> To me this seems that you're hacking the system. You're avoiding learning how to think and understand, you're just learning how to pass the class.
I can certainly grasp that these are literally different, but are they practically different for most people?
Eg I know that some of my clothes need to be washed with cold water. I don't know why, but it's never made a difference in my life.
I know my car needs oil changes, but I don't specifically know why. Some kind of lubrication, but for what and why it goes bad I have no idea.
We all do hundreds of things algorithmically without really understanding what we're doing or why. We know enough to get the answer we want and that's good enough.
The kids that do algebra algorithmically probably aren't going to be math professors, but neither am I, and there are plenty of lucrative and productive professions that are fine with getting the right answer without knowing why.
On some level, most people are doing arithmetic algorithmically anyways, based on rules structured around base 10. Ask some people you think have passable arithmetic skills to do addition and subtraction in like base 5 and watch the smoke come out of their ears. I'm not casting aspersions, I'd have a hard time too past a couple digits.
I would wager most of them can't explain why we carry numbers over, they just know it needs to happen to get the right answer. I don't think I'm much better; I'm sure there are dozens of things in basic math that I just do without really understanding why.
> That said, I cannot concieve that some kids don't understand 20-8.
I would almost put money down that it's around carrying the 1's. I've met a few people that struggled with arithmetic, and they almost always get lost around carrying over numbers.