I’m quite surprised at this guess and intrigued by your school’s methodology. I would have estimated >30 problems average across 20 weeks for myself.
My kids are still in pre-algebra, but they get way more drilling still, well over 1000 problems per semester once Zern, IReady, etc. are factored in. I believe it’s too much, but it does seem like the typical approach here in California.
For example after doing several hundred logarithms, I was eventually able to do logs to 2 decimal places in my head. (Sadly I cannot do that anymore!) I imagine if I had just done a dozen or so problems I would not have gained that ability.
Now imagine how much would your kid learn if the only input he ever received was a sequence of words?
The difference is that we don't know better methods for them, but we do know of better methods for people.
But after the training people are much more equipped to do single-shot recognition and cognitive tasks of imagery and situations they have not encountered before, e.g. identifying (from pictures) which animals is being shown, even if it is the second time of seeing that animal (the first being shown that this animal is a zebra).
So, basically, after initial training, I believe people are superior in single-shot tasks—and things are going to get much more interesting once LMMs (or something after that?) are able to do that well.
It might be that GPT-4o can actually do that task well! Someone should demo it, I don't have access. Except, of course, GPT-4o already knows what zebras look like, so something else than exactly that..
I think it was; the guesstimate I've seen is GPT-4 was trained on 13e12 tokens, that over 4 years is 8.9e9/day or about 1e5/s.
Then it's a question of how many bits per token — my expectation is 100k/s is more than the number of token-equivalents we experience, even though it's much less than the bitrate even of just our ears let alone our eyes.
The analogies get a little blurry here, but perhaps we can draw a distinction between information that an infant gets from their higher-level senses (e.g. sight, smell, touch, etc) versus any lower-level biological processes (genetics, epi-genetics, developmental processes, and so on).
The main point is that there is a fundamental difference: LLMs have very little prior knowledge [1] while humans contain an immense amount of information even before they begin learning through the senses.
We need to look at the billions of years of biological evolution, millions of years of cultural evolution, and the immense amounts of environmental factors, all which shape us before birth and before any “learning” occurs.
[1] The model architecture probably counts as hard-coded prior knowledge contained before the model begins training, but it is a ridiculously small amount of information compared to the complexity of living organisms.
Yeah, but they're seeing mostly the same thing day after day!
They aren't seeing 10k stills of 10k different dogs, then 10k stills of 10k different cats. They're seeing $FOO thousand images of the family dog and the family cat.
My (now 4.5yo) toddler did reliably tell the difference between cats and dogs the first time he went with us to the local SPCA and saw cats and dogs that were not our cats and dogs.
In effect, 2 cats and 2 dogs were all he needed to reliably distinguish between cats and dogs.
I assume he was also exposed to many images, photos and videos (realistic or animated) of cats and dogs in children books and toys he handled. In our case, this was a significant source of animal recognition skills of my daughters.
No images or photos (no books).
TV, certainly, but I consider it unlikely that animals in the animation style of pepper pig helps the classifier.
Besides which, we're still talking under a dozen cats/dogs seen till that point.
Forget about cats/dogs. Here's another example: he only had to see a burger patty once to determine that it was an altogether new type of food, different from (for example) a sausage.
Anyone who has kids will have dozens of examples where the classifier worked without a false positive off a single novel item.
To continue your example, I know I've learned calculus and was lauded at the time. Now I could only give you the vagaries, nothing practical. However I know if I was pressed, I could learn it again in short order.
Sure, but they learn a lot of labels.
> How many homework questions did your entire calc 1 class have? I'm guessing less than 100
At least 20 to 30 a week, for about 10 weeks of class. Some weeks were more, and I remember plenty of days where we had 20 problems assigned a day.
Indeed, I am a huge fan of "the best way to learn math is to do hundreds upon hundreds of problems", because IMHO some concepts just require massive amounts of repetition.
Until they encounter a similar animal and get confused, at which point you understand the implicit heuristic they were relying on. (Eg. They confused a dairy cow as a zebra, which means their heuristic was a black-and-white quadrupedal)
Doesn't this seem remarkably close to how LLMs behave with one-shot or few-shot learning? I think there are a lot more similarities here than you give it credit for.
