In the short story "The Rocking-Horse Winner" by D.H. Lawrence, the author presents a boy who,to all intents and purposes, is like any other small boy. But soon the author shows us how Paul is under stress. He is suffering the terrible strain of his mother's expectations snf hid family's limitations in meeting them. He becomes unchildlike when he internalises responsibility for all this financial pressure and tries to take satiating his mother's ambition for money and materialism and social status on his own young vulnerable shoulders. He is childlike in his belief that he can make his mother love him more, and in his belief that he has some influence over what happens in the life of his parents. For example, some children believe that a divorce happens because of them.
Paul, like all small children, seeks for approval and love from his parents. So, he does things in order to please them. This is what prompts the rocking-horse riding in the first place. He senses his mother's "cold" heart, and in a conversation with her, learns that she wants to be "lucky," or rich, more than anything else. He figures that if he can get her money, then her heart will soften, and she will finally love him.
Paul is also like a small child in the way that he trusts people. He trusts the gardener and his uncle to work with him in winning these huge amounts of money; that is pretty trusting, if you ask me. An adult would expect scamming to occur, but Paul just innocently lets the gardener handle all of the money, and deal with it. Very child-like and naive.
One other way that Paul is like other children is that he doesn't really express greed or desire for a lot of money--his desire for the money is to win his mother's love, not to get things for himself. He is very selfless and childlike about money and what it means.
Despite these traits, Paul is really more serious than most children. He is calm, serious, earnest and mature for his age. He doesn't seem to play or take things lightly. Also, he is very focused and determined. Most kids can't focus on one thing for more than ten minutes. Paul spends hours on his horse, focused and determined. Kids also tend to give up pretty easily and get frustrated when things aren't going their way--not Paul. He rides and rides until he gets the right answer, even if it takes a very long time. He is also unduly burdened with the cares of the world; his parents' desire for money weighs on him, and he takes the responsibility on himself--that is not like a child at all.
I hope that those thoughts helped; good luck!