>>672513
When in doubt, read what the Saints and Church Fathers say about the subject, especially St. John Chrysostom. He was right about almost everything.
http://newadvent.org/fathers/200152.htm
If you don't want to read the whole thing, this is the important part:
For if He had not meant to give, neither would He have given afterwards, nor would He have stopped her mouth again. But as He does in the case of the centurion, saying, "I will come and heal him," Matthew 8:7 that we might learn the godly fear of that man, and might hear him say, "I am not worthy that You should come under my roof;" Matthew 8:8 and as He does in the case of her that had the issue of blood, saying, "I perceive that virtue has gone out of me," Luke 8:46 that He might make her faith manifest; and as in the case of the Samaritan woman, that He might show how not even upon reproof she desists: John 4:18 so also here, He would not that so great virtue in the woman should be hid. Not in insult then were His words spoken, but calling her forth, and revealing the treasure laid up in her.
But do thou, I pray you, together with her faith see also her humility. For He had called the Jews "children," but she was not satisfied with this, but even called them "masters;" so far was she from grieving at the praises of others.
"Why, the dogs also," says she, "eat of the crumbs that fall from their master's table." Matthew 15:27
Do you see the woman's wisdom, how she did not venture so much as to say a word against it, nor was stung by other men's praises, nor was indignant at the reproach? Do you see her constancy? He said, "It is not meet," and she said, "Truth, Lord;" He called them "children," but she "masters;" He used the name of a dog, but she added also the dog's act. Do you see this woman's humility?
Hear the proud language of the Jews. "We be Abraham's seed, and were never in bondage to any man;" John 8:33 and, "We be born of God." John 8:41 But not so this woman, rather she calls herself a dog, and them masters; so for this she became a child. What then says Christ? "O woman, great is your faith." Matthew 15:28
Yea, therefore did He put her off, that He might proclaim aloud this saying, that He might crown the woman.
"Be it unto you even as you will." Now what He says is like this: "Your faith indeed is able to effect even greater things than these; nevertheless, Be it unto you even as you will."
This was akin to that voice that said, "Let the Heaven be, and it was." Genesis 1:3
"And her daughter was made whole from that very hour."
Do you see how this woman too contributed not a little to the healing of her daughter? For to this purpose neither did Christ say, "Let your little daughter be made whole," but, "Great is your faith, be it unto you even as you will;" to teach you that the words were not used at random, nor were they flattering words, but great was the power of her faith.
The certain test, however, and demonstration thereof, He left to the issue of events. Her daughter accordingly was straightway healed.
But mark thou, I pray you, how when the apostles had failed, and had not succeeded, this woman had success. So great a thing is assiduity in prayer. Yea, He had even rather be solicited by us, guilty as we are, for those who belong to us, than by others in our behalf. And yet they had more liberty to speak; but she exhibited much endurance.
And by the issue He also excused Himself to His disciples for the delay, and showed that with reason He had not assented to their request.