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Is it better to use punishment or rewards to change your child's behavior? I read a great New York Times article last summer about how the best choice is really something else entirely. The article is a bit long, but it has a lot of helpful information and gives specific ideas of things you should say instead of just what not to say.
What it boils down to is this: neither punishment nor rewards is the best way to discipline children. Punishment makes children afraid of you, fight you, or try to avoid the punishment instead of changing their behavior to something more positive, and rewards make children behave in certain ways for an external reward (which has to keep increasing to keep them interested) instead of choosing to behave in positive ways because of how it makes them feel and because of how it helps their relationships with others.
What we want instead are understanding, consequences, and motivation.
First, try to figure out why your child might be behaving in a certain way, and empathize with him. "You are so mad that we have to leave the playground. This is really hard." When your child is upset, it is not helpful or effective to try to reason with him. Just validate and acknowledge the feelings. But you can simply (just once, no need to hammer the point home) state the reason you are implementing a consequence, "You were running away from me in the parking lot, and that's not safe. So I am going to pick you up and carry you." Whenever you can, set your child up for success by not asking him to do something he cannot do yet, like not eating a cookie you have left on the table. Children, especially under 4, simply cannot do that; they don't have the impulse control yet. And once you know why your child is doing something, you can try to meet the need he has before he acts out or avoid those situations altogether.
Consequences for behavior, ideally, should be immediate, logical, and temporary. "If you throw sand in the sand box, you are not being safe, so you will have to leave the sandbox. You can play somewhere else, and in a few minutes, you can try again." When possible, give your child awarning of the consequences (without anticipating/expecting that he is going to misbehave: "We do not throw sand. If you throw the sand again, I will have to take you out of the sandbox." There are also natural consequences, such as having cold hands because you refuse to wear mittens. It's okay to let your child be a bit uncomfortable or unhappy as a result of a choice she has made. (As long as she is safe, of course.) If she won't eat the food you have given her for lunch, then she can get down and play. When she is hungry, she will let you know, and you can offer her the same food, in her chair. If she won't eat it there, then she won't eat until the next meal is offered. Children do not starve/freeze themselves unless you engage in a power struggle with them, in which case, they will take the opportunity to exert control.
To motivate your child without stickers or candy, try to give him the positive reinforcement ofyour attention and your positive regard, as well as pointing out how his positive behavior makes other feel good. Children are hard-wired to want to be connected to other people, and we want them to be motivated by those connections to behave in positive, pro-social ways. "Thank you for sharing your cookie with me. That makes me so happy!" or "You gave Evie a turn with the truck. Look how happy she is!" (We want to be careful, though, not to over-celebrate when a child does what is expected of him and what he can do easily. A 4-year-old does not get a party every time he cleans up his toys.) You can also help your child to notice how good he feels when he is kind or helpful to others. "That was so hard, waiting to talk to me until I was off the phone. But doesn't it feel good to know you can do that?" You can also make your child feel like part of a team--your family team--by letting him know his contributions are important to your family.
One last tip: show your child you expect her to do the right thing by using "When, then" instead of "If, then." For example, "When you get settled in your car seat, I will give you your book." This also avoids threatening consequences that you will not actually follow through on. Many, many parents tell their child who is having trouble leaving that they will leave her there. But that's not true. (Hopefully!!) So your child is either terrified that you will abandon her somewhere, or she learns that you don't mean what you say. Try just expecting that she's going to go with you and give her choices that are all acceptable to you. "Ok, time to go. Do you want me to carry you, or are you going to walk?"
This is hard, especially if it goes against the way you were raised, or the way some members of your family or community expect you to discipline your child. But keep at it--be consistent--and it will work!