wWhat's the fault to be a beaut?

Some would also debate that since it’s a title more of the words should be capitalised, but regardless it would be better to capitalise “What’s” since it’s the first letter in a sentence.

This morning, as usual, I wore over-ear headphones while doing math problems during thein class.

“Over-ear” is unnecessary - “headphones” tends to create the same image, although sometimes it is also used a more general term. “Earbuds” or “earphones” would be used for the ones that go inside the ear, with “AirPods” being a type of earphones/earbuds provided by Apple. I included AirPods in this because I think they are probably one of the more common types to see, especially nowadays. Additionally, I will say that “earphones” and “earbuds” aren’t really heard that often, or at least I personally don’t hear those very often.
“During the class” is technically right, but “in class” sounds more natural.

And this teacher Ppersonally said：“ , “you can do math problems or listen to me, just don't make noise". So I reflected on the sentence, and my action should have been admitted by him. During the class, I found the teacher staring at me , and Iwe looked at each other by no means.

I’m a little confused by what this sentence means.
“at each other” suggests that there are multiple people involved, so you should use we.
“By no means” is essentially another way of saying “not” as in “not in any way”, but in a more sophisticated way. It would go after the verb, and would usually be used after am, is, are, etc. (For example “I am by no means a professional in this field” meaning “I am not (in any way) a professional in this field”. The “in any way” part in the meaning basically emphasises that there is absolutely no chance that I am a professional in this field, so “by no means” means that there is absolutely no way that the rest of the sentence (when you take out the “by no means”) is true. However in this sentence it wouldn’t make sense. If you wanted to say that there was no meaning behind you staring at each other, then you would replace “by no means” with “meaninglessly”. Otherwise I will need some clarification on what you were trying to say.
The “my action should have been admitted by him” is also a little confusing to me.
Also make sure to have spaces after full stops, commas, exclamation marks or question marks. There’s no need to put a space before any of them.

Before a A few minutes thatbefore the class was over, he clapped my back and said to me:" , "your headphones isare too conspicuous, please wear small in-ear headphoneearphones/earbuds."

“That” isn’t necessary in either case.
“Before a few minutes the class was over” essentially means that in only a few minutes, as in “before a few minutes (had passed)”, the class was over. It indicates a much smaller jump in time between your previous sentence and this one. It also puts more focus on the class being over instead of a little before the class was over (as in it sets the sentence up in a time when the class is over or afterwards).
“A few minutes before” uses the time that the class ends as a reference time instead, and would emphasise that this sentence is being written about a time that is a little before the class ends (as in it sets the sentence up in a time while class is still in session, and only a few minutes before it ends).
“In-ear headphones” sounds weird to me, but that might only be me. Personally I would probably use earphones or earbuds instead.

I observed his haircut, it' was "Slicked-back".

Observed = past tense, so it would make more sense to say “it was” instead of “it is”.

I was prepared to saidy :" teacher , your haircut is also conspicuous and catch my eyes to let me be distracts med, please cut the Ccrew cut.”

“Prepared to say” sounds like you were getting the confidence to say it, and that you did actually end up saying it.
“Was prepared to say” on the other hand would mean that you were ready to say it, but it also implies that something stopped you and you didn’t end up saying it.
When talking to the teacher, then you would say “Mr/Mrs/Miss/Ms” (depending on gender and age/marital status, but typically for a female it’s good to go with “Ms” as a default), followed by their last name. You could also say “Sir” or “Miss”. But you would never say “teacher” because that’s not particularly polite. You would say teacher if you’re saying someone is a teacher, or when talking about a teacher (you would say “the teacher” here), but not when actually talking to the teacher.

"But I cant'ouldn’t unless I wanted to fail this course.

Technically the original was correct by itself, but you would have to make these changes to keep everything in the past tense since you’re talking about the past.

Finally, I said:" "I see, I will wear small headphones in the followingnext class."

Optional, but personally I think that “next” would be more common, although “following” might be a bit more formal.

In my opinion, the noise-canceling effect of over-ear headphones is much stronger than that of in-ear headearphones. So what's the fault of the conspicuousness?

Optional, this is mainly out of preference.

And this teacher Ppersonally said： to me: “ you can do math problems or listen to me, just don't make noise". So I reflected on the sentence, and my action should have been adpermitted by him. During the class, I founnoticed the teacher staring at me , and Iwe looked at each other by no means.

Not sure exactly what you mean by "and I looked at each other by no means".

BeforeAfter a few minutes that the class was over, he clapped my back and said to me:" "your headphones isare too conspicuous, please wear small in-ear headphones."

I prepared to said :" y: "teacher , your haircut is also conspicuous and catchught my eyes to let me beand distracted me, please cut the Ccrew cut."

"I was about to say" would sound more natural.

"But I cant't unless I want to fail this course.

Finally, I said:" "I see, I will wear small headphones in the following class."

In my opinion, the noise-canceling effect of over-ear headphones is much stronger than that of in-ear headphones. So what's the fault of problem withe conspicuousness?

This morning, as usual, I wore over-ear headphones while doing math problems during the class.

And this teacher Ppersonally said：“ “you can do math problems or listen to me, just don't make noise"." So I reflected on the sentence, and my action should have been admitted by him. During the class, I found the teacher staring at me , and Iwe looked at each other by no means.

I'm not sure what "by no means" would mean here.

Before aA few minutes that thbefore class was overended, he clapped my back and said to me:" y "Your headphones isare too conspicuous, please wear small in-ear headphones."

I prepared to said :" twas about to say "Teacher , your haircut is also conspicuous and, it catches my eyes to let me beand distracts med, please cut the Cget a crew cut."

We call teachers Mr./Mrs. Last Name. Calling them "Teacher" can be considered rude.

"But I cant't unless I want to fail this course.

Finally, I said:" "I see, I will wear small headphones in the following classfrom now on."

Sounds a bit more natural.

In my opinion, the noise-canceling effect of over-ear headphones is much stronger than that of in-ear headphones. So what's the fault oproblem if their conspicuousness?