I'm often in mentoring situations with mothers who are eager to raise godly children, and who struggle with how well they are accomplishing this task. The question of discipline comes up regularly and their concerns are whether they are being too strict or too lenient with their children. Over the years I've struggled with this myself and, like all other parenting issues, this is best dealt with by applying the Word of God to the issue of correction.
Everyone is born into the world with a sin nature. It is important to remember that those with a sin nature commit sins. This is where the law-word of God acts as a tutor for the child. He needs to be taught and reminded of the reality that he "naturally" is at war with God and needs a Savior so that he can come to peace with Him. Early on, children need to be taught that: sin is any want of conformity unto, or transgression of the law of God.
This sin nature can only be changed by an act of God through the redemption purchased by Jesus Christ and applied to an individual through the work of the Holy Spirit. We call this act of grace regeneration. However, those whose nature has been transformed through rebirth also commit sins. As parents, it is important to distinguish between those transgressions which are attempts to be obedient and miss the mark, and those which are deliberate actions carried out in defiance and/or rebellion.
When my children were young, I used the example of playing darts. The person who tries to hit the bullseye and misses and accidentally hits the dog is like the person who misses the mark. In the Greek, that sort of sin is hamartia. However, the person who is not aiming at the bullseye -- but is rather seeing if he can hit his dog is not missing the mark, but is aiming to miss. In the Greek, that sort of sin is anomia. In both cases, the dog has been hit, (sin has occurred), but the motive and circumstances dictate the appropriate correction.
The analogy can help you understand which category your child's transgression falls into. The first requires that the child take responsibility for his actions by administering first aid to the dog. It also involves a time of instruction, along with establishing certain rules and procedures when dealing with sharp objects, so something like this (or worse) doesn't happen again.
In the case of the child taking sport in hurting the animal, one is dealing with a sadistic act and discipline in the way of punishment is in order. Only after the child sees his act as sin, can repentance and instruction take root.
Of course, in order to adequately apply this principle, one must not be harsher with one's children than one is with oneself. In more colloquial terms, parents need to practice what they preach. They cannot hold their children to a higher standard than they hold themselves. Additionally, these everyday occurrences are appropriate opportunities to share the Gospel with your children as you identify where in Scripture a particular transgression is addressed (either explicitly or implicitly) and how through the blood of Jesus, one's sins are forgiven. I sometimes required an informal essay, in order to cement the point with my children.
Will your emotions or personal shortcomings sometimes contribute to you applying the incorrect punishment to a given situation? Sure. What then? Well, once you assess and confess your own sin to the Lord, it is the responsible and godly thing to make things right with your children asking them for their forgiveness if you were overly harsh OR overly lenient. That is why a systematic application of the law-word of God to every area of life and thought is so very important. Without such a standard, parents may be tempted to set up mini-dictatorships where their children have to learn how to appease the tyrant. The biblical model is to teach children the law-word of God and make sure they understand that parents are not exempt from faithful obedience. Together, these daily course corrections will be instrumental in preparing one's children for the time they will assume responsibility for themselves and their future families.