I generally buck the standard advice and memorize first, before trying to understand. Understanding is much easier for me if I can easily hold everything in my working memory.
If anything, I saw those coming from less-advanced math founder on advanced math (arithmetic to algebra, algebra to calculus, calculus to advanced subjects) because they attempted to deal with the subject based on memorization rather than grasping the basic point.
Generally, I did wind-up committing a lot of content to memory but it was and is much easier, even "effortless", when I knew the reason for each part of the content.
Of course, language is slippery and we might be talking about exactly the same thing. So I wouldn't give blanket advise - each person has to find their own learning style (or styles, if it varies by discipline).
A basketball player has memorized how to shoot a free throw, not by cramming by rote, but by practicing.
The equivalent for mathematics is solving problems that you find personally relevant and interesting and which are at the limits of your current abilities. NOT just doing piles of rote calculation. Often solving a hard problem requires a big pile of intermediate calculations, but those a tool in service to some meaningful purpose, not an end in themselves.
Forcing students to work through thousands and thousands of trivial calculation exercises is horrible pedagogy (and a very inefficient use of time).
It’s like doing nothing but shooting drills and getting quizzed on the NBA rule book for years before ever trying to play a game.
No, they start by joining a club where they get professional training, from a very young age. Just playing the game all the time won't make you any better at it. There are millions of children that spent significant part of their childhood playing basketball for fun, and never went past amateur level.
This is just a conclusion from Ericsson famous paper[1]: you need DELIBERATE practice, not just recreational playing.
Playing for fun will just develop bad habits, which you then will have to unlearn, so recreational playing might actually be an obstacle to learning.
[1] http://www.nytimes.com/images/blogs/freakonomics/pdf/Deliber...
But there are many aspects to any complicated skill, including tiny subskills, combination of various subskills, and at a higher-level meta skills like knowing when to change strategies, etc.
The problem with only sport games as a mechanism for feedback is that (a) there’s not necessarily a good way to correct and try again right away if you want to practice one particular thing, (b) situations that arise are not homogeneous enough to compare feedback across trials, (c) there are periods of waiting in between direct practice, (etc.?).
As a result, people who are training to be basketball players spend only like half their time playing realistic games, and the rest of their time on a variety of drills and workouts including general aerobic and strength training, practice running sideways and jumping, shooting / passing / dribbling drills, practice setting screens / rebounding / driving / inbounding the ball, larger-scale coordination drills, and so on. These let the players focus their conscious attention at one level at a time, and repeat a situation with analytical feedback in between.
But those who improve fastest are the ones who figure out what they need to work on (or have a mentor/coach to figure out what they need to work on) and then get lots of feedback about that particular area, in an intentional way. This can even be done within the context of a realistic full game by choosing what to direct attention/focus to.
But you’re missing my basic point, which is that all the skills in basketball are situated within a context. It’s obvious how they fit into playing a game. Students start with the high-level context (by watching / playing full games), and then break it down into small pieces they can work on separately or jointly, and then frequently practice putting those pieces fully back together in the context of the game. I doubt you’ll find anyone seriously training to play basketball who doesn’t play in a semi-realistic game-like situation at least once per week.
In classroom education, often the high-level context is completely missing, and students end up memorizing / drilling on a pile of atomized facts and trivial very low level skills without ever practicing combining those in larger settings, without ever getting a very fast or effective feedback loop going, and without ever having any motivation for going through the practice.
In the case of mathematics education, someone being taught 1-on-1 by a skilled tutor can move I’d say about 4–10 times faster than a student only watching lectures, doing completely independent book reading / trivial homework exercises / quizzes, and then getting back graded homework/quizzes a week afterward. That student can be assigned dramatically harder problems, can be helped along by limited (but well placed) hints, and can be given direct feedback/corrections at every level (from basic arithmetic computations through higher-level strategy to meta skills like strategy selection/evaluation). This is something that can’t be done in basketball because basketball is a group activity; you can’t watch a single player go through the game, and pause/backtrack everyone to the beginning of a situation whenever that player makes a mistake or suboptimal play.
That was sometimes a problem for me, if I picked up something quickly I often found it harder to retain, whereas if I had to work hard to grasp a concept then I would remember it better.
That is why we can have databases that do not fit in 'memory'. Do you see the analogy here?