Thank you for your questions, they’re proving very useful.
But it is interesting to understand, what’s happening to other children, who actually do math. Suddenly you realize, that “solving problems” for them is less energy demanding, which is awkward!
I’m not sure this is the case. We’re humans, maths is hard for everyone. I imagine it’s more about developing an ethics of work early on and being willing to delay gratification and experience unpleasant sensations for the purpose of learning something valuable. Though of course it takes a basic level of intelligence to find motivation in intellectual work. And there needs to be some specific motivation as well, i.e. math is beautiful, or math is useful.
As for the other questions… You may be getting closer than me at hitting the target here. I think the comparison between GPT-3 talk, where nothing is wrong, and “manipulation”, is central.
But “manipulation” isn’t like pattern-after-pattern, it is something different. What is it?
I think the whole thing revolves around mental models. Programming “clicks”when the stuff that you do with the code suddenly turns into a coherent mental model, so that you can even predict the result of an operation that you haven’t tried before. I became better at programming after watching a few theoretical computer science classes, because I was more proficient at building mental models of how the different systems worked. Likewise, maths clicks when you move from applying syntactical rules to building mental models of mathematical objects.
It’s easier to build mental models with programming, because the models that you’re working with are instantiated on a physical support that you can interact with. And because it’s harder to fool yourself and easier to get feedback. If you screw up, the computer will stop working and tell you. If you screw up with pen and paper, you might not even realize it.
This is not the whole story, but it’s a bit closer to what I meant to say.
This is false, there are a few genius mathematician who early in childhood proved it is easy for some humans.
I think the whole thing revolves around mental models
Exactly! There is even more specific concept in programming psychology, it is called “notional machines”. Small little machines in your head which can interpret using rules.
I think those also can transfer to math learning, as after rule-based machines concept is grasped, all the algorithmic, iterative, replacable and transitive concepts from math start making sense.
This is false, there are a few genius mathematician who early in childhood proved it is easy for some humans.
Some outliers are hypernumerate. I’m hyperlexic, so attuned to words that I was able to teach myself to read before my childhood amnesia kicked in, so I never had to learn phonics. This doesn’t mean the vast majority of humans aren’t congenitally literate or numerate. OP’s statement may be nominally false, but the exception proves the rule.
As for teaching the aesthetic beauty of math, I would give each student their own blank copy of the 10x10 multiplication table (with a zeros row and column, making it 11x11) at the start of grade 2, and teach them how to fill it in themselves. After that, they can use it in any math class that semester, but they have to make a new one at the start of each semester after that.
The inherent laziness of humanity will drive them to “cheat” by copying from lines above: filling in half the 4′s from the 2′s, half the 8′s from the 4′s, half the 6′s from the 3′s, and so on. And while they’re doing that, they’re learning in an indelible way.
Thank you for your questions, they’re proving very useful.
I’m not sure this is the case. We’re humans, maths is hard for everyone. I imagine it’s more about developing an ethics of work early on and being willing to delay gratification and experience unpleasant sensations for the purpose of learning something valuable. Though of course it takes a basic level of intelligence to find motivation in intellectual work. And there needs to be some specific motivation as well, i.e. math is beautiful, or math is useful.
As for the other questions… You may be getting closer than me at hitting the target here. I think the comparison between GPT-3 talk, where nothing is wrong, and “manipulation”, is central.
I think the whole thing revolves around mental models. Programming “clicks” when the stuff that you do with the code suddenly turns into a coherent mental model, so that you can even predict the result of an operation that you haven’t tried before. I became better at programming after watching a few theoretical computer science classes, because I was more proficient at building mental models of how the different systems worked. Likewise, maths clicks when you move from applying syntactical rules to building mental models of mathematical objects.
It’s easier to build mental models with programming, because the models that you’re working with are instantiated on a physical support that you can interact with. And because it’s harder to fool yourself and easier to get feedback. If you screw up, the computer will stop working and tell you. If you screw up with pen and paper, you might not even realize it.
This is not the whole story, but it’s a bit closer to what I meant to say.
This is false, there are a few genius mathematician who early in childhood proved it is easy for some humans.
Exactly! There is even more specific concept in programming psychology, it is called “notional machines”. Small little machines in your head which can interpret using rules.
I think those also can transfer to math learning, as after rule-based machines concept is grasped, all the algorithmic, iterative, replacable and transitive concepts from math start making sense.
Some outliers are hypernumerate. I’m hyperlexic, so attuned to words that I was able to teach myself to read before my childhood amnesia kicked in, so I never had to learn phonics. This doesn’t mean the vast majority of humans aren’t congenitally literate or numerate. OP’s statement may be nominally false, but the exception proves the rule.
As for teaching the aesthetic beauty of math, I would give each student their own blank copy of the 10x10 multiplication table (with a zeros row and column, making it 11x11) at the start of grade 2, and teach them how to fill it in themselves. After that, they can use it in any math class that semester, but they have to make a new one at the start of each semester after that.
The inherent laziness of humanity will drive them to “cheat” by copying from lines above: filling in half the 4′s from the 2′s, half the 8′s from the 4′s, half the 6′s from the 3′s, and so on. And while they’re doing that, they’re learning in an indelible way.