Also, I grew up in South Korea where early math education is highly prioritized (for better or for worse). I remember having to solve 2 dozen arithmetic problems every week after school with a private tutor. Yes, it was torture and I was miserable, but it did expose me to thousands more arithmetic questions than my American peers. All that misery paid off when I moved to the U.S. at the age of 12 and realized that my math level was 3-4 years above my peers. So yes, I think human intelligence accuracy also does improve with more training data.
Of course it didn't have to be this way - in a different language animals might be named based on size or abilities/behavior, etc.
So, your daughter wanting to label a cat-sized dog as a cat is just a reflection of her not having aligned her generalization of what you are talking about when you say "cat" vs "dog" with her own.
Not just that: people learn mathematics mainly by _thinking over and solving problems_, not by memorising solutions to problems. During my mathematics education I had to practice solving a lot of problems dissimilar what I had seen before. Even in the theory part, a lot of it was actually about filling in details in proofs and arguments, and reformulating challenging steps (by words or drawings). My notes on top of a mathematical textbook are much more than the text itself.
People think that knowledge lies in the texts themselves; it does not, it lies in what these texts relate to and the processes that they are part of, a lot of which are out in the real world and in our interactions. The original article is spot on that there is no AGI pathway in the current research direction. But there are huge incentives for ignoring this.
And almost all of it is just more text, or described in more text.
You're very much right about this. And that's exactly why LLMs work as well as they do - they're trained on enough text of all kinds and topics, that they get to pick up on all kinds of patterns and relationships, big and small. The meaning of any word isn't embedded in the letters that make it, but in what other words and experiences are associated with it - and it so happens that it's exactly what language models are mapping.
The answer is that both humans and the model are capable of reasoning, but the model is more restricted in the reasoning that it can perform since it must conform to the dataset. This means the model is not allowed to invest tokens that do not immediately represent an answer but have to be derived on the way to the answer. Since these thinking tokens are not part of the dataset, the reasoning that the LLM can perform is constrained to the parts of the model that are not subject to the straight jacket of training loss. Therefore most of the reasoning occurs in-between the first and last layers and ends with the last layer, at which point the produced token must cross the training loss barrier. Tokens that invest into the future but are not in the dataset get rejected and thereby limit the ability of the LLM to reason.
I think it's more accurate to say that they learn math by memorizing a sequence of steps that result in a correct solution, typically by following along with some examples. Hopefully they also remember why each step contributes to the answer as this aids recall and generalization.
The practice of solving problems that you describe is to ingrain/memorize those steps so you don't forget how to apply the procedure correctly. This is just standard training. Understanding the motivation of each step helps with that memorization, and also allows you to apply that step in novel problems.
> The original article is spot on that there is no AGI pathway in the current research direction.
I think you're wrong. The research on grokking shows that LLMs transition from memorization to generalized circuits for problem solving if trained enough, and parametric memory generalizes their operation to many more tasks.
They have now been able to achieve near perfect accuracy on comparison tasks, where GPT-4 is barely in the double digit success rate.
Composition tasks are still challenging, but parametric memory is a big step in the right direction for that too. Accurate comparitive and compositional reasoning sound tantalizingly close to AGI.
Simply memorizing sequences of steps is not how mathematics learning works, otherwise we would not see so much variation in outcomes. Me and Terence Tao on the same exact math training data would not yield two mathematicians of similar skill.
While it's true that memorization of properties, structure, operations and what should be applied when and where is involved, there is a much deeper component of knowing how these all relate to each other. Grasping their fundamental meaning and structure, and some people seem to be wired to be better at thinking about and picking out these subtle mathematical relations using just the description or based off of only a few examples (or be able to at all, where everyone else struggles).
> I think you're wrong. The research on grokking shows that LLMs transition from memorization to generalized circuits
It's worth noting that for composition, key to abstract reasoning, LLMs failed to generalize to out of domain examples on simple synthetic data.
