0 = 0* (-1) (0* x = 0 for all x)
= (1 + (-1))* (-1) (x + -x = 0)
= 1* (-1) + (-1)* (-1) (distributivity)
= (-1) + (-1)* (-1) (1* x = x for all x)
1 = (-1)* (-1) (equality; add 1 to both sides)
So assumption vs. definition vs. theorem can explain part of it. Another part is familiarity. 1 + 1 = 2 is ubiquitous in daily life. Depending on what you think about, (-1)* (-1) = 1 may not be. You probably have lots of interpretations of 1 + 1 = 2 (counting, number line, shopping, money, etc.) How many such interpretations of (-1)* (-1) = 1 do you have?
I highly recommend this book, written by two mathematicians:
It's now about mathematics per se, but about learning how to learn and think better. Good luck.
What I am doing is trying to relate every single new thing I am learning with real life examples, only to make sure that I have learned* that thing. Most of the times I fail at that. But I guess it's not always possible.
For example (taken from the article I mentioned on the other topic), when you think of a derivative, what is the first thing that comes to your mind?
The equation that defines it? (http://calnewport.com/blog/wp-content/uploads/2008/11/deriva...)
Or this intuitive image? http://calnewport.com/blog/wp-content/uploads/2008/11/tangen...