# Proof vs Practice

I've been brushing up on a lot of basic arithmetic, algebra, and logic as I work towards a review of calculus (and beyond), and I keep noticing that in order to fully understand many principles in the abstract, I really have to include some practice with concrete examples.

I might struggle through a rigorous proof and manage to understand each logical step, but when I step back to view the whole, it's often too much to hold at once. On the other hand, I could learn an algorithm and manage to execute it properly without understanding why it works.

But if I do my best to follow the proof, and then apply the result on some practice problems, I often find that if I revisit the proof, I understand it fully. Tracking the variables and their behavior takes less mental effort once I've seen some examples, and often a statement that was difficult to parse in abstract language is straight-forward, and even intuitive, once I've worked through some concrete examples. I might have to go back and forth between proof and practice a few times to really nail it down.

I'm sure this is nothing new, but I'm constantly astonished at how mixing these two approaches can get me over hurdles that once seemed insurmountable. Is this the conventional wisdom in math education?

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Not sure if it is a legitimate question or a rant masked a question. +1 anyway. –  Git Gud Mar 13 at 19:32
I wonder if what you call "practice" I call "examples". When I'm stuck on a proof, I often examine some specific situations in which the proof applies and others when it doesn't apply (counter-examples, in a way). If that doesn't get me unstuck, then I'm really in trouble. –  Todd Wilcox Mar 13 at 19:35
It would be very nice if all (or even many) students understood that the abstractions deal, ultimately, with concrete objects. –  André Nicolas Mar 13 at 19:39
@GitGud Yeah, that crossed my mind as I was posting, but I though I'd put it out there. I was momentarily overcome by my own enthusiasm :) –  ivan Mar 13 at 19:41
My impression is that your experience is pretty common. People seem to fall on a spectrum: some of us $-$ I’m one $-$ generally do best by starting with the abstraction and then looking at examples, but I think that we’re in a minority. Others generally do best by looking at a number of examples before ever seeing the abstraction. Most probably fall somewhere between the two extremes. But no matter where one falls, one eventually needs a fund of good examples on which to test ideas. –  Brian M. Scott Mar 13 at 21:17