Many studies in the past decades have focused on low and typical mathematics achievers, yet little is known about children with high mathematics achievement, particularly at a young age. The current study aimed to fill this gap and started from the early work of Krutetskii (1976) as a theoretical lens to study the characteristics of high mathematics achievers in primary school. Krutetskii's work was extended with more recent research on mathematical cognition. A simultaneous investigation of mathematics-specific abilities and general motivational and cognitive factors allowed us to examine their unique contributions to high mathematics achievement. Participants were 162 children, that is, 81 high mathematics achievers and 81 average achievers (8- to 10-year-olds). Children completed measures assessing their mathematical cast of mind (attention to number), flexible mental calculation (adaptive number knowledge, strategy variety), and striving for elegance (use of varying strategies for multidigit arithmetic). We also measured children's need for cognition, spatial visualization ability, and working memory. There were significant group differences on all tasks, except for the attention to number task. A logistic regression analysis revealed that strategy variety, need for cognition, and spatial visualization ability were significant predictors of group membership. These data suggest that strategy variety, need for cognition, and spatial visualization ability might be critical characteristics of high mathematics achievers in primary school. The identification of such characteristics might be an important first step in creating supportive educational environments for these children.