What is the "optimal" use of rote learning in studying (pure) math?

I feel like I wrote a decent answer to the above-titled MSE question so I'm pasting it here. 

How do you solve a jigsaw puzzle? Do you take each piece and try to figure out exactly where it must go first, piece by piece? Or do you start sorting a bunch of pieces by color, texture, until the images start clicking into place? How much of both?

Like puzzle pieces, mathematical concepts gain clarity in context -- e.g. an isolated dark spot on a puzzle piece may be identified as a roof tile only when gathered with other "roof pieces". Or maybe you could have added it with the "skyline pieces" instead. In the same way, there is often more than one way to "understand" a mathematical concept, and each way admits a different set of relations to other concepts. Look up any "intuition" question on MSE and you'll see this at play in the various answers given. It follows that if you are struggling with a particular proof or concept, it's sometimes better to cast your net wide and focus on expanding your context -- that is to say, memorize and move on.

You will find people giving similar advice when reading papers -- skim first, several times if you need to, then drill down as needed. Unfortunately, I don't have anything more specific that can apply to a general case as it even depends on how something is meant to be read, and how well it is written. The best expositors (Halmos comes to mind) mitigate the need for jumping around to some extent and try to keep things linear, but there are still lots of variables like writing style, learning style, and compatibility of background knowledge which make it difficult to give a general solution. The best thing to do, like others have mentioned, is to do the exercises to constantly evaluate your understanding and see where you might need to drill down a bit.