Krajewski and Schneider (2009) propose that on the first level of development, children learn to recite the exact number word sequence and become skilled at counting. At this stage, many of them start to use their fingers by adopting the finger counting system of the respective cultural area, even without any specific instruction to do so. In the German finger counting system, which we will refer to as the working example throughout this article, the finger counting sequence starts with the thumb for 1 and then goes on with the index finger for 2, the middle finger for 3, the ring finger for 4, the pinkie for 5, restarting the same sequence at the thumb of the other hand for 6 and so on to 10 (see Figure 1; Level I). Thus, each number word is linked to one specific finger as it is the case in most finger counting systems of Western cultures (Bender and Beller, 2011, 2012; see below for a discussion on cultural differences in finger counting). Due to this link between fingers and numbers, the counting principle of one-to-one-correspondence is easily understandable (Brissaud, 1992). On a very basic level, even the acquisition of the number words themselves might be corroborated by making use of fingers, as the finger-number association may help perceive the number words as phonological discrete items (Beller and Bender, 2011) and contributes, as some kind of marker, to memorizing them (Brissaud, 1992; Fayol and Seron, 2005; De La Cruz et al., 2014). Additionally, the counting principle of stable order and the ordinal concept of numbers might be conveyed as well (Brissaud, 1992; Fayol and Seron, 2005; Crollen et al., 2011), because the involved motor sequence during finger counting (e.g., stretching out thumb, stretching out index finger, etc.) is just as stable as the number word sequence. This might help understand that for instance “ten”, which is assigned to the ultimate finger counted, comes after “nine”, which is associated with the penultimate finger counted.

On level II of the developmental model by Krajewski and Schneider (2009), children are suggested to become aware of the quantitative meaning conveyed by each number word. The acquisition of this so called cardinal number concept can also be supported by fingers (Brissaud, 1992). During finger counting, following the German finger-counting routine, digits are stretched out one after another whereas outstretched ones are not pulled in again (see Figure 1; Level II). This procedure allows for linking each number word to the corresponding quantity and for perceiving this quantity-number association both visually as well as through tactile and even proprioceptive sensations. By nature, finger counting is thus “cardinalized counting” (Brissaud, 1992) because quantities increase steadily (one by one) with every additional finger added during counting. This visualizes not only the respective cardinal values but also their progressive summation (see Figure 1; Level II). As this summation usually occurs in a specific spatial direction, it was suggested recently that finger counting might even modulate the spatial representation of magnitude, also known as the mental number line (Fischer, 2008). For instance, Pitt and Casasanto (2014) observed that just 15 min of training finger counting in leftward direction, this means (in palm up position) from the right thumb for 1 over the left pinkie for 6 to the left thumb for 10, extinguished and partially reversed the usual association of small numbers with left and larger numbers with right which was observed in Western participants (e.g., Dehaene et al., 1993). Taken together, these considerations indicate that fingers might not only contribute to the acquisition of ordinal counting but may also corroborate the understanding of cardinal quantity-number associations and the development of spatial-numerical associations (Fischer, 2008; Tschentscher et al., 2012).

On level III of their development model, Krajewski and Schneider (2009) argue that children start to learn that numbers not only convey quantities but also allow for describing relationships between quantities. This insight enables them to compose (e.g., 2 and 3 equals 5) and de-compose numbers (e.g., 5 is de-composable into 3 and 2) as well as to quantify the precise difference between two numbers (e.g., 3 is distinct from 5 by 2). Although these abilities already reflect initial calculations, they may nevertheless be corroborated by finger-based representations. Many children use their fingers when they start to calculate but not only, as often criticized, to keep track of items while counting up (e.g., Fuson, 1988) but also to visualize and combine the involved quantities as a whole (Siegler and Shrager, 1984). To combine, for example, 3 and 2 in this way, children might first stretch out thumb, index finger, and middle finger simultaneously; then add ring finger and pinkie, just to conclude finally that the result is 5, as all fingers of one hand are stretched out in the end. The other way round, finger-based strategies can also be used to visualize number decomposition. For this purpose, children might first display the initial quantity with their fingers (e.g., all fingers of one hand for 5) and then separate two subgroups of digits from each other (e.g., thumb, index, and middle finger for 3 vs. ring finger and pinkie for 2) by putting some digits closer together (see Figure 1; Level III; left side). In addition to compositions and decompositions, fingers might even exemplify differences between numbers. When children compare, for instance, the finger magnitudes 3 and 5 with each other, they can easily see them differ by 2, namely by ring finger and pinkie (see Figure 1; Level III; right side). This indicates that even number relationships, which are required for initial calculations, can be conveyed by finger-based numerical representations.

