I am not a mathematics educator but I feel the need to chime in from the student side of this. I have taken numerous hard math classes during my BSc, and I have had my fair share of feeling hopeless, lost, and frustrated. You didn't indicate which math class you teach, and to be honest it is irrelevant.
Unlike the other answers I don't feel like this is necessarily an attempt to control you.
Point (1) sounds like something I commonly heard from fellow students who struggled not because the professor didn't try to help, but because they are taught "the solution is the goal". They get quickly discouraged, and at points frustrated enough to write smarmy marks and protest. Perhaps to address this point you could talk to the class about an anonymous assignment that said these things. Talk to them about how its more important to show how they arrived at their solution and open up the floor for a few minutes to let people tell you what they think. A quick way to find your flaws is to do this. Many complaints will be down right ignorable (this class is too hard, you're not good enough, etc) but occasionally a student will call out a character flaw worth addressing. Maybe you're not making something clear enough? Generally students don't protest unless they truly feel hopeless. A few classes will do this (mathematical statistics, proof-based math courses, etc) so it would be useful to figure out why at least. He could have came to your office to complain directly and start a dialog with you - so he gets a 0 if only to demonstrate snark wont be tolerated.
Point (2) is sort of addressed inside of point (1). You're an undergraduate educator. The student has arrived at your class having either:
- Taken an entrance exam demonstrating the minimum level of competence for the course
- Passed pre-requisites satisfactorily showing competence in the material
At any rate, people forget. There is a great book I'm sure you've heard of called How To Solve It by Polya. In the book he talks about how if a student cannot grasp the harder stuff you need to prod them with questioning until you find out what they do know and build them up from there. If you're unwilling or unable to do this, then you need to look at your department. Students struggling this much generally are being passed by easier professors or lax examiners. A certain level of struggle is acceptable, but I have been in classes that felt hopeless for that exact reason - the professor was unwilling to help us learn what we didn't know quickly, and the department pre-requisites didn't prepare us for the class at all. Many of us nearly failed, and the professor smugly proclaimed we were all incompetent. It's not a good experience being that lost, and it's worth digging into the root cause so you can address it directly either at the department level, or assisting the class with their shortcomings. Either way, there are mature ways to deal with this at the student level so he gets a 0 regardless of what you choose to do.