A proverb is a statement that generally works out. It would be nice to believe that we as parents have a lot of perfect and godly children running around. However, the reality is that all people, even our kids, have the mind to make their own choices even if we, as godly parents, did the best possible job we could do.
Yet, what never leaves the child is the teachings of Christ that always remains in the back of their brain. How many people do you know that "fell away" from God and lived a worldly lifestyle. If you talk with them, it is evident that they still know where to find the truth. They have been impacted by truth. And the truth continually reminds them of their inappropriate lifestyle (whatever it is). I believe that this is the idea behind this proverb. If you train your children in the way that they should go, the training will always be in their mind to guide them - even if they choose not to follow it.
So, with every opportunity you have with your children, train them in the godly way that they should go so if they do fall away they will know how to get back on the godly path to eternal life.
The Lord's humbled . . . John A. Reeves