Memorization is more like a shortcut. You don't need to go through the problem solving process to know the result. But with understanding, you master the heuristic factors needed to know when to take the shortcut and when to go through the problem solving route.
The Dreyfus Skill Model [0] is a good explanation. Novice typically have to memorize, then as they master the subject, their decision making becomes more heuristic based.
LLMs don't do well with heuristics, and by the times you've nailed down all the problems data, you could have been done. What they excels at is memorization, but all the formulaic stuff have been extracted into frameworks and libraries for the most popular languages.
[0]: https://en.wikipedia.org/wiki/Dreyfus_model_of_skill_acquisi...
If you're trying to expand polynomials and you constantly have to re-derive multiplication from first principles, you're never going to make any progress on expanding polynomials.
> "Just memorize it, it'll make your life so much easier."
That is because you evaluate cost of memorization to 0, because someone else is paying it. And you evaluate the cost of making mistakes due to constantly forgetting and being unable to correct to 0, because simply the kid gets blamed for not having perfect memory.
> or translate a language when they haven't memorized the vocabulary words in the lesson so they have to look up each one
Teaching language by having people translate a lot is an outdated pedagogy - it simply did not produced people capable to understand and produce the language. If the kids are translating sentences word by word, there was something going on wrongly before.
As someone who never learned my multiplication tables – it’s fine. I have a few cached lookups and my brain is fast at factoring.
8*6? Oh that’s just 4*2*6= 4*12 = 48. Easy :)
maybe they pay hits a reward centre in my brain.