Derivation and Proof

This post has two parts:
I: Derivation and Math, and II: Derivation and life...

I:
I don't know why but I (sometimes) really enjoy doing proofs and derivations. Throughout my college career I've always focused more on why something is true than on how to use it, which I guess explains why I became a math major. I feel like the concept of derivation is a philosophically satisfying one because of the following things:

1) By deriving something, you are empowered with the knowledge that has been derived because now you know why it is true, and under what conditions. This gives you a deeper understanding than you would have if you were just given the result on a silver platter. Through the process of proof/derivation, you are one of the owners of that knowledge.

2) Independent thinking. You can't create "new" knowledge (which is essentially what you're trying to do, even if it has been done before) without thinking for yourself. So to me, proofs give you faith in your reasoning abilities.

If you've ever wondered why genius physicists and mathematicians are crazy and sort of outside the scope of social norms, it's because on a daily basis they must challenge the assumptions that they and others hold about the world around them, and must use their independent, logical, critical thinking to re-evaluate how the world functions.

That's the great thing about math and (to some extent) computer science. There's no room for misinformation. If you don't think something is true, you need to go no further than your imagination to show that it is not.

3) For me personally, I can't remember something unless I know why it is true.

This is why I feel math education is sooo important, unlike some other people. It teaches people (ideally) what it takes to prove something, and gives them confidence in the validity of their independent thinking. Sure, I wouldn't say everyone needs to know calculus, but I would say everyone needs to understand how to prove something, and why the math they do know is true.

II:
I've had this idea for a while, hopefully I can do it justice here:

Let me give an example before I describe the general idea. Some people raise their children by creating a system of approval whereby the children gain the approval of the parent by acting in a desired way, for example, by being courteous. When they are not courteous, they are punished. In some cases I've seen people react by only behaving in those ways when their parent is present and able to judge their activities. I'm being really sloppy in my description here... This is "arriving" at the desired behavior without allowing the kid to "derive" it for themselves. There is a reason for courtesy, for doing chores, etc etc. If you don't let them understand that, and instead force them to act a certain way, you're not creating genuine empathy and responsibility etc.

I think developing morals and values, emotions, and many other things, can be looked at in this way... a conflict between spoon-feeding the arrived result (the end) and the derivation (the means that will eventually reach that end). I guess, in short, it's about letting people figure out why something is done a certain way, rather than just telling them how it's done.

In pedagogy this would be the banking system of education vs the problem posing method?

More examples, because this idea sounded way cooler in my head than on paper and I'm running out of stuff to say but still feel like there's more to my idea.

You can't be told what you want to do with your life, you have to derive it for yourself.

Laws are only meaningful if people follow them of their own will... so people have to derive the intent of the law and act accordingly. I feel like too often the spirit of the law is lost in meaningless verbiage, which, while intended to make the law fool-proof, does exactly the opposite.

Emotions- sometimes I feel like people feel a certain way because they know they're supposed to, not because they genuinely feel that way. The way I word this makes it sound like this is a rare event, when really everyone is guilty of this from time to time. But I think about this a lot in terms of derivation... am I just feeling something because I know I should? Or did I really reach this emotional conclusion on my own? In an earlier post, I was basically questioning myself in this way.

I pride myself on being constantly self-critical. Perceiving reality as it truly is, and me as I truly am, is important to my development, as it is for everyone. Inevitably, there will be blind spots in my self-evaluation, but reevaluating things I held as assumptions and re-deriving them is sometimes painful but healthy, right?

Edit: Even more that I meant to write about!
Deriving success in school... I feel like focusing on getting good grades or passing tests is the wrong way to go. Instead, focusing on what interests you, and trying to answer philosophical questions that come up as you digest the material, and generally just trying to actually learn the material instead of act like you've learned it, works better and is more enjoyable... at least I think so. Same thing for like building resumes, getting into college/grad school etc etc... doing it just for the paper is never going to get you as far as doing it for you. And in doing so, you'll get a better result anyway. Deriving "success" in life... I guess...

This, by the way, is why standardized testing will never be a successful way of measuring student progress. It implies, for one thing, that the value of what a student has learned depends on the opinion of someone else, when it really should be the student judging for themselves whether they have learned the material or not. Now obviously, there's an implied contradiction because what about grades and the evaluation teachers do of students? I would say that a teacher's role is more to provide a context in which the student can judge whether they have learned the material... if that makes any sense. Like you do need some feedback to be able to progress and challenge yourself, but standardized tests don't really give you that feedback, they just give you the number. So that tells you that the only importance that knowledge had was that number. That wasn't the intent but that's how it might be interpreted, and has been interpreted when you teach to the test and all that...so deriving metrics for learning, or something?