Go to school vs. go to A school

When a pupil goes to school (not A school) it is because “school” is used for its main purpose – learning. If we talk about a teacher, we say that he/she works at A school because working at a school is somehow not using it for its main purpose…

But what if a teacher just goes to (a) school. E.g. My mother is a teacher, she works at a school so every morning she goes to school – or to A school?

Thank you in advance.

We use ‘a’ school when speaking about where a teacher works because we are speaking about the establishment itself, so it would be correct to say
My mother is a teacher, she works at/in a school so every morning she goes to school.