“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?

So 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?

This is an analogy. The "bread" is referring to his teachings.

And the "children" are those who are to receive his teachings.

While the "dogs" are 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.)

Jesus is chastising the lack of compassion exhibited by his disciples. He is attempting to give them a lesson about his mission and his objectives.

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)
In other words, she submits herself in full humility and faith to the representative of God, not caring to feel offended by being compared to a dog (which Jesus was only doing to illustrate how his disciples were treating the woman).

Is Jesus being compassionate?

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

As the loving servant and representative of God, Jesus is compassionate. He cares about everyone - not just the Jews.

Jesus did not identify the person with the physical body. He knew our identity as the spirit-person within these temporary physical bodies. This is why he also said at one point:
“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.

Thus the position or family of one's body was not his concern. Rather, the condition of the person's heart. Do they have faith in the Supreme Being? Are they prepared to listen to the teachings of God’s representative and follow them?

Are they prepared to submit themselves to the Supreme Being, even gradually, through their lives, and develop their relationship with the Supreme Being?

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 upon 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)