| The purpose of this study was to examine the relationship among preschool children's numerosity estimation abilities, counting abilities, and visual-spatial cognitive abilities, based on the domestic and overseas studies on children's development of numerosity, and established in the developing relationship between preschool children's numerosity estimation abilities and other mathematical abilities.A sample of 308 children aged 3-6 (male and female in half respectively) was randomly selected from 434 children in 17 classes in three ordinary preschools from two districts in shanghai, whose numerosity estimation abilities, counting abilities and visual-spatial cognitive abilities were tested. The test tools for children's numerosity estimation abilities were adapted from the measure in previous study (Zhao, 2006). The test tools for children's visual-spatial cognitive abilities were the Blocks Pattern in the Wechsler Preschool and Primary Scale of Intelligence adapted for Chinese, and the Picture Drawing from the previous research (Dennis, 1992; Case, et al., 1996). Two months later after the pretest, 104 children were divided into two experimental groups and one controlled group, who were in junior class of preschool in pretest and now in middle class of preschool. There were about 34 children in each group (male and female in half respectively). The experimental groups were consisted of the counting ability intervention group and the visual-spatial ability intervention group in which the children had been trained through the counting activity and building blocks activity respectively for three months (once a week). Three months later, 94 children were tested about their numerosity estimation abilities, counting abilities and visual-spatial cognitive abilities again.The study examined the development of young children's numerosity estimation abilities, counting abilities and visual-spatial cognitive abilities; the development of their numerosity estimation abilities under the different circumstance of the estimation; the relationship among young children's numerosity estimation abilities, counting abilities and visual-spatial cognitive abilities. The result indicated that the preschool children's numerosity estimation abilities were in tendency of underestimation, with the frequency of the underestimation being most more than that of the overestimation; that there were significant age effect on three cognitive abilities, but no gender effect, with the difference between middle class and junior class being greater than the difference between the middle class and senior class, this implied that the developing speed of the three cognitive abilities was very faster for the children aged between 4 and 5 years; that their numerosity estimating levels were influenced by the size of the target numerosity and the circumstance of estimating, with the children's performances on numerosity estimation being more better with the reference than without the reference, and also better in regular organization than in random organization. Children's performances on estimation and the precision were all increasingly decreased with the target numerosity being increased; that the significant positive correlation emerged among young children's performance on numerosity estimation ability test, counting ability test, and visual-spatial cognitive ability test, it showed that there were close relationships among the three cognitive abilities for preschool children. The results of the intervene for children's counting ability and visual-spatial cognitive ability showed the effect of the counting ability intervened or the visual-spatial ability intervened on the numerosity estimation abilities of young children aged 4-5, it implied that the counting ability and the visual-spatial cognitive ability were all most significant factors for the development of the preschool children's numerosity estimation abilities. The implication of the results was also discussed.
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