What did Jesus mean by his apparent racial taunt to the Canaanite woman in Matt 15 when he inferred that they were "dogs"?

This is figurative language, and if you track it down it can get interesting. God's people are sheep, they are lead by the shepherd. The pagans in general are dogs, which means they can either protect the sheep or disregard the sheep. Jesus promises that the dogs which recognize the sheep and protect the sheep will not lose their reward.

Mark 9:41 King James Version (KJV)

41 For whosoever shall give you a cup of water to drink in my name, because ye belong to Christ, verily I say unto you, he shall not lose his reward.

The woman in this case recognized the shepard and desired to serve the shepherd, therefore she was accepted by Christ and indeed she was saved. Because only the saved can keep their treasures, their rewards. This can go even further though...

Goats are people who seem to be saved from the outside, but are passive on the inside. They only have intellectual assent. Pigs are pagans who not only disregard the sheep but detest them. Wolves are full on enemies of the sheep which aim to infiltrate and eat.

What Yeshua called her is minor to the lessons on overcoming obstacles and being herself, he was teaching her... as well as us today.

She approached him in the way a Jew would: Have mercy on me, O Lord, thou Son of David... He ignored her at first and the disciples misread his actions as well. However she persisted. But he answered and said, I am not sent but unto the lost sheep of the house of Israel.

Then came she and worshipped him, saying, "Lord, help me". Lesson 1: she gave up the pretense, worshipped him and just offer a simple prayer... "Lord, help me".

But he answered and said, It is not meet to take the children’s bread, and to cast it to dogs. And she said, Truth, Lord: yet the dogs eat of the crumbs which fall from their masters’ table. Lesson2: She overcame the label others placed on her so she can access the Lord's table and receive his goodness just like anyone else. (read Psalm 24:1)

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.

The actions of this woman demonstrated her great faith. Even as James says: Even so faith, if it hath not works, is dead, being alone. Yea, a man may say, Thou hast faith, and I have works: shew me thy faith without thy works, and I will shew thee my faith by my works.

Yeshua came to save the world not condemn any part of it, these are, I believe, just two of the many lessons in this brief encounter. Blessings.

Sometimes we need to use our imagination when we meet a difficult story. We only have the written words of an event; we weren't there. Imagination can fill in the context.

For example, in this story of Jesus talking to the Canaanite woman, let's imagine one extra detail. Let's imagine that Jesus winked at her as he spoke his first sentence. If so, it could change the whole feeling of the scene from a serious racial put down to a piece of light hearted banter between two people.

I'm not suggesting that this was the case. I'm only using this to illustrate that we need to be careful about the assumptions we make when all we have is a written text that doesn't necessarily have all the details. The story is true; that doesn't mean it's comprehensive.

There are actually two different Greek words for the English term “dog” that was used in the New Testament. One word is “κύων” (“kýōn”). The actual definition of this word is: “a man of impure mind, an impudent man.” This word would be the word a Jew would use if they were really calling someone else (Jew or Gentile) an “unclean” or unspiritual animal (Matthew 7:6; Luke 16:21; Philippians 3:2). This word was not the word used by Jesus in Matthew 15:26, however.

The word Jesus used was “κυνάριον” (“kynárion”), which simply means “a little dog.” If Jesus truly wanted to insult the Canaanite woman (or Gentiles in general), He would have used the word “κύων,” not “κυνάριον.”

Also, let’s use a little bit of logic here: if the woman had taken what Jesus said as an insult, her faith would have been damaged and her daughter would not have been healed. She clearly understood Jesus’s reference as nothing more than a harmless metaphor. It’s the same thing as when people are commonly described as “sheep” in the Bible.

Ultimately, Jesus frequently tested people to prove their intentions, often through response questions or challenges (see John 4:16–18; and 4:50–53). His response to the Canaanite woman is similar. In testing her, Jesus declined her request and explained that she had no legitimate expectation of His help. The woman, however, lived out the principle Jesus Himself taught in the parable of the persistent widow (Luke 18:1–8). Her response proved that she understood fully what Jesus was saying, yet had enough conviction to ask anyway (Matthew 15:27). Jesus acknowledged her faith—calling it “great”—and granted her request (Matthew 15:28). In the end, Jesus happily granted the woman’s request due to her faith.

So, according to both the context and language involved, Jesus wasn’t referring to the Canaanite woman as a “dog,” either directly or indirectly. He wasn’t using an epithet or racial slur but making a point about the priorities He’d been given by God. He was also testing the faith of the woman and teaching an important lesson to His disciples.