From: https://arxiv.org/abs/2405.15071
> The levels of generalization also vary across reasoning types: when faced with out-of-distribution examples, transformers fail to systematically generalize for composition but succeed for comparison.
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.
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).
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" operations using your fingers and counting apples was the same operation 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.
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.
I did memorize names of numbers, but that is not essential in any way to doing or understanding math, and I can remember a time where I understood addition but did not fully understand how names of numbers work (I remember, when I was six, playing with a friend at counting up high, and we came up with some ridiculous names for high numbers because we didn't understand decimal very well yet).
Addition is a thing you do on matchsticks, or fingers, or eggs, or whatever objects you're thinking about. It's merging two groups and then counting the resulting group. This is how I learned addition works (plus the invariant that you will get the same result no matter what kind of object you happen to work with). Counting up and down is one method that I learned, but I learned it by understanding how and why it obviously works, which means I had the ability to generate variants - instead of 2+8=3+7=... I can do 8+2=9+1=..., or I can add ten at a time, etc'.
Same goes for multiplication. I remember the very simple conversation where I was taught multiplication. "Mom, what is multiplication?" "It's addition again and again, for example 4x3 is 3+3+3". That's it, from that point on I understood (integer) multiplication, and could e.g. wonder myself at why people claim that xy=yx and convince myself that it makes sense, and explore and learn faster ways to calculate it while understanding how they fit in the world and what they mean. (An exception is long multiplication, which I was taught as a method one day and was simple enough that I could memorize it and it was many years before I was comfortable enough with math that whenever I did it it was obvious to me why what I'm doing here calculates exactly multiplication. Long division is a more complex method: it was taught to me twice by my parents, twice again in the slightly harder polynomial variant by university textbooks, and yet I still don't have it memorized because I never bothered to figure out how it works nor to practice enough that I understand it).
I never in my life had an ability to add 2+2 while not understanding what + means. I did for half an hour have the same for long division (kinda... I did understand what division means, just not how the method accomplishes it) and then forgot. All the math I remember, I was taught in the correct order.
edit: a good test for whether I understood a method or just memorized it would be, if there's a step I'm not sure I remember correctly, whether I can tell which variation has to be the correct one. For example, in long multiplication, if I remembered each line has to be indented one place more to the right or left but wasn't sure which, since I understand it, I can easily tell that it has to be the left because this accomplishes the goal of multiplying it by 10, which we need to do because we had x0 and treated it as x.
Current research in early mathematical education now focuses on teaching certain spatial skills to very young kids rather than (just) numbers. Mathematics is about understanding of relationships, and that is not a detached kind of understanding that we can make into an algorithm, but deeply invested and relational between the "subject" and the "object" of understanding. Taking the subject and all the relations with the world out of the context of learning processes is absurd, because that is in the exact centre of them.
Perhaps that is how you learned math, but it is nothing like how I learned math. Memorizing steps does not help, I sucked at it. What works for me us understanding the steps and why we used them. Once I understood the process and why it worked, I was able to reason my way through it.
> The practice of solving problems that you describe is to ingrain/memorize those steps so you don't forget how to apply the procedure correctly.
Did you look at the types of problems presented by the ARC-AGO test? I don't see how memorization plays any role.
> They have now been able to achieve near perfect accuracy on comparison tasks, where GPT-4 is barely in the double digit success rate.
Then lets see how they do on the ARC test? While it is possible that generalized circuits can develop in Ls with enough training but I am pretty skeptical till we see results.
Memorization is literally how you learned arithmetic, multiplication tables and fractions. Everyone starts learning math by memorization, and only later start understanding why certain steps work. Some people don't advance to that point, and those that do become more adept at math.
I understood how to do arithmetic for numbers with multiple digits before I was taught a "procedure". Also, I am not even sure what you mean by "memorization is how you learned fractions". What is there to memorize?
What did you understand, exactly? You understood how to "count" using "numbers" that you also memorized? You intuitively understood that addition was counting up and subtraction was counting down, or did you memorize those words and what they meant in reference to counting?
