Lecturer ridiculing me in front of class for making a mistake

There is nothing you can really do to prevent this behaviour. You can just report it to the university.

There should be a committee dealing with teacher behaviour; things may vary depending on your country and university, so I cannot be more specific. If you have student representatives, contact them.

Describe the abusive behaviour with as many details as you can, refrain from putting your own personal judgement and emotions into the facts, and have a colleague student support you as a witness.

There is something a bit off about your description of events. You describe that your classmates were laughing at you as your instructor verbally abused you. I can't imagine that happening. If things were as mean spirited as you say, I would expect dead silence.

I am wondering if the lecturer was just trying to insert some lighthearted humor and had no malicious intent. Instead of reporting anything to the university, maybe it would be a good idea to politely tell your instructor that you were very embarrassed by the situation (privately, of course). It may be that he had no idea and will be apologetic.

Having said all this, I have to add that public speaking is an important skill. It is worthwhile to develop the confidence to defend your ideas to an audience, and to avoid taking comments too personally. Later in your post you refer to "funny" notes made by your instructor as well as a loud public correction at the end. Remember that everyone is learning in this class and the comments may have been for the benefit of the students who are making the same mistakes (and the last public outcry was probably because a lot of them are making that particular error).

Anyway, I'm very sorry you had a bad day. I hope the situation is as I described. If talking to your instructor doesn't go well, then perhaps you should consider reporting the behavior.

Why are you letting this person get the best of you? You say:

he started to humiliate me in front of all my colleagues

I was feeling like an idiot standing there and listening to his abusive speech

This experience destroyed the lesson for me completely.

I was ashamed to talk with anybody.

These are pretty serious reactions, and I can only see them happening if you willingly participate and play his game. At the end of the day, he's just some guy talking about stuff at a blackboard. The title is "PhD", not "his majesty". If he thinks you are stupid, that's his opinion, and he has a right to it - and you have a right to dismiss his opinion as irrelevant.

• Not being good at math is not something to be ashamed of. Plenty of perfectly respectable, even great people are not good at it.
• Not being good at his particular narrow topic is nothing shameful. Nobody is born knowing everything, everybody starts somewhere.
• Not being intelligent (the fact that you were did not know something at the very beginning of the class doesn't even have to do anything with your intelligence, but nevermind) is not something to be ashamed of. The vast majority of people aren't very intelligent by most measures - so what? We don't go around mocking and shaming them, that would be ridiculous.

The way you've described them, his contentions have absolutely no basis. He asked some specialized trivia that you weren't taught, and then tried to act horrified that you didn't know it. It's a ridiculous thing to do, so why'd you take him seriously? Your response should have been, "so what?". "Yeah, I don't know this. So what?" "Yeah, I can't solve this problem easily. So what?"

It would be one thing if he gave you some homework, and you brazenly refused to put any effort into it, and then came to class complaining you can't solve the problem that the homework was meant to teach you how to solve. But this is just ridiculous. If the student is fulfilling all the duties they've been given, and still failing, the instructor is the one who should feel ashamed.

He told me that this field is "not for everybody" and that in math classes there are "only very clever people"

Well, if he thinks the class is not for you, he should kick you out of the class (as in, officially, through the school's system). He's not doing it, is he? That's because he's full of hot air. The course is literally "for everybody", evidenced by the fact that the school system as supervised by the president of the school has allowed everybody (including you) to register for the course. That's really all there is to it, if he doesn't like it, he can go petition his dean to require a mandatory IQ test to take his course, or whatever it is he wants.

So, what would you advise me to do to prevent this situation next time?

Well, if you really want to take the class (maybe you think the material is worth learning in spite of him, or you are required to by the school administration), I think you have the following options:

• Next time he tries to humiliate you, refuse to be humiliated and act as if you don't even see what's wrong. He will be forced to explain what's so worthy of humiliation, and when he explicitly says it, it will be plain to see how ridiculous and incorrect he is. It won't work, and hopefully he'll stop doing it. Beware, though, as he will probably take this as you challenging his authority, and may try to shout at you or eject you from the class - he doesn't actually have that authority, so if you stick to your guns and refuse to comply, he should eventually back off, but it may be difficult not to be suckered into complying if you can't handle people shouting you down.
• Stop going to lectures, and only show up to exams. Independently study the material yourself. In my experience, the vast majority of courses these days are better learned from the textbook anyway, and it sounds like this guy is awful and wouldn't teach you much. The danger is if he somehow builds in required attendance (whether by actually taking attendance or doing daily quizzes or something), or if the course doesn't have a textbook you can study. In that case, you're out of luck.
• Ask to speak with him privately. Firmly and clearly tell him that you disagree with his behavior, you don't appreciate it, and you won't allow it. Don't make excuses like you have in this question (eg. "I'm not from this field so it's hard for me") because you have nothing to excuse, and the excuses would legitimize his attitude. If he tries to take a childish tone with you (eg. by mocking) refuse to participate and speak like an adult. It takes a lot for someone to just say, "fine, I admit I'm being petty and childish, but you know what, I'm not ashamed, and I'm gonna do it anyway!" Even Nixon couldn't do it.

It really comes down to one of two things: Either you force him to admit he is treating you unfairly, at which point he should give up (assuming he is not truly malicious, which most academics are not), or make yourself too difficult to bully by fighting back (in which case it doesn't matter if he is malicious).

You also have the option of going through official channels, but I'm skeptical that for your situation it's likely to precipitate an immediate and perceptible improvement (the old joke about a garbage bin labeled "student complaints box"...). If bureaucracies worked, people wouldn't hate them.

But really, if you want to "prevent this situation next time" - the situation is being in a class with a bad instructor. You don't want it? Don't take classes with bad instructors. It's easier than trying to "fix" the instructor, and eventually questions will be asked about why this one instructor has so much trouble getting people to take his course (maybe even by the instructor himself).