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Numbers are of enormous significance for modern society. They form the basis of currency and economic systems, of measurement and calculation, of engineering and the natural sciences, and, as a matter of course, lie at the heart of mathematics. Therefore, teaching mathematical skills plays an important role in preschool and school education. However, before children are able to perform their first symbolic algorithms such as multi-digit addition and multiplication they must have mastered two representational number systems: a verbal system (e.g., English number words) and a notational system (e.g., Arabic digits).

The main competence that children have to acquire with these systems is to represent and handle the numerical information

Number words refer to the (theoretically) infinite set of natural numbers, the positive integers. This set defines a ratio dimension, providing

Number words are distributed representations in that they make available some of this information externally – ready to be picked up by auditory or visual processes – while other information needs to be retrieved from memory and is thus available only internally (Zhang and Norman,

Number words constitute a numerical system with distinct properties such as dimensionality and regularity (Bender and Beller,

The verbal number system, once acquired, is not only used to refer to cardinal numbers, but also for counting and calculating. The next section discusses how these processes might be supported or hindered by finger counting systems.

Counting is typically performed with recourse to the verbal number sequence. The acquisition of the counting routine takes children several years, but eventually enables them to start counting from any number, proceed forward and backward easily, and extract its numerical meaning (Fuson,

Relieving memory is even more important when it comes to calculation. Mathematical algorithms like those for multi-digit addition and multiplication are taught in the first school years. They are communicated verbally, but operate on the Arabic digits. Several of their sub-processes involve language (Dehaene,

For the numbers 1–10, a decimal system encompasses 55 basic additions and multiplications each; the highest sum is 20, the highest multiple 100. Our classic finger system with 10 as limiting number provides for only 25 additions and 15 multiplications. In order to fully cover the basic operations, the limiting number needs to be extended (for respective strategies see Guha,

Furthermore, finger counting systems are unsuitable for depicting

The acquisition of the verbal number sequence is an essential part of learning to count and an important prerequisite for mathematical education, often accompanied by finger counting strategies. Although finger counting competencies are not indispensable for the development of numerical abilities (Crollen et al., submitted), finger-training was shown to increase children's numerical performance, for instance, in quantification tasks (Gracia-Bafalluy and Noël,

Based on our analysis, however, we identified some factors that might also hinder the initial learning process: structural mismatches (e.g., in base and dimensionality) between the verbal and the finger counting sequence, the limited extent of finger counting, and too strong an association between the number words and specific objects (the fingers). While some of these problems are inherent in finger counting in general, others might be reduced by choosing an appropriate system carefully (e.g., a two-dimensional finger counting system for a two-dimensional verbal system). Such efforts appear to be worthwhile, as the prototypical finger counting systems come with at least one crucial advantage: they provide a visible and easy to manipulate set of “objects,” which helps to model and to internalize all the numerical information that is not externally represented in the arbitrary symbols of number words and digits.