This verse, together with the parallel passage in Colossians 3:21, is probably the key New Testament instruction specifically dealing with the training of children. "Nurture" is from the same Greek word that is translated "chastening" in Hebrews 12:5, 7, and "instruction" in 2 Timothy 3:16. It has particular reference to carrying out child training with both firmness and gentleness, as needed and appropriate in each particular case.
The term "admonition" is from a Greek word meaning"putting in mind." Thus, the "admonition of the Lord" implies teaching the ways of the Lord by using the Word of the Lord. There is no substitute for implanting a knowledge of God's Word in the minds of our children. Even if they should drift away for a while in later life, the Lord can use His Word in their hearts to bring them back.
Both types of training--through action and through verbal teaching--are said in this passage to be primarily the responsibility of the father. The first reference in the Bible to training children deals with Abraham's responsibility to bring up his children in the nurture and admonition of the Lord (see Genesis 18:19). Mothers, of course, also have much responsibility and ability in this ministry (see Proverbs 1:8, and the example cited by Paul himself of how Timothy's mother and grandmother had taught him--2 Timothy 1:5; 3:15). Fathers, too, sometimes delegate certain teaching responsibilities to tutors (Galatians 4:1-2), but the overall responsibilities are theirs.
And all of this training should be done in love. "Fathers, provoke not your children to anger, lest they be discouraged" (Colossians 3:21). Then we trust the Lord and pray. HMM