Preparatory abilities acquired in kindergarten are found to be strong predictors for later proficient arithmetic abilities. A research line was designed to examine if we could predict the level of children's mathematical abilities in first and second grade from their performance on preparatory abilities in kindergarten. The results pointed out that good predictions of beginning mathematical abilities can be made. Even in the long term, it was possible to make fairly good predictions of later mathematical performances, especially for the domain of mental arithmetics and number knowledge. This new book discusses the theoretical background of the study.