Comment by raincole
3 days ago
During my elementary school years, there was a teacher who told me that I didn't need to memorize it as long as I understand them. I taught he was the coolest guy ever.
Only when I got late twenties, I realized how wrong he was. Memorization and understanding go hand in hand, but if one of them has to come first than it's memorization. He probably said that because that was what kids (who were forced to do rote memorization) wanted to hear.
You could argue this is just moving the memorization to meta-facts, but I found all throughout school that if you understand some slightly higher level key thing, memorization at the level you're supposed to be working in becomes at best a slight shortcut for some things. You can derive it all on the fly.
Sort of like how most of the trigonometric identities that kids are made to memorize fall out immediately from e^iθ = cosθ+isinθ (could be taken as the definitions of cos,sin), e^ae^b=e^(a+b) (a fact they knew before learning trig), and a little bit of basic algebraic fiddling.
Or like how inverse Fourier transforms are just the obvious extension of the idea behind writing a 2-d vector as a sum of its x and y projections. If you get the 2d thing, accept that it works the exact same in n-d (including n infinite), accept integrals are just generalized sums, and functions are vectors, and I guess remember that e^iwt are the basis you want, you can reason through what the formula must be immediately.
> you can reason through what the formula must be immediately.
At least up to various factors of 2π ;-)
Those you just keep sprinkling around haphazardly until it's unitary. It's like more struts/boosters in Kerbal space program.
Probably. I hated memorization when I was a student too, because it was boring. But as soon as I did some teaching, my attitude changed to, "Just memorize it, it'll make your life so much easier." It's rough watching kids try to multiply when they don't have their times tables memorized, or translate a language when they haven't memorized the vocabulary words in the lesson so they have to look up each one.
There's things that you need to know (2*2 = 4) and there are things that you need to understand (multiplication rules). Both can happen with practice, but they're not that related.
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...
I think the problem is that in spots where the concepts build on one another, you need to memorize the lower level concepts or else it'll be too hard to make progress on the higher level concepts.
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.
I never memorized multiplication tables and was always one of those "good in math" kids. An attempt to memorize that stuff ended with me confusing results and being unable to guess when I did something wrong. Knowing "tricks" and understanding how multiplication works makes life easier.
> "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.
> It's rough watching kids try to multiply when they don't have their times tables memorized
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 :)
Smart people can get away with tricks like that.
It's a lot easier to memorize things when it's your job i find.
maybe they pay hits a reward centre in my brain.
As with most things, it depends. If you truly do understand something, then you can derive a required result from first principles. _Given sufficient time_. Often in an exam situation you are time-constrained, and having memorized a shortcut cut be beneficial. Not to mention retaining is much easier when you understand the topic, so memorization becomes easier.
Probably the best example of this I can think of (for me at least) from mathematics is calculating combinations. I have it burned into my memory that (n choose r) = (n permute r) / (r permute r), and (n permute r) = n! / (n - r)!
Can I derive these from first principles? Sure, but after not seeing it for years, it might take me 10+ minutes to think through everything and correct any mistakes I make in the derivation.
But if I start with the formula? Takes me 5 seconds to sanity check the combination formula, and maybe 20 to sanity check the permutation formula. Just reading it to myself in English slowly is enough because the justification kind of just falls right out of the formula and definition.
So, yeah, they go hand in hand. You want to understand it but you sure as heck want to memorize the important stuff instead of relying on your ability to prove everything from ZFC...
It is waaaay easier to remember when you understand. The professor had it exactly right - if you learn to understand, you frequently end up remembering. But, memorization does not lead to understanding at all.
I think we memorize the understanding. For me it also works better understanding how something works than memoryzing results. I remember in high school, in maths trigonometrics, there were a list of 20 something formulas derived from a single one. Everyboby was memorizing the whole list of formulas; i just had to memorize a simple formula and the underdtanding of how to derive the others from the fundamental one on the fly.
You don't need to memorize to understand. You can rederive it every time.
You need to memorize it to use it subconsciously while solving more complex problems. Other ways you won't fit more complex solutions into your working memory,vso whole classes of problems will be too hard for you.
Ish? I never ever memorized the multiplication tables. To this day, I don't think I know them fully. I still did quite well in math by knowing how to quiz the various equations. Not just know them, but how to ask questions about moving terms and such.