You are correct.
The chance of rolling exactly a 6 on a dice is 1/6. So to get a 6 three times in a row, your chances are (1/6) * (1/6) * (1/6) = 1/216.
I took stats last year so I'll have to dig around for my notes on this, but I know there are situations in which you do add probabilities together like your teacher did, but not in this situation.
I *THINK* what your teacher was thinking about is if you roll a dice three times, what is the probability that you will roll a 6 at least one of the times. In which case, you have (1/6) + (1/6) + (1/6) = 3/6 chance of rolling a 6 at least once.
Your teachers math of (1/6) + (1/6) + (1/6) = 3/18 is wrong. When adding fractions with like denominators, you don't add the denominators together.