Some Unanswered Questions!!

I have started thinking of the basic mathematical operations again. The thought process began when my father asked me - "why does negative number multiplied with a negative number produce a positive number?".

The basic reasoning on which we are taught multiplication does not seem to work here or if it does I just do not understand how? As per how we are taught multiplication if some one says 'a x b' then it would mean what is the value of adding 'a' 'b' number of times or what is the result of adding 'b' 'a' number of times. This scheme works perfectly when both a and b are positive. This is how we are first taught multiplication and once we understand this concept we start overlooking many things.

It works well even for if one of then is positive and the other is negative. For example 'a x -b' would imply -b added a number of times. As adding a, -b number of times seems absurd (the commutative law of multiplication seems questionable).

What if we have both the numbers negative. We all know because our teachers told that negative number multiplied with a negative number will give us a positive result. But, if I reason the way I have done before I can do no further how do I add something negative number of times. After all you can do a thing once, twice, etc.... or not do at all but how do you do it -1 time?

I am still stuck with this question. Answers and suggestions are welcome.