When I was a kid, we played a game in which we would stare into each others eyes until one gave in and blinked. We used tricks to win sometimes, but above all we would refuse to back down no matter how much our eyes were hurting. Something like that is going on between Jesus and the Canaanite woman.
There is a stand-off happening here. I think it is helpful to know that Jesus has just gone through a difficult time. He recently learned of the beheading of his cousin, John the Baptist. Jesus tries to withdraw from the crowds of people who were following him, but they won't let him escape. He ends up feeding thousands with a few loaves of bread and a couple of fish. Then there is a storm at sea and Peter's walking on the water, which is sabotaged by his fear and doubt. Everywhere Jesus looks…