Q. I read the story of Jesus and the Canaanite woman the other day and I have no idea what Jesus is talking about in the parable when he references crumbs and dogs eating the crumbs. Can you shed some light on this passage?

This story is confusing and sometimes upsetting to readers of the gospels because it appears that Jesus is not only rebuffing someone who comes to him for help, he’s actually insulting her in the process.
A Canaanite woman asks Jesus to deliver her daughter, who’s suffering at the hands of a demon, but he won’t even speak to her. When his disciples urge him to help, he replies, “I was sent only to the lost sheep of the house of Israel.” (The woman is a non-Israelite.) And when she appeals to Jesus personally, he responds, “It is not right to take the children’s bread and throw it to the dogs.”
So Jesus seems to have a very callous and insulting attitude. However, I think something different is actually going on here.
This was an oral culture whose ways were embodied in popular sayings. These were often cited in support of a particular course of action. When two people had different courses in mind, they would pit different sayings against each other until one person had to admit, “Okay, you’ve got me there.”
This kind of thing can happen in our own culture. For example, two friends might visit a new part of town on a weekend, looking for a restaurant where they can have dinner. The first place they consider says it can seat them immediately. One of them might say, “Maybe we should eat here. After all, ‘A bird in the hand is worth two in the bush.'” But if the other thinks there could be a better restaurant down the street that would be worth the wait, he might reply, “Yes, but ‘nothing ventured, nothing gained.'”
Similarly, I think Jesus is actually quoting a popular saying to the woman: “It’s not right to throw the children’s bread to the dogs.” This saying probably had a general application meaning something like, “Don’t use something expensive or valuable for a common purpose.” Jesus is applying it to the mandate he has, during his limited time on earth, to concentrate his efforts on ministry to the people of Israel, as their Messiah. (After his resurrection, his message will spread to all the people of the world from that starting point.)
The woman, however, comes up with what I think is an original saying of her own in response: “Yes, but even the dogs eat the crumbs that fall from their masters’ table.” Jesus responds, in effect, “You’ve got me there,” and he heals her daughter.
But this was not merely a battle of wits that the woman won by her cleverness and quick thinking. Rather, I believe Jesus evaluated every situation he encountered in order to discern how God might be at work in it. In the gospel of John he’s quoted as saying, “The Son can do nothing of his own accord, but only what he sees the Father doing.” So Jesus was always on the lookout for when his Father might be doing something that he could join in with.
I believe, for example, that when his mother Mary came to him at the wedding in Cana and told him that the hosts had run out of wine, while Jesus thought initially that the time hadn’t come yet for him to do “signs” in public, he ultimately recognized that Mary’s persistent and trusting faith was an indication that God was at work in the situation. And so he did his first miracle there, turning water into wine.
I believe that Jesus similarly recognized the Canaanite woman’s bold request and audacious persistence as indications that God was giving her the faith to believe her daughter could be delivered if she sought help from Jesus. It was in response to that recognition, inspired by the woman’s reply to his challenge, that Jesus acted to heal the daughter, giving an advance glimpse of how his influence would soon extend beyond the borders of Israel.