This is a problem with constructivism taken ad absurdum. Yes, understanding is better than mere memorizing. But understanding plus memorizing is often better then mere understanding, because doing lower-level operations automatically frees your mind to focus on the higher-level operations.
It's like those online articles that teach a very clever method (supposedly used by ancient Egyptians) how to calculate 6 × 7, and people write in comments "oh, why didn't they teach us this awesome thing at school?" Well, they didn't teach the method because (1) it's useless beyond 6 × 7 and maybe two or three other cases; and (2) even if you get into a situation where you need to multiply 6 by 7, by the time you have finished with your clever method, you most likely already forgot the context of the calculation, so you are looking at the number 42 not remembering how you got there. Meanwhile, your classmate automatically replaces 6 × 7 by 42 and continues along the original thread of thought.
Understanding things is good, often necessary, but "understanding vs memorization" is a false dilemma.
Agreed. I'm a big fan of spaced repetition systems now, even though I have a long way to go towards consistently using them.
By the way, I am usually a big fan of (the motte of) constructionism. Specifically with math, it makes me angry how often kids just memorize things without thinking, either because there is not enough time, or because the teachers do not understand it themselves.
But somehow, humans have the habit of taking good and reasonable ideas, making strawman versions of them, and presenting them as the true thing. (I suppose this is about signalling -- the more extreme version of X you believe, the more respected you are in the crowd that chose X as an applause light, even if that version does not make sense anymore. It's no longer about making sense, but about showing loyalty.) "Hey, if understanding things is good, then 100% of understanding with 0% of anything else (remembering, practice, etc.) must be the best thing ever, am I right?"
But somehow, humans have the habit of taking good and reasonable ideas, making strawman versions of them, and presenting them as the true thing. (I suppose this is about signalling—the more extreme version of X you believe, the more respected you are in the crowd that chose X as an applause light, even if that version does not make sense anymore. It’s no longer about making sense, but about showing loyalty.)
I think this is the root of all a lot of evil. To add to this, I'm not sure about the exact game theoretic mechanism, but signaling loyalty seems really important for human cooperation, so we can't just tell people to stop doing this. (To the extent that some people do stop using ideology for loyalty signaling upon learning about it or for other reasons, it seems to hurt their ability for collective action.) To me this is one of the biggest open problems in human group rationality.
The real-world solution seems to be leaders who preach the extreme version of the idea, but their actual actions are much more moderate. So the extremism of applause lights is balanced by the difference between what is promised and what gets actually done.
(This of course opens another bag of problems, for example it is not guaranteed that the leader will omit the stupid parts and keep the reasonable parts; it could easily be the other way round. But increasing control over the leaders would restore the original problem...)
For example, you could have a private school with a director who publicly preaches extreme constructivism, but in fact the school would use the usual repetition and memorization, only put a greater emphasis on understanding first. Most of the true believers would be happy to hear what they want to hear, and would not pay attention to details. Most non-believers would be placated with actual good outcomes. And the few ones who... -- well, you can't make everyone happy anyway.
Problem is, this solution is difficult to scale. If someone encouraged by the success would want to create a similar school in their city, they would probably go too far. And even the original director would have to carefully explain to their teachers what exactly is expected of them. (The true believers would try to shift the result in one direction, the lukewarm ones in another.) At last, the director's successor would ruin the balance, and the school would either become a crazy one, or an average one.
When I stop to think of people I support who I would peg as "extreme in words, moderate in actions", I think I feel a sense of overall safety that might be relevant here.
Let's say I'm in a fierce, conquering mood. I can put my weight behind their extremism, and feel powerful. I'm Making A Difference, going forth and reshaping the world a little closer to utopia.
When I'm in a defeatist mood, where nothing makes sense and I feel utterly hopeless, I can *also* get behind the extremism -- but it's in a different light, now. It's more, "I am so small, and the world is so big, but I can still live by what I feel is right".
