Mathematics teaching strategies have a positive impact on learning. However, there is a lack of studies on non-traditional approaches to early mathematics education in the specialized scientific literature. In this theoretical framework, a study to connect teaching methodology with the various cognitive processes implicated in learning has been designed. A total of 114 primary school students aged 74 and 84 months who were taught mathematics either with the method called Open Algorithm Based on Numbers or with the more traditional Closed Algorithm Based on Ciphers, participated in the study. After conducting a thorough examination of cognitive processes and early math performance using well-established assessment instruments, a comparative analysis was undertaken to explore the relationship between cognitive predictors of mathematical performance, while considering the mathematics teaching strategies used. Students were distributed according to their level of mathematical competence and teaching methodology and the type of schools (Charter or Public). The results from the multivariate statistical test showed that the teaching strategy was inconclusive for most of the cognitive factors studied. Significant differences according to mathematical performance were found for fluid intelligence, verbal short-term memory, and visuospatial working memory. Finally, no significant differences were found in the cognitive variables studied when considering the interaction between the teaching approach, school characteristics, and mathematical achievement as a reference.