Math Identity

The question came up: "Do you have to be obedient in math to do well at math?"  And a few of us thought, what does obedience have to do with math?

I think it means you solve problems the way the teacher modeled the problems first.  I think it means you follow a formula.  I think it means you don't extend yourself and think about what something really means or implies. 

I am often obedient this way--I tend to be procedural.  But at the same time, I think of myself as creative, discerning, and questioning.  I want to know why things work or what they really mean, and how if I know something and am able to solve a problem it is truly going to make a difference in my life.  Is understanding a complex math skill going to revolutionize how I live or the way I think?  Or is it just knowledge for knowledge sake?  If I can't form a connection and if the "fact" or procedure is too complex to spend any more time on, then most likely I am going to begin to feel frustrated and detered--and my nature, I am also not one to give up too quickly. 

So what about the problem -4 - (-5)?  How can you take away a negative value from -4?  I reasoned this using the beans, and I found a way to explain my thought by representing the numbers using a number line.  I crossed of 4 segments on the number line (as if I was taking away or canceling the -4).  Then I took one more negative number--crossed off negative 5 (the stand-alone) but reasoned you can't take something less than 0 from the equation.  I also reasoned that what you did to one side of the equation, must be done to the opposite side, only in the opposite form (so I added +1).  Now all of this will make little sense to a reader, because I know I'm not explaining very well, but in my mind, it makes perfect sense.  But I'm still slightly amused that I can operate (because I've been obedient) and I am hardly able to explain why I am doing an operation.  I always say, in a sing-song way, "Minus a negative is plus a positive" and then I change the signs.  -4 - (-5) looks like -4 +5 which is +1. 

Lauren brought up the idea that - (-5) was the same thing as -1 x -5 = +5.  But this led to an entirely different question--this one really gets me--how can we explain or reason why a negative number times a negative number is a positive number?  That seems strange to me--if you have debt and you multiply debt by debt (provided they are equal amounts of debt) shouldn't that be debt to the second power?  Why would it get smaller?  I'd imagine it would only get larger! 
But then in our own language, why is a double-negative a positive: I don't got none: really means, I have some. 

Who can explain this to me?  It seems silly and yet I've obediently followed the rule.  But I'll gladly take any creative explanations, or historical explanations.

And lastly, I am fascinated with the theories behind the creation of negative numbers--they were based on commerce and supported the idea that people could be in debt or owed debt, but there needed to be a way to express something that they didn't have or had to repay.