My experience of headinhands is that she will never, never concede a point -- she will never say, 'oh, I didn't know that', or 'ok, you have a point there', or 'yes, that's a reasonable reading of that passage'.
However, because there are other people reading, let's have a look at the Canaanite woman.
Here is the passage from Matthew, ch 15
21 Jesus left that place and went away to the district of Tyre and Sidon. 22Just then a Canaanite woman from that region came out and started shouting, ‘Have mercy on me, Lord, Son of David; my daughter is tormented by a demon.’ 23But he did not answer her at all. And his disciples came and urged him, saying, ‘Send her away, for she keeps shouting after us.’ 24He answered, ‘I was sent only to the lost sheep of the house of Israel.’ 25But she came and knelt before him, saying, ‘Lord, help me.’ 26He answered, ‘It is not fair to take the children’s food and throw it to the dogs.’ 27She said, ‘Yes, Lord, yet even the dogs eat the crumbs that fall from their masters’ table.’ 28Then Jesus answered her, ‘Woman, great is your faith! Let it be done for you as you wish.’ And her daughter was healed instantly.
So, the context of this is that the Jews considered the gentile peoples living in neighbouring areas to be unclean, and believed that God had made an especial covenant to protect the Jews (the house of Israel): they were God's chosen people. Jesus was, of course, a Jew he was brought up in his human family with these beliefs. But here, he is challenged by the Canaanite woman. He doesn't send her away, as his disciples want him to. He listens to her. And they debate: he uses an analogy to describe his mission he was sent to the lost sheep of Israel. They're not really lost sheep, but he looks after them like a shepherd does his lost sheep. Then he uses another analogy to describe the relationship of Jews and gentiles: the gentiles aren't really dogs but they come lower in the pecking order of God's family than the children, the Jews. But the woman challenges him, and says that in a family even the dogs get fed. And this argument sways Jesus: it changes his mind. She believes in him! She has faith even though she is not a Jew. He heals her daughter.
One of the great debates in early Christianity was whether it was a renewal movement within Judaism or whether the message of Christ should be spread to the Gentiles. Was Jesus the Messiah for the whole world or just for the Jews? We see Jesus here at a turning point, where he begins to change his mind and see that he has come to bring everyone, not just the chosen people, to God.
So he doesn't call her a dog. It's a figure of speech. The woman herself takes the figure of speech and uses it. She doesn't walk away saying 'you called me a dog you're a racist'. She has a more sophisticated understanding of how language works. It's a key moment when we see someone changing someone's mind by argument. And perhaps that's something we can all learn from, whether we are believers or not.