Paul writes that grace - self-motivated, condescending, reconciling, tender and forgiving mercy - has "appeared." How has grace appeared to bring salvation? In the context of Titus 2, in its broadest sense, it has appeared in the gospel of the Kingdom of God. The gospel includes the message of our great hope, the promise of Christ's return, Jesus' perfect life, and His death for the forgiveness of our sins.
The Greeks used "appeared" in their literature to describe the sun's light bursting out from the heavens onto the darkened earth. Its feminine form is used in other places to describe Christ's first and second comings. When used in the passive voice, it means "to show openly" or "shine light upon" with the sense of suddenness and unexpectedness. This is part of the sense here since we do not normally expect grace to reveal or teach us anything.
Grace, however, has a clear message that has much to do with our responsibility and growth. "Teaching" in Titus 2:12 is the Greek word paideuo, also translated as "chastens" in Hebrews 12:6. It is used in the sense of schooling, training, or disciplining. In the context of educating a child, it describes activity directed toward moral and spiritual development and influencing conscious will and action. In religious matters, paideuo means chastising to educate one to conform to divine truth. It includes instruction, as in a classroom; drilling, as in "practice, practice, practice"; and chastisement, as in spanking or rebuking to bring about correction.
God's grace teaches us by putting us under obligation negatively - to quit sinning - and positively - to grow and produce fruit. The Moffatt translation clarifies this obligation by defining the terms in more modern language. "For the grace of God has appeared to save all men, and it schools us to renounce irreligion and worldly passions and to live a life of self-mastery, of integrity, and of piety in this present world." These are the areas toward which we must turn our attention to fulfill our duty to Christ. Moffatt retains the positive and negative aspects in his version: first, the negative renouncing of "irreligion and worldly passions," then the positive living of a life of "self-mastery, integrity, and piety."