Childhood is very important moment in life, because what has been learned in this time actually can create a mind set of an adult. In this sense, it can be a crucial decision whether to encourage competitiveness or put more importance on cooperation in children.
It is good to have some sort of competitiveness to improve oneself and a society in general. If children were encouraged to compete each other, they might study harder to beat competitors and exercise more to win a sport game. This is good, because in this way the performance of all children will improve.
However, there is different option on this. Too much focus on competitiveness can lead to an individual success for a few but a failure as a whole. When children compete each other, they will not share information and will not support others who needs help. Assuming a society filled with adults with high sense of competition, it is easy to imagine a few winners take it all and many will be a loser and a society will end up unhappy in general.
Given these two different ideas, it is depend on us what kind of community we want to create in the future when our children become adults. It might not be a society winner takes it all. Personally, I have seen a lot of children who help and support other and cooperate well with and also highly productive, rather than trying to win over friends or co-workers. I believe what we need to teach to our children is cooperation more than competition.