I like to think that the Bible challenges cliche - in the sense that The New Testament Gospels show Jesus challenging a woman's understanding of who he is, and that a woman who was not an Israelite and who therefore had no reason to count herself among the people of God according to the Jewish faith, culture or tradition.
In a sense I read the passage recounted in Mark 7:24-30 as Jesus challenging the woman to think beyond the accepted morality and, by implication by faith, overcome the fact that she was not culturally positioned to inherit the kingdom of God, the same situation of many of the people Jesus met and healed (the Centurion, the widow and her son) and yet who were welcomed into it because inheritance there was not conferred by tradition or history. Similarly, as the apostle Paul went to 'gentile' or 'Greek' non-Jewish peoples and told them the gospel, many who were not the heirs according to history, became 'co-heirs in a glorious promise'.
He challenges her to think for herself, rather than allow others to impose their own morality on her - indeed first she "begged Jesus to drive the demon out of her daughter" (v26), before he posed the saying "First let the children eat all they want...for it is not right to take the children's bread and toss it to their dogs"(v27). She was desperate, one imagines, and must have been convinced that this stranger, maybe, had the power to save her daughter.
How she knew this isn't made clear; we simply aren't told in the passage any of the surrounding details, only the exchange. But she believed that he held this power, and that her daughter, in a desperate condition, could be healed if he were willing. This is a big thing to believe, but the saying Jesus may be invoking here (the IVP commentary suggests that it was a popular saying of the time) must have challenged her belief; Jesus seems to be saying that his healing ministry was for the Jewish people, which of course is true, but potentially was not for other ethnic groups.
This isn't strictly so - and you could maybe argue that the saying was indicating the future spreading of the gospel to other non-Jewish groups after Jesus' death.
Now, the woman could have slunk off at this point, despondent and hurt by Jesus' answer. But you get the impression that she may have heard the saying before, because her answer counters it: "Yes, Lord," she replied, "but even the dogs under the table eat the children's crumbs"(v28).
It might have been easier for her to go on her way and return to the horrific situation that plagued her and her daughter, accepting what Jesus said both as a man with more power than herself and therefore authority over he intellectually in that culture, and accepting that he just didn't understand.

But she didn't. Instead she challenged what he said with her own understanding, which didn't come from arrogance or the conviction that she was right, but from faith. In fact, she had so much faith in Jesus that she was willing to offer a counter argument to the argument he produced himself! What she wanted was her daughter to be healed; she identified that Jesus could do this, and wouldn't take no for an answer, however pithy the saying used to justify it was.

I suppose the point is, I wonder how many times this is still an issue for Christians, how much we have to use our critical faculties and search for the real Jesus Christ amid the particular traditions and sayings of our own churches, organisations and even friends and family. How whenever someone tells us something is 'impossible' or 'not the done thing here' we have to make the choice between accepting this against our convictions, or looking deeper into the revealed saviour's life and words to find answers there.

There is a danger, of course, that we will assert our own views to the point of arrogance, and of course this is all a delicate balance. But that isn't what's happening in this passage, and that's important to remember - the woman's convictions came from her faith in Christ, ultimately, and therefore when approaching things in this way, so should ours.