She was a Gentile, and "dog" was a term used for Gentiles, because they were separated from the commonwealth of the house of Israel and considered unclean. Jesus' purpose on earth (other than his death) was to preach to the nation of Israel, to confirm the promises of their fathers, and to perform miraculous and extraordinary wonders as evidence to them. When he told her, "It is not right to take the children's bread and throw it to the dogs," this is what he draws out. I am to do this for Israel, not for Gentiles right now. Her reply was amazing. She said, "Even dogs eat the crumbs that falls from their master's table." Very clever imo.