The numerical cognition literature offers two views to explain numerical and arithmetical development. The unique-representation view considers the approximate number system (ANS) to represent the magnitude of both symbolic and non-symbolic numbers and to be the basis of numerical learning. In contrast, the dual-representation view suggests that symbolic and non-symbolic skills rely on different magnitude representations and that it is the ability to build an exact representation of symbolic numbers that underlies math learning. Support for these hypotheses has come mainly from correlative studies with inconsistent results. In this study, we developed two training programs aiming at enhancing the magnitude processing of either non-symbolic numbers or symbolic numbers and compared their effects on arithmetic skills. Fifty-six preschoolers were randomly assigned to one of three 10-session-training conditions: (1) non-symbolic training (2) symbolic training and (3) control training working on story understanding. Both numerical training conditions were significantly more efficient than the control condition in improving magnitude processing. Moreover, symbolic training led to a significantly larger improvement in arithmetic than did non-symbolic training and the control condition. These results support the dual-representation view.

A large number of studies have pointed out the role of working memory throughout numerical development. Working memory capacities seem to be improved after training and some studies have observed an impact of working memory training on academic performance. In our study, we examined whether training visuo-spatial working memory (with Cogmed) enhances working memory abilities and numerical development in the short and middle term in 5-6 year-old children. Fourty six children were randomly assigned to the experimental condition (adaptive working memory training) or the control condition (non-adaptive, demo version). The program was implemented daily for a period of five weeks in both groups. We observed an immediate impact of the adaptive version on visuo-spatial sketchpad and visuo-spatial central executive abilities and a small impact on Arabic number comparison. No training effect was observed in verbal working memory, in counting, collection comparison and addition. Furthermore, the observed effects were not sustained ten weeks later. These results are discussed in the context of specific and general cognitive factors that support numerical development and we argue against the idea of developing general cognitive factors to efficiently boost numerical development.

Executive functions and social cognition competences are associated with many important areas of life, such as school readiness, academic success or sociability. Numerous intervention programs aiming to improve these capacities have emerged and have been shown to be effective. As inhibition in particular, is closely related with social cognition competences, we developed a training program that targets both abilities and implemented it in kindergarten and lower primary school classes for 6 months. We evaluated its effectiveness at improving inhibition and social cognition as well as its possible impact on academic performance. The results showed that tackling inhibition and social cognition in the classroom at an early age improved inhibition, visual attention and flexibility as well as Theory of Mind and social information processing skills. However, the impact on academic learning was weak; a slight effect on a mathematical task was observed.

Working memory capacities are associated with mathematical development. Many studies have tried to improve working memory abilities through training. Furthermore, the central executive has been shown to be the component of working memory, which is the most strongly related to numerical and arithmetical skills. Therefore, we developed a training program that specifically targeted the central executive to examine the possible impact on kindergarteners' working memory and numerical abilities. Thirty-four 5-6 year olds were randomly assigned to the experimental or the control conditions. Both groups received 16 30-minute sessions of their respective training program. The experimental group was trained with games tapping the central executive and the control group received a vocabulary-learning program. The results showed an immediate training effect on one central executive task and an effect of training appeared on another central executive task at follow-up, 6 months later. However, no impact was observed on numerical and arithmetical skills.

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