"22 And, behold, a woman of Canaan came out of the same coasts, and cried unto him, saying, Have mercy on me, O Lord, thou son of David; my daughter is grievously vexed with a devil.
23 But he answered her not a word. And his disciples came and besought him, saying, Send her away; for she crieth after us.
24 But he answered and said, I am not sent but unto the lost sheep of the house of Israel.
25 Then came she and worshipped him, saying, Lord, help me.
26 But he answered and said, It is not meet to take the children's bread, and to cast it to dogs.
27 And she said, Truth, Lord: yet the dogs eat of the crumbs which fall from their masters' table.
28 Then Jesus answered and said unto her, O woman, great is thy faith: be it unto thee even as thou wilt. And her daughter was made whole from that very hour."(Matthew 15: 22-28)
The thing I get from Jesus' interaction, with the Canaanite woman, is that He was sent to deal with the Jews as a first priority. They were the in a Covenant relationship with God. They worshipped him. They studied scripture. Made sacrifices, were blessed and cursed according to the law.
The Canaanites were not in a Covenant relationship with God, did not worship Him, did not observe or necessarily know the law. Hence the comparison to a pet animal, who doesn't know our laws, have our knowledge understanding and appreciation. Although we will feed and keep a pet animal if we can afford to.
People tend to prioritise the needs of people over animals? Why, animals are living creatures? But we do not share knowledge and understanding with them. Would we expect them to appreciate as we do?
Then Jesus discounts this type of consideration, sees that the woman can appreciate Him, although she will have not studied scripture, will have little formal knowledge, and carries out her request to heal her daughter.