7:25 For a certain woman, whose young daughter had an unclean spirit, heard of him, and came and fell at his feet:
7:26 The woman was a Greek, a Syrophenician by nation; and she besought him that he would cast forth the devil out of her daughter.
7:27 But Jesus said unto her, Let the children first be filled: for it is not meet to take the children's bread, and to cast it unto the dogs.
7:28 And she answered and said unto him, Yes, Lord: yet the dogs under the table eat of the children's crumbs.
7:29 And he said unto her, For this saying go thy way; the devil is gone out of thy daughter.
Because the woman was not a jew ("the children" - obviously he wasn't talking about children children because she was imploring him for her child), it would be as if he were feeding a dog if he were to feed her request.
And when she responded with a clever retort, he condescended to drive the devil from her daughter.
Any problems with my interpretation?
Calling someone a dog in that culture was one of the most potent insults one could issue forth.