No abstract

Children's sense of numbers before formal education is thought to rely on an approximate number system based on logarithmically compressed analog magnitudes that increases in resolution throughout childhood. School-age children performing a numerical estimation task have been shown to increasingly rely on a formally appropriate, linear representation and decrease their use of an intuitive, logarithmic one. We investigated the development of numerical estimation in a younger population (3.5- to 6.5-year-olds) using 0-100 and 2 novel sets of 1-10 and 1-20 number lines. Children's estimates shifted from logarithmic to linear in the small number range, whereas they became more accurate but increasingly logarithmic on the larger interval. Estimation accuracy was correlated with knowledge of Arabic numerals and numerical order. These results suggest that the development of numerical estimation is built on a logarithmic coding of numbers--the hallmark of the approximate number system--and is subsequently shaped by the acquisition of cultural practices with numbers.

Greater skill in solving single-digit multiplication problems requires a progressive shift from a reliance on numerical to verbal mechanisms over development. Children with math learning disability (MD), however, are thought to suffer from a specific impairment in numerical mechanisms. Here we tested the hypothesis that this impairment might prevent MD children from transitioning towards verbal mechanisms when solving single-digit multiplication problems. Brain activations during multiplication problems were compared in MD and typically developing (TD) children (3rd to 7th graders) in numerical and verbal regions which were individuated by independent localizer tasks. We used small (e.g. 2 × 3) and large (e.g. 7 × 9) problems as these problems likely differ in their reliance on verbal versus numerical mechanisms. Results indicate that MD children have reduced activations in both the verbal (i.e. left inferior frontal gyrus and left middle temporal to superior temporal gyri) and the numerical (i.e. right superior parietal lobule including intra-parietal sulcus) regions suggesting that both mechanisms are impaired. Moreover, the only reliable activation observed for MD children was in the numerical region when solving small problems. This suggests that MD children could effectively engage numerical mechanisms only for the easier problems. Conversely, TD children showed a modulation of activation with problem size in the verbal regions. This suggests that TD children were effectively engaging verbal mechanisms for the easier problems. Moreover, TD children with better language skills were more effective at engaging verbal mechanisms. In conclusion, results suggest that the numerical and language related processes involved in solving multiplication problems are impaired in MD children.

The operational momentum (OM) effect describes a cognitive bias whereby we overestimate the results of mental addition problems while underestimating for subtraction. To test whether the OM emerges from psychophysical characteristics of the mental magnitude representation we measured two basic parameters (Weber fraction and numerical estimation accuracy) characterizing the mental magnitude representation and participants' performance in cross-notational addition and subtraction problems. Although participants were able to solve the cross-notational problems, they consistently chose relatively larger results in addition problems than in subtraction problems, thus replicating and extending previous results. Combining the above measures in a psychophysical model allowed us to partially predict the chosen results. Most crucially, however, we were not able to fully model the OM bias on the basis of these psychophysical parameters. Our results speak against the idea that the OM is due to basic characteristics of the mental magnitude representation. In turn, this might be interpreted as evidence for the assumption that the OM effect is better explained by attentional shifts along the mental magnitude representation during mental calculation.

A milestone in numerical development is the acquisition of counting principles which allow children to exactly determine the numerosity of a given set. Moreover, a canonical left-to-right spatial layout for representing numbers also emerges during preschool. These foundational aspects of numerical competence have been extensively studied, but there is sparse knowledge about the interplay between the acquisition of the cardinality principle and spatial mapping of numbers in early numerical development. The present study investigated how these skills concurrently develop before formal schooling. Preschool children were classified according to their performance in Give-a-Number and Number-to-position tasks. Experiment 1 revealed three qualitatively different groups: (i) children who did not master the cardinality principle and lacked any consistent spatial mapping for digits, (ii) children who mastered the cardinality principle and yet failed in spatial mapping, and (iii) children who mastered the cardinality principle and displayed consistent spatial mapping. This suggests that mastery of the cardinality principle does not entail the emergence of spatial mapping. Experiment 2 confirmed the presence of these three developmental stages and investigated their relation with a digit comparison task. Crucially, only children who displayed a consistent spatial mapping of numbers showed the ability to compare digits by numerical magnitude. A congruent (i.e., numerically ordered) positioning of numbers onto a visual line as well as the concept that moving rightwards (in Western cultures) conveys an increase in numerical magnitude mark the mastery of a spatial mapping principle. Children seem to rely on this spatial organization to achieve a full understanding of the magnitude relations between digits.