how much did you pay a month/for a month/per month?

Of the three, a native-speaker would be least likely to say "pay for a month," except perhaps in a particular context.

None of these expressions is "ungrammatical." Saying "per month" does not have any particular scientific sound, i have no idea how you come to that conclusion.

I would say that a native-speaker would be most likely to say "how much did you pay a month?" and not unlikely to say "how much did you pay per month?" In the specific context you mentioned, i think a native-speaker would be very unlikely to say "how much did you pay for a month?"

All of which being said, in the United States, a person's university education is most often calculated in credit hours--if you take a course that offers three hours of credit, and you're paying $300 per credit hour, it will cost $900. I can't imagine anyone in the United States asking someone else how much they pay a month for their education, it just wouldn't make any sense.

Thank you, Setanta~

I got the idea of PER being formal/scientific in some books that I've forgotten the name of.

I did not know in the US people pay for their tertiary education not monthly, but in the Far East, we do.... A new piece of knowledge, thank you~

As a general rule, in the United States, people pay by the credit hour. So, for example, if you are taking a course in English literature, it would probably be worth three credit hours. You would attend class three days a week, for one hour. Sometimes a course might be offered as a one day, three hour session, but that would be uncommon. You need a certain number of credit hours to graduate--for a bachelor's degree in arts (BA) or in science (BS), you would usually need 120 credit hours. If you were to take five, three-hour courses each semester (or quarter--some schools divide the school year into semesters, others into quarters), or fewer courses per quarter, although you would enroll for three quarters in a year (and you could enroll for all four quarters), you would have roughly the same number of credit hours in a year, about 30 credit hours per year. About $300 or $400 per credit hour is common (or until recently, was common) meaning that a four year course of study leading to a bachelor's degree in arts or sciences would cost you from about $35,000 to $50,000. The cost may have gone up since the last time i paid attention to the matter.

Do you have to pay for US history/propaganda classes?

No, JTT, we don't have to pay for American history/propaganda classes.. Since you're as American as the rest of us, I'd have thought you remembered. The government pays for all the courses, they're compulsory through graduate school, and their content is strictly controlled. There are compulsory tests at least monthly. If you don't get at least 98% "correct", your driver's license is revoked and you lose your job. If you get 98% or more, you get a lollipop or a tootsie-pop, your choice. You do remember, don't you?

You might pay for a month to participate in any activity that lasts one month. "How much did you pay for a month of fitness classes?"

That must be why USians can't think for themselves.

A sentence: What my friend paid less than a pound a day for last year she had to pay two dollars a day at a major Reading establishment last Christmas.

Is FOR in FOR LAST YEAR used with PAID in PAID LESS THAN, or FOR used with LAST YEAR to show the purpose of something related to last year?

You pay for something. Whatever that "something" is in this case, is what the "for" refers to--the word order is different, but that's what it's talking about, Doesn't refer to the year. Question is, why was she paying in pounds during the year and in dollars at Christmas?