localizationist account (Dehaene, Piazza, Pinel, & Cohen, 2003)
co-occurrence of symptoms related to finger and number processing in neuropsychological cases (e.g., lesions
in left angular gyrus; Gerstmann, 1940) and evidence for neural overlap between number and finger processing
are due to high anatomical proximity of crucial neural resources for finger and number processing, and not due
to a functional relationship between the finger and number processing systems.
According to the evolutionary account, the finger sensorimotor system is involved in number processing
because of a general evolutionary phenomenon named “neural reuse” (Anderson, 2010). Given that the human
brain has not gone through a significant evolutionary change to accommodate new cognitive skills, like verbal
language and mathematics, the neural reuse theory asserts that new cognitive skills rely on and reuse existing
bodily systems, which originally evolved to realize other, more bodily functions (e.g., for fingers, visually guided
fine motor movements). In the case of number processing, the finger sensorimotor circuits, which allow
independent multimodal representation of fingers, are reused to represent and process numerical quantities
(Penner-Wilger & Anderson, 2013). The evolutionary account is not necessarily mutually exclusive with the
developmental account. While the finger and mathematics relation might have some genetic bases, the
development and fine tuning of the number processing network might still heavily rely on bodily experiences
during development.
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 (Dehaene & Cohen 1997; Dehaene
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et al., 2003). First, a non-verbal magnitude representation system, located bilaterally in the intraparietal sulcus
(IPS), is involved in processing of numerical magnitudes (e.g., size and distance relations between numbers).
Second, a visual number representation system, supports recognizing and encoding Arabic numerals, and is
located in bilateral posterior superior parietal areas. Finally, a third verbal representation system, mainly located
in the left angular gyrus (AG), is involved in processing verbal number information (e.g., retrieving arithmetic
facts from phonological memory). In a later study, left inferior frontal gyrus (IFG) and bilateral supplementary
motor areas, were proposed to be included in TCM for their supportive role for magnitude processing and
application of rule-based heuristics in mathematical operations (e.g., as in arithmetic), respectively (Klein et al.,
2016).
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, 1940). It was these studies by Gerstmann in the
1920s that first linked fingers to number processing at the neural level. Since then, a number of studies have
replicated Gerstmann’s findings, with patients with parietal lesions (Mayer et al., 1999). In addition, two
magnetic stimulation studies with healthy adults parallel findings from patient studies, showing that stimulation
of angular gyrus leads to disruptions both in number processing and finger gnosis (Roux, Boetto, Sacko,
Chollet, & Tremoulet, 2003; Rusconi et al., 2005).
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 (Rusconi, Pinel, Dehaene, & Kleinschmidt,
2010). However, the position against Gerstmann’s syndrome can be considered a case of argument from
ignorance. The lack of obvious sub-functions that underlie the four affected abilities, might be partially due to
our lack of understanding of the neural mechanisms for these abilities. One potential explanation, at least to
explain the relationship between finger gnosis and number processing, is that number processing uses finger
schemas. Dehaene et al. (2003) originally dismissed this explanation, proposing that “… the syndrome may
represent a happenstance conjunction of distinct, but dissociable, deficits that frequently co-occur due to a
common vascularisation, and that are only loosely connected at the functional level due to the overarching
spatial and sensorimotor functions of the parietal lobe” (p. 493). According to this localizationist approach, there
is no functional (casual) relation between finger and number representations, and the neuropsychological
findings are due to large lesions affecting neural correlates of both finger and number representations, which
are in close proximity in the parietal lobe.
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 (Sato et al., 2007). In another study, enumeration of dots on
the screen both with numbers and letters were found to increase the corticospinal excitability of hand muscles
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(Andres, Seron & Olivier, 2007). Because the effect was found both for enumeration with numbers and letters,
the authors proposed that finger circuits are involved whenever a set of items have to be matched with the
elements of an ordered series. In a dual-task study Michaux et al. (2013) showed that finger movements
interfered more with arithmetic, compared to feet movements, and that the interference of finger movements on
addition and subtraction was more than it was on multiplication, with adult participants. They proposed that the
finger counting-based strategies used in childhood for addition and subtraction, but not for multiplication, leads
to an early grounding of addition and subtraction processes in finger representations. Soylu and Newman
(2016) followed up on these findings and studied the interaction between arithmetic difficulty and tapping
complexity in a dual-task fMRI study. They reported differential interference of finger movements on single-digit
addition, compared to double-digit addition, and traced this effect to the modulation of activity in the left angular
gyrus. Additionally, Berteletti and Booth (2015) investigated activation of finger somatosensory and motor areas
during single-digit subtraction and multiplication tasks, with children (8 to 13 years of age). They found
significant activations in motor areas only for subtraction and not multiplication, suggesting reliance on finger
counting strategies. They also reported greater somatosensory activation for subtraction problems with larger
operands, indicating greater reliance on finger representations. Together these studies provide further support
for a functional relation between fingers and number processing, possibly due to a developmental association
that can be traced back to early finger counting experiences.
