I studied a lot of math, pure and applied,
taught it, applied it, published research in it, etc.
so developed some ideas relevant to the
OP.<p>For<p>> To the mathematician this material,
together with examples showing why the
definitions chosen are the correct ones
and how the theorems can be put to
practical use, is the essence of
mathematics.<p>Is good, but more is needed.<p>(1) Plan to go over the material more than
once. The early passes are just to get a
general idea what is going on.<p>In such passes, for the proofs, they are
usually the near the end of what to study
and not the first.<p>(2) When get to the proofs, for each proof
and each of the hypotheses (<i>givens</i>,
assumptions), try to see where the proof
uses the hypothesis.<p>Next, try to see what are the more
important earlier theorems used in the
proof. So, sure, in this way might begin
to see some of how one result leads to or
depends on another and have something of a
<i>web</i>, acyclic directed graph, of results.<p>And try to see what are the core, clever
ideas used in the proof.<p>(3) For still more if you have time, and
likely you will not, can use the P. Halmos
advice, roughly,<p>"Consider changes in the hypotheses and
conclusions that make the theorem false or
still true."<p>(4) But, note that to solve exercises or
apply or extend the theory, need some
ideas. So, where do such ideas come from?
In my experience, heavily the ideas come
from intuitive views of the subject.<p>So, my best suggestion is to try to
develop some intuitive ideas about the
material. Definitely be willing to draw
pictures, maybe on paper, maybe only in
your head.<p>In the end, a solution or proof does not
depend on intuitive ideas, but finding a
solutions or proof can make use of a lot
in intuitive ideas.<p>For research, most of the above applies,
but IMHO there are more techniques needed.