Those are really emotionally powerful and salient times for me, and ones that have a profound effect on my sense of loyalty to certain causes. But most of the time, I'm puttering along and happy to be in the world of moderation. Intellectually, I understand that moderation is almost always going to be the best way forward; emotionally, it's another story entirely.
Upon first reading, I had the thought that a lot of people don't notice the extreme/moderate dichotomy of most of their leaders. I still think that's true. And then a lot of people do learn of that dichotomy, and they become disgusted by it, and turn away from anyone who falls in that camp. Which makes sense, honesty is a great virtue, why can't they just say what they mean? But then I look at myself, and while it doesn't feel *optimal* to me, it does feel like just another element of playing the game of power. There's this skill of reading between the lines that I think most people know is there, but they're a little reluctant to look straight at it.
I have single-digit multiplications just kinda cached as ingrained associations between any two given digits and their product, but now I'm curious what algorithm you were using for them. Repeated addition maybe?
I would picture them as rectangles and count. Like, 2x3 would look like
xxx
xxx
in my head, and for small numbers I could use the size of it to feel whether I was close. I remember doing really well with ratios and fractions and stuff for that reason.
For larger numbers, like 8x8, I would often subdivide into smaller squares (like 4x4 or 2x2), and count those. Then it would be easy to subdivide the larger one and repeat-add. I would often get a sour taste if the answer just "popped" into my head and I would actively fight against it, so I think there was a part of me that really just viscerally hated the idea of letting 'mere' memorization into my learning at all.
Incidentally, my past memories are saying that's why 6x7 and 7x7 gave me such trouble in particular; there was no "easy" way to decompose that in my head, it just looked like a square and another almost-maybe-a-square.
When you first try working in a base you're not familiar with, you get to see how much you rely on memorization.
76 * 89 =
(76*90)-76 = (760-76)-76
I have a distinct memory of being 8 years old, or so, and being handed one of those worksheets where they ask you to multiply numbers up through 12x12, and being viscerally disgusted by the implied pedagogy of it. That was over a hundred things you were asking me to memorize. On my own time. The whole reason I rush through my school work is so I don't have to do anything when I get home. I don't know if eight year old me swore, but this was definitely a "Screw you" moment for him.
But he actually ended up being able to do that sheet pretty quickly, at least compared to most of the rest of the class. There were a few kids who were faster than me, but I got the impression they were dumb enough to have to practice this instead of watching Ed, Edd 'n' Eddy at home. Or worse, they actually did memorize this stuff, instead of practice to get quick with the multiply-numbers-in-your-head algorithm like I did. (Because of course nobody else in the class would be doing it the same way I did, just much faster. But eight-year-olds aren't known to have particularly nuanced concepts of self that can gracefully accept that there are other people naturally much better than them at what they do best.)
Later on, we moved up to multiplying arbitrary two-digit-by-one-digit numbers, and then two-digit-by-two-digit numbers. (I didn't piece together how uncommon this was until a few years later.) Everyone who outpaced me in the times-tables speed tests were now far, far below me; meanwhile, I just had to chain my little "multiply-small-numbers" mental motion to a few "add-up-the-sums" motions. 76 * 89 = 7*8*100 + 6*8*10 + 7*9*10 + 6*9. I felt like I was so clever. I started to take pride in the fact that I was now leading the pack, even though I had told myself before that I didn't care!
That is, of course, until the kids who were originally faster than me also realized how to perform that mental motion, and then they leapt past me in speed with the combined force of split-second memory of times tables and a quick ability to perform algorithms.
I think by the time we were finished with the lightning round worksheet practice, I was in the bottom quarter of the class for speed, and when I did push myself to speed up, I'd start making careless mistakes like mixing up which one of 6*7 and 7*7 was 42 and which was 49, again?
Later in my mathematical pedagogy, I am taking a Real Analysis course. There are two midterms in this course. The first one I did not prepare for at all, falling into my old 8-year-old failure mode: "If I can't just compute the answer on the spot to the question, I sort of deserve to fail, don't I?" I got a B-, in the lower half of the class.
The second one, I reminded myself of the times tables kids. I got an A.