What's the difference between 我是老师 or 我当老师? Sometimes when I take a 滴滴打车 taxi to the university campus, the driver asks me if I'm a student, and I reply something along the lines of:


  我是老师
  I'm a teacher
  Wǒ shì lǎoshī


As far as I can tell, this is grammatically correct.  However, more than once, the driver replied:


  你当老师
  You're a teacher
  Nǐ dāng lǎoshī


It feels almost like they're correcting me, but I don't really understand why.  It's possible 我是老师 sounds more along the lines of what a child might say as it's very simple.  (It's also possible I simply misheard them.)

Question: What's the difference between 我是老师 or 我当老师?

Among the zillions of meanings of 当, one is "to act as", so maybe 我当老师 has a meaning closer to "I perform the role of a teacher".  But I still see no concrete reason that 我是老师 is wrong or inferior.
我是老师 = I am a teacher" It is completely correct

当 is a verb that means 'to be/ working as/ take on the role or position of' 

你当老师= you being a teacher

你来当老师 = you'll be a teacher

你去当老师 = you go and be a teacher

你当上了老师 = you've became a teacher


  The driver's question "你当老师?" (you being a teacher?) was a rhetorical one. That imply "It is hard to believe people as young as you would be a teacher." He could had asked "你是老师?" and the meaning would be the same.