How should instructors deal with the fact that the students who know the least are precisely the ones who think they know everything?
NOTE: This question is not and never was intended to be primarily about mathematics instruction, but was intended to be equally applicable to all subjects to which it can be applied (which probably means all subjects). It has been edited by seemingly many people. I hope I've now expunged the most offensive edits from it. The examples below are about mathematics for the obvious reason. END OF SPECIAL NOTE
Someone takes high-school math courses in which all that is required of them is that they learn algorithms for solving assigned problems, and they think that learning those is what learning math consists of (in other words, they learned no math) and since they always worked hard and got straight "A+"s, they think they're really good at math. Then they say they're entitled to take a course requiring only learning technical skills and algorithms and not requiring them to understand anything, and they think it's outrageous that an instructor would think there's something more that should be required of them. You can't get more ignorant of mathematics than to think that learning procedures to follow mechanically in solving assigned problems is what learning mathematics consists of (or at least you probably can't get more ignorant than that and still get admitted to a university), nor more arrogant than to think that an instructor needs instruction from people who think that.
High-school courses in which students effectively learn (without being told so) that learning mathematics consists of learning procedures to apply mechanically to solve assigned problems are the result of official decrees that everybody must learn mathematics. So ignorance of mathematics results from decrees intended to have the opposite effect.
In one case a student wrote on a test something like "Are you kidding? u-substitution? We shouldn't see that until next semester!" An entirely inexperienced person would have laughed out loud at a student mistakenly thinking under the circumstances that they were being asked to use the technique informally called "u-substitution", and moreover the class had been told in advance that that particular problem would be there. And sometimes they're not angry (as in the example in my previous comment) but they neglect a topic of which they are completely ignorant because they think they already know everything about it.