Is it bad to tell my school that I’m looking elsewhere?

I currently had an interview for a DOE school. They asked if they could get in touch with my supervisor or principal for a reference. I was honest with them and told them that my school does not know that I have been applying. They are a private school and not affiliated with the DOE.

Should I tell my supervisor that I am applying to ask if she could be a reference. I’ve had positive experiences at this school and have done well.