Discrete methods

I'm a -huge- fan of discrete methods:
that is, looking at "differences" of series (qua discrete
derivatives), discrete convolutions (over $\bZ$ before $\bR$)
and so forth.
...summation as discrete integration, etc.
[Incidentally, note that differentiation is represented by convolving
with $-\delta'$ b/c convolution negative-commutes,
and the identity is convolving with $\delta$.]

This is a very useful, beautiful and interesting analogy,
and helps bridge the "continuous vs. discrete" in a very nice way.

In particular, discrete stuff is very concrete, and even kids can play
with it. Further, it clarifies some complicated stuff (convolution is
my key EG: it took me a -while- to understand it, but the discrete
picture is so easy!)
[if my memory were better, I bet I'd be able to see integration
 better from this POV: integration over a compact interval is the
 limiting case of finite sums.
 The problem with series is that they start out infinite, which is
 tricky (here they're analogous to -improper- integrals, integration
 over non-compact, non-finite volume spaces)]
