This study was made to examine different point of views about young children`s counting. Mathematics education for young children has some difficulties. Numbers does not exist in the real world. There is no perceptable reference in the number concept. However young children need perceptable information to get knowledge for their developmental characteristics` sake. Those difficulties raise fundermantal questions how to teach mathematics effectively for young children. Piaget who thought mathematics as series of relationship brought up a new direction to teach mathematics, while traditional approach emphasies just skills, for example rote memory. Piaget argued that preschoolers are in pre-operational stage and generally they don`t understand basic number concept, like number conservation. Consequently, counling education has no meaning when children do not have number conservation concept. Piaget`s theory had great influences on practical early childhood education. Many of his followers have given more attention to classification, order, and one-to-one correspondence than to counting as prerequisite skills to number concept. However several recent studies disagree with Piaget`s theory. They argued that Piaget underestimated preschoolers` abilities who are in pre-operational stage. They also pointed out that Piaget`s conservation task has some problems. Gelman, Gallistel and other researchers argued that counting is a prerequisite to have number concept, while Piaget asserted counting is not a prerequisite. Gelman said that we should change our point of view from `children lack certain abilities` to `they have certain abilities`. According to Piaget, we should wait to teach mathematics until children have basic number concept, and we can`t teach preschoolers mathematics, because they are in preoperational stage. However Gelman`s study presented that we don`t have to consider such developmental stage to teach mathematics. In spite of Gelman`s hopeful explanations, generally preschoolers can`t do Piaget`s conservation task better than older children. In conclusion, we need more exact understanding about pre-operational stage, and only with that understanding, we can supply more proper mathematical environment to the preschoolers.