The proverb "Give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime" is, like most proverbs, inadequate because it sets up a false dichotomy.
First, we need to remember that the "fish" is simply any person's need, whether it be food or a skill, or an occupation, or health.
The two options in helping someone in need aren't just giving or teaching. In both those options, someone knows how to live life and other doesn't. But we might also:
Give people who already know how to fish the opportunity to.
Fish with someone who is too afraid to fish because of past experiences.
Give someone a fishing hole, because they've been kicked out of all the other ones.
Give them an opportunity to help you give fish to the community as a partner.
If all you want to do is give or to teach, then you are assuming the person in need is less than a person than you. Because somehow, you think they can only be fulfilled if you do are over them in some way. The best way to help a needy person is to come alongside them and support them to do better as an equal, not as a philanthropist or a mentor.