The article shows a confusion between training (committing things to muscle memory, or to visual patterns that can be recognized or executed) and teaching (a deeper understanding of a thing). Training is always more testable than teaching, yet for the most part the things we test people on (e.g. multiplication of small numbers, solving quadratic equations) are only evidence that something like mathematical skill and understanding is building up rather than the desired thing itself.
Rote memorization is a skill that's quite useful, but it's only one skill among many others that people should take away from school or university. And just as it's wrong to dismiss certain teaching as "just" rote memorization (e.g. knowing vocabulary in a foreign language) it's also wrong to just omit the teaching part altogether and train people to do well on standardized tests.
Even more, it confuses unconsciously learned skills and skills learned by repetition. Repetition is not always the best or the only way to learn those skills that get mastered at an unconscious level (though it's definitely an important way to achieve unconscious mastery at some things). These sorts of skills often involve physical activity, movement, writing and coordination generally. If a person begins with "poor form" in such a skill, repeating the activity only makes the poor form more ingrained.
Alternative method include something directly guiding student physically and practicing in such a way that you get immediate feedback if you are wrong.
I've been top 50 a few times at a moba I play, and hang out around the top 0.1% of players when I'm active in my game mode. I just have losing streaks these days though since I only play every 3 months. All that fluency is "like dust in the wind" when you stop using it.
Edit: to clarify, I'm probably in the top 35-45% when I'm rusty, which is a massive drop. I can't play top .1% without spending a few months grinding
At least with memory you can recall enough to relearn quickly.
The core knowledge of intelligent thinking, in mathematics and beyond, is the rules of mathematical derivation in the most abstract and universally applicable form. Those rules can be applied in a myriad of daily situations. This universal applicability in problem solving makes the basis of what others consider an intelligent person. If properly formulated and represented for learning, these rules can be memorized in a standard way; in other words, memorization can be a way toward intelligence!"
But... you have to know what a derivative is for in order to be motivated to learn how to do it. If it's all just mindless manipulation, how will one know whether to apply this or that mindless manipulation?
For me it was cause I wanted to beat the shit out of my curve and get scholarship money and letters for grad school. Similar sentiments are found in Hardy’s book I think. If you spend enough time around math professors you will certainly hear a story about so-and-so who is now an acclaimed mathematician who showed up for some undergraduate course when he was 17 or something extremely confused about what the “meaning” of the material was but figured out how to solve all the problems and get an A cause he cared about being good at it (and had the ability to do so).
Rote memorization is a skill that's quite useful, but it's only one skill among many others that people should take away from school or university. And just as it's wrong to dismiss certain teaching as "just" rote memorization (e.g. knowing vocabulary in a foreign language) it's also wrong to just omit the teaching part altogether and train people to do well on standardized tests.