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> Simply memorizing sequences of steps is not how mathematics learning works, otherwise we would not see so much variation in outcomes Everyone starts by memor

> Simply memorizing sequences of steps is not how mathematics learning works, otherwise we would not see so much variation in outcomes Everyone starts by memorizing how to do basic arithmetic on numbers, their multiplication tables and fractions. Only some then advance to understanding why those operations must work as they do. > It's worth noting that for composition, key to abstract reasoning, LLMs failed to generalize to out of domain examples on simple synthetic data. Yes, I acknowledged that when I said "Composition tasks are still challenging". Comparisons and composition are both key to abstract reasoning. Clearly parametric memory and grokking have shown a fairly dramatic improvement in comparative reasoning with only a small tweak. There is no evidence to suggest that compositional reasoning would not also fall to yet another small tweak. Maybe it will require something more dramatic, but I wouldn't bet on it. This pattern of thinking humans are special does not have a good track record. Therefore, I find the original claim that I was responding to("there is no AGI pathway in the current research direction") completely unpersuasive.

11101010001100 2y ago
The point is the memorization exercise requires orders of magnitude fewer examples for bootstrapping.
- naasking 2y ago
  Does it though? It's a common claim but I don't think that's been rigourously established.
SonOfLilit 2y ago
I started by understanding. I could multiply by repeat addition (each addition counted one at a time with the aid of fingers) before I had the 10x10 addition table memorized. I learned university level calculus before I had more than half of the 10x10 multiplication table memorized, and even that was from daily use, not from deliberate memorization. There wasn't a day in my life where I could recite the full table. Maybe schools teach by memorization, but my mom taught me by explaining what it means, and I highly recommend this approach (and am a proof by example that humans can learn this way).
- naasking 2y ago
  > I started by understanding. I could multiply by repeat addition How did you learn what the symbols for numbers mean and how addition works? Did you literally just see "1 + 3 = 4" one day and intuit the meaning of all of those symbols? Was it entirely obvious to you from the get-go that "addition" was the same as counting using your fingers which was also the same as counting apples which was also the same as these little squiggles on paper? There's no escaping the fact that there's memorization happening at some level because that's the only way to establish a common language.
  - SonOfLilit 2y ago
    There's a difference between memorizing meanings of words (addition is same as counting this and then the other thing, "3" means three things) and memorizing methods (table of single digit addition/multiplication to do them faster in your head). You were arguing the second, I'm a counterexample. I agree about the first, everyone learns language by memorization (some rote, some by use), but language is not math.
    - naasking 2y ago
      > You were arguing the second, I'm a counterexample. I still don't think you are. Since we agree that you memorized numbers and how they are sequential, and that counting is moving "up" in the sequence, addition as counting is still memorizing a procedure based on this, not just memorizing a name: to add any two numbers, count down on one as you count up on the other until the first number number reaches zero, and the number that counted up is the sum. I'm curious how you think you learned addition without memorizing this procedure (or one equivalent to it). Then you memorized the procedure for multiplication: given any two numbers, count down on one and add the other to itself until the counted down number reaches one. This is still a procedure that you memorized under the label "multiplication". This is exactly the kind of procedure that I initially described. Someone taught you a correct procedure for achieving some goal and gave you a name for it, and "learning math" consists of memorizing such correct procedures (valid moves in the game of math if you will). These moves get progressively more sophisticated as the math gets more advanced, but it's the same basic process. They "make sense" to you, and you call it "understanding", because they are built on a deep foundation that ultimately grounds out in counting, but it's still memorizing procedures up and down the stack. You're just memorizing the "minimum" needed to reproduce everything else, and compression is understanding [1]. The "variation in outcomes" that an OP discussed is simply because many valid moves are possible in any given situation, just like in chess, and if you "understand" when a move is valid vs. not (eg. you remember it), then you have an advantage over someone who just memorized specific shortcuts, which I suspect is what you are thinking I mean by memorization. [1] https://philpapers.org/rec/WILUAC-2 https://philpapers.org/rec/WILUAC-2

11101010001100 2y ago

The point is the memorization exercise requires orders of magnitude fewer examples for bootstrapping.

naasking 2y ago
Does it though? It's a common claim but I don't think that's been rigourously established.

SonOfLilit 2y ago

I started by understanding. I could multiply by repeat addition (each addition counted one at a time with the aid of fingers) before I had the 10x10 addition table memorized. I learned university level calculus before I had more than half of the 10x10 multiplication table memorized, and even that was from daily use, not from deliberate memorization. There wasn't a day in my life where I could recite the full table. Maybe schools teach by memorization, but my mom taught me by explaining what it means, and I highly recommend this approach (and am a proof by example that humans can learn this way).

naasking 2y ago
> I started by understanding. I could multiply by repeat addition How did you learn what the symbols for numbers mean and how addition works? Did you literally just see "1 + 3 = 4" one day and intuit the meaning of all of those symbols? Was it entirely obvious to you from the get-go that "addition" was the same as counting using your fingers which was also the same as counting apples which was also the same as these little squiggles on paper? There's no escaping the fact that there's memorization happening at some level because that's the only way to establish a common language.
- SonOfLilit 2y ago
  There's a difference between memorizing meanings of words (addition is same as counting this and then the other thing, "3" means three things) and memorizing methods (table of single digit addition/multiplication to do them faster in your head). You were arguing the second, I'm a counterexample. I agree about the first, everyone learns language by memorization (some rote, some by use), but language is not math.
  - naasking 2y ago
    > You were arguing the second, I'm a counterexample. I still don't think you are. Since we agree that you memorized numbers and how they are sequential, and that counting is moving "up" in the sequence, addition as counting is still memorizing a procedure based on this, not just memorizing a name: to add any two numbers, count down on one as you count up on the other until the first number number reaches zero, and the number that counted up is the sum. I'm curious how you think you learned addition without memorizing this procedure (or one equivalent to it). Then you memorized the procedure for multiplication: given any two numbers, count down on one and add the other to itself until the counted down number reaches one. This is still a procedure that you memorized under the label "multiplication". This is exactly the kind of procedure that I initially described. Someone taught you a correct procedure for achieving some goal and gave you a name for it, and "learning math" consists of memorizing such correct procedures (valid moves in the game of math if you will). These moves get progressively more sophisticated as the math gets more advanced, but it's the same basic process. They "make sense" to you, and you call it "understanding", because they are built on a deep foundation that ultimately grounds out in counting, but it's still memorizing procedures up and down the stack. You're just memorizing the "minimum" needed to reproduce everything else, and compression is understanding [1]. The "variation in outcomes" that an OP discussed is simply because many valid moves are possible in any given situation, just like in chess, and if you "understand" when a move is valid vs. not (eg. you remember it), then you have an advantage over someone who just memorized specific shortcuts, which I suspect is what you are thinking I mean by memorization. [1] https://philpapers.org/rec/WILUAC-2 https://philpapers.org/rec/WILUAC-2