Your instructor is alive within the Tao. :)
Having read through most of this thread, I'm a little confused. On one hand, you want to understand the fundamentals, but on the other you are only concerned with math that is useful for geology and geography.
If you really want to grok proofs- how to read and write them- you need to get comfortable with meaninglessness. Math is a game where we make up the rules and see what happens. Formal proofs are what we use to make sure our rules, and their alleged consequences, are consistent with each other according to some more basic rules of logic, which are themselves made up. If it sounds like castles upon castles in the air, that's because it is. But guess what- in the end, we can typically use this abstract machinery to solve real problems.
If you are only interested in this for the purpose of knowing which extra useless notation to put on your test, then frankly you're probably better off letting those points go, passing the class with a B (or whatever), and thinking about other things.
If instead you want to know more about how to read, write, and come up with proofs, this book is one of my favorites.