Bridging Mathematics Education and Neurocognitive Research:
A Roadmap for Future Research
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.
The Relation Between Finger Gnosis, Perceptual Units and Counting Types
Earlier in this paper we detailed how Steffe, von Glasersfeld and colleagues (von Glasersfeld, 1981; Steffe et
al., 1988, 1983; Steffe & von Glasersfeld, 1985) provided an account for how the fundamental perceptual ability
to perceive discrete “things,” by extracting aspects of the sensorimotor experience, lays out the foundation for
understanding whole-part relations, and eventually for different forms of counting (i.e., motoric, figural,
abstract). Even though there is accumulating evidence for a relation between finger gnosis and numerical
ability, what the origin of this relation is, and how far we can trace it back in the developmental process is not
clear. Steffe, von Glasersfeld and colleagues’ work point us in one direction. If extracting discrete unitary items
from the fuzzy stream of sensorimotor experience, and applying these heuristics to construct an understanding
of plurality and numerosity lead to development of number sense, then early sensorimotor experiences with
fingers might have a crucial role for development of numerical skills. Von Glasersfeld (1981) defined the initial
step of perceiving “a thing” as extracting multimodal features from a background and integrating these features:
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“We do divide our visual, auditory, and tactual fields of experience into separate parts which, in our cognitive
organization, then become individual items or ‘things.’” (p. 86). Each finger is a distinct sensorimotor item. One
of the first, if not the first, set of discrete items infants encounter are their fingers. At birth the multimodal
networks that allow coordination of sensory (tactile), motor, and visual modalities are not fully developed yet.
These reentrant networks (Edelman, 1987; Smith & Gasser, 2005) develop during diverse multimodal bodily
experiences. Piaget (1954) provided a vivid example of how this happens with an eight month old infant:
“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 (Newman, 2016; Noël, 2005; Reeve &
Humberstone, 2011) and fine motor skills (Luo, Jose, Huntsinger, & Pigott, 2007) correlate with and predict
some numerical abilities, starting with kindergarten age, the ontogeny of the these relations is not clear. If the
finger and number relation goes back to early and pre-verbal stages of development, then there is need for
longitudinal studies tracing how early developments in the sensorimotor system set the stage for later
numerical competencies.
What Can Finger Counting Habits Tell Us?
In several lab studies finger counting habits (the way one counts on her fingers from one to ten) showed
significant effects in performance measures. Newman and Soylu (2014) reported higher addition performance
for both adults and children (all right-handed) who were right starters (start counting with their right hands;
numbers one to five are matched with fingers on the right hand, whereas numbers six to ten match with the
left), compared to left-starters. Additionally, right-starter adults had significantly higher digit-span scores. The
researchers explained the results in terms of hemispherical lateralization differences between right- and left-
starters: while right-starters had a more left-dominant number processing network (which is the case for most
right-handed individuals), left-starters were hypothesized to have a more bilateral number representation, which
increases reliance on communication across the two hemispheres, thereby decreasing performance. In an
earlier study, Tschentscher et al. (2012) provided supporting evidence for this explanation. In an fMRI study
with adult participants, they found that in a categorization task (numbers vs. meaningless words or symbols),
both number words and Arabic numerals (from one to nine) activated the finger motor network contralateral to
the hand, matching with the finger counting configuration (e.g., right-starters activated the left motor areas,
whereas left-starters activated right motor areas for numbers from one to five).
Domahs et al. (2010) compared the performance of deaf German signers, and non-deaf German and Chinese
adults in a number comparison task using Arabic numerals. The German and Chinese groups differed in their
finger counting habits in that while German participants (both impaired and hearing) used one hand for
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numbers smaller than six, and two hands for number larger than five, the Chinese participants used only one
hand to sign all numbers smaller than and equal to 10. Both German groups (impaired and hearing) showed a
decrease in performance in the comparison task, not predicted by the magnitude of the numbers alone, when
at least one of the numbers required use of both hands for signing. This effect was not observed for Chinese
participants. This study supports the idea that, independent of whether number signing is still actively used
(which is the case for hearing-impaired German signers but not for the hearing German group), finger counting
habits modulate performance.
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. Domahs, Krinzinger, and Willmes (2008)
found above chance level split-five errors with first-grade children during addition and subtraction problems
involving two double-digit numbers. Split-five errors are characterized by errors deviating by exactly ±5 from the
correct result (e.g., 33 – 17 = 11). This result implies that children’s mental representations for large numbers
inherit the sub-base-five property of their hands.
