Mathematical performance of young Chinese children in Taiwan as compared to American children beginning as early as first grade has been well-documented. A similar study by Huntsinger, et al, 1995, demonstrated the same phenomenon in comparisons of Chinese-American young children with Euro-American children. Parents influence young children's developing competence by the experiences they provide and the activities they choose for their children, and cultural belief systems influence parents' ideas concerning the relative importance of activities. This study is a longitudinal comparison of 36 second-generation Chinese-American children and 40 Euro-American young children (from well-educated families in the suburban Chicago are) in their family environments, providing a synopsis of the data collected in 1995 when the children were in first and second grades. The data were collected in order to: (1) investigate whether the Chinese-American early mathematics advantage is maintained after children are exposed to formal mathematics teaching in first grade; (2) further identify differences in the early socialization practices of parents in the two cultural groups; and (3) examine relationships between mathematics achievement, parental socialization practices, and perceptions of children's competence. A total of 36 second-generation Chinese-Americans from Chicago suburbs participated in this second phase of data collection, including 95 percent from the first phase (1993). Each child was given the Sequential Assessment of Mathematics Inventories (SAMI) and other assessments. According to the results, Chinese-American children continued to demonstrate higher mathematics performance in Phase 2. Parents provided different experiences for their young children, who reflected different parental values and appeared to foster different competencies. Contains 7 tables of data and 14 references. (BGC)