The rote learner learned nothing more than the ability to regurgitate a mechanical process while the academic learner learned understanding, in my view. You have to work through some examples to solidify it but that should be totally secondary to understanding the meaning and concepts. Like a physics teacher of mine put it, rote math learning is like teaching sea lions to honk horns for fish - students give the right memorized answer and get the grades without really learning anything.
In my experience, it's easy to think one has understanding without actually understanding. The process of applying that tentative understanding to a problem to test it is essential to figuring out what the flaws are in the tentative understanding so that real understanding can eventually be achieved. Once it gets to the point that a person can crank through problems without significant errors, then I think further problems don't do much good except as an occasional refresher.
Hence the importance of word problems, I think. They force a synthesis of abstract and concrete that really tests/enhances understanding on all levels.
Everyone can learn to solve equations quickly if they’re forced to do it 1000 times. They might forget it the day after the test, the might not know why they’re doing anything, but you can be damn certain you know what you’re doing if just for that brief window in time.
It’s also clear if you’re succeeding or failing. I’ve noticed that private school parents in the US seem to hate homework. The kids get three optional problems a week and no tests. All the time that they might’ve spent thinking on their own about problems is instead spent in class, which really limits how much new stuff can be taught. The kids think they know everything because there are no grades. The kids that actually perform at grade level have private tutors giving them homework, while the rest are just playing fortnite and unable to do fractions.
> They might forget it the day after the test, the might not know why they’re doing anything, but you can be damn certain you know what you’re doing if just for that brief window in time.
What was the point then? It's a complete waste of time in that scenario.
Applying reasoning to a similar but more familiar domain is a good sanity check for said reasoning. Let's then apply what you said to programming. Would you say that the "academic learner" who read a lot of CS and programming books but haven't written a single program on an actual computer learned understanding, while a "rote learner" who keeps going through hundreds of small programming exercises learned to "regurgitate a mechanical process"? Doesn't pass the sniff test to me.
The way I see it, doing the thing gives you ample opportunity to figure out theory on your own - repeating a menial task prompts the brain to invent shortcuts. Meanwhile I can't imagine someone learning only theory to not fail spectacularly at practical tasks - because theory doesn't capture everything; there are lots of practical, intangible things that crop up when you start doing the work. On top of that, it's easy to feel you understand something without actually understanding it.
The best way, IMO, is what philipov's father said - you have to alternate between theory and practice. It's how you can build understanding faster while continuously validating it.
I did say you need to do some exercises, just not an extreme amount or at the expense of theory. And these math exercises are different from programming - they amount to applying some process to different numbers over and over again while at least there would be some degree of problem solving if you wrote a bunch of genuinely different programs.
