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The reason Jesus was amazed was because the woman, in a very obtuse way, had referenced the promise of Abraham that was to the nations (faith). She was part of that, but in that culture of the time, those outside the Mosaic Covenant were termed "dogs".
The dogs, us Gentiles, got the crumbs at the time being outside of the Mosaic Law that was well-known and operating in the region. Christ was amazed she invoked something little known.
Recall how difficult it was for even Peter to preach to the nations. It took a vision from God to clear the way (again) for him to visit Cornelius and open the door to the kingdom.
Today no one should be amazed that the message of salvation by faith in what God promised is for all.
Vines EDofNTW Dog: is used in two senses, (a) natural, Mat 7:6; Luk 16:21; 2Pe 2:22; (b) metaphorical, Phl 3:2; Rev 22:15, of those whose moral impurity will exclude them from the New Jerusalem. The Jews used the term of Gentiles, under the idea of ceremonial impurity. Among the Greeks it was an epithet of impudence. Lat., canis, and Eng., "hound" are etymologically akin to it.
Upvote:2
Jesus in His life of faith would have gone only to places that were His Father's will. Some schools of thought highlight that Jesus went to the borders of Tyre and Sidon specifically for the Syrophenician women since no other event of significance was recorded there.
Jesus would have been well acquainted with the burdens of the women before she even approached Him. He responded to her in a way that a Jewish person in His time would, but instead of discouragement or pride rising to the surface, the women's response was full of humility and faith. "God resisteth the proud and gives grace to the humble" (James 4:6). Her response was in sharp contrast to many of the responses He received from people of His own nation, many of whom that had rejected Him. Her response was very precious in His sight.
Ellen White in her Bible commentary on Jesus' life describes:
The woman urged her case with increased earnestness, bowing at Christ's feet, and crying, “Lord, help me.” Jesus, still apparently rejecting her entreaties, according to the unfeeling prejudice of the Jews, answered, “It is not meet to take the children's bread, and to cast it to dogs.” This was virtually asserting that it was not just to lavish the blessings brought to the favored people of God upon strangers and aliens from Israel. This answer would have utterly discouraged a less earnest seeker. But the woman saw that her opportunity had come. Beneath the apparent refusal of Jesus, she saw a compassion that He could not hide. “Truth, Lord,” she answered, “yet the dogs eat of the crumbs which fall from their masters’ table.” While the children of the household eat at the father's table, even the dogs are not left unfed. They have a right to the crumbs that fall from the table abundantly supplied. So while there were many blessings given to Israel, was there not also a blessing for her? She was looked upon as a dog, and had she not then a dog's claim to a crumb from His bounty?
Jesus had just departed from His field of labor because the scribes and Pharisees were seeking to take His life. They murmured and complained. They manifested unbelief and bitterness, and refused the salvation so freely offered them. Here Christ meets one of an unfortunate and despised race, that has not been favored with the light of God's word; yet she yields at once to the divine influence of Christ, and has implicit faith in His ability to grant the favor she asks. She begs for the crumbs that fall from the Master's table. If she may have the privilege of a dog, she is willing to be regarded as a dog. She has no national or religious prejudice or pride to influence her course, and she immediately acknowledges Jesus as the Redeemer, and as being able to do all that she asks of Him.
Jesus was the promised seed by whom "all nations was to be blessed." (Gen 22:18). In fact, many of Jesus' followers who expressed the greatest faith were non-Jews, such as the Centurion and the Samaritan woman. All of them received blessings and compassion from their Saviour.