Numbers, space, and spatial attention: Operational momentum in non-symbolic and symbolic mental arithmetic, and number-line estimation in preschool children

Mental arithmetic and estimating number position on the number-line are proven predictors of success in mathematics education, and tap on the basic properties of numerical representations and processes: spatial organization and scaling. However, little is known about the interdependences between these abilities in the preschool period, when the child uses an approximate numerical representation and only acquires the symbolic system. We tested 119 preschoolers aged 3-7 with two non-symbolic arithmetic tasks (+/- 1 and +/- 5), symbolic +/- 1 arithmetic, and non-symbolic and symbolic number-line estimation in 1-9 range. Some form of operational momentum indicating the use of spatial processing appeared in all arithmetic tasks, partly replicating previous studies, however these effects were not correlated across tasks, and only the symbolic arithmetic correlated with number line estimation (nonsymbolic and symbolic). We hypothesize that mental arithmetic involves spatial processing, but scale and spatial arrangement are not inherent properties of number representation but taskspecific constructions. Educational relevance statement: In our research, we dealt with the performance of mental arithmetic and estimating the position of a number on a number-line in both non-symbolic (sets of dots) and symbolic (Arabic numerals) versions in children aged 3-7. We were more interested in specific biases than accuracy in completing tasks - the logarithmic scaling of the number line, and tendencies to overestimate/underestimate the result in addition/subtraction (so called "magnitude operational momentum") or linking addition with shifting attention to the right and subtraction to the left ("directional operational momentum"). Such biases, and the relation between them and the accuracy measures in the given task, may shed light on the most basic numerical representations and processes in children, and in particular the spatial properties of the mental representation of number and the use of "recycled" spatial attention processes for its processing. Understanding such basic representations and processes is crucial to planning early math education. Our results showed that in preschoolers, new symbolic representations of number link to more primitive, non-symbolic ones, and that mapping number to space to adapt spatial attention processes for number processing may be crucial for early numerical development.