There are two definitions of childish: something pertaining to a child and lacking maturity. Adam demonstrates both of these in the opening chapter of April Morning by Howard Fast.
Adam acts like a child, particularly toward his father. He believes his father is unbending and impatient, unwilling to listen seriously to anything his son has to say. We see two incidents in which Adam does a childish thing that gets him in trouble. First, after his father scolds him, Adam turns away and mutters under his breath; of course, he is not as quiet as he thinks he is and his father hears him. Naturally Adam gets in trouble for this petulant and childish act. Second, he remembers a time when, as his father was punishing him, he taunted his father about being superstitious; naturally, he got his punishment (a whipping) more than doubled for talking back to his dad.
His relationship with his brothers is also quite childish; they bicker and argue over virtually nothing. Adam also has to be told to do his household chores, such as fetching water for his mother. In another sense, though, Adam is a typical child because he has thoughts about death and is too afraid to share them with anyone.
I thought of death and was full of fear, and I just wanted to sit down somewhere and put my face in my hands and give in to the terrible frightened feeling I had; but I didn't. I have all kinds of strange thoughts and feelings of that sort, and I guess I never talked to anyone about them, except perhaps a little to Granny, because I didn't really believe that anyone on the world ever had just the same kind of thoughts.
Adam also lacks maturity. He tries to demonstrate his maturity by swearing, but trying to get attention by using curse words is not a mature act. He calls his brother a "bastard" and uses "damned" in front of his grandmother. His mother's standards of swearing are little stricter, and she chides him for saying "good heavens." Clearly he is bent on using colorful words to get a reaction from the people around him.
His belief in curses is another sign of immaturity. As he brings up the water from the well, he even says the spell which will take the curse off of the water.
Adam foolishly, as an act of defiance, tells his grandmother that he is not interested in education or being a God-fearing man, two things for which the Cooper men have always been known.
Finally, Adam is certain that everyone is out to get him, or at the very least to keep him from doing the things he wants to do. This is an indication of immaturity.
In other ways, of course, Adam does demonstrate maturity. For example, he wants to be involved in the political action going on around him. No one lets him, at least not yet, but at least he is interested.