Hi! I wrote grammar corrections in red and stylistic suggestions in blue.

Should children be taught to compete or to cooperate?

"Some people think that a sense of competition in children should be encouraged. Others believe that children who are taught to cooperate rather than compete become more useful adults.
Discuss both these views and give your own opinion."

..........

One of the problems which raises concerns among parents as well as educationalists these days is the question of how best to teach children in the best way. Various statements have been made, and one claim says that children should be taught to compete with each other to encourage them to make every effort to study better. This proves to be a piece of reasonable advice.

The people who advocate the view mentioned earlier can justify their belief, and the main argument is that only with competitions can children try their best to study hard to catch up with their friends in school, due to the fact that these children often have a tendency to feel frustrated when they get lower scores than others. One more plausible explanation is that they will not be anxious or less confident when they have to face the big competitions in the future because they will have been familiar with them since they were in school.

There are, just the same, many who believe that in order to make children become the useful adults in the future, they should be taught to cooperate rather than compete. And they can give the convincing arguments somehow (this should be a seperate sentence. Perhaps you mean, "They also give convincing arguments."), their strongest point is that these children will tend not to ignore their friends when they are in the midst of difficulties, and furthermore will try to help them solve those problems. Or to put it another way, they will not leave their partners back in the competition. Another view worth taking into account is that the children who work in supportive a way, for definite (This phrase isn't nesessary, but if you wanted to keep it, I think you mean either "definitely" or "for sure"), usually have many good friends, and support each other to deal with any troubles and problems as well in their lives.

All things considered, the answer to the question of whether children should be taught to compete or cooperate depends on children 's (do you mean child's character or children's characters?) character. I personally think that children should be led to be cooperating people to contribute more to the society.

Nice paper! My main comment is that you use many words where fewer could be more effective. For instance, the phrase "as often as not" could be replaced with "usually". 

Should children be taught to compete or to cooperate?

"Some people think that a sense of competition in children should be encouraged. Others believe that children who are taught to cooperate rather than compete become more useful adults.
Discuss both these views and give your own opinion."

..........

One of the problems that raises concerns among parents as well as educationalists these days is the question of how to teach children in the best way. Various statements have been made, and one claim says that children should be taught to compete each other to encourage them to make every effort to study in a better way. This proves to be a piece of reasonable advice.

The people who advocate the view mentioned earlier can justify their belief and the main argument can be that only with competitions can children try their best to study hard to catch up with their friends in school, due to the fact that these children more often than not have a tendency to feel frustrated when they get lower scores than others. One more plausible explanation is that they will not be anxious or less confident when they have to face the big competition in the future because they have been familiar with them since they are in school.

On the other hand, many believe that in order to make children become the useful adults in the future, they should be taught to cooperate rather than compete. While they can give the convincing arguments somehow, their strongest point is that these children will tend not to ignore their friends when they are in the midst of difficulties and furthermore, try to help them solve those problems. To put it another way, they will not leave their partners back in the competition. Another view worth taking into account is that the children who work in supportive a way, definitely, as often as not do have many good friends so they may support each other to deal with any troubles and problems as well in their lives.

All things considered, the answer to the question as to whether children should be taught to compete or cooperate depends on children 's character. I personally think that children should be led to be cooperating people to contribute more to the society.

<em>[Very good.  The question you have to ask yourself is "Do I spend most of my adult life competing with others or cooperating with them?"  Children should be taught to be cooperative and to compete - your answer to that question tells you which one you should emphasize.]</em>