> Also, I am not even sure what you mean by "memorization is how you learned fractions". What is there to memorize?
The procedure to add or subtract fractions by establishing a common denominator, for instance. The procedure for how numerators and denominators are multiplied or divided. I could go on.
I do have the single digit multiplication table memorized now, but there was a long time where that table had gaps and I would use my understanding of how numbers worked to to calculate the result rather than remembering it. That same process still occurs for double digit number.
Mathematics education, especially historically, has indeed leaned pretty heavily on memorization. That does mean thats the only way to learn math, or even a particularly good one. I personally think over reliance on memorization is part of why so many people think they hate math.
Sure, I did that plenty too, but that doesn't refute the point that memorization is core to understanding mathematics, it's just a specific kind of memorization that results maximal flexibility for minimal state retention. All you're claiming is that you memorized some core axioms/primitives and the procedures that operate on them, and then memorized how higher-level concepts are defined in terms of that core. I go into more detail of the specifics here:
https://news.ycombinator.com/item?id=40669585
I agree that this is a better way to memorize mathematics, eg. it's more parsimonious than memorizing lots of shortcuts. We call this type of memorizing "understanding" because it's arguably the most parsimonious approach, requiring the least memory, and machine learning has persuasively argued IMO that compression is understanding [1].
Yes. All that learning is feeding off one another. They're learning how reality works. Every bit of new information informs everything else. It's something that LLMs demonstrate too, so it shouldn't be a surprising observation.
> Once they have the basics down concept acquisition time shrinks rapidly
Sort of, kind of.
> and kids can easily learn their new favorite animal in as little as a single example.
Under 5 they don't. Can't speak what happens later, as my oldest kid just had their 5th birthday. But below 5, all I've seen is kids being quick to remember a name, but taking quite a bit longer to actually distinguish between a new animal and similarly looking ones they already know. It takes a while to update the classifier :).
(And no, they aren't going to one-shot recognize an animal in a zoo that they saw first time on a picture hours earlier; it's a case I've seen brought up, and I maintain that even most adults will fail spectacularly at this test.)
> Compare this to LLMs which can one-shot certain tasks, but only if they have essentially already memorized enough information to know about that task. It gives the illusion that these models are learning like children do, when in reality they are not even entirely capable of learning novel concepts.
Correct, in the sense that the models don't update their weights while you use them. But that just means you have to compare them with ability of humans to one-shot tasks on the spot, "thinking on their feet", which for most tasks makes even adults look bad compared to GPT-4.
> How many homework questions did your entire calc 1 class have? I'm guessing less than 100 and (hopefully) you successfully learned differential calculus.
I don't believe someone could learn calc in 100 exercises or less. Per concept like "addition of small numbers", or "long division", or "basic derivatives", or "trivial integrals", yes. Note that in-class exercises count too; learning doesn't happen primarily by homework (mostly because few have enough time in a day to do it).
This simply is not true as stated in the article. ARC-AGI is a one-shot task test that humans reliably do much, much better on than any AI model.
> I don't believe someone could learn calc in 100 exercises or less.
I learned the basics of integration in a foreign language I barely understood by watching a couple of diagrams get drawn out and seeing far less than 100 examples or exercises.
I have kids so I'm presuming I'm allowed to have an opinion here.
This is ignoring the fact that babies are not just learning labels, they're learning the whole of language, motion planning, sensory processing, etc.
Once they have the basics down concept acquisition time shrinks rapidly and kids can easily learn their new favorite animal in as little as a single example.
Compare this to LLMs which can one-shot certain tasks, but only if they have essentially already memorized enough information to know about that task. It gives the illusion that these models are learning like children do, when in reality they are not even entirely capable of learning novel concepts.
Beyond just learning a new animal, humans are able to learn entirely new systems of reasoning in surprisingly few examples (though it does take quite a bit of time to process them). How many homework questions did your entire calc 1 class have? I'm guessing less than 100 and (hopefully) you successfully learned differential calculus.