> You can smooth things out with quality teaching, but there is a human hardware limit, and you reach it sooner with those two.
Shmaybe. But I doubt many people ever approach it. What they hit is, instead, a "software lockout" - a limit of the kind that some hardware vendors (such as those making oscilloscopes or farming machines) put in their products, where the hardware is already there, but its capability is restricted in software, and you need to pay to have the restrictions lifted.
If you'd seen me as a kid, you'd say I am one of those people with rather low "hardware limit" when it comes to math. But then, when I was around 13, that limit suddenly shot up so high and so fast, it may as well have disappeared. What happened? I started to learn to code, with an intent to make videgames. Eventually I reached the point when I was trying to figure out how to rotate some sprites, and made a mental connection between what I want (understanding how to implement rotating images) and what I was taught at school (trigonometry). Immediately, math switched from being something I had to learn, and became something I wanted to learn. So, what you'd think was a severe hardware limit, was actually a purely software one - and it went away the moment I found the magic unlock code (which, in my case, was "find something that makes math seem relevant and exciting, instead of a bullshit chore").
And yes, I'll admit that I do have a hardware limit relevant here - most kids do well at math without having to find a motivational "cheat code", but people like me, with impaired executive functions, can't naturally power through bullshit, and need workarounds to keep up. But this is a different kind of limit. It's also one we're getting increasingly good at understanding, recognizing and overcoming, and it teaches us that what you see as hardware skill limit, may just be a software limit induced by another problem.
Or, in short, if someone told the twelve year old me the things you wrote here today, it would have derailed my life. Fortunately, nobody did, and this lazy and dumb-ish kid ended up handling undergrad-level math perfectly fine.
Shmaybe. But I doubt many people ever approach it. What they hit is, instead, a "software lockout" - a limit of the kind that some hardware vendors (such as those making oscilloscopes or farming machines) put in their products, where the hardware is already there, but its capability is restricted in software, and you need to pay to have the restrictions lifted.
If you'd seen me as a kid, you'd say I am one of those people with rather low "hardware limit" when it comes to math. But then, when I was around 13, that limit suddenly shot up so high and so fast, it may as well have disappeared. What happened? I started to learn to code, with an intent to make videgames. Eventually I reached the point when I was trying to figure out how to rotate some sprites, and made a mental connection between what I want (understanding how to implement rotating images) and what I was taught at school (trigonometry). Immediately, math switched from being something I had to learn, and became something I wanted to learn. So, what you'd think was a severe hardware limit, was actually a purely software one - and it went away the moment I found the magic unlock code (which, in my case, was "find something that makes math seem relevant and exciting, instead of a bullshit chore").
And yes, I'll admit that I do have a hardware limit relevant here - most kids do well at math without having to find a motivational "cheat code", but people like me, with impaired executive functions, can't naturally power through bullshit, and need workarounds to keep up. But this is a different kind of limit. It's also one we're getting increasingly good at understanding, recognizing and overcoming, and it teaches us that what you see as hardware skill limit, may just be a software limit induced by another problem.
Or, in short, if someone told the twelve year old me the things you wrote here today, it would have derailed my life. Fortunately, nobody did, and this lazy and dumb-ish kid ended up handling undergrad-level math perfectly fine.