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You have, I would suggest, several different goals here--which will be more easily achieved if they are considered separately. The first goal is simply to prevent your 3 year old from hurting his 15 month old sister by shoving her or pushing her or intimidating her psychologically. Since both children are quite young, the only way to reliably prevent a 3 year old from hurting a 15 month old is to be RIGHT THERE supervising them, so that you can step in quickly and say, "Oh no, no! We don't pull hair! That would hurt, so we don't pull hair" in a calm, friendly manner. Getting between the children physically may be necessary to intervene at times. Here the goal is not to explain or teach or mold character. The goal is just to prevent the smaller child from being hurt. It might help you keep your cool at these times if you assume that the 3 year old really doesn't know better. It might be the case that a 3 year old, some of the time, actually does know better. But because he is only 3, his capacity for self-discipline is very limited--so when he is feeling cross or annoyed or tired or angry, "knowing better" is no use to him. A grown-up has to step in, to prevent injury.
The second goal, as you have suggested, is to help your 3 year old along the long pathway to self-knowledge, self-discipline, and self-expression. Over time (over a very long time!), children learn to observe their own emotions, to name them, and to express their feelings in words rather than in direct action with their arms and legs. When he is feeling hateful, a 3 year old may bite or hit; when he is more mature, he may say, "I hate you." This would be great progress, to be sure, but it isn't the full story here either.
The third goal is to reduce the amount of hate. This will not be accomplished by taking away your 3 year old's toys or putting him into time out--since these punishments only increase the amount of hate that he experiences from others and within himself. Reducing the amount of hate is a matter of helping your 3 year old experience less jealousy of the losses that he has experienced through your pregnancy, delivery, and raising of his little sister. Your own world may have been filled with joy at the arrival of a little daughter, but it is likely that your son's world was filled with confusion and sorrow--at least in part. Whereas he was once the center of your universe, he has been displaced from this paradise. He is now in time out, while you coo at his tiny rival. You cannot, of course, push back the clock to a time when he, alone, was the apple of your eye. All the same, trying to imagine how frustrated the 3 year old must often feel can help you make adjustments which can help counteract his sense of loss. Your expressions of love, gestures of devotion, and moments of intimacy with your son can help him feel less deserted and alone.
Helping your son recapture a sense of shared joy in his relationship with you will turn down the fuel of his hate, and--in addition--smooth the pathway to his identification with you as a loving, protective, sharing person. Like all small children, he wants at moments to be an indulged infant and at other moments to be a powerful grown-up, someone capable of indulging and taking care of others. Part of your son wants to cherish and protect his little sister, and you will be gladdened bit by bit as the growth of this grown-up, protecting person begins to express itself more powerfully in his personality.
The answers from our experts are for educational purposes only. Please always refer to your child's pediatrician and mental health expert for more in-depth advice.