Taken together, these considerations strongly suggest that finger-based numerical representations are well-suited to corroborate the acquisition of basic numerical concepts such as counting as well as the understanding of cardinality and number relations.

While above considerations clearly argue for a specific role of finger-based representations in children’s numerical development, there also seem to be constraints and limitations to this account (e.g., Bender and Beller, 2011; Previtali et al., 2011 for discussions of this point).

A first point to consider is whether finger-based representations are a necessary step in children’s numerical development. This would be a very strong claim and hard to proof. Actually, Butterworth et al. (2011) observed that some indigene Australian cultures do not at all use their fingers in numerical contexts but are nevertheless able to perform simple calculations. On the other hand, Poeck (1964) reported the case of a girl born without forearms, who counted the fingers of her phantom hands to solve simple arithmetic problems. Additionally, Crollen et al. (2011) found that even blind children use their fingers to count and calculate, yet less often and systematically. Against this background, we are confident that finger-based numerical representations can corroborate the acquisition of basic numerical concepts, although we do not wish to claim that they are mandatory or even necessary to develop basic numerical concepts.

A second point to consider is whether above mentioned advantages of finger-based numerical representations may be generalizable across culturally differing finger counting routines. Between cultures, finger counting routines vary, for instance, (i) in the finger on which counting is started (i.e., thumb for 1 in Germany, index finger for 1 in the US, little finger for 1 in Iran); (ii) how the finger counting sequence continues from 6 to 10 (i.e., with the same finger sequence on the second hand in most Western cultures or with the first hand using finger symbolic gestures in China). In our opinion, it seems reasonable to assume that the acquisition of basic numerical concepts (i.e., counting, understanding of cardinality and number relations) may be corroborated best, when fingers and numbers are associated in 1-to-1-correspondence and a stable finger sequence is used. Thus, it should not matter which finger (on which hand) is used to begin and how the finger counting sequence continues because these two preconditions allow for both, associating (i) a specific finger with a specific number; and (ii) a specific finger pattern with a specific cardinality. In case of the Chinese finger counting system, for instance, this is only fulfilled for numbers up to 5. For numbers exceeding 5, which are represented symbolically (see above), the order of the counting sequence is still stable but there is no 1-to-1-correspondence between fingers and numbers. Therefore no finger is specifically associated with number 7, nor is the cardinality of 7 reflected in its associated finger pattern, which makes decompositions and compositions impossible (see also Domahs et al., 2012 for the case of the Korean finger counting system).

Finally, recent evidence indicates that finger counting habits vary not only across cultures but also on the individual level. Wasner et al. (2014a) observed, for instance, that whether German participants started counting on their left or right hand was influenced reliably by whether both of their hands were equally available. Additionally, Wasner et al. (2014b) found that finger patterns differed—at least for specific numbers (e.g., 4)—when participants were asked to either count to that number (i.e., by thumb, index, middle, and ring finger) or to show the respective number as a spontaneous finger pattern (i.e., index, middle, ring finger, and pinky). Irrespective of the fact that there tends to be some flexibility in adults’ finger counting habits, it seems reasonable that children benefit most, if they stick to a stable motor sequence when learning to count on fingers (for similar results on object counting see Kamawar et al., 2010). Importantly, this is corroborated by a recent study on cognitive robotics, in which De La Cruz et al. (2014) observed “that learning the number words in sequence along with [stable] finger configurations helps the fast building of the initial representation of number in the robot” and “the internal representations of the finger configurations themselves […] sustain the execution of basic arithmetic operations” (p. 1).

Nevertheless, it should be noted that all studies on the differences of finger-based representations across individuals as well as cultures were conducted primarily with adult participants. To the best of our knowledge, there is currently no study investigating the flexibility of children’s finger counting habits. Thus, further research is needed to clarify how children develop finger-counting habits and whether different finger counting routines impact on numerical development differentially.