The focus of Jesus' ministry is Israel, but in his encounter with a Canaanite woman Matthew reminds us that the issue of a person being ceremonially clean or unclean, of being a Jew or a Gentile, is irrelevant. It was the quality of the woman's faith that saved her not any religious credentials.
This story is extremely important in that it heralds a change. The Pharisees are defective in their understanding of scripture and offended by the conduct of Jesus and his disciples. The focus moves to someone outside Israel, a pagan woman, a descendant of the ancient enemies of Israel.
Jesus left Galilee, and went to the border district of Tyre and Sidon. Possibly she lived in the border region and came out of her home to meet Jesus. Jesus keeps silent and the disciples ask that she be sent away.
He answered her that he had come to save the lost sheep of Israel, lost in the sense of perishing .
the woman came and knelt before him. He said is it right to deprive the children and toss it to the dogs, probably a house dog in contrast to a street dog. The imagery here is of children = Jews, and dogs = Gentiles. Jews viewed Gentiles as dogs.
The woman disagrees with what Jesus has said, explaining that actually the dogs do get to eat, the scraps that fall from the table.
Jesus exclaims at her faith. Your request will be granted. From that hour her daughter was healed.