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?
