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Basic numerical skills are fundamental to daily living in modern society and support achievement in school mathematics. From very early on, infants can represent numerical quantities in their mind and discriminate between different quantities. Later on, children learn to count arrays of objects and to carry out arithmetic calculations. Children with Down syndrome (DS) often experience difficulties in these activities. They can have major difficulties in counting (e.g. Nye et al., 2001) and discriminating numerosities, particularly for quantities within the subitizing range (e.g. Sella et al., 2013). In this chapter, we first describe the acquisition of basic number skills in typically-developing children, then move on to numerical skills in DS. Given the reported contradictory findings, we also discuss how study designs and methods can influence the results in this setting. Bearing in mind how important math skills are in everyday life, we believe that fostering these abilities in children with DS should become a major goal. The last part of the chapter is consequently dedicated to a review of the available literature on intervention tools, and to the description of two single cases analyzed in our lab.

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.

It has long been known that the diameter of human pupil enlarges with increasing effort during the execution of a task. This has been observed not only for purely mechanical effort but also for mental effort, as for example the computation of arithmetic problems with different levels of difficulty. Here we show that pupil dilation reflects changes in visuospatial awareness induced by attentional load during multi-tasking. In the single-task condition, participants had to report the position of lateralized, briefly presented, masked visual targets ("right", "left", or "both" sides). In the multitasking conditions, participants also performed additional tasks, either visual or auditory, to increase the attentional load. Sensory stimulation was kept constant across all conditions to rule out the influence of low-level factors. Results show that event-related pupil dilation strikingly increased with task demands, mirroring a concurrent decrease in visuospatial awareness. Importantly, pupil dilation significantly differed between two dual-task conditions that required to process the same number of stimuli but yielded differed levels of accuracy (difficulty). In contrast, pupil dilation did not differ between two conditions which were equally challenging but differed both in the modality of the dual task (auditory vs. visual) and in the number of stimuli to be attended. We conclude that pupil dilation genuinely reflects the top-down allocation of supramodal attentional resources.

Publisher Summary The distinction between links based on short-term memory associations (STM-based links) and links based on long-term memory associations (LTM-based links) is a very important feature that deserves further elaboration. One primary source for spatially corresponding associations between stimulus location and response location are the oculomotor responses to lateralized visual or auditory stimulations and the reaching/grasping manual responses to objects that are present in the environment. Instances of spatially corresponding associations outnumber instances of spatially noncorresponding associations. In the Barber and O’Leary's activational model, there is the explicit distinction between LTM links and STM links. This distinction will be likely to prove very useful in modeling stimulus–response compatibility (SRC) phenomena. Most stimulus–response (S–R) tasks require arbitrary associations, which are set up according to experimental instructions. In contrast, LTM links are pre-existing links. One such link would be that between the spatial attributes of the stimulus and the (spatially organized) response domain that is involved in executing key-press responses to the stimuli.