Some have interpreted this verse and others like it (e.g. Proverbs 13:24) as a justification--if not admonition--to use corporal punishment to discipline children. There is evidence in the scriptures that this is indeed a valid interpretation (such as Proverbs 23:13-14, which explicitly mentions beating children with a rod in order to save them from hell). I would, however, offer the following considerations when it comes to how followers of God should discipline our children.
First, it's important to remember that God uses language that His people will understand. At the time that the Proverbs were written, that included a reference to corporal punishment. Corporal punishment includes all methods of deliberately inflicting pain or discomfort in order to discourage bad behavior, though usually not with the intent of causing lasting injury. It was a very common form of correction for centuries. However, modern studies have shown that it is not an effective form of discipline and can actually cause more inappropriate behavior rather than discouraging it. (Exploring the negative effects of corporal punishment and their causes is beyond the scope of this article, but many scholarly sources exist, including this one for those who'd like a place to start.) Why then does the Bible seem to encourage the practice? It is probable that these verses are referring to discipline in general. Rather than being prescribed a specific method of discipline, we are being commanded to teach our children the right way and correct their behavior when needed.
Second, let us consider other possible meanings of the word "rod" in the scriptures. It can refer to a shepherd's rod, used to guide sheep, giving them a gentle nudge in the right direction or even pulling them away from an incorrect or dangerous path (see Psalm 23:4). It can mean the rod of a ruler, which represents authority, either that of God or His chosen servants (see Exodus 4:2-5). It sometimes refers to a measuring rod used for determining lengths and distances (see Ezekiel 40:3). And finally, Latter-day Saints are very familiar with the symbolism of a rod representing the Word of God (see 1 Nephi 11:25). Any of these interpretations could be applied to the teaching and correction of our children. Like shepherds, we must watch over and protect our children, sometimes (or rather, often!) pointing them in the right direction and pulling them away from physical or spiritual danger until they reach the level of maturity where they can find the right path for themselves. We should teach our children to respect parental authority, as well as the authority of God. We should not shy away from regularly measuring our children's behavior against what God has commanded so we know when correction is necessary. And finally, we must remember to always nourish our children with the good word of God and teach them to hold firmly to it.
Finally, let us not forget that we are all children. All of us, no matter how old and learned, still make mistakes and need to be corrected from time to time. And we are all children of God, who is our Heavenly Father. He is perfectly wise, kind, just, patient, and merciful; in short, the perfect Parent. We needn't doubt that He corrects His children in whatever ways will be most effective at helping those willing to be taught so that they may learn, grow, and eventually obtain lasting happiness. In order to be successful parents ourselves, we would do well to consider the ways in which we have been corrected by God and how we can apply those principles to the teaching of our own children.
Make no mistake, we absolutely must discipline our children. We must correct them when needed, and sometimes that correction takes the form of appropriate punishment. But we must not forget that the ultimate reason for doing so is to help them learn to make correct choices of their own free will as they mature rather than doing so out of fear. And the best way to accomplish that is to correct with love.