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I'm advocating for a mixed approach that serves more kids, and arguing that you had such a mixed approach yourself but don't seem to acknowledge it.

Memorization (done properly, that is - I invoked "spaced repetition", an evidence-based learning technique from the field of education, you're the one talking about corporal punishment from nuns) is effective in precisely this and related domains having tons of minutiae.

It's not that learning the process is inefficient, that's not what I meant - learning only the process and not focusing on rote memorization as well leaves you with only the process to rely on when learning further math (your experience sounds like you got both, regarding multiplication).

Relying on only rules/processes to complete intermediate steps that are not the subject under instruction is what is inefficient. Using rules to reach simple multiplication facts when trying to learn algebra or even just long division is brutal for kids with any attention difficulty whatsoever. By the time they've solved the multiplication answer they wanted, they've lost the thread on the new concept. Rote memorization reduces the effort needed to use multiplication when learning everything else. It doesn't feel that you're reading very carefully here, but it could be me who failed to make myself plain.

I myself am a process guy and high on pattern-seeking. I write software for a living and live in abstractions layered on abstractions - even the physics is invisible lol, nothing (but fans and I guess HDD heads where still used) ever moves. It all feels like pretend!

My point is that understanding processes and relationships in the space of numbers can arise FROM being forced to learn many small truths over and over. A student can identify patterns (the shortcuts) from just learning the facts. Similarly you can get to the facts if you understand the process - like most math there's a lovely symmetry there that you seem unwilling to agree with me about. They both inform and train the brain differently and you seem to have benefitted from that yourself.

We need both, and rote memorization is especially useful in a small number of domains, irreplaceable. Anyone who has gone through an Anatomy & Physiology class successfully will agree too, and I can give more examples. There's no "process" or rules involved.

Anyway, I think we're mostly talking past each other and probably mostly agree.