What can we learn from finger counting habits for populations with mathematics learning disabilities and other
cognitive disorders? So far studies on how finger counting habits interact with mathematics performance have
mostly focused on children and adults without diagnosed disabilities. As we discussed earlier, children with
mathematics learning difficulties have a harder time transitioning from finger counting to verbal counting and
retrieval strategies (Bryant, 2005; Geary, 2004; Jordan & Hanich, 2000; Jordan et al., 2008), however, to date
there are no studies investigating patterns of specific finger counting habits (e.g., right vs. left starter, symmetric
vs. asymmetric configurations, anatomical vs. non-anatomical ordering) in children with mathematics learning
disabilities, or other cognitive disorders that can impact mathematics learning and performance. If there are
specific patterns, these can help diagnose and characterize different mathematics learning disabilities, and help
design new interventions.
For example, patients diagnosed with autism (Gunter, Ghaziuddin, & Ellis, 2002), ADHD (Roessner,
Banaschewski, Uebel, Becker, & Rothenberger, 2004), dyslexia (von Plessen et al., 2002), and developmental
language disorder (Herbert et al., 2004) were found to have abnormalities in the size and function of their
corpus callosum (the sets of nerve fibers connecting the left and right hemispheres; see van der Knaap & van
der Ham, 2011 for a review) and lateralization patterns. It is possible that the abnormalities with corpus
callosum, which affect interhemispheric communication, lead to differences in finger counting habits as well. If
so, these can be of diagnostic value.
Development of Finger Skills in the Preschool Period
Findings from disparate studies show that, in addition to finger counting, fine motor skills (Luo et al., 2007), as
well as finger gnosis (Noël, 2005; Reeve & Humberstone, 2011; Wasner et al., 2016) predict various
mathematical competencies. We don’t know enough about how these skills co-develop. We need longitudinal
studies to investigate how finger gnosis, finger counting, fine motor abilities and mathematical skills co-develop
and change between the ages of 4 and 6, when children transition from mere counting to arithmetic with finger
counting, and eventually to fact retrieval and complex arithmetic calculations. Such studies can provide us with
new insights on how the development of finger skills is linked to the development of numerical competency, and
the developmental origins of mathematics learning disabilities. Insights gained can also shift our perspective
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about reasons for extended reliance on finger counting in children with mathematics disabilities. It is possible
that the early developmental trajectory of finger skills signal future problems with mathematics development.
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 (Tandon, Zhou, Lozano, & Christakis, 2011). Screen time correlates with obesity, attentional and
behavioral problems, and low school performance (Laurson et al., 2008; Swing, Gentile, Anderson, & Walsh,
2010). Apart from the more general effects of screen time, an unexplored area is how opportunities to develop
manual dexterity and finger skills during play are being replaced with passive viewing (e.g., TV) and alternative
modes of interaction with a wide range of devices (e.g., smartphones, game consoles). Given that clicking,
scrolling and dragging can hardly match the richness of sensorimotor experiences that other forms of
interaction with the physical world can afford, there is desperate need for research that can provide a critical
evaluation of how interaction with a wide range of technologies affect finger gnosis and fine motor skills, and
how differences in the development of these skills affect early numerical learning.
The research on the relation between fingers and numerical development can also guide development of new
learning technologies. For example, Touch Counts is one such instructional program that harnesses
affordances of new modes of interaction, touchscreen in this case, to provide learning experiences aligned with
research on embodied cognition and the relevance of finger-based interactions for numerical development
(Sinclair & Heyd-Metzuyanim, 2014). Touch Counts aims to support number sense by allowing enhanced
engagement with number concepts through rich finger-based interactions. For example to improve counting
skills learners can touch to the screen with multiple fingers to create numbered discs, one for each finger. As
long as the learner’s fingers stay on the screen the disc stays attached to it, when let go the discs fall to the
bottom of the screen, due to the virtual gravity. Every time one or more fingers touch to the screen a matching
number of discs are created (added on top of the existing ones), allowing representation of numbers more than
10. Touch Counts also targets associating multiple representations for numbers; finger-based, symbolic, non-
symbolic, and auditory, by providing multi-modal interactions that involve these representations. Further
interpretation of research findings for learning design efforts can lead to new instructional programs, software,
and interventions to facilitate numerical development.
The Impact of the Home Environment
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., Niklas &
Schneider, 2013) and mathematical skills (Anders et al., 2012; Melhuish et al., 2008). However, little is known
about how home environment affects use of fingers during numerical development. Just as early exposure to
literacy is important to the development of phonology and then reading skills, early exposure to numeracy is
important to mathematical development, including the development of the neural pathways that support
mathematical achievement. Part of this early exposure involves such experiences as observing a parent
pointing to a quantity with finger gestures, or imitating finger counting by vocalizing number words, initially in
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arbitrary ways; until realizing that each vocalization matches with the movement of a single finger. The amount
of exposure to experiences like these are likely to vary across different home environments and further
research is needed to explore the impact of early finger-based numerical experiences on development of
mathematical skills.
