Actually, the 6,000 year number comes from a calculation done by Bishop James Ussher in 1633 by using a chronology from the Bible based on linking Biblical events to events with other historical attribution.
It was actually a very academic and careful undertaking, and like most bishops, Ussher was a very educated man who today might well have even accepted the Theory of Evolution.
The problem is that he lived in the 17th Century and no one at all knew what evolution was, and he was a Protestant bishop to boot. Such people, failing other reasonable alternatives, will go to the Bible for answers.
However, taken for what it is, his Bible chronology is quite defensible, albeit not the only possible reading of the Bible text.
So, while it is possible that the Young Earth Creationists just don't like Evolution, they didn't have to make it up to make their point. It was very good Bible scholarship, but only if you insist (as fundamentalists do) that the Bible is unerring and not allegory at all, ever. Most Christians do not believe that the whole Bible is literally consisting of the actual words God spoke. Some parts of it are solid, if slanted, history that you can base good archaeological digs on. Some parts of it need to seriously be taken on faith or accepted as stories which interpret the big questions in ways that a person in ancient times would have interpreted them.
Of course, there is always the possibility that the YE Creationists are right. It is entirely possible that there is a metagame out there where rules allow the universe as it is today to not be the universe as it once was. That is generally discounted, because its completely useless to any sort of practical application, but with an all powerful Creator God, who is above all physical laws and even logic itself, you can literally have *anything* happen. In that scenario, the only way you know it is different is if someone tells you it was different. And then you have to believe them. There's no other choice. This is what we call "Faith" with a capital "F".
Of course, you don't need that sort of maddening, useless, mind-bending scenario to have an actual deity, but it can never be ruled out because it is completely untestable, and it isn't even illogical. Absolute power makes anything possible.
This is why science is never going to equal truth. Science is useful because it confines itself to the observable and the testable, but something does not need to be observable to be true, nor will all true things be testable.
So, the answer to the Young Earth Creationists is not that they are wrong (although my gut says that they probably are), but that we can't derive any policy or theories based on untestable truths. Evolution does not have to contradict Creationism, but YE Creationism is not really useful for such subjects as genetics or anthropology or whatever. It does not match what we have tested and observed.
I would move all untestable theories to the Philosophy class, including Creationism and whatever is going for the atheist hypothesis about how we ultimately ended up existing. They're both untestable and that's where you can have the two fight it out in debates and leave the good science for the Science classes.