What were the twelve aware of when they asked Jesus to send away the Canaanite woman?
The twelve were aware of Jesus' earlier marching orders to them in Matthew 10: not to go into the way of the Gentiles or any Samaritan city, but to go only to the lost sheep of the house of Israel. So they asked Him to send the Canaanite woman away, not from bigotry but under God's control.
Les explains that in Matthew 15:22 a woman of Canaan came out of the same borders and cried to Jesus for mercy because her daughter was grievously vexed with a demon. Jesus answered her not a word, and His disciples came and besought Him, saying, "Send her away, for she crieth after." Les points out that these were the twelve, and they were aware of what Jesus had told them back in Matthew chapter 10: they were to have nothing to do with Gentiles and go only to the lost sheep of the house of Israel. They were under God's control, not being bigoted Jews.