I'd be fine with a random anesthesiologist teaching my grade schooler, but I'd hesitate to have a random elementary school teacher giving him anesthesia.
Elementary school teaching: a teaching certificate.
University professor: PhD, pre-tenure work & published research (Note that there are less-qualified people teaching in universities that are compensated much less.)
Nurse practitioner: Master's degree (undergraduate plus 2 years)
Anesthesiologist: Undergraduate + medical degree (4 years) + residency (4 years) + possible specialty training.