Vriska said:
I studied calculus (equivalent to calc 2 perhaps) back in school last year but it was just procedure, patters and tricks. I wanted to do more so I picked up spivaks calculus and it's taking me far too long.
One learns calculus over a lifetime. It's usual to begin by learning routine procedures and pick up the rigorous content later. But the rigorous content can't be completely appreciated by reading one book because the rigorous content of one level of calculus is usually just a special case of more general mathematics for higher levels of calculus - e.g. material in the first course of calculus is generalized to tackle "calculus on manifolds".
To a great extent, learning the rigorous content of mathematics is a matter of acclimating to particular atmospheres of abstractions. For example, a calculus student can eventually become accustomed to doing complicated epsilon-delta reasoning, but when that student takes a first course in abstract algebra, its a different ball game to deal with homomorphisms, quotient groups etc. Each new atmosphere of abstractions causes a certain level of discomfort until one gets used to it.
Each branch and each level of mathematics also has its own bag of mechanic procedures and tricks.
How thoroughly one should study math and science texts is an interesting question. For example, since the big hurdle is acclimating yourself to new atmospheres of abstractions and bags of tricks, you could conceivable study a mathematical text in spotty manner, skipping over some things you don't understand, and acclimate yourself to the new type of thinking that is required. If you do this, then you might be as well off as someone studied the text thoroughly a few years ago and has forgotten some of the material.
People who thoroughly "master" the material in a textbook sometimes become so attached to the ideas presented in the textbook that they increase their difficulty in understanding generalizations of the material or different approaches to it. For example, someone who begins to think that the continuity of a function is exclusively presented by the treatment of the continuity of a real valued function of a real variable may find it hard to acclimate to the concept of the continuity of a function in a metric space or the continuity of a function in a topological space.
(An amusing example I encounter is talking to people who have concentrated on the theory of electronic signal processing. Every real life problem gets translated to some scenario that involves filters, spectrum, and bandwith.)
If your main interest in in math puzzles and math olympiad-type problems, you obviously want to master the techniques useful in solving those types of problems. I don't know how much of the material in Spivak's books is relevant to that task.
