“It is not right to take the children’s bread and toss it to their dogs.” (Matthew 15:26)

A Canaanite woman from that vicinity came to him, crying out, “Lord, Son of David, have mercy on me! My daughter is demon-possessed and suffering terribly.” Jesus did not answer a word. So his disciples came to him and urged him, “Send her away, for she keeps crying out after us.” He answered, “I was sent only to the lost sheep of Israel.” The woman came and knelt before him. “Lord, help me!” she said. He replied, “It is not right to take the children’s bread and toss it to the dogs.” “Yes it is, Lord,” she said. “Even the dogs eat the crumbs that fall from their master’s table.” (Matthew 15:22-27)

What do the 'children's bread' and 'dogs' symbolize?

Jesus is responding to his disciples’ remarks about sending the woman away. Why is he talking about taking children’s bread and tossing it to the dogs then?

We should note that Jesus is chastising his disciples for their lack of compassion on the woman. He is attempting to teach them a lesson about his mission and his objectives, using a metaphor or analogy.

What does this analogy mean?

The "bread" is referring to his teachings - God's mercy. Jesus is introducing the Supreme Being. This is God's mercy, because through Jesus' teachings we are able to better come to know and love the Supreme Being.

And the "children" are those who receive his teachings. Those who take Jesus' teachings into their hearts. Some might interpret this as those who were Jewish, but through this conversation and others, we find that Jesus was perfectly willing to teach those who were not Jewish in ancestry or faith.

The "dogs" symbolize those who are not ready or unwilling to receive his teachings. (During those times, people did not feed dogs their dinners as they do today.)

To Jesus’ statement above about the dogs, the humble woman says:
"Yes, Lord, but even the dogs eat the crumbs that fall from their masters' table." (Matt 15:27)
This is stating that even those who are not that interested in receiving Jesus' teachings - symbolized by the "dogs" - also will benefit because they will receive small portions of wisdom by being in proximity to his teachings.

Is Jesus being compassionate?

The woman has submitted herself with humility and faith to Jesus. She also did not feel offended by being compared to a dog - which Jesus was only doing to illustrate how his disciples were treating the woman.

We find that Jesus praises her humility and faith, and heals her daughter without any hesitation (see Matthew 15:28).

As the loving servant and messenger of God, Jesus is compassionate. He cares about everyone. This is illustrated with this exchange.

Jesus also did not identify the woman with her physical body. Jewish tradition restricted a rabbi's contact with those of the opposite sex. 

But Jesus saw the person within her body. He knew her identity as the spirit-person within a temporary physical body. Jesus indicated this concept elsewhere:
“Do not be afraid of those who kill the body but cannot kill the soul. Rather, be afraid of one who can destroy both soul and body in hell." (Matthew 10:28)
The physical body will live for a few decades at the most and then die. It is the spirit-person within that Jesus was considering, and who he was teaching this to.

Furthermore, being a Canaanite woman made her even more of a challenge in terms of Jewish tradition, as rabbis were also restricted from mixing with women of other cultures.

But the position or family of one's body was not an issue for Jesus. The condition of the person's heart was. Do they have faith in the Supreme Being? Are they prepared to listen to the teachings about God and follow them?

This is the ultimate in mercy and compassion. Caring about someone and working to help that person, as long as they will accept the help. (As opposed to those who thought they were just fine and didn't need Jesus' help.)  

These are the issues Jesus was focused upon, rather than the issues of the temporary physical body - the body that will die and decompose in the ground.

Jesus wants us to focus on spiritual matters. Specifically, he wants us to learn to love God and love others:
“ ‘Love the Lord your God with all your heart and with all your soul and with all your mind.’ This is the first and greatest commandment. And the second is like it: ‘Love your neighbor as yourself.’" (Matthew 22:37-39)