Mat 9:16 No man puts a patch of new cloth onto an old garment, for that patch lifts up from the garment, and the hole is made worse.

This verse seems to contrast the idea of mending old ideas with new ones in the original translation, but it doesn't mean that at all in Greek.

The term translated as "new" is agnaphos, which means "unmilled," "unfulled," "undressed," and "unprocessed." "Cloth" is rhakos, which means a "rag," "remnant," or a "tattered garment." So Christ wasn't saying "a patch of new cloth" but "a patch from an unprocessed remnant." The "old garment" is palaios ("old," "ancient," "worn") himation, which is an outer garment of cloth, like a mantle or cloak.

The term for "lifts us" is airo, which means "to elevate," "to bear," and "to carry off." The term for "hole" is schisma, which means "division" and "dissention." The idea is that is you use an the unprocessed remnant for a patch, it will shrink as the mantle is washed, making the hole you are trying to patch worse than before.

So what is Christ talking about here? He is talking about an old cloak with a hole in it, but the cloak is good quality cloth. He is saying that you don't patch that old, good cloak with a remnant of unmilled cloth because that will make its hole worse. The larger context is the question about the difference between Christ and John's the Baptist's teachings. John the Baptist's teaching about the coming of the kingdom of heaven was the patch and the old garment of the traditional practices of Judaism, which include fasting. Then the larger message would be that John the Baptist had only a part of the message ("remnant") and that it was unfinished. Judaism isn't made better by trying to patch it with this unfinished part but made weaker.

A minor note: The term translated as "puts" is epiballo, which means "to cast on," or "to lay on." The term translated as "patch" is epiblema, which means "that which is thrown over" and it is used to refer to patches, bedspreads, headcoverings, tapestries, etc. The two terms a play on world. Kind of "You don't throw on a throw over." The feel of this reinforces what Christ was saying about not using a hasty patch on Judaism.