Jesus went away from there, and withdrew into the district of Tyre and Sidon. And a Canaanite woman from that region came out and began to cry out, saying, "Have mercy on me, Lord, Son of David; my daughter is cruelly demon-possessed." But He did not answer her a word. And His disciples came and implored Him, saying, "Send her away, because she keeps shouting at us." But He answered and said, "I was sent only to the lost sheep of the house of Israel." But she came and began to bow down before Him, saying, "Lord, help me!" And He answered and said, "It is not good to take the children's bread and throw it to the dogs." But she said, "Yes, Lord; but even the dogs feed on the crumbs which fall from their masters' table." Then Jesus said to her, "O woman, your faith is great; it shall be done for you as you wish." And her daughter was healed at once.
The Syrophoenician woman clouds what is otherwise a clear-cut theology concerning the role of faith during prayer. The clear-cut answer says you must have a specific promise to apply to your specific situation. Then and only then may you believe God for this promise. You can't just "believe" God for anything. You need content to tie off on.
Enter Syrophoenician woman. Not only does she not have any specific promise that applies to her specific situation, she is a Gentile and therefore outside of God's covenant people! And more than this - she can't even get Christ to talk to her. When she finally does get Him to speak He calls her a dog! Yet she persisted, and Christ commended her for having "great" faith.
This account in no way negates the need for content in order to believe. She had plenty of content. She knew Christ was compassionate and able to help the helpless. But she had no specific promise that God would help her daughter.
I think our caution in this area may come more from a reaction against the Charismatic "name it and claim it" practice than honestly dealing with the Bible's teaching. Yet I confess I don't have the answers on how this works out in practice. I simply know that I'm asking God to make Mark 11:24 more of a reality in my life.