If a child is "left to himself," where is his training coming from? Obviously, in this case, mom and dad are not having a great impact on their child. The training must then be coming from society, most likely from the child's peers. Because "foolishness is bound up in the heart of a child" (Proverbs 22:15), a child left to himself brings shame—he is bound to get into trouble if his training is haphazard or undirected, or if he is not drilled or disciplined. The flipside of this verse says, "But the rod of correction will drive it far from him." The rod symbolizes that someone has taken an interest in the outcome of this child's life. He is giving direction, correction, instruction, to steer this child where he is to go. The training, the teaching, makes all the difference in the world.
An example from the life of David illustrates this proverb. "And [Adonijah's] father had not rebuked him at any time by saying, 'Why have you done so?' He was also a very good-looking man. His mother had borne him after Absalom" (I Kings 1:6). David was very old and was very shortly to die. His family and his close advisors probably knew that he intended to pass his crown to Solomon. But Adonijah tried to prevent that. He made a political move to grab the throne before Solomon had a secure grip on it. His ploy failed because Bathsheba, Solomon's mother, and David's faithful advisors appealed to the king, and he made it very clear whom he had chosen to succeed him.
David, though a man after God's own heart, did not take an active hand in teaching Adonijah. In this verse God states why Adonijah rebelled. In essence, David actually encouraged his son to rebel by not taking an interest in rearing him. David failed to train him in the way he should go, so that he would not depart from it. Instead, David trained him in a way that was bound to produce rebellion. This flaw of David's shows up in others of his children: Absalom, Amnon, and others. It does not matter whether one is a child of God having His Spirit or not. If a parent does not carry through with the right kind of training, then the results will surface in his children.