Sometimes, the correct response to a question is unclear. For instance, if I ask you: "So, you didn't get to go to the mall yesterday?" (and you did not go), the correct response, natively spoken, would be, "no".
The common logic would be that, "no, I didn't get to go" is a double negative statement, which makes it a positive statement as the two negative words-"no" added to "didn't (did not)"-would cancel each other, and means that you DID actually get to go to the mall (a positive statement).
I know this is what you are taught here, and normally it is true.
In this case however, the person responding, would say, "no", because they are simply confirming the statement, not adding to it. i.e.:
So, you didn't get to go to the mall yesterday?"
"no" or "no, I didn't"
The second response, "no, I didn't" makes your answer clearer-that you are confirming what the other person asked.
Please don't stress too much when you hear a native do this; even WE sometimes ask for clarification when all we get is a one word answer, because we do know about the double negative rule.
Responding by saying, "no, I didn't", shows that you are confirming their assumption, not adding to their sentence.
I hope this is clear. If not, ask me here and I'll try to help more. (When I get on here next. See my profile and the SpeakAlley blog for more.)