Finger-Based Interventions to Improve Mathematical Performance
In the reading literature there are several studies that examine the impact of phonological training on reading
skill (Temple et al., 2003; Wagner, Torgesen, & Rashotte, 1994). For instance, Keller and Just (2009) examined
a group of, non-dyslexic, poor readers who underwent 100 hours of remediation focused on word-level
decoding skills with a strong phonological skill focus. They found significant improvements pre- and post-
training in word recognition skills. Additionally, the brain structure itself was altered as a result of training. Post-
training, white matter tracts (denoting connectivity across primary regions for reading) in poor readers changed
to look more similar to that of good readers. The many phonological training studies show that training on a
component process that is fundamental to reading can improve reading performance. A similar approach can
be applied to the relation between finger skills and numerical performance. Training on finger processing (i.e.,
finger gnosis, fine motor ability, finger counting) might improve numerical and mathematical competency in a
similar way to how training on phonology improves reading performance.
There are a few studies examining the impact of finger training on arithmetic performance. Gracia-Bafalluy and
Noël (2008) performed a finger training study with first graders (6-7 year olds) and found improvements both in
finger gnosis and measures of numerical competence after training. However, these results should be
considered with caution. Fischer (2010) pointed out that the results Gracia-Bafalluy and Noël (2008) reported
might be, at least partially, due to simple regression toward the mean. Additionally, Zafranas (2004) reported
improvements in hand movements, spatial memory, and arithmetic performance in a study of piano keyboard
training. While it is not clear whether the music training or the keyboard/finger training led to better arithmetic
performance, the results are consistent with the hypothesis that finger training is linked to arithmetic
performance. It is possible that among other factors (Anvari, Trainor, Woodside, & Levy, 2002; Vaughn, 2000),
development of finger skills (both tactile and motor) mediate the benefits observed from music training on
mathematical skills. No research so far explicitly focused on how music training affects mathematics
development and performance through development of finger skills, and this is a topic ripe for research, with
important implications for music and mathematics curricula at schools.
Conclusion
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
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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 (Maravita & Iriki, 2004) to gesture based communication (Arbib, 2005). In
addition, our tactile experiences, which also involve motor and visual modalities, provide rich information (e.g.,
temperature, texture, hardness) about different characteristics of the objects and other living things around us.
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 (Moeller et al., 2012; Wasner, Moeller,
Fischer, & Nuerk, 2015).
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 (von
Glasersfeld, 1981; Steffe et al., 1988, 1983; Steffe & von Glasersfeld, 1985), showed how number concepts are
constructed as a result of bodily interactions with the world, and how these concepts go through
representational transitions through development. Interactions with our hands and fingers set the foundation for
concept formation, understanding of one-to-one correspondence, and quantity. Fingers also act as figural
representations for quantity, and are used both for number gestures, and for finger counting. Mathematics
education also provides us with accounts of how finger counting strategies for arithmetic evolve over time,
eventually to be replaced by more memory- and rule-based mental calculation strategies. For the most part,
this body of research does not explain how the brain systems for number and finger processing change through
development, and how they relate to each other. It also does not focus extensively on how fingers relate to
number processing in adulthood, likely because the relation between fingers (and the body in general) and
mathematics is assumed to become less relevant in adulthood, due to the development of more abstract
representations and calculation strategies.
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
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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 (Treiman & Zukowski,
1991) and deficits in phonological awareness are typical in dyslexia (Temple et al., 2003). Additionally, while
congenitally hearing-impaired individuals (who are unaware of the speech sounds) do learn to read, it is often
laborious and many have poor reading skills (Holt, 1994). Analogously, early interactions with the world may
help the emergence of networks in the brain linking visuospatial, tactile and motor modalities for hands and
fingers and, later, figural use of fingers to represent and process quantitates through number gestures. In short,
finger skills may be a doorway to mathematical competence in the same way that phonology is for reading. The
dependency of numerical development on finger skills, and the sustained involvement of the finger
sensorimotor system in mathematical cognition, extending into adulthood, paints a different picture for the
finger and mathematics relation than is often assumed. Taken together, mathematics education and
neurocognitive research paint a more comprehensive picture of the finger and mathematics relation, and the
embodiment of mathematics in general.
Funding
The authors have no funding to report.
Competing Interests
The authors have declared that no competing interests exist.
Acknowledgments
The authors have no support to report.
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