There is a certain danger in trying to learn maths with this mindset though. A lot of topics seem intuitively easy, but the proofs only come somewhere near the end of a degree level course.
It's a very healthy mindset to have, but it's a blessing and a curse. Some people are just very good at just abstracting away the details and getting on with things; what seems like "clicking" is not always the same as intuition. Probably it's good to operate in both modes. It's a really really useful skill to know when you need to care about the internals and when you can safely ignore them.
I’ve definitely learned that now after 10 years of development experience. But as a curious kid who didn’t just want to memorize and regurgitate it was quite a challenge. I’ve debated restarting mathematics all the way from basic algebra and geometry to see if I’d do any better these days.
I noticed my high school math(s) teachers were very bad at explaining why and what we were doing.
Arbitrary example:
One teacher kept saying "f(x)" but couldn't explain what a function is. He just said "it's anything", then "don't worry about it". If he had even said "a function is like a machine that takes number(s) as input, changes them with a formula and outputs the new number(s)", I think it would have helped us grok.
I think he understood math(s) so well that he couldn't relate to someone who didn't know what a function was.
> If he had even said "a function is like a machine that takes number(s) as input, changes them with a formula and outputs the new number(s)", I think it would have helped us grok.
Helpful but not quite right and one of the common misconceptions students seem to carry over from high school.
A function is something that takes inputs out of a set (its domain, e.g. people in this class) and gives you exactly one output with a prescribed data type (e.g. a date. The function than for instance being person -> birthdate). Neither numbers nor a formula are needed.
> I think he understood math(s) so well that he couldn't relate to someone who didn't know what a function was.
I think he probably could, but the syllabus said explicitly not discuss this ("too abstract") and there is time pressure. High school maths mostly is somewhat handwavy and stringing "definitions" together by examples. So you would mostly see lots of examples of functions and the "definition" of 'function' is then simply "things like that".
It's a very healthy mindset to have, but it's a blessing and a curse. Some people are just very good at just abstracting away the details and getting on with things; what seems like "clicking" is not always the same as intuition. Probably it's good to operate in both modes. It's a really really useful skill to know when you need to care about the internals and when you can safely ignore them.