My question comes from Shabbat 30b (Koren Talmud Bavli). Two contradictory proverbs are mentioned:
Answer not a fool according to his folly, lest you also be like him.
Answer a fool according to his folly, lest he be wise in his own eyes.
The Gemara concludes:
..this, when one should answer a fool, is referring to the case where the fool is making claims concerning Torah, whereas that, where one should not answer him is referring to the case where the fool is making claims about mundane matters."
My question is, why they took this decision. Why should I make a fool feel wise when it comes to important questions regarding Torah? Are there some other sources supporting this attitude? I'm a teacher and I mostly proceed according to the former proverb when it comes to important things.