She didn't; we consider it part of the First Commandment.
The Commandments are found in two places in the Bible: Exodus 20:1-17 and Deuteronomy 5:6-21. If you read those two accounts, you will notice that neither one is numbered. The Bible states elsewhere that those verses contain ten distinct commands (Exodus 34:28), but it does not specifically enumerate them. As a consequence, Jews and various Christian groups each number the commandments slightly differently, particularly the first two.
Jews consider the words "I am the Lord thy God, which have brought thee out of the land of Egypt, out of the house of bondage" to be the First Commandment, and the proscriptions against other gods and graven images to be the Second. Eastern Orthodox Christians and most Protestants (except Lutherans) consider the words "Thou shalt have no other gods before me" to be the First Commandment, and the part against graven images the Second.
Catholics and Lutherans consider the proscription against other gods and graven images together to be the First Commandment, even as the Jews consider them one commandment (though it's the Second in their reckoning). So we do not exclude the command against graven images, we simply include it in the First Commandment along with "Thou shalt have no other gods before me".
For more on the numbering of the Ten Commandments, see the following article by James Akin: http://www.cin.org/users/james/files/numberng.htm.
On the surface, the text seems to say that the Lord punishes children for the sins of parents to the third or fourth generation. But it can't mean that because God elsewhere says that he will not punish us for the sins of our parents (Ezekiel 18:2-20).
Theologians tell us that the proper sense of Exodus 20:5-6 is that a child born of sinful parents will have two difficulties:
God does not cause this to happen, but He does permit it. He allows us to suffer the negative consequences of our actions, and that sometimes affects others as well. So we must avoid sin not only for ourselves, but also for the sake of others.
Exodus 20:5-6 occurs toward the end of the First Commandment, against idolatry and worship of false gods. God seems to be saying that such false worship will have deleterious effects on future generations, particularly by way of a bad example.
When God made his covenant with Abraham, He promised that by Abraham's "seed" all the nations would be blessed. The New Testament says that this is a reference to Jesus, the Messiah, a descendant of Abraham. With so many pagan nations practicing idolatry and immorality, God wanted to prepare a holy and faithful nation into which His Son could be born. That is why He chose and blessed the Israelites, the offspring of his friend Abraham.
The Messiah, though Jewish by nationality, would nonetheless be the Savior of the whole world (John 4:42). Even in the Old Testament we see hints of God's intention to extend salvation to all nations. Besides the promise to Abraham mentioned above, there are Psalms which call upon the whole world to praise God (Ps. 67 for instance). There is also the prophecy that one day "the earth will be full of the knowledge of the Lord as the waters cover the sea" (Isaiah 11:9), and that all nations will seek the "root of Jesse", ie. the Messiah (verse 10).
These prophecies are fulfilled in the New Testament. God has not exactly had a "change of heart"; He always intended to lead all nations to Himself through the Jewish people, and particularly though the Jewish Messiah, Who was to be Savior of Jews and Gentiles alike (Ephesians 2:11-19). The New Testament is the realization of that goal.
We could say that God specially chose the Jews so that He could bless the whole world through them. There is therefore complete harmony between the Old and New Testaments.
These verses occur in the story of Jesus' encounter with the Canaanite woman. His seeming reluctance to help her was not a refusal, but a test of her faith. (Of course, He knew how the test would work out; it was more for the benefit of His followers, and for us).
Here are the verses in question:
26 (Jesus) answered, "It is not fair to take the children's bread and
throw it to the dogs."
27 She said, "Yes, Lord, yet even the dogs eat the crumbs that fall from their masters' table."
Though He appears to be harsh, Jesus was not really insulting her. He was kind of "prodding" her, to show the depth of her faith in his ability. She made a pretty clever comeback, to which Jesus responded "Great is your faith!" (her persistence was a sign of her faith in Him). Then he performed the miracle she requested.
Now, she was a non-Jew. Some Jews at the time looked down on Gentiles, since they did not have the Scriptures and the Covenant. Jesus was showing His followers (who were Jewish) that Gentiles are also capable of faith. A similar event would be the healing of the centurian's servant (Mt 8:5-13).
These are lessons for us. Faith is a gift of God, and God can give it to anyone, regardless of ethnicity. This Canaanite woman was a forerunner of many Gentiles who would believe in Jesus and become members of His Church.
The Bible says that all who are in Christ are one body: "There is neither Jew nor Greek...for all are one in Christ Jesus" (Galatians 3:28). Gentile believers can no longer be called "dogs" under the master's table; they have been exalted to sonship. God is their Father in Christ.
We see evidence in the Hebrew Scriptures of "righteous Gentiles", that is, men who were not descendents of Abraham yet who served the true God. Noah, of course, was one; but after him came Job, Jethro the priest of Midian (Exodus 2:16), and possibly Melchizedek (some believe that he was a theopany of Christ, based on an interpretation of Hebrews 7:1-3, but it is likely that he is just a type of Christ, not Christ Himself). Even the pagan magician Balaam (who could hardly be called "righteous") knew of the Most High God and even prophesied in His name (Numbers 22-24).
The fact that some non-Hebrews worshipped the one true God shows that God did not totally abandon the pagans, but revealed Himself to any who had hearts open to Him. This may well have occurred in non-Middle Eastern cultures also. I see no reason why God would find righteous pagans in only one geographical area!
As for the idolatry of many pagans, Saint Paul said that before the coming of Christ God overlooked their ignorance (Acts 17:30), but now commands all to repent. So if God pardoned their ignorance there was perhaps hope for their salvation; Christ could have chosen to save those who tried to live by their consciences (Romans 2:14:16).
Interestingly, Paul's statement in Acts 17 comes during his sermon to the Athenians about their altar to the "unknown God". Paul told them that when they worshipped at that altar they were unknowingly worshipping the one true God (vvs. 22-24)!
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