Nonsymbolic Arithmetic in Adults and Young Children

Human infants and animals can process numerical quantities through approximate representations of their magnitude. Here we consider whether these nonsymbolic numerical representations might serve as a building block of uniquely human, learned mathematics by investigating whether adults and preschool children can perform simple arithmetic calculations on nonsymbolic numerosities. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Nonsymbolic numerical representations therefore are computationally functional and may provide a foundation for formal mathematics. Numerical abilities observed in animals, human infants, and human adults share many characteristics, including a signature ratio limit on discriminability (1-5). Several lines of evidence support the hypothesis that this preverbal “number sense,” or rough representation of numerical magnitude, provides a foundation for formal mathematics. First, many tasks that deal explicitly with exact symbolic numerosities automatically activate nonsymbolic number representations (6, 7). Second, human adults are able to perform rapid approximate arithmetic on Arabic digits, apparently utilizing a cerebral circuit distinct from that involved with the retrieval of exact numerical facts (8). Third, neurological patients with impairments to the nonverbal number sense show impaired symbolic calculation abilities (5). However, a key prediction of the hypothesis remains untested: if nonsymbolic representations of numerical quantity serve as the foundation for formal mathematics then it should be possible to perform arithmetic operations on numerical representations without engaging verbal/symbolic processes (9). Here we investigate whether human adults and preschool children can calculate using nonsymbolic quantity information. In the first study, adults performed numerical comparison and addition tasks on large sets of 10 to 70 elements, either within a single sensory modality or across two modalities. In the within-modality comparison task, participants judged which of two dot arrays contained more dots. In the across-modality comparison task, participants judged which of two heterogeneous displays (an array of dots and a sequence of tones) contained more elements. In the within-modality addition task, participants mentally added 2 dot arrays and compared the resulting sum to a third dot array (Fig. 1A). For the across-modality addition task, participants mentally added a dot array to a tone sequence and compared the sum to a third display of dots or tones. In all tasks, dot sizes and tone durations were varied in order to discourage reliance on continuous quantitative variables other than number; these tasks were also designed to preclude strategies based on sensitivity to the ranges of numerosities involved (10). Performance levels, as measured by both accuracy and reaction time, were above chance for each of these four conditions. Further, there was no performance cost for addition compared to comparison or for addition across sensory modalities compared to addition within the visual modality (Fig. 1B). This study suggests that adults can indeed perform addition on nonsymbolic number representations. Nonsymbolic addition resulted