Question

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.

Answer 1

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.