Wisdom in aging

Yeah,the math discovery process is still going on.It is still a struggle between how much to push the methodology,and how much, the reasoning behind it.
I was wondering about how much I got the reasoning behind the math I was doing back when I was in school-aeons ago!
Frankly,not much I think.Also back home there weren't too many surprises thrown at you-you were given the methodology and followed it blind.I remember being tutored for math in XII standard.
Small digression before the math story-my son's piano teacher went into a rage because he did not know his Roman numerals!!She is Ukrainian. That is not considered very important here right?We are so used to writing our class numbers as Roman numerals that it almost becomes second nature-atleast until XII!
Going back to my tutoring-my teacher gave us short cuts on how you could skip an entire unit if you could do this, this and this with the other units and how the question paper would have probably this kind of question and not that-so basically you did not have to use your mind much.If you had the methodology down,you were good to go.Not that the logic behind it wasn't explained to you.You just weren't expected to extrapolate it into anything else.
Does it work still?When people get to college,it is mostly application based academics-no one methodology to follow and it becomes a struggle-unless by that time your intelligence has finally caught up with your age!!Mine still hadn't at college.Besides I hated doing Engineering,so maybe that contributed to my inability to still catch up.
So what is a solid foundation?Knowing both methodology and reasoning or a little bit of both,or more of one than the other?Because at some point the two clash-if the methodology is not well ensconced in the mind, application becomes difficult-you don't know what to do.But if you have the methodology down pat without the reasoning behind it,then applying what  you have learnt is a no go.Add to it that the ability/intelligence between kids varies dramatically,and you have a true dilemma-I am experiencing that right now.Wish me luck.I don't want to mess my kid up.I will have no one but myself to blame.
I wonder how people who have more than 4 kids and homeschool all of them ( we know quite a few) handle that!!Or maybe they have far more diligent kids?Poor Emmy!!Not his fault,I assure you!!