1. What did you learn in math class today?

2. She is a music teacher. She is very strict.

3. He is a career teacher. He is kind-hearted.

4. He is a Chinese characters teacher. He is humorous.
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Are they all correct? Do we have to capitalize "career"? It was used a subject name. • English - Writeacher, Tuesday, March 14, 2017 at 8:35am
All sentences look good.

Usually, when someone teaches a class in the US in which students learn about careers, it's called Careers -- with capital C and in plural form.

I'd rephrase #4 like this:
He is a teacher of Chinese characters.
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Thank you. then, which expressions are correct?

5. He is a career teacher.
6. He is a Careers teacher.
7. He is a teacher of career.
8. He is a teacher of Careers.

If "Careers" is the name of the class, then, yes, it should be capitalized. Above, #5, if it was used as an adjective to describe what kind of teacher he was/is, it is not capitalized.

Out of the four expressions you provided, #6 and #8 are correct.

#6 "He is a Careers teacher" is correct because it correctly uses the plural form of "career" when referring to a subject name. In this case, "Careers" is a subject or class where students learn about different career paths.

#8 "He is a teacher of Careers" is also correct because it states that the person is a teacher who teaches the subject or class of Careers.

On the other hand, #5 "He is a career teacher" and #7 "He is a teacher of career" are not commonly used or not as natural in English. It is more common to use the plural form "Careers" when referring to a subject name rather than using the singular form "career." Additionally, when using the phrase "teacher of" it is more common to specify the subject or field of study, such as "teacher of English" or "teacher of mathematics."

So in summary, expressions #6 "He is a Careers teacher" and #8 "He is a teacher of Careers" are the correct ones to use.