Teach the rule or teach how to derive the rule

So after class on Thursday a 'fundamentals' student asked me a couple of questions that I now think are worth exploring a bit on the blog because they can lead to some interesting philosophical questions about math, although at the time I was too hungry to answer clearly.

"Why is a negative of a negative a positive?" and "why is a negative number times a negative number a positive number?" My first thought when she asked me these was 'because' and my second thought was 'chicken sandwich now or a donut to tide me over until I get home?' It was particularly ironic that she was asking me these questions because on the first day of the 'Fundamentals' class I ask at least one similar question to gauge how well students are able to respond as part of their math background (and I probably got a few answers like 'chicken sandwich').

These are probably types of questions that teachers of basic math get asked all the time. Mathematicians encounter statements like this all the time too and we need to justify them in terms of the axioms.

To answer the question "Why is a negative of a negative a positive?" we need to know what 'negative' means: for any element x, -x is the element for which x + (-x) = 0.

This also means that -(-x) + (-x) = 0, so x = 0 + x = (-(-x) + (-x)) + x = -(-x) + ((-x) + x) = -(-x) + 0 = -(-x).

The second question, "why is a negative number times a negative number a positive number?" can be justified using the fact that a negative of a negative is a positive: 0 = a*0 = a*(b + (-b)) = a*b + a*(-b)
so a*(-b) = -(a*b) because a*(-b) is the thing you add to a*b to get 0.
Now (-a)*(-b) = -((-a)*b) = -(b*(-a)) = -(-(a*b)) = a*b

When I was asked this question, the 'fundamentals' student stated it as "Why is 3-(-5) = 3+5?" And to me the good answer to tell a elementary/high schooler would be "well, because that is the rule" unless I really wanted to drive them away from math. To me students just have to follow the rules of basic arithmetic and they have to learn them forwards and backwards. Her explanation was something like "We know 3-(-5) = 3+(-1)*(-5) = 3+1*5 = 3+5." And this explanation was no more convincing to me than my "because that is the rule" answer.

The more I think about it, the more I realize I am not sure how I would respond when asked by an elementary or high school student.

So here is where I get all philosophical (or perhaps I am getting not philosophical, but practical). Should you teach your students how to derive the basic rules from the definitions so that they are easier to remember, or do you just tell them the answer is "because that is the rule" ? As a teacher, do you need to know enough basic algebra to derive these rules yourself or can you accept them "because that is the rule" ?